![ACKNOWLEDGMENTS
We are grateful to the National Academy of Sciences of
Ukraine for the financial support of the Target Scienti-
fic Research Programs “Structure and composition of the
Universe, hidden mass and dark energy” (2007—2009),
“Astrophysical and cosmological problems of hidden mass
and dark energy of the Universe” (2010—2012) [“CosmoMi-
croPhysics”], within which this monograph was originated.
We are thankful to academician Valerij Shulga, the editor
of this monograph, for the suggestion of its writing, partici-
pation in the formation of its content, and organization of
its publication. We also thank academician Yaroslav Yatskiv
for his generous support of these Programs. We also appreci-
ate helpful and highly professional assistance from the staff
of the academic publisher “Akademperiodyka”.
AUTHORS
Kyiv, June, 2013](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-13-320.jpg)
![C H A P T E R
GRAVITATIONAL LENSING
AS A KEY TO SOLVING THE DARK
MATTER PROBLEM
V.M. Shulga, A.A. Minakov, V.G. Vakulik,
G.V. Smirnov, V.S. Tsvetkova
1.1. Gravitational lensing: general conception
11
The second part of the XX century occurred to be very suc-
cessful for astronomers, astrophysicists and cosmologists. In
the first 60ths, the most remote sources of radiation, — the
quasars — were discovered [85]. A few years later, pulsa-
ting sources — the pulsars — were detected [38]. And fi-
nally, a decade later existence of space mirages — gravi-
tational lenses (GL) — was confirmed [124]. Discovery of
the first GL has stimulated dedicated searches for other
lenses. To do this, most powerful instruments of the opti-
cal and radio wavelength ranges were being used, and even
some new instruments dedicated to the search for mani-
festations of the gravitational lensing phenomenon have
been created 1. Several international programs dedicated to
the GL problems have been approved, which remain to be
operational so far.
The interest to the space “mirages” is due, first of all,
to a possibility of solving a set of topical astrophysical and
cosmological problems inherent in this phenomenon. They
are, for example, determination of the Hubble constant, as
well as studies of the fine structure of the quasars’ emitting
regions with the highest resolution that is as yet unachieva-
ble for ground-based and even space-based observations.
Last time, the GL phenomenon turned out to be very
efficient in searching for the hidden mass in the Universe
1
Sometimes this phenomenon is also referred to as the “gravitati-
onal focusing”.](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-15-320.jpg)
![CHAPTER 1. Gravitational lensing
and estimating its abundance. As it became evident about three quarters of
a century ago, the observed rotation curves of galaxies can only be explained
by an assumption that the visible matter is immersed into a massive extended
but invisible formation, which was named a dark matter halo. A considerable
attention has been paid to find out the halo constituents, and several moni-
toring programs that used the gravitational lensing has been carried out to
clear up what fraction of the halo mass is formed by MACHOs — Massive
Compact Halo Objects. The answer is that the halo predominantly consists of
the non-baryonic dark matter. On the other hand, a lot of compact objects has
been found by using gravitational lensing, however not in the halo, but in the
Galactic disk. These objects may include brown dwarfs, white dwarfs, planets
with masses 10−5M⊙ M 10−3M⊙, neutron stars and even low-mass black
holes. A question about the relative content of these objects is presently vital
for solving the problem of the Universe origin and evolution.
The essence of the gravitational lensing phenomenon is deflection of the
light rays in the gravitational field of a massive object. I. Newton was the first
to ask a question: can it be that bodies act upon the light at some distance
to bend the light rays, and would this action be (all factors being the same)
the strongest at the least distance? A history of developing the theory and
observations of the GL phenomenon can be found in the books by P. Bliokh
and A. Minakov [10], P. Schneider, J. Ehlers and E. Falco [107] and A. Zakha-
rov [147].
In 1804, German astronomer J. Zoldner determined the angle of deviation
of the light beam from a star in its transition near the edge of the solar disc
and obtained 0′′.87 [107]. A hundred and ten yeas later, A. Einstein considered
the effect of the light ray deflection in the gravitational field of the Sun in the
framework of his general relativity theory (GRT). His first result, obtained in
1911, did not differ from that of Zoldner. In 1915, however, a new work by
A. Einstein appeared, where a revised value was presented, which is twice as
much as compared to the previous estimate. The first experimental confirmati-
on of gravitational deflection of the light rays was obtained in observations of
the total solar eclipse on the 29th of May, 1919.
By now a number of various manifestations of the effects of GL in the
optical and radio wave length ranges at various spatial scales reaches hundreds.
Three types of gravitational lensing are presently differentiated, each of which
has its intrinsic potentials as concerned to solving the dark matter problem:
weak lensing, microlensing and strong lensing.
The term “weak lensing” is used when a lens produces a single slightly
distorted image of a particular source. Weak lensing reveals itself in observati-
ons of vast sky areas of very distant objects (quasars, galaxies) through the
gravitational fields of massive foreground clusters of galaxies. In addition to
small deformations of shapes of the distant source images, minor changes of
12](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-16-320.jpg)
![1.1. Gravitational lensing: general conception
their brightness and distribution over the sky occur. In principle, peculiarities
of these distortions provide a clue to restore the pattern of the gravitational
potential of the cluster. To reveal the dark matter, the restored distribution of
the gravitational potential can be then compared to that caused by distribution
of the visible (luminous) matter.
Addressing to the weak lensing phenomenon with the aim to detect the
hidden mass usually implies statistical analysis of very large sky areas obtai-
ned in the deep surveys. For example, the well known work by Jee et al. [42]
reports detection of a ring-like dark matter substructure in the rich galaxy
cluster Cl 0024 + 17 at a radius of r ≈ 75′′ surrounding a soft, dense core at
r ≤ 50′′. To restore distribution of the total gravitational potential of the
cluster, the authors fulfilled statistical analysis of 1300 images of background
objects produced by both weak and strong lensing. They interpret this ring-like
substructure as the result of a line-of-sight collision of two massive clusters.
Another impressive result of detecting the peculiar dark matter pattern with
the use of the weak lensing is the Bullet cluster 1E0657-558 [18], which consists
of two colliding clusters of galaxies.
Microlensing is divided into a “near” microlensing (called also “galactic”
microlensing) and a “distant” (extragalactic) one. The galactic microlensing is
meant when a star located in the Galaxy or in the Magellanic clouds changes
its brightness due to gravitational focusing by a compact object of our Galaxy
crossing the line of sight between the star and observer. The extragalactic
microlensing is spoken about when a compact object of a lensing galaxy passes
close to the line of sight resulting in essential brightness change of a particular
lensed quasar image.
In 1986, B. Paczyński [83] proposed a peculiar method to investigate a
population of the halo of our Galaxy, which implies observations of the gravi-
tational lensing events of stars in the Milky Way by other stars. Since the
probability of observing an individual event of such a kind is very small
(P ≈ 10−6 events per year), Paczyński proposed to monitor several milli-
on stars simultaneously, for example, in the Magellanic Clouds. Analogous
programs dealt with monitoring of stars near the nucleus of the Milky
Way. Several dedicated programs have been launched, such as MACHO,
EROS and, somewhat later, OGLE, which resulted in detecting several
hundred microlensing events. The amplitudes and durations of light curves
of these events contain information about masses of compact bodies and their
velocities.
As was noted above, microlensing events may be produced by compact
bodies not only in our Galaxy, but also in the other galaxies, which lens remote
quasars. Observations of gravitationally lensed quasars show that microlensing
is a phenomenon of a rather high occurrence, — almost all of them demonstrate
microlensing activity that is different in different objects. Of all the known
13](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-17-320.jpg)
![CHAPTER 1. Gravitational lensing
gravitationally lensed quasars, the highest microlensing activity is observed in
the quadruple lens system Q 2237 + 0305, the Einstein Cross.
Strong lensing is spoken about when either multiple, or arc-like, or ring-
like images of a remote source (a quasar) are formed in gravitational focusing.
It takes place when a sufficiently massive galaxy occurs to be close to the path
of light rays from a distant quasar to the observer. The observer will see then,
instead of one source, several distorted images of the same quasar formed by
the gravitational field of the galaxy.
The phenomena of strong lensing and microlensing provide several app-
roaches to solving the dark matter problem which are based on the long-term
observations of transient events in images of gravitationally lensed quasars. We
mention here some of them.
1. The method of histograms. Each of the images of a gravitationally lensed
quasar (macroimage) may change its brightness because of possible microlen-
sing events, which are signaling that a compact object (a star or a planet) is
moving near the line of sight corresponding to a particular macroimage. The
method of histograms is based on studying the magnification probability di-
stributions for macroimages due to microlensing events. Numerical simulation
shows that such distributions are sensitive to the relationship between the mass
fraction in compact objects and that one in the uniformly distributed matter,
or in the objects with extremely small masses.
2. Flux ratio anomaly method. It has been long noticed that flux ratios of
the lensed quasar components are poorly reproduced by the lens models with
smooth (regular) distribution of gravitational potential (the problem of flux
ratio anomalies). Since 2001, the anomalies of mutual fluxes observed in many
quasar lenses are thought to be caused by the presence of the dark matter
substructures in lensing galaxies. Existence of these substructures is predicted
by the scenario of hierarchical formation of structures in the Universe.
3. The method of time delays proposed in 1964 by S. Refsdal [94] to determi-
ne the Hubble constant, is based on the fact that fluctuations of the intrinsic
brightness of a quasar are seen in its lensed images with the delays, which
are determined — at a given system geometry — by the surface mass density
distribution in a lens galaxy and by the Hubble constant value.
Below the basic results of observations and theoretical studies of the GL
phenomenon will be presented, which were being carried out by a joint Kharkiv
and Kiev group consisting of researches from the Institute of Radio Astronomy
of the National Academy of Sciences of Ukraine, Institute of Astronomy of
V.N. Karazin National University of Kharkiv and Astronomical Observatory
of Taras Shevchenko National University of Kyiv. The works were supported
by the target Program of the National Academy of Sciences of Ukraine “Investi-
gation of the Universe structure and composition, hidden mass and dark
energy” (“Cosmomicrophysics”).
14](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-18-320.jpg)
![1.1. Gravitational lensing: general conception
1.1.1. Basic equations
of the gravitational focusing phenomenon
In the theory of GL, a system of coordinates presented in
Fig. 1.1 is usually used, which is convenient for a terrestrial observer. The
OZ axis passes through the point of observations P and the mass center of
a gravitating object O. In this coordinate system, a source of radiation with
small angular dimensions (a “point source”) is situated in the source plane with
angular coordinates y.
When considering the effect of lensing produced by gravitational fields
of complicated objects, such as galaxies and stellar clusters, one should take
into account that their masses are contained not only in the large-scale dif-
fusely distributed structures (e.g. dust or gas clouds, dark matter), but also in
compact objects. A number of such objects inside a cluster is large enough. For
example, a number of stars in a spiral galaxy similar to the Galaxy can reach
N ≈ 1010 ÷ 1011. It should be also noted that in addition to stars, star-like
and planet-like bodies can act as microlenses, and their number is also large
enough inside a cluster or a galaxy.
As a rule, the effects of microlensing are investigated theoretically with
the use of a two-dimensional model for distribution of stars in a cluster. In
doing so, a uniform in the average distribution of the microlens stars is consi-
dered, with the correlation between their positions being neglected. Though
such representation does not allow the “fine” effects to be predicted from
the observed brightness fluctuations of microlensed images, it is justified by
the following. Firstly, we use the minimal number of free parameters in this
representation, and secondly, our ignorance of the actual spatial distributions
of stars in the clusters makes the model refinements resulting in minor quanti-
tative corrections to be senseless, (see, e.g. [10,107]).
The gravitational field averaged over large interstellar volumes of matter
distribution provides a global lensing effect of the cluster as a whole. Such
lens was called a macrolens in literature. Compact bodies randomly distri-
buted inside the cluster produce the lensing effect too, though of a smaller
scale. The phenomenon of focusing in gravitational fields of compact “low-mass”
objects was named microlensing. Though the spatial scales of microlenses are
small, their effect results in significant brightness variations of macro-images of
sources — quasars. An extended large-scale mass constituent in the cluster will
be further referred to as a “diffuse” one, while compact masses will be named
a “stellar” constituent.
The approximations of quasi-statics, geometrical optics and a thin phase
screen form the basis of the theory. A validity of applying the method of the
phase screen (MPS) is usually justified by the following considerations. Excep-
ting heavily populated regions around galactic nuclei, the spatial density of
15](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-19-320.jpg)
![1.1. Gravitational lensing: general conception
where x and y are the angular positions of the image and the source, Dd, Ds
are angular diameter distances from the observer to the lens and to the source,
respectively, and Dds is that from the lens to the source. Because typical angles
in GLS are small, vectors x = (x1, x2) and y = (y1, y2) can be viewed as
Cartesian coordinates near the origin 2. Equation (1.1) is the well-known lens
equation, or the aberration equation 3.
The angle of gravitational deflection of rays Θg is determined in the MPS
approximation with the use of projected surface mass density σ(x) as
Θg(x) = −
4GDd
c2
∫
Σd
d2
x′ x − x′
|x − x′|2 σ
(
x′
)
, (1.2)
where G is the gravitational constant, c is the velocity of light in the vacuum,
and the integration is performed over the whole deflector area projected on
the sky. It should be noted that the sign of the two-dimensional deviation
angle Θg in equation (1.2) is opposite to that commonly used in the literature
(e.g., [107]) (though after substitution into (1.1) we get the equivalent result).
To our opinion, the minus sign is more appropriate because it reflects the real
physical situation. Indeed, the lensing mass attracts the light from the source
(not “repulses”), and the bending angle must be counted off the optical axis,
not in the reverse direction.
Equation (1.2) is often written in another form. Using the so called critical
surface mass density σcr = Dsc2/4πGDdDds and normalized surface mass
density κ = σ/σcr (microlensing optical depth), we obtain
y = x + η(x), (1.3)
where a two-dimensional vector of a “normalized” gravitational deflection is
introduced
η(x) = −
1
π
∫
Σd
d2
x′ x − x′
|x − x′|2 κ
(
x′
)
. (1.4)
Now let us consider focusing of radiation of a source with small angular
dimensions (a “point” source), around one isolated image x0 of a point source
at y0 = x0 + η(x0). For a smooth mass distribution the function η(x0) can
be linearized around x0. Then we choose the coordinates origins: at x0 in the
2
Correspondingly, we speak about the lens plane of coordinates (x1, x2) and the source
plane of (y1, y2). Thus the left-hand side of (1.1) realizes mapping of the lens plane onto the
source plane.
3
Sometimes the lens equation is written in terms of linear distances, i.e. Cartesian
coordinates in the real lens and source planes as it is shown in (1). This form of the lens
equation is obtained from (1.1) by rescaling x → Dd x, y → Ds y.
17](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-21-320.jpg)
![CHAPTER 1. Gravitational lensing
lens plane and at y0 in the source plane. The linearized lens equation is then
y = Q x, (1.5)
where Q is a so-called (two-dimensional) amplification matrix with the
elements
Qij = δij +
∂ηi
∂xj
, i, j = 1, 2.
It can be shown that (see, e.g., Schneider et al. [106])
∂η1
∂x2
=
∂η2
∂x1
,
∂η1
∂x1
+
∂η2
∂x2
= −2κ(x).
Therefore the spur tr Q = 2(1 − κ). In a proper reference frame
Q = diag [λ−, λ+] =
(
1 − κ − γ, 0
0, 1 − κ + γ
)
, (1.6)
λ± = 1 − κ ± γ being the eigenvalues of the amplification matrix, and we
introduced the shear γ = (λ+ − λ−)/2.
It is known from the theory of nonabsorbing lenses (including GLs) that
the brightness along an infinitely thin light ray does not change in focusing.
Therefore, change of the source magnitude as observed through the lens is due
to the change of angular dimensions. The amplification factor µj for the j-th
image of a point source can be determined from comparison of the image area
and the source area. The result can be expressed by means of the determinant
of the amplification matrix [10,107]:
µj = (1 − κ)2
− γ2
−1
x=xj
. (1.7)
The total amplification of a gravitational lens is equal to the sum of µj for
individual images:
µ =
N
∑
j=1
µj. (1.8)
The quantity
µ (x) = [1 − κ (x)]2
− γ2
(x)
−1
(1.9)
can be regarded as an amplification field for images of some “point” source.
Sometimes a magnitude sign is omitted in considering µ(x), that is, a relati-
onship µ(x) = {[1 − κ(x)]2 − γ2(x)}
−1
is merely considered. In doing so,
images with µ 0 are called direct, and those with µ 0 are referred to
as inverted ones.
The field of amplification µ(x) may reach infinitely large values at some
points x = xcr of the lens plane which are called critical points. The locus of
18](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-22-320.jpg)
![1.1. Gravitational lensing: general conception
points where µ(x) → ∞ is called a critical curve. Substituting xcr into the lens
equation yields the caustics in the source plane:
ycs = xcr + η (xcr). (1.10)
When the point source approaches the caustic curve from the caustic “light”
side (y → ycs), some of the observed images (called critical) are coming closer.
When the source is projected exactly at the caustic (y = ycs), the images
approach each other and merge to form a single image situated exactly at the
lens critical curve. The magnitude of the merging pair is growing infinitely in
the process. As the source is moving further (a transit to the caustic “shadow”),
the merged images are gradually disappearing.
Besides amplification of an image, there is another important characteristic
of GLS: the time delay of signals coming from a source and seen in its lensed
images (macroimages). S. Refsdal [94] was the first to notice that the time
delays, which are different for different quasar images, can be used to derive
the estimates of the Hubble constant, which is one of the most important
cosmological parameters determining the age and scale of the Universe. The
analysis presented for the first time in [21] has shown that for cosmological
distances the formulas for propagation times of signals differ from those for
the flat space-time only by a multiplier (1 + zd), which takes into account
the Universe expansion during the time as the signal comes to an observer.
Omitting the details of derivation (for this see, e.g., in [75, 107]), we present
the final expression for the time delay difference ∆t(xj, xm) between the signals
from two visible quasar images xj and xm coming to the observer [30,107]:
c∆t (xj, xm) = (1 + zd) Dd
xj
∫
xm
{
Θg
(
x′
)
−
1
2
[Θg (xj) + Θg (xm)]
}
dx′
. (1.11)
The integration is performed in the lens plane along an arbitrary path connecti-
ng points xj and xm.
The expressions presented above form a basis not only for a geometrical-
optics analysis of the GL phenomenon, but also play an important role in
solving the inverse problem of recovering parameters of gravitating masses from
deformations of images of remote sources as observed through the gravitational
fields.
19](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-23-320.jpg)
![CHAPTER 1. Gravitational lensing
1.1.2. Recovery of gravitating objects’
parameters from observations of the GL phenomenon
Before coming to specific results of investigations in the fra-
mework of the program “Cosmomicrophysics”, some peculiarities of solving the
inverse problems should be mentioned, which concern recovering parameters
of gravitating objects from observations of the GL phenomenon.
In studying characteristics of the GL phenomenon provided by observati-
ons, the latter can be divided into two groups: 1) observations of distorted
images of remote objects, and 2) observations of gravitating masses which
produce the lensing effect. The observables for images are their redshifts, shapes
and angular positions, relative amplifications, as well as differential time delays.
It should be noted that, because of the GL effect, observations of undistorted
source images are impossible. For deflectors, the principal observables are their
redshifts, velocity dispersions and positions of centroids of their brightness di-
stributions. For nearby galaxies, however, more detailed characteristics of mass
distributions can be obtained from observations (certainly, without a contri-
bution from a “hidden” constituent).
It was as early as in 1964 that S. Refsdal proposed a simple method to inde-
pendently determine the Hubble constant and the mass of a gravitating object
[94]. The idea of his method can be demonstrated with the use of the afore-
cited expression (1.11) for the time delays between the signals seen in different
lensed images. The angular coordinates xj of compact images, redshifts of
the source zs and lens zd, as well as the time delays ∆tjm(xj, xm) are the
measured quantities in (1.11). The mass distribution in the galaxy and angular
diameter distances, which enter the lens equation (1.1), (1.2), are unknown
quantities. Dependence of the angular diameter distance on the source redshift
is determined, first of all, by the cosmological model of the Universe. In the
theory of gravitational lensing produced by cosmological objects, the models by
Friedman—Lemaitre with Robertson—Walker metrics (FLRW) are commonly
used. This model is characterized by several principal parameters. This is,
first of all, the value of the Hubble constant at the present epoch, H0, that
is often written, in view of the measure of uncertainty, in the form H0 =
= 100 · h [km/s Mpc], where the dimensionless parameter h lies within 0.6
h 0.9. In standard ΛCDM cosmological model, in addition to H0, we have
a parameter of cold matter density, ΩM , then ΩΛ related with the cosmological
constant Λ, and finally, the curvature parameter Ωk associated with the current
curvature radius Rk of space. It follows from the Friedmann equations that
these three parameters must satisfy a condition ΩM + ΩΛ + Ωk = 1 for a
uniform and isotropic model. According to the ideas based on the data of
observations, we have the following values of parameters at the present epoch:
ΩM + ΩΛ ≈ 1 (ΩM ≈ 0.23; ΩΛ ≈ 0.75), that is, Ωk ≈ 0, [14,86,112]. Equality
20](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-24-320.jpg)
![1.1. Gravitational lensing: general conception
of parameter Ωk ≈ 0 to zero means that R−1
k ≈ 0, and the three-dimensional
space is Euclidean with a high degree of accuracy. For a model with Ωk = 0,
the angular diameter distance between two objects with redshifts z1 and z2
(z2 ≥ z1) is determined as follows [39]:
D1−2 (z1, z2) =
c
H0
1
1 + z2
z2
∫
z1
dz′
√
ΩM (1 + z′)3
+ ΩΛ
. (1.12)
Setting z1 = 0 (observer), and z2 = zd (deflector), one will have D1−2 = Dd
(the angular diameter distance between the observer and GL). For z1 = 0 and
z2 = zs (source), we have D1−2 = Ds for the angular diameter distance from
the source to observer. And finally, if z1 = zd, and z2 = zs then D1−2 = Dds
(the angular diameter distance from the lens to source). For angular diameter
distances for other cosmological FLRW models see, e.g., Kayser et al. [45].
With all the aforesaid taken into account, the initial expression can be
written in the following general form:
H0 = Υ(zd, zs; the Universe model) ×
× T(xj, xm; deflector model) /∆t (xj, xm), (1.13)
where Υ (zd, zs; the Universe model) and T (xj, xm; deflector model) are di-
mensionless functions, which depend on particular values of redshifts and on
the accepted model of the Universe, as well as on the observed angular coordi-
nates of the lensed images and the assumed deflector model, respectively. If
the cosmological parameters (the Universe model) and surface mass distri-
bution in a deflector, including its hidden constituent (deflector model) were
known precisely enough we would easily determine the unknown quantity H0
having measured zd, zs, xj, xm and ∆t(xj, xm). For a simple GL model, e.g.
central point mass model, [94], the deflector mass can also be determined from
Equation (1.2). Unfortunately, an attractive idea of Refsdal had met a number
of formidable difficulties in reality. Without going into details, note briefly the
following.
• Mathematically, the inverse problem is an ill-posed one. The detailed
analysis made by Falco et al. [30] has shown that there exists a class of ma-
thematical transformations, which remain the data of optical observations wi-
thout changes. This means that an unambiguous determination of the desired
parameters from observations of only GL phenomenon is impossible. Involving
additional data on interaction of matter and on gravitational fields in a broad
wavelength range is necessary.
• Determining the time delay is often difficult because of a high level of
various random noises. Microlensing events play a particular role in this case.
They occur when star-like compact bodies intersect, in their random motions in
the lens galaxy, the light rays corresponding to individual quasar images. The
21](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-25-320.jpg)
![CHAPTER 1. Gravitational lensing
microlensing events result in random variations of brightness of the observed
source images.
• Serious problems occur also in the attempts to account for the hidden
mass, for which the distribution in galaxies and clusters of galaxies in the
wide range of spatial scales is unknown so far. Though the hidden mass has
a gravitational effect on propagation of radiation from a distant source, only
indirect estimates of a spatially local distribution of the invisible matter can
be made from the GL phenomenon.
• In fact, we cannot even formulate the inverse problem in its strict state-
ment. Cosmologically, we deal with the data of observations referring to a “sing-
le” point of the space, “single” direction of arriving the radiation, and “single”
time moment. Estimating parameters of the Universe and sources of radiation,
and determining the total mass is only possible in the framework of solving
direct problems with the use of simulation. The final result is determined by
the available initial data of observations and by intuition of a researcher.
Summarizing, we may conclude that the inverse problem of determining
the parameters of the Universe and searching for the hidden mass with the use
of the GL phenomenon is an intricate problem which needs joint efforts both
from observers and theoreticians working in various fields.
1.2. Spatial structure
of quasars from microlensing studies
As was noted above, images of a gravitationally lensed quasar
(macroimages) may change its brightness because of possible microlensing
events. The effect of microlensing is totally analogous to the phenomenon
of scintillations on the medium inhomogeneities, which is well known in op-
tics and radio physics. While the effects of scintillation in optics and radio
astronomy are produced by moving irregularities of the atmosphere or space
plasma, in gravitational focusing we deal with irregularities of gravitational
fields produced by compact objects (microlenses). Various applications of this
phenomenon are known from the theory of scintillations. In particular, valuable
information about spatial scales and velocities of the medium irregularities can
be obtained from the statistical analysis of the data of observations, and fi-
ne structure of the sources of radiation can be examined with a high angular
resolution. Similarly, the analysis of observations of microlensing is capable
of providing information about mass function of microlenses and about their
velocities in galaxies. Also, a possibility emerges to estimate dimensions of
quasar structure with a resolution, which is presently inaccessible for observati-
ons. The recent work by Minakov and Vakulik [75] is dedicated to applicati-
on of the methods of statistical radio physics to the analysis of gravitational
microlensing.
22](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-26-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
Fig. 1.2. Images of the gravitati-
onally lensed quasar Q 2237 + 0305
(the Einstein Cross) obtained from
the Maidanak Observatory at six
different epochs. The magnitudes
of the components caused by mi-
crolensing events are clearly seen to
change in time
The following inferences from observations of the “near” and “distant”
microlensing can be regarded as most important: 1) only a minor fraction
of the Galaxy dark matter (about 20 %) is represented by compact masses;
2) the masses responsible for the microlensing effects observed are rather small:
M ∼ 10−1M⊙.
As was already noted, the phenomenon of microlensing provides the un-
precedented possibility to study spatial structure of remote quasars at micro-
arcsecond angular scales. This is a vital issue for astrophysics in itself, but in
addition, parameters of the quasar structure model provide important const-
raints to adequately interpret transient events in gravitationally lensed quasars.
In particular, as will be demonstrated in section 1.3, the source quasar dimensi-
ons have an effect on amplification probability density distributions caused by
microlensing, which are assumed to be diagnostic of the dark matter abundance
in lensing galaxies.
The quadruply lensed quasar Q 2237 + 0305 (Fig. 1.2) is the most promi-
sing object for microlensing studies, as was noted immediately after its dis-
covery [43, 44]. This is explained by the high optical density for microlen-
sing and by proximity of the lensing galaxy to the observer. Large amount of
observational data in the optical, IR, radio wavelengths, and in the X-rays is
assembled for this object, which have become the basis to determine the effecti-
ve source sizes in different spectral ranges and to infer the principal macro- and
microlensing parameters. The Kharkov group made a noticeable contribution
into the total database on this object, having provided it with the monitoring
data in three filters. The data cover the time period from 1997 till at least 2008.
In the next two sections, we show how these data can be used to investigate
the spatial structure of the Q 2237 + 0305 quasar and to estimate abundance
of smoothly distributed (dark) matter in the lensing galaxy.
The approaches to infer microlensing parameters from the light curves
may be divided into two classes. One of them is based upon the analysis of
23](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-27-320.jpg)
![CHAPTER 1. Gravitational lensing
individual microlensing events interpreted as the source crossing of a caustic
fold or cusp (e.g., [33,110,132,142]). Another approach referred hereafter as the
statistical one, utilizes all the available observational data to infer the statistical
parameters of interest. This approach is represented, e.g., by the structure
function analysis by Lewis and Irwin [65], or the analysis of distribution of
the Q 2237 light curve derivatives by Wyithe, Webster Turner [138–140].
Recently, Kochanek [58] applied a method of statistical trials to analyse the
well-sampled light curves of Q 2237 + 0305 obtained in the framework of the
OGLE monitoring campaign.
Both approaches have their intrinsic weak points and advantages. In parti-
cular, in analysing an individual microlensing event, it is necessary to presume
that the source actually crosses a single caustic, and that the source size
is significantly smaller than the Einstein ring radius of typical microlenses.
Moreover, there must be some complexity caused by the unknown vector dif-
ference between the microlens trajectory and the macrolens shear.
In applying the statistical approach, the microlensing parameters are ob-
tained through the analysis of the light curves as a whole, where a variety
of microlensing events with different circumstances may contribute simulta-
neously. This approach may encounter the problem of insufficiency of statis-
tics, however, and Q 2237 + 0305 is not an exclusion, in spite of its uniqueness:
according to [126] and [132], light curves of more than 100 years in duration
are needed to obtain reliable statistical estimates of microlensing parameters.
It should be noted here that in analysing microlensed light curves, one deals
with the well-known degeneracy between the principal parameters derived from
the observed microlensing light curves, namely, between the source size, mi-
crolens mass and transverse velocity. In fact, at least one of this parameters
should be fixed to determine the rest. To do this, some additional considerati-
ons are involved, which are not always applicable to an individual microlen-
sing event. This may be an important reason to give preference to statistical
methods. A significant complication inherent in both approaches is due to the
fact that the actual brightness distribution over the source quasar can not be
restored from the microlensed light curves, but instead, one may estimate only
a characteristic size parameter describing a certain photometric model of the
quasar adopted in simulations, e.g., full width on half-maximum (FWHM),
or parameter σ for a Gaussian brightness profile, etc. This is a well-known
consequence of difficulties associated with solving the inverse problems, noted
in the previous section.
24](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-28-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
1.2.1. The problem of spatial structure of quasars
The mechanism of accretion onto the massive black hole is
presently believed to provide the most efficient power supply in AGNs and
quasars, and effectively all researchers uses various accretion disk models when
interpreting microlensing events in gravitationally lensed quasars. However,
with the accretion disc being generally accepted as a central engine in quasars,
the difficulties in explaining the amplitudes of the long-term microlensed light
curves still remain, as well as in interpreting polarization and spectral properti-
es of quasar radiation and their variety.
It has long been understood that introducing of some additional structural
elements could resolve these discrepancies. In 1992, Jaroszyński, Wambsganss
and Paczyński [41] admitted existence of an outer feature of the quasar, that
reprocesses emission from the disk and may contribute up to 100% light in B
or V spectral bands. A bit later, Witt Mao [137] demonstrated in their si-
mulations of microlensed light curves of Q 2237, that a source model consisting
of a small central source surrounded by a much larger halo structure, would
better explain the observed amplitudes of the Q 2237 light curves.
Various candidates for the extended structural elements have been propo-
sed, such as, e.g., an envelope of high-velocity clouds or wind re-emitting the
hard X-ray radiation from the central engine as a network of broad emission
lines; gas outflows, which are believed to be launched from the central engine;
an equatorial torus containing the dark clouds that re-absorb the radiation
emanating in some directions (e.g., [6,13,29] and references therein).
There are many observational evidences for existence of these extended
structures in the Q 2237 + 0305 quasar. Mid-infrared observations of Q 2237 +
+ 0305 by Agol et al. [1] favor existence of a shell of hot dust extending between
1 pc and 3 pc from the quasar nucleus and intercepting about half of the quasar
luminosity. The flux ratios of the four Q 2237 macroimages measured at 3.6 cm
and 20 cm by Falco et al. [31] were also interpreted as originating in a source
much larger than that radiating in the optical wavelengths. Observations in
the broad emission lines (BEL) also suggest that they originate in a very large
structure, much larger than that emitting the optical continuum, [66,71,92,98,
111,131]. This is consistent with determinations of spatial scales of the broad-
line regions in AGNs obtained through reverberation mapping, [87]. Recently,
Pooley et al. [89] found out, through a comparison of the flux ratio anomalies in
the X-ray and optical bands for ten quadruply lensed quasars, that the optical
emission regions of quasars are much (by factors of 10—100) larger than the
basic disc models predict.
Microlensing light curves of these complicated source structures may noti-
ceably differ from those for a simple source structure represented by an accreti-
on disc alone. In particular, the accretion disc alone cannot reproduce in si-
25](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-29-320.jpg)
![CHAPTER 1. Gravitational lensing
mulations the observed amplitudes of the Q 2237 + 0305 light curves. While
providing good fits for the peaks, which are most sensitive to the effect of
the central source, it fails to provide the actual amplitudes of the rest of light
curves, [41]. In this respect, the results by Yonehara [142], Shalyapin et al. [110],
and Gil-Merino et al. [33], who analysed the regions of the light curves near the
peaks of HME, provide successful estimates of the central source, but ignore the
effect of a possible quasar outer feature. In 2003, Schild Vakulik [104] have
shown how the double-ring model of the quasar surface brightness distributi-
on, resulting from the Elvis’s [29] quasar spatial structure model, successfully
explains the rapid low-amplitude brightness fluctuations in light curves of the
First Lens Q0957 + 561. Interestingly, recent analysis [67] of brightness records
of 55 radio-quiet quasars monitored by the MACHO project indicates the
presence of large-scale outflow structures consistent with the Elvis [29] and
‘dusty torus’ [6] models of quasars.
A current concept of spatial structures of active galactic nuclei (AGN)
and quasars implies that there is a dusty torus lying outside the accretion disc
on the scale of several tenths of kiloparsec, with a broad-line emitting region
(BLR) in between. According to the recent inferences of Agol et al. [2] made
from their IR observations of Q 2237 + 0305, the accretion disc and the torus
may contribute in the total luminosity almost equally near one micron.
Polarimetric observations (e.g. [54]) raised a question about existence of
an additional electron scattering region (ESR) in quasars, [46], which has been
earlier detected around some AGNs and is presently believed to be responsible
for reprocessing emission from the accretion disk into polarized radiation [6,
113]. The exact geometry of the ESR, its dimensions and mutual location with
respect to the BLR are still poorly constrained at present, and seem to differ
in different objects. Future spectropolarimetric observations at different phases
of microlensing events are expected to be diagnostic concerning this issue.
In the sections to follow, we present the results of our analysis of two
most extensive monitoring data sets of Q 2237 + 0305 — those of the OGLE
(Optical Gravitational Lensing Experiment) group, obtained in filter V and
covering the time period from 1997 to 2008, and the data of monitoring the
Einstein Cross from the Maidanak observatory in filters V , R and I during
2001—2008, but with slightly lower sampling rate. We analyse these data
sets to test a two-component model of the Q 2237 quasar structure and to
determine parameters of this source model. In contrast to the ring model
proposed earlier, [104], we used a simplified model, consisting of a compact
central source and an extended outer structure with a much smaller surface
brightness. Such a model, being much easier for calculations, possesses the
principal property of the ring model to produce sharp peaks of the simulated
light curves, while damping the amplitudes of the entire microlensed light
curves.
26](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-30-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
Thus, our basic approach is to accept existence of the inner and outer
structural elements as detailed above, and to derive from parameter fitting only
the size of the inner luminous feature and the fraction of the total UV-optical
energy from the extended outer feature as compared to the luminosity origina-
ting in the compact central feature. We will show that the structural elements
of this two-component quasar model satisfactorily explain the observed light
curves amplitudes of the Q 2237 + 0305 image components and variations of
their color indices caused by microlensing.
1.2.2. Testing of the two-component
quasar model: application to Q 2237 + 0305
In the vicinity of a selected macroimage, the equation (1.5)
describes only a smooth part of the lens equation. In order to take into account
an effect of a highly inhomogeneous gravitational field due to stars (as well
as other compact objects) on the line of sight one must add an ensemble of
microlenses that may be assumed randomly distributed in the lens plane [43,
84]. This yields
y =
(
1 − κc − γ, 0
0, 1 − κc + γ
)
x −
∑
i
mi
x − xi
|x − xi|2
,
mi =
4GMi
c2
DdDds
Ds
,
(1.14)
κc is the microlensing optical depth of a smooth background, and Mi are
microlens masses.
Using equation (1.14) and the ray tracing method [107], it is possible to
calculate the distribution of magnification rate M(y) for a small (quasi-point)
source for all its possible locations y — the so-called magnification map. Magni-
fication of an extended source with a surface brightness distribution B(y) can
be calculated from the formula:
µ(y) =
∫
B(y′)M(y − y′)d2y′
∫
B(y′)d2y′
. (1.15)
For moving source we have a set of consecutive shifts of B(y) along the
source trajectory which generate dependence µ(t) and the light curve m(t) =
= −2.5 lg [µ(t)] + C. Once calculated, the magnification map can be used to
produce a large set of simulated light curves for various models of surface light
distribution B(y) over the source.
We simulated microlensing of a two-component source, with one of them
the compact, central luminous source having a surface brightness distributi-
on B1(y). The other, outer, structure, is associated with the larger structural
elements — a shell, a torus, Elvis’s biconics, [29] — and is characterized by
27](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-31-320.jpg)
![CHAPTER 1. Gravitational lensing
substantially lower surface brightness B2(y). For such a source the magnifi-
cation µ12 can be written as [120]:
µ12(y) =
µ1(y) + εµ2(y)
1 + ε
. (1.16)
The magnifications µ1 and µ2 are calculated according to (1.15) for the surface
brightness distributions B1 and B2, while ε is determined as a ratio of the
integral luminosities of these structures:
ε =
∫
B2(y) d2y
∫
B1(y) d2y
. (1.17)
The characteristic time-scale of the observed Q 2237 microlensing brightness
fluctuations is known to be almost a year. We infer from known cosmological
transverse velocities that such a scale is due to microlensing of the compact
inner quasar structure. Since the predicted spatial scale of the outer structure is
more than an order of magnitude larger as compared to the inner part [29,104],
the expected time scale of its microlensing brightness variations must exceed
ten years. So, because of the large dimensions of the extended structure, the
amplitudes of its magnification in microlensing must be noticeably less, as
compared to microlensing of the compact structure. Thus we conclude, that
on time-scales near 4 years, the magnification rate µ2(y) is almost invariable
and does not differ noticeably from the average magnification rate of the j-th
image component µj resulting from macrolensing: µ2(y) ≈ ⟨µ2(y)⟩ ≈ µj (j =
= 1, 2, 3, 4). Under these assumptions, equation (1.16) can be rewritten:
µ12(y) =
µ1(y) + εµj
1 + ε
. (1.18)
It is clear that, under such assumptions, microlensing of the extended
(outer) structure does not produce noticeable variations of magnification or bri-
ghtness fluctuations on the observationally sampled time-scales, and therefore
is effectively a brightness plateau above which the inner structure brightness
fluctuations are seen. So the observed inner region brightness fluctuations are
reduced by 1/(1 + ε).
Therefore, when analysing the light curves, we did not attempt to esti-
mate the size of the extended structure, and the accepted value of ε was the
only parameter which characterized the outer structure. The inner compact
structure of the source was simulated by the Gaussian surface brightness di-
stribution, and with its characteristic size expressed in units of the Einstein
ring radius of a typical microlens rs = r/rE at the one-sigma level as a fitted
parameter. We used this simple central source model because it is more easy for
computation. In doing so, we relied on the work by Mortonson et al. [80], who
28](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-32-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
examined the effect of the source brightness profile on the observed magnitude
fluctuations in microlensing. They used a variety of accretion disc models,
including Gaussian disk, and concluded that the statistics of microlensing
fluctuations depends mainly on the effective radius of the source, while being
relatively insensitive to a particular light distribution over the source disc.
In producing magnification maps, we accepted the values for microparame-
ters κ∗ and γ (shear) taken from Kochanek [58]: 0.392, 0.375, 0.743, and 0.635
for κ∗, and 0.395, 0.390, 0.733, 0.623 for γ, for the A, B, C and D components,
respectively. For each of the four Q 2237 + 0305 components, we calculated five
magnification maps with dimensions of 30 × 30 microlens Einstein radii, (a
pixel scale of 0.02 rE). We assumed here that the entire mass is concentrated
only in stars — that is, κc equals zero. To simplify computations, all the stars
were accepted as having the same mass. We are well aware that this assumption
is rather artificial, but in doing so, we refer to the works by Wambsganss [128]
and Lewis Irwin [64], who demonstrated, having used various mass functions,
that the resulting magnification probability distributions are independent of
the mass function of the compact lensing objects. For the sake of completeness,
the more recent works by Schechter, Wambsganss Lewis [101] and Lewis
Gil-Merino [63] should be mentioned, where simulations with two populations
of microlenses with noticeably differing masses were carried out to show that
the magnification probability distributions do depend on the mass function.
It is not surprising that such an exotic case has demonstrated the effect the
authors wanted to demonstrate. But it is of little relevance for our work, since
the more relevant calculation of Lewis Irwin [64] was made for more realistic
mass functions — for microlenses in 0.3M⊙, 10.0M⊙ stars, and for the Salpeter
mass function; these do not show appreciable dependence of the magnification
probability distribution on the adopted mass function.
For our analysis, we used the Q 2237 + 0305 light curves obtained in the V
filter by the OGLE group in 1997—2000. High sampling rate (2—3 datapoints
weekly) and low random errors are inherent in the photometric data from this
program. To compare with the simulated light curves, results of the OGLE
photometry were averaged within a night, thus providing 108 data points in
the light curve of each image component. For every set of the source model
parameters, we estimated the probability to produce good approximations to
the observed brightness curves. The values of the source model parameters
providing the maximum probabilities, were accepted as the parameter esti-
mates. To estimate consistency of the results, the analysis was carried out for
each of the four image components separately.
Quasars are known to be variable objects, and their luminosity may change
noticeably on time-scales of several years, months, and even days [26, 35, 91].
If a variable source is macrolensed, the intrinsic brightness variations will be
observed in each lensed image with some time delays. This is just the fact
29](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-33-320.jpg)
![CHAPTER 1. Gravitational lensing
that allows the Hubble constant to be determined from measurement of the
time delays. In analysing microlensing, however, variability of the source is an
interfering factor, which needs to be taken into account. In the Q 2237 + 0305
system, because of an extreme proximity of the lensing galaxy, (zd = 0.04),
and because of almost symmetric locations of the macroimages with respect
to the lens galaxy center, the expected time delays do not exceed a day, (e.g.
[105,129]). This is why the intrinsic brightness variations of the source would
reveal themselves as almost synchronous variations of brightnesses of all lensed
images. This was observed in 2003 [119], when the microlensing activity was
substantially subdued in all the four image components.
Generally, separation of the intrinsic brightness variations of the source
from the light curves containing microlensing events is a poorly defined and
intricate task. In our attempts to obtain the intrinsic source light curve for
Q 2237 + 0305, we introduced the following assumptions:
1. No effects of microlensing on the brightnesses of the components were
observed during the time interval from January to June, 2002 (JD 2090—
2250), when the magnitudes of all the components were almost unchanged. The
magnitude of each component was accepted as a zero level, and its brightness
variations were analysed relative to this level.
2. Relative to this zero level, we regarded that the closer these light curves
were to each other, the higher the probability that the components are not
microlensed within this time interval, while almost synchronous variations of
their brightness are due to changes of the quasar brightness. And vice versa,
the more the light curve of a component deviates from others, the larger the
probability that the component undergoes microlensing, which veils and di-
storts the source variations.
Thus, the weighted average variations of the component brightnesses with
respect to their zero levels were adopted as the estimate of the source bri-
ghtness curve, with the statistical weight for variations of every component
being selected depending on how close this brightness variation to variations
of other components is. The quasar intrinsic brightness curve, obtained on the
basis of these assumptions, as well as the OGLE light curves for the indivi-
dual images, reduced to their zero levels, are shown in Fig. 1.3. We expect the
largest source of error in estimating this intrinsic source brightness history to
be encountered during the time interval from August 1998 to December 1999,
when, possibly, all the components underwent microlensing.
To characterize similarity of the simulated and observed light curves, a χ2
statistics for each image component was chosen:
χ2
j =
NS
∑
j
[mj(ti) − Mj(ti, p)]2
σ2
j
, (1.19)
30](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-34-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
Fig. 1.3. The OGLE light curves obtained in the V filter in 1997—2005. The light curves
are reduced to the same (zero) level at the time interval JD 2090—2250. The solid grey line
is our estimate of brightness variations of the source quasar
where mj(ti) is the observed light curve of the j-th component, and Mj(ti, p)
is one of the simulated light curves, produced from a source trajectory at the
magnification map, and NS is a number of points in the observed light curve.
The magnification map was calculated for the source model described by a
set of parameters p, which could be varied. The quantity σ2
j characterizes the
errors of the observed light curve measurements, which are 0.032m, 0.039m,
and 0.038m for the A, B and C components, and 0.057m for the faintest D
component.
The probability that, for a given set of parameters p, a simulated light
curve will be close enough to the observed light curve, — that is, the value of
χ2 will happen to be less than some boundary value χ2
0, — such a probability
will be:
P(χ2
χ2
0) = lim
N→inf
Nχ2 χ2
0
Ntot
, (1.20)
where Nχ2 χ2
0
is a number of successful trials, and Ntot is a total number of
trials. The boundary value χ2
0/NS = 3 was adopted for calculations.
The direction of an image motion with respect to the shear is an important
parameter, which affects the probabilities noticeably. In [58], directions of moti-
on of each component were chosen randomly and independently of directions
of other components. This is not quite correct, since the directions of moti-
on of components are not independent, and are determined by the motion of
the source. Therefore, specifying the motion of one of the components must
automatically specify motions of other components, if the bulk velocities and
31](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-35-320.jpg)
![CHAPTER 1. Gravitational lensing
velocity dispersion of microlenses can be neglected [130, 139]. As a result of
almost perfect symmetry of Q 2237 + 0305, for any direction of the source moti-
on, directions of the opposite components with respect to the shear direction
must coincide, while motion for the two other images must be perpendicular to
the shear direction. That’s why, unlike the work by Kochanek [58], we analysed
trajectories for two selected directions at the magnification maps, — when the
A and B components are moving along the shear, with the C and D moving
transversely to it, and vice versa, when A and B are moving transversely to
the shear. Also, we did not undertake the local optimization of trajectories, as
in [58], since this may distort the estimates of probabilities.
Thus, for each magnification map, and for each of the two selected di-
rections, a map of distribution of the initial points of the trajectories can be
calculated, for which χ2 χ2
0. The probability (1.20) can then be calculated
as the relative area of such regions on the map.
To reduce computing time, the map of χ2 was calculated initially with a
coarse mesh, (∼0.3 r/rE), to localize the regions with low values of χ2. Then,
more detailed calculations with a finer mesh were carried out for only these
regions.
In our simulation, the following parameters could be varied: the radius
of the central compact feature rs, expressed in units of a microlens Einstein
ring radius rE; the brightness ratio, ε, expressing the ratio of the total outer
structure brightness to the total inner structure brightness; and the relative
transverse velocity of the source, Vt(rE/year), expressed in the units of the
Einsten ring radius of a microlens per year, which is also a scaling factor for
simulated light curves.
The search of probabilities for our three fitting parameters, r/rE, ε and Vt
is a rather complicated task, which needs much computing time. We attempted
to simplify it in the following way. Assuming that the effect of the outer
structure on the characteristic time scales of microlensing brightness variati-
ons is insignificant, we put ε = 0 at the first stage. Hence, a dependence of
probabilities on two parameters, — the scaling factor and relative dimension
of the compact feature, — was evaluated at the first stage. In Fig. 1.4, the di-
agrams are presented, which demonstrate distributions of probabilities to find
simulated light curves, which would be close to the observed ones, — dependi-
ng on the scaling factor and the compact feature dimension. The diagrams are
built for all the four components for two directions of the source motion: A and
B along the shear, C and D transversely — at the left, and C and D along the
shear, A and B transversely — at the right. Overall, a nearly linear dependence
of the scaling factor from the source dimension is seen. The largest values of
probability occur for r/rE ≈ 0.4 in the first case, and for r/rE ≈ 0.3 in the
second case. Since the maximum of probability distribution is, on average, at
the source radius of 0.35 rE and the velocity of 1.82 rE per year (see Fig. 1.4),
32](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-36-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
Fig. 1.6. Some of the most successful simulated light curves plotted against the observed
light curves. Microlensing brightness fluctuations beyond the time interval of fitting are
approximately of the amplitude and duration observed. The time scale is in the Julian
dates
fitting, and no unacceptably large brightness fluctuations are observed outside
this interval, unlike the results in fig. 10 of Kochanek [58].
Thus, the two-component source model consisting of a compact inner
structure and much larger outer structure with lower surface brightness, allows
to successfully model the brightness monitoring data and, importantly, to avoid
the effect from the standard accretion disc model that large amplitudes of mi-
crolensing brightness fluctuations are predicted but not observed [41,58].
The proposed source model consisting of two structures, — an inner
compact structure and an extended outer region, — provides higher values
of probability to find “good” simulated light curves as compared to the central
compact source alone, and produces better fits to long-term light curves.
The calculated distributions of the joint probabilities has well-marked
maxima, and their locations in the domain of parameters ε and r/rE allow
reasonable confidence in their determined values which provide the best fit
of the simulated light curves to the observed ones. The range for the most
probable values of the relative luminosity ε of the extended feature, determi-
ned from probability distributions of the individual macroimages, is between 1
and 3, while the estimate of the relative size of the compact central feature of
the quasar varies within a range of 0.1 r/rE 0.45. When determined from
distributions of the joint probabilities, the values of ε equal 2 in both cases,
while r/rE is about 0.4 for A and B motion perpendicular to γ, and 0.15 for
A and B moving parallel to the shear γ.
35](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-39-320.jpg)
![CHAPTER 1. Gravitational lensing
Very significantly, the simulated light curves calculated for the proposed
two-component source model with the parameters indicated above, do not tend
to unacceptably increase their amplitudes outside the time interval where they
were objectively selected according to the χ2 χ2
0 criterion, as is seen in fig. 10
from the work by Kochanek [58].
For better comparison with other authors, we adopted, following [58], a
probable projected cosmological transverse source velocity of Vt = 3300 km/s
to determine a linear size of the compact central source of r ≈ 2 · 1015 cm,
(1.2 · 1015 cm r 2.8 · 1015 cm). This size was estimated by Kochanek to
be between r ≈ 1.4 · 1015h−1 cm and 4.5 · 1015h−1 cm for the accretion disc
model and for the same transverse velocity, (h = 100/H0, where H0 is the
excepted value of the Hubble constant). For the relative size of the source of
0.3rE, (0.1rE r 0.45rE), the estimate for the average microlens mass is
⟨m⟩ = 1.88 · 10−3h2M⊙, (3.08 · 10−4h2M⊙ ⟨m⟩ 3.3 · 10−2h2M⊙).
1.2.3. Two-component quasar model
in interpreting chromatic microlensing
Though the gravitational lensing phenomenon is known to
be achromatic, the importance of observations at several spectral bands has
become understood long ago. As far back as in 1986, Kayser, Refsdal Stabell
[43] suggested that chromatic phenomena can be expected for microlensing of
a source with a radial temperature gradient. This possibility was later confir-
med in simulations by Wambsganss Paczyński [127]. In 1992, Jaroszyński,
Wambsganss and Paczyński [41] demonstrated in simulations that microlensing
maxima tend to be “bluer” than the rest of the light curve, with the expected
(B − R) microlensing colour changes in Q 2237 + 0305 as large as a few tenths
of a magnitude.
Rix et al. [95] and Corrigan et al. [22] were the first to conclude indepen-
dently that the colour indices of the Q 2237 + 0305 components seem to vary in
time. Their suggestion was later confirmed independently by Vakulik et al. [122]
and Burud et al. [12]. The first detailed analysis of variations of colour indices
in Q 2237 + 0305 was made by Vakulik et al. [118], who presented their long-
term V , R and I observations of Q 2237 + 0305 from the Maidanak Observatory
with the subsequent statistical analysis. Variations of the V − I colour indices
turned out to be correlated with the brightness changes. A tendency of the
components to become bluer towards the microlensing peaks predicted in [127]
and [41], has been confirmed. Moreau et al. [78] built the (V −R) vs V diagrams
from the data of the GLITP collaboration [3], and noted their similarity with
that reported in [118].
Observations of Q 2237 + 0305 in spectral ranges other than optical conti-
nuum (IR, radio and in the quasar emission lines, see references in section
1.2.1) — all indicate that the Q 2237 quasar structures emitting in these spectral
36](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-40-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
regions are almost nonsusceptible to microlensing and thus, must be much
larger than those emitting the UV and optical continuum.
Very interesting results concerning microlensing chromatic phenomena in
Q 2237 + 0305 have been recently reported in [27,28,81,111] and [4]. Mosquera
et al. [81] detected a chromatic microlensing event in image A of Q 2237 + 0305
through a single-epoch narrow-band photometry in eight different filters cove-
ring the wavelength interval 3510—8130 Å. Considering a Gaussian brightness
profile for the accretion disc (in a simple thin-disc model) they found from
simulations that the effective disc size varies in wavelength according to the
Rλ ∝ λ4/3 law, which follows from the thin accretion disc model by Shakura—
Syunyaev [109]. They also confirmed that the emission line regions of quasars
are much larger than that emitting the UV and visual light.
Eigenbrod et al. [27,28] presented the results of spectroscopic monitoring
of Q 2237 + 0305 during October 2004—December 2006, with the wavelength
coverage 4000—8000 Å. They conclude that the continuum and broad line
regions of the quasar are subjected to microlensing. They found that, accor-
ding to theoretic predictions, microlensing brightness variations are stronger at
shorter wavelengths, and restored the energy profile of the accretion disc. The
relative sizes of the accretion disc emitting at different wavelengths turned out
to follow the Rλ ∝ λβ law, with β = 1.2 ± 0.3 for the UV/optical continuum,
that is close to the result of Mosquera et al. [81].
Similar results for wavelength dependence of the effective source size are
reported by Anguita et al. [4], who used their g’- and r’-band photometry at
the Apache Point Observatory 3.5-m telescope together with the OGLE V -
band data to obtain 1.2—1.45 for the ratio of the source sizes in the r’-band to
that in the g’-band, with the 0.5—0.9 uncertainty, however.
Thus, as compared to the single-band observations, multi-colour observati-
ons are capable of providing a more comprehensive idea of a quasar’s spatial
structure and thus, of physical processes responsible for the observed properties
of quasar radiation.
Our data obtained at the Maidanak Observatory allow to see the signatures
of chromatic phenomena directly from comparison of the V , R and I light
curves of a particular microlensing event. This is illustrated by Fig. 1.7, where
the V , R and I light curves of the high-magnification event in image A started
in 2005 are shown. A tendency of amplitudes to decrease towards the longer
wavelengths is clearly seen in this picture.
However, a single microlensing event, no matter how prominent it may
be, is of a lower diagnostic value as compared to the statistics of all available
microlensing history of all the four Q 2237 image components. Below, we find
and analyse statistical relationships between the observed variations of colour
indices and magnitudes, and compare them to the results of microlensing si-
mulation fulfilled for a set of the quasar structure parameters.
37](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-41-320.jpg)
![CHAPTER 1. Gravitational lensing
Fig. 1.7. Brightness changes of the A component of Q 2237 + 0305 in the
V , R and I filters during the high magnification event started in 2005
(observations from the Maidanak Observatory). The amplitudes of the
event are clearly seen to decrease consecutively from filter V to R and I
To properly investigate microlensing phenomena, one has to disentangle
them from the intrinsic variability of quasars and from the effects of dust
extinction in lensing galaxies, which are known to act simultaneously and in a
similar way.
Yonehara, Hirashita Richter [143] have made an attempt to evaluate
relative contributions from these three factors. They conclude that the intrin-
sic quasar variability is hardly a dominant factor for the observed chromatic
phenomena in gravitationally lensed quasars from their sample counting about
25 systems. They claim that both dust extinction and microlensing are capable
of producing the observed behavior of colour indices of the lensed qua-
sar images.
We regard that one should be more careful concerning this issue. Accor-
ding to [134] and [34], variability of quasars increases towards the shorter
wavelengths, and most of quasars exhibits hardening of their spectra in the
bright phases. Giveon et al. [34] undertook an extensive statistical analysis
of variability properties of 42 quasars from the Palomar Green (PG) sample
having used the results of the 7-years monitoring in the Johnson-Cousins B
and R bands with the Wisa Observatory 1-m telescope. The main goal of their
study was to search for correlations between the variability properties and
other parameters of quasars, such as luminosity, redshift, radio loudness, etc.
38](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-42-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
Fig. 1.8. Q 2237 + 0305 light curves in filter V from 2001 to 2008; grey symbols — the OGLE
data, the Maidanak data are shown in darker symbols. The D light curve is 0.5 mag shifted
for better view
They confirmed previous finding that the spectra of quasars become harder
(bluer) at brighter phases, but found out that this trend holds for only a half
of their subsample. They presented the diagrams which demonstrate a rather
high correlation between variations of the B − R colours and variations of the
B and R magnitudes, with a regression line slope of approximately 0.25.
The Q 2237 + 0305 system is usually referred to as that one, where bri-
ghtness changes of all the four lensed quasar images are dominated by mi-
crolensing events. Recent observations have shown, however, that quasar vari-
ability may contribute noticeably to the Q 2237 + 0305 light curves. It is seen
especially during 2003—2006 (Fig. 1.8), when the microlensing activity was
somewhat subdued for images A, C and D (for image B, a noticeable microlensi-
ng event has happened during this time period). Recall that this fact had made
it possible to estimate the time delays in Q 2237 + 0305 [119]. Observations of
this time period have shown that the contribution of the source variations to
the observed light curves can not be neglected. To be sure of this, we esti-
mated contributions from the microlensing and quasar intrinsic variabilities
quantitatively, having applied a simple statistical approach to the whole R
light curve during 1997—2008. Having made use of the fact that the time
39](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-43-320.jpg)
![CHAPTER 1. Gravitational lensing
delays in Q 2237 + 0395 are negligibly small as compared to the variability
time-scales, we analysed the sums and differences of light curves for all the
six combinations of pairs of images to obtain the following values of the RMS
fluctuations: σS = 0.10 ± 0.03 mag for the intrinsic quasar brightness variati-
ons, and σmicr = 0.16±0.04 mag for microlensing variability (averaged over all
the four image components). Similar results were obtained from the analysis
of the OGLE V light curves.
Thus, the intrinsic brightness fluctuations of the Q 2237 + 0305 quasar do
contribute significantly to the observed light curves of the lensed images, and
therefore one must find a possibility to exclude the quasar constituent to study
the net microlensing colour and magnitude variations.
The most simple way to exclude the quasar variability is to analyse the dif-
ference light curves in the units of stellar magnitudes, where these two consti-
tuents are additive. Thus, we may form the difference light curves ∆RA−B,
∆RA−C, ∆RA−D, ∆RB−C, ∆RB−D, ∆RC−D, and the corresponding difference
colour curves, ∆(V − I), where ∆ means, similar to [118], deviations of the
corresponding quantities from their average values over the time interval under
consideration.
At first, we built the ∆(V − I) vs. ∆R diagram for all the four quasar
components from our V RI photometry of 2001—2008 and compared it to the
similar diagram for the 1995—2000 data shown in fig. 2 from [118]. The di-
agrams turned out to be very similar both in the regression line slopes and
in correlation factors, with the difference well within the error bars. However,
both the V −I colour indices and magnitudes in R varied within a larger range
for the 1995—2000 data as compared to the observations of 2001—2008. This
can be naturally explained by two unprecedented high-magnification events
happened to images A and C in 1999. Taking into account obviously different
mutual contributions of the microlensing variability and quasar intrinsic vari-
ability to the observed light curves during these two periods, the likeliness of the
diagrams can be regarded as an indication to the similar statistical characteri-
stics of these two types of variability (compare the B − R vs. R regression line
slope of 0.25 for PG quasars in fig. 6 from [34] with 0.28 in [118]).
Thus, to analyse the net microlensing statistics, we will further address
the difference colour and magnitude variations curves. Besides, it is reasonable
to expect that the difference curves calculated as described above will be less
affected by the inevitable photometry errors, both random, such as, e.g., night-
to-night error, and systematic ones. The difference ∆(V − I) vs. ∆R diagram
built for all possible combinations of pairs of images is shown in Fig. 1.9. Here
we used the photometry results obtained in observations from the Maidanak
Observatory during the time period from 2001 to 2008.
The regression line slope for these difference data points is insignificantly
larger than that in fig. 2 from the work by Vakulik et al. [118], while the
40](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-44-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
Fig. 1.9. A diagram showing correlation between variations of the (V − I)
colour indices (vertical axis) and variations of magnitudes in filter R,
calculated from the V RI observations in 2001—2008 with the 1.5-m
telescope at Maidanak Observatory
correlation index turned out to be noticeably higher, in accordance with
expectations noted above. Hence, we have a diagram representing statistics of
microlensing variations of colour indices exempted from the interfering effects
of the quasar variability, and therefore suitable for the further interpreting
in simulations. We argue that the regression line slope and correlation factor
of the ∆(V − I) vs. ∆R diagram are important parameters of microlensing
statistics, which are diagnostic for a spatial structure of the source quasar.
Wambsganss Paczynski [127] seem to be the first to simulate statisti-
cal dependencies between variations of colour indices and variations of bri-
ghtness for Q 2237 + 0305 caused by microlensing of a source with a radial
temperature gradient. They simulated the B and R microlensing light curves
for Q 2237 + 0305 for the Gaussian brightness profile of the quasar with the
half width radius varying between 2 · 1014 cm and 3.2 · 1015 cm, and found out
that |∆(B − R)| may be approximately proportional to |∆B| for some source
dimensions and for the ratios of the source sizes at the two spectral bands of
at least 1.3 or larger.
Similar to the previous section, we considered a photometric model of the
Q 2237 + 0305 quasar consisted of a compact central source at some brightness
pedestal as a case that is much more simple computationally, but results in
similar effects. We, again, did not attempt to specify the size and brightness
41](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-45-320.jpg)
![CHAPTER 1. Gravitational lensing
profile of the extended feature in our simulations, but intended to find the
relative energy contribution of the extended feature into the total quasar lumi-
nosity that would provide the best fit of the simulated ∆(V −I) vs. ∆R diagram
to that obtained from our V RI photometry.
The magnification maps were calculated using the inverse ray tracing
method, with the local lensing parameters κ∗ and γ accepted, similar to the
previous section, to be the same as in Kochanek [58]. To utilize all the statisti-
cs available from the generated magnification maps, we did not select source
trajectories and thus, did not use the corresponding light curves in the further
analysis, but instead, operated directly with data points of the maps themselves
to find the desired parameters. In this way, we excluded the unknown direction
of the source motion over the map, as well as its relative velocity.
To decrease the amount of calculations while having the size of magnifi-
cation maps sufficient enough to provide reasonable accuracy of the calculated
model parameters, we produced two sets of magnification maps differing in
their scales. For the source radii from rs = 0.05rE to rs = 1.0rE, three maps of
(46×46)rE in dimension with a 44pix/rE scale were calculated for each image
component, while for rs ranging from 0.5rE to 3rE, one (205 × 205)rE map
with a scale of 10pix/rE was used for each image component.
Since the correlation coefficient of the colour-magnitude diagram must be
sensitive to errors in the values of V − I and R, we added a random noise to
our magnification maps, thus making our simulations more realistic. The RMS
values of the imitated errors were accepted to equal those estimated for the
actual errors of our V RI photometry: 0.019m, 0.037m, 0.042m and 0.039m for
images A, B, C and D, respectively, and were made to be distributed normally.
As was shown in the previous section, the value of magnification µ12 in
microlensing of the two-component source consisting of a compact central
structure and an extended outer structure, can be represented by equati-
on (1.18).
The average magnifications can be adopted to equal those determined by
the local lensing parameters. With these taken from [105], and using the known
relationship between the normalized surface density in stars, normalised shear
parameter and macroamplification, (see equation (1.7) in section 1.1.1), we
have the following values of amplifications for the A, B, C and D components:
µA ≈ 4, µB ≈ 4.3, µC ≈ 2.45 and µD ≈ 4.9.
As was noted in section 1.2.1, some authors, e.g. Mortonson et al. [80],
claim that the effective size is a universal parameter of a source model for
microlensing simulations, and that a particular type of the brightness profile
virtually does not affect fluctuations of the simulated light curves. To check
if this is valid for simulation of colour fluctuations, we considered both the
accretion disc and a source with Gaussian surface brightness distribution as the
quasar models. We adopted a generalized power law for wavelength dependence
42](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-46-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
from the extended feature (three upper panels in the right-hand column), and
for the two-component source model with two sets of ελ (right-hand panels
in two bottom rows). For the central compact source alone, both Gaussian
and accretion disc, the parameters ρ and a of the ∆(V − I) vs. ∆R diagram
are seen to approach those of actually observed ones only for unrealistically
large sources (indicated in the plot). For the two-component source model, the
simulated diagrams are consistent with the observed ones for quite reasonable
values of the central source size.
It is interesting to note a horizontal strip-like condensation of data poi-
nts seen in the uppermost diagrams both in Fig. 1.10 and Fig. 1.11, that is
stretched parallel to the magnitude axis against the zero colour indices. A si-
milar condensation can also be seen in the ∆(V −I) vs. ∆V diagram simulated
by Mosquera et al. [81] for image A of Q 2237 + 0305 and shown in their fig. 5.
The origin of the strip becomes evident from the color curves in the upper left
panels of our Fig. 1.11, showing variations of the (V − I) colour index for a
very small source (Rs = 0.05rE) moving along a certain trajectory over the
magnification map, (the appropriate light curves in filters V and I are plotted
above the colour curves). The color curves clearly shows a high probability for
a small source to have low values of ∆(V − I), close to zero. Our simulations
show that the strip is gradually disappearing as the source size is increasing.
Since there is no signs of such a strip in the diagram built from the data of
observations (Fig. 1.9), its presence in the simulated diagrams can serve as an
indication to the unrealistically small source size.
One more important feature of the simulated ∆(V − I) vs. ∆V diagrams
should be noted. We overlaid the individual ∆(V − I) vs. ∆R curves resulted
from a particular source trajectory over the amplification map against the
clouds of points, which form the simulated diagrams (right-hand column in
Fig. 1.11). We see the entangled curves at the diagrams, which demonstrate
that relationship between variations of colour indices and magnitudes in mi-
crolensing is ambiguous. This curves demonstrate why the analysis of a single
event may produce wrong inferences about a quasar structure and thus, provi-
de a strong argument in favour of statistical approach to studying chromatic
events in microlensing.
To quantitatively interpret our multi-colour data, we abandoned simulati-
on of many individual V − I vs. R diagrams calculated for a set of specific
source parameters, but instead, chose to analyse behaviors of the diagram
parameters, namely, the correlation factor ρ and regression line slope a as
functions of the source parameters. At first, we analyzed if Gaussian source
alone can reproduce the observed ρ and a pair of the diagram parameters.
This can be seen in Fig. 1.12, where three curves illustrate three sets of the
ρ and a pairs calculated for three different values of β in the rλ ∼ λβ law
adopted for the Gaussian source profile (indicated in the plot). Each data
45](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-49-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
Fig. 1.13. Sequence of values for the pair of parameters ρ and a of the
∆(V − I) vs. ∆R diagram simulated for the accretion disc as a model for
the Q 2237 quasar, with the effective source radius rs varying in wavelength
according to the rs(λ) ∝ λ4/3
law. The effective source size grows from
0.05rE to 3rE from left to right again
Also, a tendency of the regression line slope a to approach the observed value
a = 0.30 ± 0.03 for very large source dimensions is observed again.
Thus, both Gaussian and accretion disc do not reproduce in simulations
the parameters of the ∆(V −R) vs. ∆R diagram (Fig. 1.9), built from our V RI
photometry, — a = 0.3 ± 0.03, ρ = 0.82 ± 0.10. As has been shown above and
presented in [120], a two-component photometric model of the source structure
provides much better fit to the observed amplitudes of the V light curves of
Q 2237 + 0305 as compared to the single central source model. The central
source size was estimated in that work to lie within 0.15rE and 0.4rE. This
is consistent with determinations of the Q 2237 + 0305 source size obtained by
other authors, from the early works, such as e.g., [125,126,132], and up to the
recent studies by, e.g., Kochanek et al. (2004) [58], Eigenbrod et al. (2008) [27],
Mosquera et al. (2009) [81]. Summarizing all available estimates of the central
source radius, we may adopt r = (0.3 ± 0.1)rE as the most probable value.
Therefore, our next step was to simulate microlensing ∆(V −I) vs. ∆R dia-
grams for the two-component source model and to find the model parameters,
which would provide a and ρ of the simulated diagram consistent with those
actually observed. Fig. 1.14 shows behaviors of the a and ρ pair simulated
for the two-component source model with three different values of the relative
47](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-51-320.jpg)
![1.2. Spatial structure of quasars from microlensing studies
Having averaged these differences over the whole magnification map, we
can calculate the RMS variations σµ
12 and σµ
1 for µ12 and µ1. Both magnifi-
cations and their RMS variations are understood to be functions of the central
source size r: σµ
1 = σµ
1 (r) and σµ
12 = σµ
12(r). This will result in ε being a
function of r too and thus, we can write for ε(r):
ε(r) =
σµ
1 (r)
σµ
12(r)
− 1. (1.23)
When checking the hypothesis of the two-component source model, σµ
12(r)
should be treated as the actually observed RMS flux variations σobs of quasar
images, while σµ
1 (r) = σsim(r) are those obtained from simulations with a
single-source model for a set of the central source sizes. Applying Eq. (1.23) to
the data taken in different filters, we have:
εV (r) =
σsim
V (r)
σobs
V
− 1; εR(r) =
σsim
R (r)
σobs
R
− 1; εI(r) =
σsim
I (r)
σobs
I
− 1, (1.24)
where the observed flux variations in filters V , R and I were obtained from
our light curves to equal σobs
V = 0.167, σobs
R = 0.142 σobs
I = 0.124. Using these
observational constraints and the values of σsim(r) obtained from simulation,
we can calculate the corresponding values of the relative integral brightness
of the extended structure as functions of the central source size. The result is
presented in Fig. 1.15.
For all the three filters the relative contributions of the extended feature
needed to explain parameters of the ∆(V −I) vs. ∆R diagram of Q 2237 + 0305
are seen to decrease as the central source radius increases. Now, to obtain
a reasonable solution, we must fix the size of the central source. Adopting
r = (0.3 ± 0.2)rE mentioned above for the compact source radius, we obtain
εV ≈ 1.3, εR ≈ 1.5, εI ≈ 1.6 for the relative energy contributions of the
extended structure of our two-component quasar model. The uncertainties are
±0.5, ±0.6 and ±0.8, respectively.
As is seen, the values of ε obtained with the approach based on compari-
son of fluctuations of the observed and simulated light curves expressed by
equations (1.24) are larger than those selected in a rather random manner in
direct simulation to fit the observed values of a and ρ (Fig. 1.14). This can
easily be understood from Eq. (1.24): the values of σobs
V , σobs
R and σobs
I were
obtained from the observed light curves, which, in contrast to the simulated
ones, cannot be regarded as the representative samples of microlensing bri-
ghtness variations (see, e.g. [84]). This must result in underestimation of σobs
and thus, in overestimation of ε from expressions (1.24). Therefore, the values
εV , εR, εI indicated above should be considered as the upper limits for the
relative energy contributions of the extended feature in filters V , R and I.
49](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-53-320.jpg)
![CHAPTER 1. Gravitational lensing
Fig. 1.15. Relative brightness of the extended feature in filters V , R and
I as a function of the quasar compact structure relative size expressed in
units of a microlens Einstein ring radius, rs/rE
Indeed, simulations show that lower values of εV , εR and εI much better
reproduce the observed ρ and a pair. Moreover, simulations show that a family
of solutions exists diverging not only in the values of ε, but also in the ratios
of ε in different spectral bands. Therefore, additional constraints are needed,
which would be based on more reliable data about the spectral properties of
the supposed extended feature. At the present level of our knowledge, we can
only argue that a contribution of the extended structure into the total quasar
emission grows towards the longer wavelengths, with the upper limits in the
V , R and I bands of εV ≈ 1.3, εR ≈ 1.5, εI ≈ 1.6
Thus, to explain the parameters of our ∆(V −I) vs. ∆R diagram, we, again,
had to admit existence of an extended structure around a compact central
source, with a rather large contribution into the total quasar emission. It is
obvious that the effect of such structure, when treated in terms of the Rλ ∝ λβ
law, is equivalent to the steeper wavelength dependence of the effective source
size as compared to that expected for the accretion disc. Meanwhile, the most
recent observations of chromatic microlensing in Q 2237 + 0305 [2, 5, 28, 81]
and in other lensed quasars give the value of β in the λβ law varying around
4/3 predicted by the thin accretion disc model.
To comment discrepancy between our results and the results of the authors
cited above, we would like to refer to the recent work by Mosquera et al. [81],
50](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-54-320.jpg)
![This ebook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg License included with this
ebook or online at www.gutenberg.org. If you are not located in the
United States, you will have to check the laws of the country where
you are located before using this eBook.
Title: Hell's Hatches
Author: Lewis R. Freeman
Release date: January 9, 2014 [eBook #44632]
Most recently updated: October 23, 2024
Language: English
Credits: Produced by Greg Bergquist, Ernest Schaal, and the Online
Distributed Proofreading Team at http://www.pgdp.net
(This
file was produced from images generously made available
by The Internet Archive/American Libraries.)
*** START OF THE PROJECT GUTENBERG EBOOK HELL'S HATCHES
***](https://image.slidesharecdn.com/18628186-250604184525-c4803c77/85/Dark-Matter-Astrophysical-Aspects-Of-The-Problem-Shulga-Vm-60-320.jpg)
Dark Matter Astrophysical Aspects Of The Problem Shulga Vm
- 1. Dark Matter Astrophysical Aspects Of The Problem Shulga Vm download https://ebookbell.com/product/dark-matter-astrophysical-aspects- of-the-problem-shulga-vm-37256372 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Visible And Dark Matter In The Universe A Short Primer On Astrophysical Dynamics Giuseppe Bertin https://ebookbell.com/product/visible-and-dark-matter-in-the-universe- a-short-primer-on-astrophysical-dynamics-giuseppe-bertin-47399530 Manifestations Of Dark Matter And Variations Of The Fundamental Constants In Atoms And Astrophysical Phenomena 1st Edition Dr Yevgeny V Stadnik Auth https://ebookbell.com/product/manifestations-of-dark-matter-and- variations-of-the-fundamental-constants-in-atoms-and-astrophysical- phenomena-1st-edition-dr-yevgeny-v-stadnik-auth-6617054 Knowledge In A Nutshell Astrophysics The Complete Guide To Astrophysics Including Galaxies Dark Matter And Relativity Sten Odenwald https://ebookbell.com/product/knowledge-in-a-nutshell-astrophysics- the-complete-guide-to-astrophysics-including-galaxies-dark-matter-and- relativity-sten-odenwald-11023650 Dark Matter Michelle Paver https://ebookbell.com/product/dark-matter-michelle-paver-47767298
- 3. Dark Matter A Ghost Story Michelle Paver https://ebookbell.com/product/dark-matter-a-ghost-story-michelle- paver-48664962 Dark Matter 5 By Dust Consumed 1st Edition Don Bassingthwaite https://ebookbell.com/product/dark-matter-5-by-dust-consumed-1st- edition-don-bassingthwaite-49075576 Dark Matter Independent Filmmaking In The 21st Century Michael Winterbottom https://ebookbell.com/product/dark-matter-independent-filmmaking-in- the-21st-century-michael-winterbottom-50673762 Dark Matter Credit Philip T Hoffman Gilles Postelvinay Jeanlaurent Rosenthal https://ebookbell.com/product/dark-matter-credit-philip-t-hoffman- gilles-postelvinay-jeanlaurent-rosenthal-50730882 Dark Matter Of The Mind The Culturally Articulated Unconscious Daniel L Everett https://ebookbell.com/product/dark-matter-of-the-mind-the-culturally- articulated-unconscious-daniel-l-everett-51438378
- 6. NATIONAL ACADEMY OF SCIENCES OF UKRAINE INSTITUTE of RADIO ASTRONOMY MAIN ASTRONOMICAL OBSERVATORY TARAS SHEVCHENKO NATIONAL UNIVERSITY OF KYIV V.N. KARAZIN KHARKIV NATIONAL UNIVERSITY НАЦIОНАЛЬНА АКАДЕМIЯ НАУК УКРАЇНИ РАДIОАСТРОНОМIЧНИЙ IНСТИТУТ ГОЛОВНА АСТРОНОМIЧНА ОБСЕРВАТОРIЯ КИЇВСЬКИЙ НАЦIОНАЛЬНИЙ УНIВЕРСИТЕТ iменi ТАРАСА ШЕВЧЕНКА ХАРКIВСЬКИЙ НАЦIОНАЛЬНИЙ УНIВЕРСИТЕТ iменi В.Н. КАРАЗIНА
- 8. UDK 524.8, 539 BBK 22.6, 22.3 D20 Reviewers: I.L. ANDRONOV, Dr. Sci., Prof., Head of Department of the Odesa National Maritime University O.L. PETRUK, Dr. Sci., Leading researcher of the Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NASU Approved for publication by: Scientific Council of the Institute of Radio Astronomy of NASU (June, 2013) Scientific Council of the Main Astronomical Observatory of NASU (June, 2013) Publication was made possible by a State contract promoting the production of scientific printed material D20 Dark energy and dark matter in the Universe: in three volumes / Editor V. Shulga. — Vol. 2. Dark matter: Astrophysical aspects of the problem / Shulga V.M., Zhdanov V.I., Alexandrov A.N., Berczik P.P., Pavlenko E.P., Pavlenko Ya.V., Pilyugin L.S., Tsvetkova V.S. — K. : Akademperiodyka, 2014. — 356 p. ISBN 978-966-360-239-4 ISBN 978-966-360-253-0 (vol. 2) This monograph is the second issue of a three volume edition under the general title “Dark Energy and Dark Matter in the Universe”. It concentrates mainly on astrophysical aspects of the dark matter and invisible mass problem including those of gravitational lensing, mass distribution, and chemical abundance in the Universe, physics of compact stars and models of the galactic evolution. The monograph is intended for science professionals, educators and graduate students, specializing in extragalactic astronomy, cosmology and general relativity. UDC 524.8, 539 BBK 22.6, 22.3 ISBN 978-966-360-239-4 ISBN 978-966-360-253-0 (vol. 2) c ⃝ Shulga V.M., Zhdanov V.I., Alexandrov A.N., Berczik P.P., Pavlenko E.P., Pavlenko Ya.V., Pilyugin L.S., Tsvetkova V.S., 2014 c ⃝ Akademperiodyka, design, 2014
- 9. CONTENTS 5 1 CHAPTER 2 CHAPTER 3 CHAPTER FOREWORD OF THE EDITOR . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Gravitational lensing as a key to solving the dark matter problem Shulga V.M., Minakov A.A., Vakulik V.G., Smirnov G.V., Tsvetkova V.S. 1.1. Gravitational lensing: general conception. . . . . . . . . . . . . . . 11 1.2. Spatial structure of quasars from microlensing studies 22 1.3. Dark matter content from probability density distributi- ons of microlensing amplifications. . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.4. Flux ratio anomalies as a key to detect dark matter substructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.5. Time delay lenses: impact on the Hubble constant and/or the dark matter problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Qualitative problems in gravitational microlensing Zhdanov V.I., Alexandrov A.N., Fedorova E.V., Sliusar V.M. 2.1. Gravitational lensing: a short overview. . . . . . . . . . . . . . . . . 85 2.2. Approximate solutions of the lens equation in the vicinity of the high amplification events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.3. Amplification of extended sources near the fold . . . . . . . . 99 2.4. Astrometric gravitational microlensing. . . . . . . . . . . . . . . . 111 Chemical evolution of late type galaxies of different masses Pilyugin L.S. 3.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.2. The chemical abundances in nearby galaxies . . . . . . . . . . 137 3.3. The maximum attainable value of the oxygen abundan- ce in spiral galaxies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
- 10. Contents 3.4. The redshift evolution of oxygen and nitrogen abundan- ces in late type galaxies. Galaxy downsizing . . . . . . . . . . . . . . . 159 3.5. Galaxy downsizing and the origin of the scatter in the N/H—O/H diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.6. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4 C H A P T E R Dissipative N-body gasodynamical model of the triaxial protogalaxy collapse Berczik P.P., Kravchuk S.G., Spurzem R., Hensler G. 4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4.2. Dissipative N-body code for galaxy evolution . . . . . . . . . 193 4.3. Galaxy as dynamical system with accreting cold gas halo 200 4.4. Gasodynamical model of the triaxial protogalaxies col- lapse. Isothermal and adiabatic solutions . . . . . . . . . . . . . . . . . . 209 4.5. Chemo-dynamical SPH code for evolution of star formi- ng disk galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.6. Chemo-photometric evolution of star forming disk ga- laxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 4.7. Multi-phase chemo-dynamical hardware accelerated SPH code for galaxy evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 4.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 5 C H A P T E R Cataclysmic variables — the faint interacting close binary stars at the late stage of evolution Pavlenko E.P. 5.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 5.2. Observations and results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 6 C H A P T E R Ultracool dwarfs in our Galaxy Pavlenko Ya.V. 6.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 6.2. M-dwarfs of later spectral classes . . . . . . . . . . . . . . . . . . . . . 320 6.3. Brown dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6.4. Low mass objects of spectral classes L-T-Y . . . . . . . . . . . 335 6.5. Exoplanets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 6.6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 ABOUT THE AUTHORS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
- 11. FOREWORD OF THE EDITOR 7 At present it is no surprise that the major part of matter in the Universe is either invisible or dark. More surprising is the fact that first evidence for the existence of dark matter and dark energy (and the overall matter-energy content in the Universe) have been established by the methods of astronomy, i.e. by observing emissions from baryonic objects which are bright objects. Since the discovery of the dark Uni- verse dominated by dark matter and dark energy, astronomi- cal observations continue to be the main tool for studying the composition and distribution of dark matter, and the effect of dark energy on the evolution of the Universe. The present book is the second in a three volume mono- graph under the general title “Dark Energy and Dark Matter of the Universe”. It concentrates mainly on astrophysical aspects of the dark matter problem. The aim of this book is to provide a self-contained description of the data which underlie our present understanding of the physical processes in the baryonic Universe containing dark matter. The book is divided into several parts according to the content of the material. Chapters 1 and 2 summarize a sig- nificant fraction of the theoretical and observational results obtained by two research teams from Kharkov and Kyiv in Ukraine. They are dedicated to problems of gravitatio- nal lensing effects that provide a direct proof of the exi- stence of dark matter. Chapter 3 presents material on the composition of galactic objects, focusing on the formation and evolution of galaxies. Chapter 4 contains special to- pics relating to the evolution of galactic and multi-galactic systems, using the techniques of hydrodynamic simulation and N-body numerical modeling, with account of certain
- 12. Foreword of the Editor cosmological assumptions. Chapters 5 and 6 concern compact galactic objects that form a most numerous population of small sized stars and are an important part of the poorly observable baryonic matter. In Chapter 5 close binary stars are discussed with emphasis on the late stage of their evolution, including such compact objects as dwarf stars and neutron stars, or black holes. Studies of physical conditions and evolution processes in low-mass stars of another kind are presented in Chapter 6, also discussing the formation of spectra of ultra cold and brown dwarfs. The list of contributors to Chapter 1 includes the names of Prof. Anatoly Minakov and Victor Vakulik, two researchers who were pioneers of strong gravi- tational lensing studies in Ukraine. Both of them left this world too early, passing away before this book has been completed. We will remember. V. SHULGA
- 13. ACKNOWLEDGMENTS We are grateful to the National Academy of Sciences of Ukraine for the financial support of the Target Scienti- fic Research Programs “Structure and composition of the Universe, hidden mass and dark energy” (2007—2009), “Astrophysical and cosmological problems of hidden mass and dark energy of the Universe” (2010—2012) [“CosmoMi- croPhysics”], within which this monograph was originated. We are thankful to academician Valerij Shulga, the editor of this monograph, for the suggestion of its writing, partici- pation in the formation of its content, and organization of its publication. We also thank academician Yaroslav Yatskiv for his generous support of these Programs. We also appreci- ate helpful and highly professional assistance from the staff of the academic publisher “Akademperiodyka”. AUTHORS Kyiv, June, 2013
- 15. C H A P T E R GRAVITATIONAL LENSING AS A KEY TO SOLVING THE DARK MATTER PROBLEM V.M. Shulga, A.A. Minakov, V.G. Vakulik, G.V. Smirnov, V.S. Tsvetkova 1.1. Gravitational lensing: general conception 11 The second part of the XX century occurred to be very suc- cessful for astronomers, astrophysicists and cosmologists. In the first 60ths, the most remote sources of radiation, — the quasars — were discovered [85]. A few years later, pulsa- ting sources — the pulsars — were detected [38]. And fi- nally, a decade later existence of space mirages — gravi- tational lenses (GL) — was confirmed [124]. Discovery of the first GL has stimulated dedicated searches for other lenses. To do this, most powerful instruments of the opti- cal and radio wavelength ranges were being used, and even some new instruments dedicated to the search for mani- festations of the gravitational lensing phenomenon have been created 1. Several international programs dedicated to the GL problems have been approved, which remain to be operational so far. The interest to the space “mirages” is due, first of all, to a possibility of solving a set of topical astrophysical and cosmological problems inherent in this phenomenon. They are, for example, determination of the Hubble constant, as well as studies of the fine structure of the quasars’ emitting regions with the highest resolution that is as yet unachieva- ble for ground-based and even space-based observations. Last time, the GL phenomenon turned out to be very efficient in searching for the hidden mass in the Universe 1 Sometimes this phenomenon is also referred to as the “gravitati- onal focusing”.
- 16. CHAPTER 1. Gravitational lensing and estimating its abundance. As it became evident about three quarters of a century ago, the observed rotation curves of galaxies can only be explained by an assumption that the visible matter is immersed into a massive extended but invisible formation, which was named a dark matter halo. A considerable attention has been paid to find out the halo constituents, and several moni- toring programs that used the gravitational lensing has been carried out to clear up what fraction of the halo mass is formed by MACHOs — Massive Compact Halo Objects. The answer is that the halo predominantly consists of the non-baryonic dark matter. On the other hand, a lot of compact objects has been found by using gravitational lensing, however not in the halo, but in the Galactic disk. These objects may include brown dwarfs, white dwarfs, planets with masses 10−5M⊙ M 10−3M⊙, neutron stars and even low-mass black holes. A question about the relative content of these objects is presently vital for solving the problem of the Universe origin and evolution. The essence of the gravitational lensing phenomenon is deflection of the light rays in the gravitational field of a massive object. I. Newton was the first to ask a question: can it be that bodies act upon the light at some distance to bend the light rays, and would this action be (all factors being the same) the strongest at the least distance? A history of developing the theory and observations of the GL phenomenon can be found in the books by P. Bliokh and A. Minakov [10], P. Schneider, J. Ehlers and E. Falco [107] and A. Zakha- rov [147]. In 1804, German astronomer J. Zoldner determined the angle of deviation of the light beam from a star in its transition near the edge of the solar disc and obtained 0′′.87 [107]. A hundred and ten yeas later, A. Einstein considered the effect of the light ray deflection in the gravitational field of the Sun in the framework of his general relativity theory (GRT). His first result, obtained in 1911, did not differ from that of Zoldner. In 1915, however, a new work by A. Einstein appeared, where a revised value was presented, which is twice as much as compared to the previous estimate. The first experimental confirmati- on of gravitational deflection of the light rays was obtained in observations of the total solar eclipse on the 29th of May, 1919. By now a number of various manifestations of the effects of GL in the optical and radio wave length ranges at various spatial scales reaches hundreds. Three types of gravitational lensing are presently differentiated, each of which has its intrinsic potentials as concerned to solving the dark matter problem: weak lensing, microlensing and strong lensing. The term “weak lensing” is used when a lens produces a single slightly distorted image of a particular source. Weak lensing reveals itself in observati- ons of vast sky areas of very distant objects (quasars, galaxies) through the gravitational fields of massive foreground clusters of galaxies. In addition to small deformations of shapes of the distant source images, minor changes of 12
- 17. 1.1. Gravitational lensing: general conception their brightness and distribution over the sky occur. In principle, peculiarities of these distortions provide a clue to restore the pattern of the gravitational potential of the cluster. To reveal the dark matter, the restored distribution of the gravitational potential can be then compared to that caused by distribution of the visible (luminous) matter. Addressing to the weak lensing phenomenon with the aim to detect the hidden mass usually implies statistical analysis of very large sky areas obtai- ned in the deep surveys. For example, the well known work by Jee et al. [42] reports detection of a ring-like dark matter substructure in the rich galaxy cluster Cl 0024 + 17 at a radius of r ≈ 75′′ surrounding a soft, dense core at r ≤ 50′′. To restore distribution of the total gravitational potential of the cluster, the authors fulfilled statistical analysis of 1300 images of background objects produced by both weak and strong lensing. They interpret this ring-like substructure as the result of a line-of-sight collision of two massive clusters. Another impressive result of detecting the peculiar dark matter pattern with the use of the weak lensing is the Bullet cluster 1E0657-558 [18], which consists of two colliding clusters of galaxies. Microlensing is divided into a “near” microlensing (called also “galactic” microlensing) and a “distant” (extragalactic) one. The galactic microlensing is meant when a star located in the Galaxy or in the Magellanic clouds changes its brightness due to gravitational focusing by a compact object of our Galaxy crossing the line of sight between the star and observer. The extragalactic microlensing is spoken about when a compact object of a lensing galaxy passes close to the line of sight resulting in essential brightness change of a particular lensed quasar image. In 1986, B. Paczyński [83] proposed a peculiar method to investigate a population of the halo of our Galaxy, which implies observations of the gravi- tational lensing events of stars in the Milky Way by other stars. Since the probability of observing an individual event of such a kind is very small (P ≈ 10−6 events per year), Paczyński proposed to monitor several milli- on stars simultaneously, for example, in the Magellanic Clouds. Analogous programs dealt with monitoring of stars near the nucleus of the Milky Way. Several dedicated programs have been launched, such as MACHO, EROS and, somewhat later, OGLE, which resulted in detecting several hundred microlensing events. The amplitudes and durations of light curves of these events contain information about masses of compact bodies and their velocities. As was noted above, microlensing events may be produced by compact bodies not only in our Galaxy, but also in the other galaxies, which lens remote quasars. Observations of gravitationally lensed quasars show that microlensing is a phenomenon of a rather high occurrence, — almost all of them demonstrate microlensing activity that is different in different objects. Of all the known 13
- 18. CHAPTER 1. Gravitational lensing gravitationally lensed quasars, the highest microlensing activity is observed in the quadruple lens system Q 2237 + 0305, the Einstein Cross. Strong lensing is spoken about when either multiple, or arc-like, or ring- like images of a remote source (a quasar) are formed in gravitational focusing. It takes place when a sufficiently massive galaxy occurs to be close to the path of light rays from a distant quasar to the observer. The observer will see then, instead of one source, several distorted images of the same quasar formed by the gravitational field of the galaxy. The phenomena of strong lensing and microlensing provide several app- roaches to solving the dark matter problem which are based on the long-term observations of transient events in images of gravitationally lensed quasars. We mention here some of them. 1. The method of histograms. Each of the images of a gravitationally lensed quasar (macroimage) may change its brightness because of possible microlen- sing events, which are signaling that a compact object (a star or a planet) is moving near the line of sight corresponding to a particular macroimage. The method of histograms is based on studying the magnification probability di- stributions for macroimages due to microlensing events. Numerical simulation shows that such distributions are sensitive to the relationship between the mass fraction in compact objects and that one in the uniformly distributed matter, or in the objects with extremely small masses. 2. Flux ratio anomaly method. It has been long noticed that flux ratios of the lensed quasar components are poorly reproduced by the lens models with smooth (regular) distribution of gravitational potential (the problem of flux ratio anomalies). Since 2001, the anomalies of mutual fluxes observed in many quasar lenses are thought to be caused by the presence of the dark matter substructures in lensing galaxies. Existence of these substructures is predicted by the scenario of hierarchical formation of structures in the Universe. 3. The method of time delays proposed in 1964 by S. Refsdal [94] to determi- ne the Hubble constant, is based on the fact that fluctuations of the intrinsic brightness of a quasar are seen in its lensed images with the delays, which are determined — at a given system geometry — by the surface mass density distribution in a lens galaxy and by the Hubble constant value. Below the basic results of observations and theoretical studies of the GL phenomenon will be presented, which were being carried out by a joint Kharkiv and Kiev group consisting of researches from the Institute of Radio Astronomy of the National Academy of Sciences of Ukraine, Institute of Astronomy of V.N. Karazin National University of Kharkiv and Astronomical Observatory of Taras Shevchenko National University of Kyiv. The works were supported by the target Program of the National Academy of Sciences of Ukraine “Investi- gation of the Universe structure and composition, hidden mass and dark energy” (“Cosmomicrophysics”). 14
- 19. 1.1. Gravitational lensing: general conception 1.1.1. Basic equations of the gravitational focusing phenomenon In the theory of GL, a system of coordinates presented in Fig. 1.1 is usually used, which is convenient for a terrestrial observer. The OZ axis passes through the point of observations P and the mass center of a gravitating object O. In this coordinate system, a source of radiation with small angular dimensions (a “point source”) is situated in the source plane with angular coordinates y. When considering the effect of lensing produced by gravitational fields of complicated objects, such as galaxies and stellar clusters, one should take into account that their masses are contained not only in the large-scale dif- fusely distributed structures (e.g. dust or gas clouds, dark matter), but also in compact objects. A number of such objects inside a cluster is large enough. For example, a number of stars in a spiral galaxy similar to the Galaxy can reach N ≈ 1010 ÷ 1011. It should be also noted that in addition to stars, star-like and planet-like bodies can act as microlenses, and their number is also large enough inside a cluster or a galaxy. As a rule, the effects of microlensing are investigated theoretically with the use of a two-dimensional model for distribution of stars in a cluster. In doing so, a uniform in the average distribution of the microlens stars is consi- dered, with the correlation between their positions being neglected. Though such representation does not allow the “fine” effects to be predicted from the observed brightness fluctuations of microlensed images, it is justified by the following. Firstly, we use the minimal number of free parameters in this representation, and secondly, our ignorance of the actual spatial distributions of stars in the clusters makes the model refinements resulting in minor quanti- tative corrections to be senseless, (see, e.g. [10,107]). The gravitational field averaged over large interstellar volumes of matter distribution provides a global lensing effect of the cluster as a whole. Such lens was called a macrolens in literature. Compact bodies randomly distri- buted inside the cluster produce the lensing effect too, though of a smaller scale. The phenomenon of focusing in gravitational fields of compact “low-mass” objects was named microlensing. Though the spatial scales of microlenses are small, their effect results in significant brightness variations of macro-images of sources — quasars. An extended large-scale mass constituent in the cluster will be further referred to as a “diffuse” one, while compact masses will be named a “stellar” constituent. The approximations of quasi-statics, geometrical optics and a thin phase screen form the basis of the theory. A validity of applying the method of the phase screen (MPS) is usually justified by the following considerations. Excep- ting heavily populated regions around galactic nuclei, the spatial density of 15
- 20. CHAPTER 1. Gravitational lensing Fig. 1.1. Mutual locations of a source quasar (S), observer (P) and a lens galaxy (GL), where the microlens-stars are present microlens stars in the galaxy is small enough. In this case, the effects of radi- ation re-scattering from one microlens to another can be neglected, and this allows the thin screen approximation to be used. All microlenses are assumed in the theory of microlensing as being located in one the same plane coinci- ding with that passing through the mass center of the macrolens normally to axis OZ, which connects the observer P and the macrolens mass center at the origin z = 0 (Fig. 1.1). The geometrical-optics description of the gravitational focusing phenome- non is as follows. As a rule, a source and observer are both at large distances from a gravitating mass, and deflection of light rays occurs only within a small region around the gravitating mass. These two circumstances make it possible to substitute the actual curved rays by their “linear” asymptotes. So the light rays from the source are assumed to reach the phase screen (z = −0) travelling along the “straight” lines and after the phase screen (z = +0) the light wave acquires an additional phase shift that produces a sharp bending of the ray at an angle Θg that will be treated as a two-dimensional vector on the unit sphere. The rays refracted at the screen propagate again along the “straight” lines into the right semi-space (z 0). According to Fig. 1.1, a vector equation can be written for a point-like element of the source surface. This equation allows to select, of the whole set of rays refracted at the screen, only those ones passing through the point of observation: y = x + Dds Ds Θg(x), (1.1) 16
- 21. 1.1. Gravitational lensing: general conception where x and y are the angular positions of the image and the source, Dd, Ds are angular diameter distances from the observer to the lens and to the source, respectively, and Dds is that from the lens to the source. Because typical angles in GLS are small, vectors x = (x1, x2) and y = (y1, y2) can be viewed as Cartesian coordinates near the origin 2. Equation (1.1) is the well-known lens equation, or the aberration equation 3. The angle of gravitational deflection of rays Θg is determined in the MPS approximation with the use of projected surface mass density σ(x) as Θg(x) = − 4GDd c2 ∫ Σd d2 x′ x − x′ |x − x′|2 σ ( x′ ) , (1.2) where G is the gravitational constant, c is the velocity of light in the vacuum, and the integration is performed over the whole deflector area projected on the sky. It should be noted that the sign of the two-dimensional deviation angle Θg in equation (1.2) is opposite to that commonly used in the literature (e.g., [107]) (though after substitution into (1.1) we get the equivalent result). To our opinion, the minus sign is more appropriate because it reflects the real physical situation. Indeed, the lensing mass attracts the light from the source (not “repulses”), and the bending angle must be counted off the optical axis, not in the reverse direction. Equation (1.2) is often written in another form. Using the so called critical surface mass density σcr = Dsc2/4πGDdDds and normalized surface mass density κ = σ/σcr (microlensing optical depth), we obtain y = x + η(x), (1.3) where a two-dimensional vector of a “normalized” gravitational deflection is introduced η(x) = − 1 π ∫ Σd d2 x′ x − x′ |x − x′|2 κ ( x′ ) . (1.4) Now let us consider focusing of radiation of a source with small angular dimensions (a “point” source), around one isolated image x0 of a point source at y0 = x0 + η(x0). For a smooth mass distribution the function η(x0) can be linearized around x0. Then we choose the coordinates origins: at x0 in the 2 Correspondingly, we speak about the lens plane of coordinates (x1, x2) and the source plane of (y1, y2). Thus the left-hand side of (1.1) realizes mapping of the lens plane onto the source plane. 3 Sometimes the lens equation is written in terms of linear distances, i.e. Cartesian coordinates in the real lens and source planes as it is shown in (1). This form of the lens equation is obtained from (1.1) by rescaling x → Dd x, y → Ds y. 17
- 22. CHAPTER 1. Gravitational lensing lens plane and at y0 in the source plane. The linearized lens equation is then y = Q x, (1.5) where Q is a so-called (two-dimensional) amplification matrix with the elements Qij = δij + ∂ηi ∂xj , i, j = 1, 2. It can be shown that (see, e.g., Schneider et al. [106]) ∂η1 ∂x2 = ∂η2 ∂x1 , ∂η1 ∂x1 + ∂η2 ∂x2 = −2κ(x). Therefore the spur tr Q = 2(1 − κ). In a proper reference frame Q = diag [λ−, λ+] = ( 1 − κ − γ, 0 0, 1 − κ + γ ) , (1.6) λ± = 1 − κ ± γ being the eigenvalues of the amplification matrix, and we introduced the shear γ = (λ+ − λ−)/2. It is known from the theory of nonabsorbing lenses (including GLs) that the brightness along an infinitely thin light ray does not change in focusing. Therefore, change of the source magnitude as observed through the lens is due to the change of angular dimensions. The amplification factor µj for the j-th image of a point source can be determined from comparison of the image area and the source area. The result can be expressed by means of the determinant of the amplification matrix [10,107]: µj = (1 − κ)2 − γ2 −1 x=xj . (1.7) The total amplification of a gravitational lens is equal to the sum of µj for individual images: µ = N ∑ j=1 µj. (1.8) The quantity µ (x) = [1 − κ (x)]2 − γ2 (x) −1 (1.9) can be regarded as an amplification field for images of some “point” source. Sometimes a magnitude sign is omitted in considering µ(x), that is, a relati- onship µ(x) = {[1 − κ(x)]2 − γ2(x)} −1 is merely considered. In doing so, images with µ 0 are called direct, and those with µ 0 are referred to as inverted ones. The field of amplification µ(x) may reach infinitely large values at some points x = xcr of the lens plane which are called critical points. The locus of 18
- 23. 1.1. Gravitational lensing: general conception points where µ(x) → ∞ is called a critical curve. Substituting xcr into the lens equation yields the caustics in the source plane: ycs = xcr + η (xcr). (1.10) When the point source approaches the caustic curve from the caustic “light” side (y → ycs), some of the observed images (called critical) are coming closer. When the source is projected exactly at the caustic (y = ycs), the images approach each other and merge to form a single image situated exactly at the lens critical curve. The magnitude of the merging pair is growing infinitely in the process. As the source is moving further (a transit to the caustic “shadow”), the merged images are gradually disappearing. Besides amplification of an image, there is another important characteristic of GLS: the time delay of signals coming from a source and seen in its lensed images (macroimages). S. Refsdal [94] was the first to notice that the time delays, which are different for different quasar images, can be used to derive the estimates of the Hubble constant, which is one of the most important cosmological parameters determining the age and scale of the Universe. The analysis presented for the first time in [21] has shown that for cosmological distances the formulas for propagation times of signals differ from those for the flat space-time only by a multiplier (1 + zd), which takes into account the Universe expansion during the time as the signal comes to an observer. Omitting the details of derivation (for this see, e.g., in [75, 107]), we present the final expression for the time delay difference ∆t(xj, xm) between the signals from two visible quasar images xj and xm coming to the observer [30,107]: c∆t (xj, xm) = (1 + zd) Dd xj ∫ xm { Θg ( x′ ) − 1 2 [Θg (xj) + Θg (xm)] } dx′ . (1.11) The integration is performed in the lens plane along an arbitrary path connecti- ng points xj and xm. The expressions presented above form a basis not only for a geometrical- optics analysis of the GL phenomenon, but also play an important role in solving the inverse problem of recovering parameters of gravitating masses from deformations of images of remote sources as observed through the gravitational fields. 19
- 24. CHAPTER 1. Gravitational lensing 1.1.2. Recovery of gravitating objects’ parameters from observations of the GL phenomenon Before coming to specific results of investigations in the fra- mework of the program “Cosmomicrophysics”, some peculiarities of solving the inverse problems should be mentioned, which concern recovering parameters of gravitating objects from observations of the GL phenomenon. In studying characteristics of the GL phenomenon provided by observati- ons, the latter can be divided into two groups: 1) observations of distorted images of remote objects, and 2) observations of gravitating masses which produce the lensing effect. The observables for images are their redshifts, shapes and angular positions, relative amplifications, as well as differential time delays. It should be noted that, because of the GL effect, observations of undistorted source images are impossible. For deflectors, the principal observables are their redshifts, velocity dispersions and positions of centroids of their brightness di- stributions. For nearby galaxies, however, more detailed characteristics of mass distributions can be obtained from observations (certainly, without a contri- bution from a “hidden” constituent). It was as early as in 1964 that S. Refsdal proposed a simple method to inde- pendently determine the Hubble constant and the mass of a gravitating object [94]. The idea of his method can be demonstrated with the use of the afore- cited expression (1.11) for the time delays between the signals seen in different lensed images. The angular coordinates xj of compact images, redshifts of the source zs and lens zd, as well as the time delays ∆tjm(xj, xm) are the measured quantities in (1.11). The mass distribution in the galaxy and angular diameter distances, which enter the lens equation (1.1), (1.2), are unknown quantities. Dependence of the angular diameter distance on the source redshift is determined, first of all, by the cosmological model of the Universe. In the theory of gravitational lensing produced by cosmological objects, the models by Friedman—Lemaitre with Robertson—Walker metrics (FLRW) are commonly used. This model is characterized by several principal parameters. This is, first of all, the value of the Hubble constant at the present epoch, H0, that is often written, in view of the measure of uncertainty, in the form H0 = = 100 · h [km/s Mpc], where the dimensionless parameter h lies within 0.6 h 0.9. In standard ΛCDM cosmological model, in addition to H0, we have a parameter of cold matter density, ΩM , then ΩΛ related with the cosmological constant Λ, and finally, the curvature parameter Ωk associated with the current curvature radius Rk of space. It follows from the Friedmann equations that these three parameters must satisfy a condition ΩM + ΩΛ + Ωk = 1 for a uniform and isotropic model. According to the ideas based on the data of observations, we have the following values of parameters at the present epoch: ΩM + ΩΛ ≈ 1 (ΩM ≈ 0.23; ΩΛ ≈ 0.75), that is, Ωk ≈ 0, [14,86,112]. Equality 20
- 25. 1.1. Gravitational lensing: general conception of parameter Ωk ≈ 0 to zero means that R−1 k ≈ 0, and the three-dimensional space is Euclidean with a high degree of accuracy. For a model with Ωk = 0, the angular diameter distance between two objects with redshifts z1 and z2 (z2 ≥ z1) is determined as follows [39]: D1−2 (z1, z2) = c H0 1 1 + z2 z2 ∫ z1 dz′ √ ΩM (1 + z′)3 + ΩΛ . (1.12) Setting z1 = 0 (observer), and z2 = zd (deflector), one will have D1−2 = Dd (the angular diameter distance between the observer and GL). For z1 = 0 and z2 = zs (source), we have D1−2 = Ds for the angular diameter distance from the source to observer. And finally, if z1 = zd, and z2 = zs then D1−2 = Dds (the angular diameter distance from the lens to source). For angular diameter distances for other cosmological FLRW models see, e.g., Kayser et al. [45]. With all the aforesaid taken into account, the initial expression can be written in the following general form: H0 = Υ(zd, zs; the Universe model) × × T(xj, xm; deflector model) /∆t (xj, xm), (1.13) where Υ (zd, zs; the Universe model) and T (xj, xm; deflector model) are di- mensionless functions, which depend on particular values of redshifts and on the accepted model of the Universe, as well as on the observed angular coordi- nates of the lensed images and the assumed deflector model, respectively. If the cosmological parameters (the Universe model) and surface mass distri- bution in a deflector, including its hidden constituent (deflector model) were known precisely enough we would easily determine the unknown quantity H0 having measured zd, zs, xj, xm and ∆t(xj, xm). For a simple GL model, e.g. central point mass model, [94], the deflector mass can also be determined from Equation (1.2). Unfortunately, an attractive idea of Refsdal had met a number of formidable difficulties in reality. Without going into details, note briefly the following. • Mathematically, the inverse problem is an ill-posed one. The detailed analysis made by Falco et al. [30] has shown that there exists a class of ma- thematical transformations, which remain the data of optical observations wi- thout changes. This means that an unambiguous determination of the desired parameters from observations of only GL phenomenon is impossible. Involving additional data on interaction of matter and on gravitational fields in a broad wavelength range is necessary. • Determining the time delay is often difficult because of a high level of various random noises. Microlensing events play a particular role in this case. They occur when star-like compact bodies intersect, in their random motions in the lens galaxy, the light rays corresponding to individual quasar images. The 21
- 26. CHAPTER 1. Gravitational lensing microlensing events result in random variations of brightness of the observed source images. • Serious problems occur also in the attempts to account for the hidden mass, for which the distribution in galaxies and clusters of galaxies in the wide range of spatial scales is unknown so far. Though the hidden mass has a gravitational effect on propagation of radiation from a distant source, only indirect estimates of a spatially local distribution of the invisible matter can be made from the GL phenomenon. • In fact, we cannot even formulate the inverse problem in its strict state- ment. Cosmologically, we deal with the data of observations referring to a “sing- le” point of the space, “single” direction of arriving the radiation, and “single” time moment. Estimating parameters of the Universe and sources of radiation, and determining the total mass is only possible in the framework of solving direct problems with the use of simulation. The final result is determined by the available initial data of observations and by intuition of a researcher. Summarizing, we may conclude that the inverse problem of determining the parameters of the Universe and searching for the hidden mass with the use of the GL phenomenon is an intricate problem which needs joint efforts both from observers and theoreticians working in various fields. 1.2. Spatial structure of quasars from microlensing studies As was noted above, images of a gravitationally lensed quasar (macroimages) may change its brightness because of possible microlensing events. The effect of microlensing is totally analogous to the phenomenon of scintillations on the medium inhomogeneities, which is well known in op- tics and radio physics. While the effects of scintillation in optics and radio astronomy are produced by moving irregularities of the atmosphere or space plasma, in gravitational focusing we deal with irregularities of gravitational fields produced by compact objects (microlenses). Various applications of this phenomenon are known from the theory of scintillations. In particular, valuable information about spatial scales and velocities of the medium irregularities can be obtained from the statistical analysis of the data of observations, and fi- ne structure of the sources of radiation can be examined with a high angular resolution. Similarly, the analysis of observations of microlensing is capable of providing information about mass function of microlenses and about their velocities in galaxies. Also, a possibility emerges to estimate dimensions of quasar structure with a resolution, which is presently inaccessible for observati- ons. The recent work by Minakov and Vakulik [75] is dedicated to applicati- on of the methods of statistical radio physics to the analysis of gravitational microlensing. 22
- 27. 1.2. Spatial structure of quasars from microlensing studies Fig. 1.2. Images of the gravitati- onally lensed quasar Q 2237 + 0305 (the Einstein Cross) obtained from the Maidanak Observatory at six different epochs. The magnitudes of the components caused by mi- crolensing events are clearly seen to change in time The following inferences from observations of the “near” and “distant” microlensing can be regarded as most important: 1) only a minor fraction of the Galaxy dark matter (about 20 %) is represented by compact masses; 2) the masses responsible for the microlensing effects observed are rather small: M ∼ 10−1M⊙. As was already noted, the phenomenon of microlensing provides the un- precedented possibility to study spatial structure of remote quasars at micro- arcsecond angular scales. This is a vital issue for astrophysics in itself, but in addition, parameters of the quasar structure model provide important const- raints to adequately interpret transient events in gravitationally lensed quasars. In particular, as will be demonstrated in section 1.3, the source quasar dimensi- ons have an effect on amplification probability density distributions caused by microlensing, which are assumed to be diagnostic of the dark matter abundance in lensing galaxies. The quadruply lensed quasar Q 2237 + 0305 (Fig. 1.2) is the most promi- sing object for microlensing studies, as was noted immediately after its dis- covery [43, 44]. This is explained by the high optical density for microlen- sing and by proximity of the lensing galaxy to the observer. Large amount of observational data in the optical, IR, radio wavelengths, and in the X-rays is assembled for this object, which have become the basis to determine the effecti- ve source sizes in different spectral ranges and to infer the principal macro- and microlensing parameters. The Kharkov group made a noticeable contribution into the total database on this object, having provided it with the monitoring data in three filters. The data cover the time period from 1997 till at least 2008. In the next two sections, we show how these data can be used to investigate the spatial structure of the Q 2237 + 0305 quasar and to estimate abundance of smoothly distributed (dark) matter in the lensing galaxy. The approaches to infer microlensing parameters from the light curves may be divided into two classes. One of them is based upon the analysis of 23
- 28. CHAPTER 1. Gravitational lensing individual microlensing events interpreted as the source crossing of a caustic fold or cusp (e.g., [33,110,132,142]). Another approach referred hereafter as the statistical one, utilizes all the available observational data to infer the statistical parameters of interest. This approach is represented, e.g., by the structure function analysis by Lewis and Irwin [65], or the analysis of distribution of the Q 2237 light curve derivatives by Wyithe, Webster Turner [138–140]. Recently, Kochanek [58] applied a method of statistical trials to analyse the well-sampled light curves of Q 2237 + 0305 obtained in the framework of the OGLE monitoring campaign. Both approaches have their intrinsic weak points and advantages. In parti- cular, in analysing an individual microlensing event, it is necessary to presume that the source actually crosses a single caustic, and that the source size is significantly smaller than the Einstein ring radius of typical microlenses. Moreover, there must be some complexity caused by the unknown vector dif- ference between the microlens trajectory and the macrolens shear. In applying the statistical approach, the microlensing parameters are ob- tained through the analysis of the light curves as a whole, where a variety of microlensing events with different circumstances may contribute simulta- neously. This approach may encounter the problem of insufficiency of statis- tics, however, and Q 2237 + 0305 is not an exclusion, in spite of its uniqueness: according to [126] and [132], light curves of more than 100 years in duration are needed to obtain reliable statistical estimates of microlensing parameters. It should be noted here that in analysing microlensed light curves, one deals with the well-known degeneracy between the principal parameters derived from the observed microlensing light curves, namely, between the source size, mi- crolens mass and transverse velocity. In fact, at least one of this parameters should be fixed to determine the rest. To do this, some additional considerati- ons are involved, which are not always applicable to an individual microlen- sing event. This may be an important reason to give preference to statistical methods. A significant complication inherent in both approaches is due to the fact that the actual brightness distribution over the source quasar can not be restored from the microlensed light curves, but instead, one may estimate only a characteristic size parameter describing a certain photometric model of the quasar adopted in simulations, e.g., full width on half-maximum (FWHM), or parameter σ for a Gaussian brightness profile, etc. This is a well-known consequence of difficulties associated with solving the inverse problems, noted in the previous section. 24
- 29. 1.2. Spatial structure of quasars from microlensing studies 1.2.1. The problem of spatial structure of quasars The mechanism of accretion onto the massive black hole is presently believed to provide the most efficient power supply in AGNs and quasars, and effectively all researchers uses various accretion disk models when interpreting microlensing events in gravitationally lensed quasars. However, with the accretion disc being generally accepted as a central engine in quasars, the difficulties in explaining the amplitudes of the long-term microlensed light curves still remain, as well as in interpreting polarization and spectral properti- es of quasar radiation and their variety. It has long been understood that introducing of some additional structural elements could resolve these discrepancies. In 1992, Jaroszyński, Wambsganss and Paczyński [41] admitted existence of an outer feature of the quasar, that reprocesses emission from the disk and may contribute up to 100% light in B or V spectral bands. A bit later, Witt Mao [137] demonstrated in their si- mulations of microlensed light curves of Q 2237, that a source model consisting of a small central source surrounded by a much larger halo structure, would better explain the observed amplitudes of the Q 2237 light curves. Various candidates for the extended structural elements have been propo- sed, such as, e.g., an envelope of high-velocity clouds or wind re-emitting the hard X-ray radiation from the central engine as a network of broad emission lines; gas outflows, which are believed to be launched from the central engine; an equatorial torus containing the dark clouds that re-absorb the radiation emanating in some directions (e.g., [6,13,29] and references therein). There are many observational evidences for existence of these extended structures in the Q 2237 + 0305 quasar. Mid-infrared observations of Q 2237 + + 0305 by Agol et al. [1] favor existence of a shell of hot dust extending between 1 pc and 3 pc from the quasar nucleus and intercepting about half of the quasar luminosity. The flux ratios of the four Q 2237 macroimages measured at 3.6 cm and 20 cm by Falco et al. [31] were also interpreted as originating in a source much larger than that radiating in the optical wavelengths. Observations in the broad emission lines (BEL) also suggest that they originate in a very large structure, much larger than that emitting the optical continuum, [66,71,92,98, 111,131]. This is consistent with determinations of spatial scales of the broad- line regions in AGNs obtained through reverberation mapping, [87]. Recently, Pooley et al. [89] found out, through a comparison of the flux ratio anomalies in the X-ray and optical bands for ten quadruply lensed quasars, that the optical emission regions of quasars are much (by factors of 10—100) larger than the basic disc models predict. Microlensing light curves of these complicated source structures may noti- ceably differ from those for a simple source structure represented by an accreti- on disc alone. In particular, the accretion disc alone cannot reproduce in si- 25
- 30. CHAPTER 1. Gravitational lensing mulations the observed amplitudes of the Q 2237 + 0305 light curves. While providing good fits for the peaks, which are most sensitive to the effect of the central source, it fails to provide the actual amplitudes of the rest of light curves, [41]. In this respect, the results by Yonehara [142], Shalyapin et al. [110], and Gil-Merino et al. [33], who analysed the regions of the light curves near the peaks of HME, provide successful estimates of the central source, but ignore the effect of a possible quasar outer feature. In 2003, Schild Vakulik [104] have shown how the double-ring model of the quasar surface brightness distributi- on, resulting from the Elvis’s [29] quasar spatial structure model, successfully explains the rapid low-amplitude brightness fluctuations in light curves of the First Lens Q0957 + 561. Interestingly, recent analysis [67] of brightness records of 55 radio-quiet quasars monitored by the MACHO project indicates the presence of large-scale outflow structures consistent with the Elvis [29] and ‘dusty torus’ [6] models of quasars. A current concept of spatial structures of active galactic nuclei (AGN) and quasars implies that there is a dusty torus lying outside the accretion disc on the scale of several tenths of kiloparsec, with a broad-line emitting region (BLR) in between. According to the recent inferences of Agol et al. [2] made from their IR observations of Q 2237 + 0305, the accretion disc and the torus may contribute in the total luminosity almost equally near one micron. Polarimetric observations (e.g. [54]) raised a question about existence of an additional electron scattering region (ESR) in quasars, [46], which has been earlier detected around some AGNs and is presently believed to be responsible for reprocessing emission from the accretion disk into polarized radiation [6, 113]. The exact geometry of the ESR, its dimensions and mutual location with respect to the BLR are still poorly constrained at present, and seem to differ in different objects. Future spectropolarimetric observations at different phases of microlensing events are expected to be diagnostic concerning this issue. In the sections to follow, we present the results of our analysis of two most extensive monitoring data sets of Q 2237 + 0305 — those of the OGLE (Optical Gravitational Lensing Experiment) group, obtained in filter V and covering the time period from 1997 to 2008, and the data of monitoring the Einstein Cross from the Maidanak observatory in filters V , R and I during 2001—2008, but with slightly lower sampling rate. We analyse these data sets to test a two-component model of the Q 2237 quasar structure and to determine parameters of this source model. In contrast to the ring model proposed earlier, [104], we used a simplified model, consisting of a compact central source and an extended outer structure with a much smaller surface brightness. Such a model, being much easier for calculations, possesses the principal property of the ring model to produce sharp peaks of the simulated light curves, while damping the amplitudes of the entire microlensed light curves. 26
- 31. 1.2. Spatial structure of quasars from microlensing studies Thus, our basic approach is to accept existence of the inner and outer structural elements as detailed above, and to derive from parameter fitting only the size of the inner luminous feature and the fraction of the total UV-optical energy from the extended outer feature as compared to the luminosity origina- ting in the compact central feature. We will show that the structural elements of this two-component quasar model satisfactorily explain the observed light curves amplitudes of the Q 2237 + 0305 image components and variations of their color indices caused by microlensing. 1.2.2. Testing of the two-component quasar model: application to Q 2237 + 0305 In the vicinity of a selected macroimage, the equation (1.5) describes only a smooth part of the lens equation. In order to take into account an effect of a highly inhomogeneous gravitational field due to stars (as well as other compact objects) on the line of sight one must add an ensemble of microlenses that may be assumed randomly distributed in the lens plane [43, 84]. This yields y = ( 1 − κc − γ, 0 0, 1 − κc + γ ) x − ∑ i mi x − xi |x − xi|2 , mi = 4GMi c2 DdDds Ds , (1.14) κc is the microlensing optical depth of a smooth background, and Mi are microlens masses. Using equation (1.14) and the ray tracing method [107], it is possible to calculate the distribution of magnification rate M(y) for a small (quasi-point) source for all its possible locations y — the so-called magnification map. Magni- fication of an extended source with a surface brightness distribution B(y) can be calculated from the formula: µ(y) = ∫ B(y′)M(y − y′)d2y′ ∫ B(y′)d2y′ . (1.15) For moving source we have a set of consecutive shifts of B(y) along the source trajectory which generate dependence µ(t) and the light curve m(t) = = −2.5 lg [µ(t)] + C. Once calculated, the magnification map can be used to produce a large set of simulated light curves for various models of surface light distribution B(y) over the source. We simulated microlensing of a two-component source, with one of them the compact, central luminous source having a surface brightness distributi- on B1(y). The other, outer, structure, is associated with the larger structural elements — a shell, a torus, Elvis’s biconics, [29] — and is characterized by 27
- 32. CHAPTER 1. Gravitational lensing substantially lower surface brightness B2(y). For such a source the magnifi- cation µ12 can be written as [120]: µ12(y) = µ1(y) + εµ2(y) 1 + ε . (1.16) The magnifications µ1 and µ2 are calculated according to (1.15) for the surface brightness distributions B1 and B2, while ε is determined as a ratio of the integral luminosities of these structures: ε = ∫ B2(y) d2y ∫ B1(y) d2y . (1.17) The characteristic time-scale of the observed Q 2237 microlensing brightness fluctuations is known to be almost a year. We infer from known cosmological transverse velocities that such a scale is due to microlensing of the compact inner quasar structure. Since the predicted spatial scale of the outer structure is more than an order of magnitude larger as compared to the inner part [29,104], the expected time scale of its microlensing brightness variations must exceed ten years. So, because of the large dimensions of the extended structure, the amplitudes of its magnification in microlensing must be noticeably less, as compared to microlensing of the compact structure. Thus we conclude, that on time-scales near 4 years, the magnification rate µ2(y) is almost invariable and does not differ noticeably from the average magnification rate of the j-th image component µj resulting from macrolensing: µ2(y) ≈ ⟨µ2(y)⟩ ≈ µj (j = = 1, 2, 3, 4). Under these assumptions, equation (1.16) can be rewritten: µ12(y) = µ1(y) + εµj 1 + ε . (1.18) It is clear that, under such assumptions, microlensing of the extended (outer) structure does not produce noticeable variations of magnification or bri- ghtness fluctuations on the observationally sampled time-scales, and therefore is effectively a brightness plateau above which the inner structure brightness fluctuations are seen. So the observed inner region brightness fluctuations are reduced by 1/(1 + ε). Therefore, when analysing the light curves, we did not attempt to esti- mate the size of the extended structure, and the accepted value of ε was the only parameter which characterized the outer structure. The inner compact structure of the source was simulated by the Gaussian surface brightness di- stribution, and with its characteristic size expressed in units of the Einstein ring radius of a typical microlens rs = r/rE at the one-sigma level as a fitted parameter. We used this simple central source model because it is more easy for computation. In doing so, we relied on the work by Mortonson et al. [80], who 28
- 33. 1.2. Spatial structure of quasars from microlensing studies examined the effect of the source brightness profile on the observed magnitude fluctuations in microlensing. They used a variety of accretion disc models, including Gaussian disk, and concluded that the statistics of microlensing fluctuations depends mainly on the effective radius of the source, while being relatively insensitive to a particular light distribution over the source disc. In producing magnification maps, we accepted the values for microparame- ters κ∗ and γ (shear) taken from Kochanek [58]: 0.392, 0.375, 0.743, and 0.635 for κ∗, and 0.395, 0.390, 0.733, 0.623 for γ, for the A, B, C and D components, respectively. For each of the four Q 2237 + 0305 components, we calculated five magnification maps with dimensions of 30 × 30 microlens Einstein radii, (a pixel scale of 0.02 rE). We assumed here that the entire mass is concentrated only in stars — that is, κc equals zero. To simplify computations, all the stars were accepted as having the same mass. We are well aware that this assumption is rather artificial, but in doing so, we refer to the works by Wambsganss [128] and Lewis Irwin [64], who demonstrated, having used various mass functions, that the resulting magnification probability distributions are independent of the mass function of the compact lensing objects. For the sake of completeness, the more recent works by Schechter, Wambsganss Lewis [101] and Lewis Gil-Merino [63] should be mentioned, where simulations with two populations of microlenses with noticeably differing masses were carried out to show that the magnification probability distributions do depend on the mass function. It is not surprising that such an exotic case has demonstrated the effect the authors wanted to demonstrate. But it is of little relevance for our work, since the more relevant calculation of Lewis Irwin [64] was made for more realistic mass functions — for microlenses in 0.3M⊙, 10.0M⊙ stars, and for the Salpeter mass function; these do not show appreciable dependence of the magnification probability distribution on the adopted mass function. For our analysis, we used the Q 2237 + 0305 light curves obtained in the V filter by the OGLE group in 1997—2000. High sampling rate (2—3 datapoints weekly) and low random errors are inherent in the photometric data from this program. To compare with the simulated light curves, results of the OGLE photometry were averaged within a night, thus providing 108 data points in the light curve of each image component. For every set of the source model parameters, we estimated the probability to produce good approximations to the observed brightness curves. The values of the source model parameters providing the maximum probabilities, were accepted as the parameter esti- mates. To estimate consistency of the results, the analysis was carried out for each of the four image components separately. Quasars are known to be variable objects, and their luminosity may change noticeably on time-scales of several years, months, and even days [26, 35, 91]. If a variable source is macrolensed, the intrinsic brightness variations will be observed in each lensed image with some time delays. This is just the fact 29
- 34. CHAPTER 1. Gravitational lensing that allows the Hubble constant to be determined from measurement of the time delays. In analysing microlensing, however, variability of the source is an interfering factor, which needs to be taken into account. In the Q 2237 + 0305 system, because of an extreme proximity of the lensing galaxy, (zd = 0.04), and because of almost symmetric locations of the macroimages with respect to the lens galaxy center, the expected time delays do not exceed a day, (e.g. [105,129]). This is why the intrinsic brightness variations of the source would reveal themselves as almost synchronous variations of brightnesses of all lensed images. This was observed in 2003 [119], when the microlensing activity was substantially subdued in all the four image components. Generally, separation of the intrinsic brightness variations of the source from the light curves containing microlensing events is a poorly defined and intricate task. In our attempts to obtain the intrinsic source light curve for Q 2237 + 0305, we introduced the following assumptions: 1. No effects of microlensing on the brightnesses of the components were observed during the time interval from January to June, 2002 (JD 2090— 2250), when the magnitudes of all the components were almost unchanged. The magnitude of each component was accepted as a zero level, and its brightness variations were analysed relative to this level. 2. Relative to this zero level, we regarded that the closer these light curves were to each other, the higher the probability that the components are not microlensed within this time interval, while almost synchronous variations of their brightness are due to changes of the quasar brightness. And vice versa, the more the light curve of a component deviates from others, the larger the probability that the component undergoes microlensing, which veils and di- storts the source variations. Thus, the weighted average variations of the component brightnesses with respect to their zero levels were adopted as the estimate of the source bri- ghtness curve, with the statistical weight for variations of every component being selected depending on how close this brightness variation to variations of other components is. The quasar intrinsic brightness curve, obtained on the basis of these assumptions, as well as the OGLE light curves for the indivi- dual images, reduced to their zero levels, are shown in Fig. 1.3. We expect the largest source of error in estimating this intrinsic source brightness history to be encountered during the time interval from August 1998 to December 1999, when, possibly, all the components underwent microlensing. To characterize similarity of the simulated and observed light curves, a χ2 statistics for each image component was chosen: χ2 j = NS ∑ j [mj(ti) − Mj(ti, p)]2 σ2 j , (1.19) 30
- 35. 1.2. Spatial structure of quasars from microlensing studies Fig. 1.3. The OGLE light curves obtained in the V filter in 1997—2005. The light curves are reduced to the same (zero) level at the time interval JD 2090—2250. The solid grey line is our estimate of brightness variations of the source quasar where mj(ti) is the observed light curve of the j-th component, and Mj(ti, p) is one of the simulated light curves, produced from a source trajectory at the magnification map, and NS is a number of points in the observed light curve. The magnification map was calculated for the source model described by a set of parameters p, which could be varied. The quantity σ2 j characterizes the errors of the observed light curve measurements, which are 0.032m, 0.039m, and 0.038m for the A, B and C components, and 0.057m for the faintest D component. The probability that, for a given set of parameters p, a simulated light curve will be close enough to the observed light curve, — that is, the value of χ2 will happen to be less than some boundary value χ2 0, — such a probability will be: P(χ2 χ2 0) = lim N→inf Nχ2 χ2 0 Ntot , (1.20) where Nχ2 χ2 0 is a number of successful trials, and Ntot is a total number of trials. The boundary value χ2 0/NS = 3 was adopted for calculations. The direction of an image motion with respect to the shear is an important parameter, which affects the probabilities noticeably. In [58], directions of moti- on of each component were chosen randomly and independently of directions of other components. This is not quite correct, since the directions of moti- on of components are not independent, and are determined by the motion of the source. Therefore, specifying the motion of one of the components must automatically specify motions of other components, if the bulk velocities and 31
- 36. CHAPTER 1. Gravitational lensing velocity dispersion of microlenses can be neglected [130, 139]. As a result of almost perfect symmetry of Q 2237 + 0305, for any direction of the source moti- on, directions of the opposite components with respect to the shear direction must coincide, while motion for the two other images must be perpendicular to the shear direction. That’s why, unlike the work by Kochanek [58], we analysed trajectories for two selected directions at the magnification maps, — when the A and B components are moving along the shear, with the C and D moving transversely to it, and vice versa, when A and B are moving transversely to the shear. Also, we did not undertake the local optimization of trajectories, as in [58], since this may distort the estimates of probabilities. Thus, for each magnification map, and for each of the two selected di- rections, a map of distribution of the initial points of the trajectories can be calculated, for which χ2 χ2 0. The probability (1.20) can then be calculated as the relative area of such regions on the map. To reduce computing time, the map of χ2 was calculated initially with a coarse mesh, (∼0.3 r/rE), to localize the regions with low values of χ2. Then, more detailed calculations with a finer mesh were carried out for only these regions. In our simulation, the following parameters could be varied: the radius of the central compact feature rs, expressed in units of a microlens Einstein ring radius rE; the brightness ratio, ε, expressing the ratio of the total outer structure brightness to the total inner structure brightness; and the relative transverse velocity of the source, Vt(rE/year), expressed in the units of the Einsten ring radius of a microlens per year, which is also a scaling factor for simulated light curves. The search of probabilities for our three fitting parameters, r/rE, ε and Vt is a rather complicated task, which needs much computing time. We attempted to simplify it in the following way. Assuming that the effect of the outer structure on the characteristic time scales of microlensing brightness variati- ons is insignificant, we put ε = 0 at the first stage. Hence, a dependence of probabilities on two parameters, — the scaling factor and relative dimension of the compact feature, — was evaluated at the first stage. In Fig. 1.4, the di- agrams are presented, which demonstrate distributions of probabilities to find simulated light curves, which would be close to the observed ones, — dependi- ng on the scaling factor and the compact feature dimension. The diagrams are built for all the four components for two directions of the source motion: A and B along the shear, C and D transversely — at the left, and C and D along the shear, A and B transversely — at the right. Overall, a nearly linear dependence of the scaling factor from the source dimension is seen. The largest values of probability occur for r/rE ≈ 0.4 in the first case, and for r/rE ≈ 0.3 in the second case. Since the maximum of probability distribution is, on average, at the source radius of 0.35 rE and the velocity of 1.82 rE per year (see Fig. 1.4), 32
- 37. 1.2. Spatial structure of quasars from microlensing studies Fig. 1.4. Probability distributions to find “good” simulated light curves as functions of the scaling factor and compact source size. Directions of the source motion are indicated in the left upper corner. The probability scale is shown in levels of grey and taking into account a nearly linear dependence between these values, we adopted the following expression for the further calculations: Vt = 1.82+1.18 −0.52 r/rE 0.35 . (1.21) Here, the source velocity Vt is expressed in the units of the microlens Einstein ring radius per year. At the second stage of our simulation, the distribution of the probabiliti- es to find simulated light curves similar to the observed ones was estimated, depending on the source’s compact feature dimension rs = r/rE, and on the relative integral brightness of the outer feature ε. The scaling factor Vt was determined according to (1.21) for each determination of the source dimension. The diagrams constructed for two different directions of the source motion, — along the shear γ and transversely to it for each component, — are shown in Fig. 1.5. Joint probability distributions for all the four components, calculated as Pall = P(A)P(B)P(C)P(D), are also shown in the third row in this figure. It is very significant that the probability maxima for all the components are found for values of ε larger than zero. This means that the outer quasar structures must noticeably contribute to the total quasar brightness in the opti- cal wavelengths. The outer structure decreases the amplitudes of microlensing brightness fluctuations and is at the core of the conundrum that in Q 2237, observed microlensing events are lower in amplitude than inferred for simple luminous accretion disc models. Interestingly, we also see from Fig. 1.5 that the statistics and locati- ons of probability maxima in the domain of parameters ε and r/rE found 33
- 38. CHAPTER 1. Gravitational lensing Fig. 1.5. Probability distributions to find “good” simulated light curves as functions of the compact structure dimension, r/rE, and of the ratio ε of the integral brightnesses of the outer and inner source structures. Directions of the source motion are indi- cated in the right upper corner. The 3-d row shows the joint probability distributions for each of the components se- parately, differ for different di- rections of motion of the image components with respect to the lens shear, γ. Examining four upper and four bottom panels of our Fig. 1.5, we see that the values of probabilities for the A and B components at their maxima are almost the same for the two selected di- rections of the source motion, while the C and D components both exhibit higher probabili- ties for the source to move parallel to the line connecting C and D rather than A and B. However, the joint proba- bilities calculated for all the four components, (the third row panels of Fig. 1.5), though giving slightly differing values of ε and r/rE, favor neither of these two cases in terms of the maximal values of pro- babilities. Much larger statis- tics is needed to solve this im- portant problem, which is be- yond our current computatio- nal resources. In Fig. 1.6, some of the most successful simulated light curves are shown together with the corresponding observed light curves reduced to their zero level. (The quasar light va- riations have been subtracted as described previously.) It should be noted that the si- mulated light curves reproduce the observed ones well enough within the time interval of the 34
- 39. 1.2. Spatial structure of quasars from microlensing studies Fig. 1.6. Some of the most successful simulated light curves plotted against the observed light curves. Microlensing brightness fluctuations beyond the time interval of fitting are approximately of the amplitude and duration observed. The time scale is in the Julian dates fitting, and no unacceptably large brightness fluctuations are observed outside this interval, unlike the results in fig. 10 of Kochanek [58]. Thus, the two-component source model consisting of a compact inner structure and much larger outer structure with lower surface brightness, allows to successfully model the brightness monitoring data and, importantly, to avoid the effect from the standard accretion disc model that large amplitudes of mi- crolensing brightness fluctuations are predicted but not observed [41,58]. The proposed source model consisting of two structures, — an inner compact structure and an extended outer region, — provides higher values of probability to find “good” simulated light curves as compared to the central compact source alone, and produces better fits to long-term light curves. The calculated distributions of the joint probabilities has well-marked maxima, and their locations in the domain of parameters ε and r/rE allow reasonable confidence in their determined values which provide the best fit of the simulated light curves to the observed ones. The range for the most probable values of the relative luminosity ε of the extended feature, determi- ned from probability distributions of the individual macroimages, is between 1 and 3, while the estimate of the relative size of the compact central feature of the quasar varies within a range of 0.1 r/rE 0.45. When determined from distributions of the joint probabilities, the values of ε equal 2 in both cases, while r/rE is about 0.4 for A and B motion perpendicular to γ, and 0.15 for A and B moving parallel to the shear γ. 35
- 40. CHAPTER 1. Gravitational lensing Very significantly, the simulated light curves calculated for the proposed two-component source model with the parameters indicated above, do not tend to unacceptably increase their amplitudes outside the time interval where they were objectively selected according to the χ2 χ2 0 criterion, as is seen in fig. 10 from the work by Kochanek [58]. For better comparison with other authors, we adopted, following [58], a probable projected cosmological transverse source velocity of Vt = 3300 km/s to determine a linear size of the compact central source of r ≈ 2 · 1015 cm, (1.2 · 1015 cm r 2.8 · 1015 cm). This size was estimated by Kochanek to be between r ≈ 1.4 · 1015h−1 cm and 4.5 · 1015h−1 cm for the accretion disc model and for the same transverse velocity, (h = 100/H0, where H0 is the excepted value of the Hubble constant). For the relative size of the source of 0.3rE, (0.1rE r 0.45rE), the estimate for the average microlens mass is ⟨m⟩ = 1.88 · 10−3h2M⊙, (3.08 · 10−4h2M⊙ ⟨m⟩ 3.3 · 10−2h2M⊙). 1.2.3. Two-component quasar model in interpreting chromatic microlensing Though the gravitational lensing phenomenon is known to be achromatic, the importance of observations at several spectral bands has become understood long ago. As far back as in 1986, Kayser, Refsdal Stabell [43] suggested that chromatic phenomena can be expected for microlensing of a source with a radial temperature gradient. This possibility was later confir- med in simulations by Wambsganss Paczyński [127]. In 1992, Jaroszyński, Wambsganss and Paczyński [41] demonstrated in simulations that microlensing maxima tend to be “bluer” than the rest of the light curve, with the expected (B − R) microlensing colour changes in Q 2237 + 0305 as large as a few tenths of a magnitude. Rix et al. [95] and Corrigan et al. [22] were the first to conclude indepen- dently that the colour indices of the Q 2237 + 0305 components seem to vary in time. Their suggestion was later confirmed independently by Vakulik et al. [122] and Burud et al. [12]. The first detailed analysis of variations of colour indices in Q 2237 + 0305 was made by Vakulik et al. [118], who presented their long- term V , R and I observations of Q 2237 + 0305 from the Maidanak Observatory with the subsequent statistical analysis. Variations of the V − I colour indices turned out to be correlated with the brightness changes. A tendency of the components to become bluer towards the microlensing peaks predicted in [127] and [41], has been confirmed. Moreau et al. [78] built the (V −R) vs V diagrams from the data of the GLITP collaboration [3], and noted their similarity with that reported in [118]. Observations of Q 2237 + 0305 in spectral ranges other than optical conti- nuum (IR, radio and in the quasar emission lines, see references in section 1.2.1) — all indicate that the Q 2237 quasar structures emitting in these spectral 36
- 41. 1.2. Spatial structure of quasars from microlensing studies regions are almost nonsusceptible to microlensing and thus, must be much larger than those emitting the UV and optical continuum. Very interesting results concerning microlensing chromatic phenomena in Q 2237 + 0305 have been recently reported in [27,28,81,111] and [4]. Mosquera et al. [81] detected a chromatic microlensing event in image A of Q 2237 + 0305 through a single-epoch narrow-band photometry in eight different filters cove- ring the wavelength interval 3510—8130 Å. Considering a Gaussian brightness profile for the accretion disc (in a simple thin-disc model) they found from simulations that the effective disc size varies in wavelength according to the Rλ ∝ λ4/3 law, which follows from the thin accretion disc model by Shakura— Syunyaev [109]. They also confirmed that the emission line regions of quasars are much larger than that emitting the UV and visual light. Eigenbrod et al. [27,28] presented the results of spectroscopic monitoring of Q 2237 + 0305 during October 2004—December 2006, with the wavelength coverage 4000—8000 Å. They conclude that the continuum and broad line regions of the quasar are subjected to microlensing. They found that, accor- ding to theoretic predictions, microlensing brightness variations are stronger at shorter wavelengths, and restored the energy profile of the accretion disc. The relative sizes of the accretion disc emitting at different wavelengths turned out to follow the Rλ ∝ λβ law, with β = 1.2 ± 0.3 for the UV/optical continuum, that is close to the result of Mosquera et al. [81]. Similar results for wavelength dependence of the effective source size are reported by Anguita et al. [4], who used their g’- and r’-band photometry at the Apache Point Observatory 3.5-m telescope together with the OGLE V - band data to obtain 1.2—1.45 for the ratio of the source sizes in the r’-band to that in the g’-band, with the 0.5—0.9 uncertainty, however. Thus, as compared to the single-band observations, multi-colour observati- ons are capable of providing a more comprehensive idea of a quasar’s spatial structure and thus, of physical processes responsible for the observed properties of quasar radiation. Our data obtained at the Maidanak Observatory allow to see the signatures of chromatic phenomena directly from comparison of the V , R and I light curves of a particular microlensing event. This is illustrated by Fig. 1.7, where the V , R and I light curves of the high-magnification event in image A started in 2005 are shown. A tendency of amplitudes to decrease towards the longer wavelengths is clearly seen in this picture. However, a single microlensing event, no matter how prominent it may be, is of a lower diagnostic value as compared to the statistics of all available microlensing history of all the four Q 2237 image components. Below, we find and analyse statistical relationships between the observed variations of colour indices and magnitudes, and compare them to the results of microlensing si- mulation fulfilled for a set of the quasar structure parameters. 37
- 42. CHAPTER 1. Gravitational lensing Fig. 1.7. Brightness changes of the A component of Q 2237 + 0305 in the V , R and I filters during the high magnification event started in 2005 (observations from the Maidanak Observatory). The amplitudes of the event are clearly seen to decrease consecutively from filter V to R and I To properly investigate microlensing phenomena, one has to disentangle them from the intrinsic variability of quasars and from the effects of dust extinction in lensing galaxies, which are known to act simultaneously and in a similar way. Yonehara, Hirashita Richter [143] have made an attempt to evaluate relative contributions from these three factors. They conclude that the intrin- sic quasar variability is hardly a dominant factor for the observed chromatic phenomena in gravitationally lensed quasars from their sample counting about 25 systems. They claim that both dust extinction and microlensing are capable of producing the observed behavior of colour indices of the lensed qua- sar images. We regard that one should be more careful concerning this issue. Accor- ding to [134] and [34], variability of quasars increases towards the shorter wavelengths, and most of quasars exhibits hardening of their spectra in the bright phases. Giveon et al. [34] undertook an extensive statistical analysis of variability properties of 42 quasars from the Palomar Green (PG) sample having used the results of the 7-years monitoring in the Johnson-Cousins B and R bands with the Wisa Observatory 1-m telescope. The main goal of their study was to search for correlations between the variability properties and other parameters of quasars, such as luminosity, redshift, radio loudness, etc. 38
- 43. 1.2. Spatial structure of quasars from microlensing studies Fig. 1.8. Q 2237 + 0305 light curves in filter V from 2001 to 2008; grey symbols — the OGLE data, the Maidanak data are shown in darker symbols. The D light curve is 0.5 mag shifted for better view They confirmed previous finding that the spectra of quasars become harder (bluer) at brighter phases, but found out that this trend holds for only a half of their subsample. They presented the diagrams which demonstrate a rather high correlation between variations of the B − R colours and variations of the B and R magnitudes, with a regression line slope of approximately 0.25. The Q 2237 + 0305 system is usually referred to as that one, where bri- ghtness changes of all the four lensed quasar images are dominated by mi- crolensing events. Recent observations have shown, however, that quasar vari- ability may contribute noticeably to the Q 2237 + 0305 light curves. It is seen especially during 2003—2006 (Fig. 1.8), when the microlensing activity was somewhat subdued for images A, C and D (for image B, a noticeable microlensi- ng event has happened during this time period). Recall that this fact had made it possible to estimate the time delays in Q 2237 + 0305 [119]. Observations of this time period have shown that the contribution of the source variations to the observed light curves can not be neglected. To be sure of this, we esti- mated contributions from the microlensing and quasar intrinsic variabilities quantitatively, having applied a simple statistical approach to the whole R light curve during 1997—2008. Having made use of the fact that the time 39
- 44. CHAPTER 1. Gravitational lensing delays in Q 2237 + 0395 are negligibly small as compared to the variability time-scales, we analysed the sums and differences of light curves for all the six combinations of pairs of images to obtain the following values of the RMS fluctuations: σS = 0.10 ± 0.03 mag for the intrinsic quasar brightness variati- ons, and σmicr = 0.16±0.04 mag for microlensing variability (averaged over all the four image components). Similar results were obtained from the analysis of the OGLE V light curves. Thus, the intrinsic brightness fluctuations of the Q 2237 + 0305 quasar do contribute significantly to the observed light curves of the lensed images, and therefore one must find a possibility to exclude the quasar constituent to study the net microlensing colour and magnitude variations. The most simple way to exclude the quasar variability is to analyse the dif- ference light curves in the units of stellar magnitudes, where these two consti- tuents are additive. Thus, we may form the difference light curves ∆RA−B, ∆RA−C, ∆RA−D, ∆RB−C, ∆RB−D, ∆RC−D, and the corresponding difference colour curves, ∆(V − I), where ∆ means, similar to [118], deviations of the corresponding quantities from their average values over the time interval under consideration. At first, we built the ∆(V − I) vs. ∆R diagram for all the four quasar components from our V RI photometry of 2001—2008 and compared it to the similar diagram for the 1995—2000 data shown in fig. 2 from [118]. The di- agrams turned out to be very similar both in the regression line slopes and in correlation factors, with the difference well within the error bars. However, both the V −I colour indices and magnitudes in R varied within a larger range for the 1995—2000 data as compared to the observations of 2001—2008. This can be naturally explained by two unprecedented high-magnification events happened to images A and C in 1999. Taking into account obviously different mutual contributions of the microlensing variability and quasar intrinsic vari- ability to the observed light curves during these two periods, the likeliness of the diagrams can be regarded as an indication to the similar statistical characteri- stics of these two types of variability (compare the B − R vs. R regression line slope of 0.25 for PG quasars in fig. 6 from [34] with 0.28 in [118]). Thus, to analyse the net microlensing statistics, we will further address the difference colour and magnitude variations curves. Besides, it is reasonable to expect that the difference curves calculated as described above will be less affected by the inevitable photometry errors, both random, such as, e.g., night- to-night error, and systematic ones. The difference ∆(V − I) vs. ∆R diagram built for all possible combinations of pairs of images is shown in Fig. 1.9. Here we used the photometry results obtained in observations from the Maidanak Observatory during the time period from 2001 to 2008. The regression line slope for these difference data points is insignificantly larger than that in fig. 2 from the work by Vakulik et al. [118], while the 40
- 45. 1.2. Spatial structure of quasars from microlensing studies Fig. 1.9. A diagram showing correlation between variations of the (V − I) colour indices (vertical axis) and variations of magnitudes in filter R, calculated from the V RI observations in 2001—2008 with the 1.5-m telescope at Maidanak Observatory correlation index turned out to be noticeably higher, in accordance with expectations noted above. Hence, we have a diagram representing statistics of microlensing variations of colour indices exempted from the interfering effects of the quasar variability, and therefore suitable for the further interpreting in simulations. We argue that the regression line slope and correlation factor of the ∆(V − I) vs. ∆R diagram are important parameters of microlensing statistics, which are diagnostic for a spatial structure of the source quasar. Wambsganss Paczynski [127] seem to be the first to simulate statisti- cal dependencies between variations of colour indices and variations of bri- ghtness for Q 2237 + 0305 caused by microlensing of a source with a radial temperature gradient. They simulated the B and R microlensing light curves for Q 2237 + 0305 for the Gaussian brightness profile of the quasar with the half width radius varying between 2 · 1014 cm and 3.2 · 1015 cm, and found out that |∆(B − R)| may be approximately proportional to |∆B| for some source dimensions and for the ratios of the source sizes at the two spectral bands of at least 1.3 or larger. Similar to the previous section, we considered a photometric model of the Q 2237 + 0305 quasar consisted of a compact central source at some brightness pedestal as a case that is much more simple computationally, but results in similar effects. We, again, did not attempt to specify the size and brightness 41
- 46. CHAPTER 1. Gravitational lensing profile of the extended feature in our simulations, but intended to find the relative energy contribution of the extended feature into the total quasar lumi- nosity that would provide the best fit of the simulated ∆(V −I) vs. ∆R diagram to that obtained from our V RI photometry. The magnification maps were calculated using the inverse ray tracing method, with the local lensing parameters κ∗ and γ accepted, similar to the previous section, to be the same as in Kochanek [58]. To utilize all the statisti- cs available from the generated magnification maps, we did not select source trajectories and thus, did not use the corresponding light curves in the further analysis, but instead, operated directly with data points of the maps themselves to find the desired parameters. In this way, we excluded the unknown direction of the source motion over the map, as well as its relative velocity. To decrease the amount of calculations while having the size of magnifi- cation maps sufficient enough to provide reasonable accuracy of the calculated model parameters, we produced two sets of magnification maps differing in their scales. For the source radii from rs = 0.05rE to rs = 1.0rE, three maps of (46×46)rE in dimension with a 44pix/rE scale were calculated for each image component, while for rs ranging from 0.5rE to 3rE, one (205 × 205)rE map with a scale of 10pix/rE was used for each image component. Since the correlation coefficient of the colour-magnitude diagram must be sensitive to errors in the values of V − I and R, we added a random noise to our magnification maps, thus making our simulations more realistic. The RMS values of the imitated errors were accepted to equal those estimated for the actual errors of our V RI photometry: 0.019m, 0.037m, 0.042m and 0.039m for images A, B, C and D, respectively, and were made to be distributed normally. As was shown in the previous section, the value of magnification µ12 in microlensing of the two-component source consisting of a compact central structure and an extended outer structure, can be represented by equati- on (1.18). The average magnifications can be adopted to equal those determined by the local lensing parameters. With these taken from [105], and using the known relationship between the normalized surface density in stars, normalised shear parameter and macroamplification, (see equation (1.7) in section 1.1.1), we have the following values of amplifications for the A, B, C and D components: µA ≈ 4, µB ≈ 4.3, µC ≈ 2.45 and µD ≈ 4.9. As was noted in section 1.2.1, some authors, e.g. Mortonson et al. [80], claim that the effective size is a universal parameter of a source model for microlensing simulations, and that a particular type of the brightness profile virtually does not affect fluctuations of the simulated light curves. To check if this is valid for simulation of colour fluctuations, we considered both the accretion disc and a source with Gaussian surface brightness distribution as the quasar models. We adopted a generalized power law for wavelength dependence 42
- 47. 1.2. Spatial structure of quasars from microlensing studies Fig. 1.10. Diagrams ∆(V − I) vs. ∆R simulated for image A of Q 2237 + 0305 as described above for a central source with Gaussian brightness profile. The calculations were made for three different effective sizes of the source and three values of the parameter β in the rλ ∼ λβ law (shown in the left upper corners). The resulting values of the correlation coefficient ρ and regression line slope a are indicated in the right bottom corner of each panel of the Gaussian source effective radius, rλ ∝ λβ. The value of β varied from 0.5 to 2.5 for the Gaussian source brightness profile, with β = 4/3 valid for the Shakura—Syunyaev accretion disc. Then, we simulated the colour-brightness diagrams for our two-component source structure, with the accretion disc as the compact central source surrounded by an extended feature with its relative energy contribution ελ varying in wavelength. Fig. 1.10 is presented to qualitatively illustrate appearances of the ∆(V − − I) vs. ∆R diagrams simulated for a compact source with Gaussian brightness profile for three values of the effective radius and for three different values of β. None of the resulting regression line slopes and correlation factors (shown in the plots) can be regarded as consistent with those of the color-magnitude diagram built from the observations. Fig. 1.11 demonstrates typical appearances of colour-magnitude diagrams for three different sizes of the classical accretion disc with no contribution 43
- 48. CHAPTER 1. Gravitational lensing Fig. 1.11. Illustration of the effect of the source model on appearance of the ∆(V − I) vs. ∆R diagrams simulated for image A of Q 2237 + 0305. The light curves in filters V and I (denoted as ∆V and ∆I) and the curves (∆(V − I) corresponding to a particular source trajectory, are in the left column. The time scale is presented by pixels on the magnification map. The source model parameters are also indicated in the left panels. The ∆(V − I) vs. ∆R diagrams built from the whole amplification maps as described in the text are presented in the right column. The entangled curves at the diagrams correspond to particular source trajectories and are shown to illustrate the ambiguity of the color-magnitude relationship produced by microlensing 44
- 49. 1.2. Spatial structure of quasars from microlensing studies from the extended feature (three upper panels in the right-hand column), and for the two-component source model with two sets of ελ (right-hand panels in two bottom rows). For the central compact source alone, both Gaussian and accretion disc, the parameters ρ and a of the ∆(V − I) vs. ∆R diagram are seen to approach those of actually observed ones only for unrealistically large sources (indicated in the plot). For the two-component source model, the simulated diagrams are consistent with the observed ones for quite reasonable values of the central source size. It is interesting to note a horizontal strip-like condensation of data poi- nts seen in the uppermost diagrams both in Fig. 1.10 and Fig. 1.11, that is stretched parallel to the magnitude axis against the zero colour indices. A si- milar condensation can also be seen in the ∆(V −I) vs. ∆V diagram simulated by Mosquera et al. [81] for image A of Q 2237 + 0305 and shown in their fig. 5. The origin of the strip becomes evident from the color curves in the upper left panels of our Fig. 1.11, showing variations of the (V − I) colour index for a very small source (Rs = 0.05rE) moving along a certain trajectory over the magnification map, (the appropriate light curves in filters V and I are plotted above the colour curves). The color curves clearly shows a high probability for a small source to have low values of ∆(V − I), close to zero. Our simulations show that the strip is gradually disappearing as the source size is increasing. Since there is no signs of such a strip in the diagram built from the data of observations (Fig. 1.9), its presence in the simulated diagrams can serve as an indication to the unrealistically small source size. One more important feature of the simulated ∆(V − I) vs. ∆V diagrams should be noted. We overlaid the individual ∆(V − I) vs. ∆R curves resulted from a particular source trajectory over the amplification map against the clouds of points, which form the simulated diagrams (right-hand column in Fig. 1.11). We see the entangled curves at the diagrams, which demonstrate that relationship between variations of colour indices and magnitudes in mi- crolensing is ambiguous. This curves demonstrate why the analysis of a single event may produce wrong inferences about a quasar structure and thus, provi- de a strong argument in favour of statistical approach to studying chromatic events in microlensing. To quantitatively interpret our multi-colour data, we abandoned simulati- on of many individual V − I vs. R diagrams calculated for a set of specific source parameters, but instead, chose to analyse behaviors of the diagram parameters, namely, the correlation factor ρ and regression line slope a as functions of the source parameters. At first, we analyzed if Gaussian source alone can reproduce the observed ρ and a pair of the diagram parameters. This can be seen in Fig. 1.12, where three curves illustrate three sets of the ρ and a pairs calculated for three different values of β in the rλ ∼ λβ law adopted for the Gaussian source profile (indicated in the plot). Each data 45
- 50. CHAPTER 1. Gravitational lensing Fig. 1.12. Sequence of values for the pair of parameters ρ and a of the ∆(V − I) vs. ∆R diagram simulated for the Gaussian central source alone with the effective source radius rs varying in wavelength according to the rs(λ) ∝ λβ law. The data points in the curves (crosses, triangles and squares) are the values of the ρ and a pair simulated for source sizes that increase discretely from 0.05rE to 3rE from left to right. The actually observed ρ and a are marked with a black square above the curves point in this plot (denoted by crosses, triangles and squares) corresponds to the ρ and a pair calculated by averaging over the whole magnification maps for a particular source size, which increases progressively from left to right. The error bars demonstrate the RMS deviations of the estimates from their average values. An isolated point in the upper part of the plot corresponds to the values of ρ and a of the diagram obtained from our V RI photometry (Fig. 1.9). The curves calculated for β = 1.33 and β = 2.25 do provide the needed value of a, but fail to provide the correlation factor ρ actually observed: it reaches its maximal values for unrealistically large source radii rs, though remaining to be lower than that obtained from observations. A similar set of values for the ρ and a pair was also calculated for the standard accretion disc model with the source size varying from 0.05rE to 3rE, (Fig. 1.13). The squares along the curve correspond to the source size growing from left to right again. General trend of this set resembles that one obtained for the Gaussian source with the power index β ≈ 1.33. Though slightly larger values of the correlation index are obtained for this model, they still remain to be less than ρ = 0.82 ± 0.1 of the diagram built from the data of observations. 46
- 51. 1.2. Spatial structure of quasars from microlensing studies Fig. 1.13. Sequence of values for the pair of parameters ρ and a of the ∆(V − I) vs. ∆R diagram simulated for the accretion disc as a model for the Q 2237 quasar, with the effective source radius rs varying in wavelength according to the rs(λ) ∝ λ4/3 law. The effective source size grows from 0.05rE to 3rE from left to right again Also, a tendency of the regression line slope a to approach the observed value a = 0.30 ± 0.03 for very large source dimensions is observed again. Thus, both Gaussian and accretion disc do not reproduce in simulations the parameters of the ∆(V −R) vs. ∆R diagram (Fig. 1.9), built from our V RI photometry, — a = 0.3 ± 0.03, ρ = 0.82 ± 0.10. As has been shown above and presented in [120], a two-component photometric model of the source structure provides much better fit to the observed amplitudes of the V light curves of Q 2237 + 0305 as compared to the single central source model. The central source size was estimated in that work to lie within 0.15rE and 0.4rE. This is consistent with determinations of the Q 2237 + 0305 source size obtained by other authors, from the early works, such as e.g., [125,126,132], and up to the recent studies by, e.g., Kochanek et al. (2004) [58], Eigenbrod et al. (2008) [27], Mosquera et al. (2009) [81]. Summarizing all available estimates of the central source radius, we may adopt r = (0.3 ± 0.1)rE as the most probable value. Therefore, our next step was to simulate microlensing ∆(V −I) vs. ∆R dia- grams for the two-component source model and to find the model parameters, which would provide a and ρ of the simulated diagram consistent with those actually observed. Fig. 1.14 shows behaviors of the a and ρ pair simulated for the two-component source model with three different values of the relative 47
- 52. CHAPTER 1. Gravitational lensing Fig. 1.14. Sequence of values for the pair of parameters ρ and a of the ∆(V − I) vs. ∆R diagram simulated for the two-component model of the Q 2237 quasar with three different sets of εV , εR, εI . The effective radius of the central source decreases from left to right in this case from 0.05rE to 1.6rE energy contribution from the extended source in filters V , R and I (shown in the sidebar). Similar to Fig. 1.12 and Fig. 1.13, the data points fix the a and ρ pairs calculated for the central source size increasing consecutively from left to right. We see from Fig. 1.14 that the two-component source model provides much better fit to the a and ρ pair obtained from observations as compared to the single central source model, with both Gaussian and accretion disc profiles. However, we cannot find a unique solution from the analysis of this plot, because in contrast to the single source model, the two-component model is described by three additional parameters, εV , εR and εI, which we need to be constrained in some way. To search for necessary observational constraints, we addressed Eq. (1.18) that describes a relationship between amplifications of the two-component (µ12) and single-component (µ1) source models, resulted from microlensing. Taking into account that the extended structure is almost insusceptible to microlensing (µ2(y) ≈ ⟨µ2(y)⟩ ≈ µj, see section 1.2.2), — we can write for deviations of the corresponding magnifications µ12 and µ1 from their values ⟨µ12⟩ and ⟨µ1⟩ averaged over the ensemble under consideration: µ12 − ⟨µ12⟩ = µ1 − ⟨µ1⟩ 1 + ε . (1.22) 48
- 53. 1.2. Spatial structure of quasars from microlensing studies Having averaged these differences over the whole magnification map, we can calculate the RMS variations σµ 12 and σµ 1 for µ12 and µ1. Both magnifi- cations and their RMS variations are understood to be functions of the central source size r: σµ 1 = σµ 1 (r) and σµ 12 = σµ 12(r). This will result in ε being a function of r too and thus, we can write for ε(r): ε(r) = σµ 1 (r) σµ 12(r) − 1. (1.23) When checking the hypothesis of the two-component source model, σµ 12(r) should be treated as the actually observed RMS flux variations σobs of quasar images, while σµ 1 (r) = σsim(r) are those obtained from simulations with a single-source model for a set of the central source sizes. Applying Eq. (1.23) to the data taken in different filters, we have: εV (r) = σsim V (r) σobs V − 1; εR(r) = σsim R (r) σobs R − 1; εI(r) = σsim I (r) σobs I − 1, (1.24) where the observed flux variations in filters V , R and I were obtained from our light curves to equal σobs V = 0.167, σobs R = 0.142 σobs I = 0.124. Using these observational constraints and the values of σsim(r) obtained from simulation, we can calculate the corresponding values of the relative integral brightness of the extended structure as functions of the central source size. The result is presented in Fig. 1.15. For all the three filters the relative contributions of the extended feature needed to explain parameters of the ∆(V −I) vs. ∆R diagram of Q 2237 + 0305 are seen to decrease as the central source radius increases. Now, to obtain a reasonable solution, we must fix the size of the central source. Adopting r = (0.3 ± 0.2)rE mentioned above for the compact source radius, we obtain εV ≈ 1.3, εR ≈ 1.5, εI ≈ 1.6 for the relative energy contributions of the extended structure of our two-component quasar model. The uncertainties are ±0.5, ±0.6 and ±0.8, respectively. As is seen, the values of ε obtained with the approach based on compari- son of fluctuations of the observed and simulated light curves expressed by equations (1.24) are larger than those selected in a rather random manner in direct simulation to fit the observed values of a and ρ (Fig. 1.14). This can easily be understood from Eq. (1.24): the values of σobs V , σobs R and σobs I were obtained from the observed light curves, which, in contrast to the simulated ones, cannot be regarded as the representative samples of microlensing bri- ghtness variations (see, e.g. [84]). This must result in underestimation of σobs and thus, in overestimation of ε from expressions (1.24). Therefore, the values εV , εR, εI indicated above should be considered as the upper limits for the relative energy contributions of the extended feature in filters V , R and I. 49
- 54. CHAPTER 1. Gravitational lensing Fig. 1.15. Relative brightness of the extended feature in filters V , R and I as a function of the quasar compact structure relative size expressed in units of a microlens Einstein ring radius, rs/rE Indeed, simulations show that lower values of εV , εR and εI much better reproduce the observed ρ and a pair. Moreover, simulations show that a family of solutions exists diverging not only in the values of ε, but also in the ratios of ε in different spectral bands. Therefore, additional constraints are needed, which would be based on more reliable data about the spectral properties of the supposed extended feature. At the present level of our knowledge, we can only argue that a contribution of the extended structure into the total quasar emission grows towards the longer wavelengths, with the upper limits in the V , R and I bands of εV ≈ 1.3, εR ≈ 1.5, εI ≈ 1.6 Thus, to explain the parameters of our ∆(V −I) vs. ∆R diagram, we, again, had to admit existence of an extended structure around a compact central source, with a rather large contribution into the total quasar emission. It is obvious that the effect of such structure, when treated in terms of the Rλ ∝ λβ law, is equivalent to the steeper wavelength dependence of the effective source size as compared to that expected for the accretion disc. Meanwhile, the most recent observations of chromatic microlensing in Q 2237 + 0305 [2, 5, 28, 81] and in other lensed quasars give the value of β in the λβ law varying around 4/3 predicted by the thin accretion disc model. To comment discrepancy between our results and the results of the authors cited above, we would like to refer to the recent work by Mosquera et al. [81], 50
- 55. Another Random Scribd Document with Unrelated Content
- 59. The Project Gutenberg eBook of Hell's Hatches
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Hell's Hatches Author: Lewis R. Freeman Release date: January 9, 2014 [eBook #44632] Most recently updated: October 23, 2024 Language: English Credits: Produced by Greg Bergquist, Ernest Schaal, and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.) *** START OF THE PROJECT GUTENBERG EBOOK HELL'S HATCHES ***
- 61. HELL'S HATCHES NEW FICTION THE CURTAIN By Alexander Macfarlan THE SYRENS By Dot Allan OLD MAN'S YOUTH By William de Morgan THE PURPLE HEIGHTS By M. C. Oemler HAGAR'S HOARD By George Kibbe Turner THE VILLA OF THE PEACOCK By Richard Dehan IN CHANCERY By John Galsworthy SNOW OVER ELDEN By Thomas Moult EUDOCIA By Eden Phillpotts LONDON: WILLIAM HEINEMANN 21, Bedford Street, W.C. 2
- 62. CONTENTS
- 63. CHAPTER PAGE I A Reputation Questioned 1 II Hard-Bit Derelicts 10 III The Girl Herself 25 IV Slant Allen Retires Again 38 V A Ship of Death 50 VI Compulsory Volunteering 65 VII Rona Comes Aboard 80 VIII I Leave the Island 93 IX A Grim Tale of the Sea 106 X Art and Suspense 124 XI A Hero's Homecoming 142 XII A Bad Man's Plea 180 XIII The Scene of the Final Drama 193 XIV Hell's Hatches Off 206 XV The Face 220 XVI A Sudden Visitor 231 XVII Down the Flume 255 XVIII The Masterpiece 268 XIX After All 282
- 64. HELL'S HATCHES
- 65. CHAPTER I A REPUTATION QUESTIONED Slant Allen and I, between us, had been monopolizing a good share of the feature space in the Queensland and New South Wales papers for a week or more—he as the Hero-Ticket-of-Leave-Man and I as the gifted Franco-American painter whose brilliant South Sea marines have taken the Australian art world by storm—and now that it was definitely reported that he had left Brisbane on his way to connect with the reception the boyhood home from which he had been shipped in disgrace five years before had prepared for him, I knew it was but a matter of hours before he would be doing me the honour of a call. He simply had to see me, I figured; that was all there was to it: for with Bell and the girl dead (that much seemed certain, both from the newspaper accounts of the affair and from what I had been able to pick up in the few minutes I had been ashore during the stop of my southbound packet at Townsville) I was the only living person who knew he was not the hero of the astonishing Cora Andrews affair, the audacious daring and almost sublime courage characterizing which had touched the imagination of the whole world; that, far from having volunteered to navigate a shipload of plague-stricken blacks through some hundreds of miles of the worst reef-beset—and likewise the most ill-charted—waters of the Seven Seas on the off chance of saving the lives of perhaps one in ten of them, he had been brought off and forced to mount the gangway of that ill-fated schooner at the point of a knife in the hands of a slender slip of a Kanaka girl. To be sure, two or three of the blacks who were hanging over the rail at the end of that accursed afternoon may have been among the
- 66. survivors (for it could have been only the strongest of them that had been able to fight their way up to the air when Bell chopped open the hatches they had been battened under ever since the Cora's officers had succumbed who knows how many hours before); but, even so, their rolling, bloodshot eyes could have fixed on nothing to have led them to believe that the greasy shawl of Chinese embroidery the girl appeared to have thrown affectionately over the shoulder of the belated passenger in the leaking outrigger concealed the diminutive Malay kris whose point she was pressing into the fleshy part of his neck above the jugular. No, there could be no doubt that I was all that stood between Slant Allen, Ticket-of-Leavester, beachcomber, black-birder, pearl- pirate and (more or less incidentally to all of the foregoing) murderer, and the Hon. Hartley Allen, second son of the late James Allen, Bart., racing man, polo player and once the greatest gentleman jockey on the Australian turf. Pardon for the comparative peccadilloes—a pulled horse or two, a money fraud in connection with a sweep, and the rather rough treatment of a chorus girl, who had foolishly asked for time to consider his proposal that she come to him at once from the Queensland stockman who was only just finishing refurnishing her George Street flat—which, cumulatively, had been responsible for his being packed off to The Islands, was already assured, and it looked as though more was to come—that his spectacular and self-sacrificing heroism was going to wipe out the unpleasant memories that had barred him from sporting and social circles even before the law stepped in. A sporting writer in that morning's Herald had speculated as to whether or not he would be seen again riding Number 1 for the unbeaten Boomerang Four, with whom he had qualified for his handicap of 8, still standing as the highest ever given an Australian polo player; and the racing column of the latest Bulletin had devoted a good part of its restricted space to a discussion of the possibility that the weight he had put on in his years of easy life in 'The Islands' might force him to confine his riding to steeplechases. Of the record which had made the name of Slant Allen a byword for all that was desperate and
- 67. devilish from Port Moresby to Papeete, from Yap to Suva, little seemed to be known and nothing at all was said. But then, that old beach-combers' maxim to the effect that What a man does in 'The Islands' don't figure in St. Peter's 'dope sheet,' was one from which even I myself had been wont to extract no little solace. With nothing but my fever-wracked and absinthe-soaked (I may as well confess at the outset that I was in the grip of the green at this time) anatomy standing between, on the one hand, and Allen more despicable than even I, who was fairly familiar with the lurid swath he had cut across Polynesia, had ever dreamed he could be, and, on the other hand, an Allen who might easily become more the idol of sporting (which is, of course, the real) Australia than he had ever been at the zenith of his meteoric career as a turfman and athlete, it was plain enough that he would not—nay, could not— ignore for long my presence in a city that was standing on tiptoe to acclaim him as a native son whose deed had done it honour in the eyes of the world. It was something like that the Telegraph had it, I believe. Where a word from me (and Allen would know that my friendship for Bell, to say nothing of the girl, would impel me to speak it in my own good time) would dash him from the heights to depths which even he had not yet sounded—there were degrees of treachery which The Islands themselves would not stand for—it was only to be expected that a man of his stamp would make some well- thought-out move calculated to impose both immediate and eventual silence upon me. If we were still north of twenty-two I would have had no doubt what form that move would take, and even here in the heart of the Antipodean metropolis—well, that I was leaving no unnecessary loop-holes of attack open was attested by the fact that I was awaiting his coming wearing a roomy old shooting jacket, in the wide pockets of which a man's fingers could work both freely and unobtrusively. I had shot away a good half-dozen patch pockets from that old jacket in practising unostentatious self-defence, and when a man gets to a point where he can spatter a sea-slug at five
- 68. paces from his hip he really hasn't a great deal to fear from the frontal attack of anyone—or anything—that hunts by daylight. Yes, though I hardly expected to have to shoot Allen, at least on this first showdown, I was quite prepared to do so if he gave me any excuse at all for it; indeed, I may as well admit that I was going to be disappointed if he did not furnish me such an excuse. There need be nothing on my conscience, that was sure, for, if the fellow had had his deserts according to civilized law, he would have been put out of the way something like twenty times already. I had heard him make that boast himself one night in Kai, just before he went under Jackson's table as a consequence of trying to toss off three-fingers of Three Star for every man he claimed to have killed. Moreover, I had a sort of a feeling that old Bell would have liked to have seen his score evened up that way, for he, more than almost anyone I could recall, had marvelled at what he called the tricks I had tucked away in my starboard trigger pocket. But—I may as well own it—my principal reason for hoping for a decisive showdown straightaway was that I felt sure I could see my way through an affair of that kind, even with so cool and resourceful a hand as I knew Allen to be. As an absinthe drinker, what I dreaded was to have the crisis postponed, knowing all the while that during only about from four to six hours of the twenty-four would I be fit in mind or body to oppose a child, let alone a man who, for five years and among as desperate a lot of cut-throats as the South Pacific had ever known, had lived up to his boast that he drew the line at no act under heaven to gain his end. It had struck me as just a bit providential that Allen almost certainly would be coming to see me in the early afternoon—the very time at which, physically and mentally, I would be best prepared for him. It varies somewhat with different addicts of the drug, but with me the hour of strength—the interval of the swinging back of the pendulum, when all the faculties are as much above normal as they have been below it during the preceding interval of depression—was mid-afternoon. From about ten in the
- 69. morning I was just about my natural self—just about at the turn of the tide between weakness and strength—for three or four hours; but from about three to five, when the renewed cravings began to stir and it had long been my custom to pour my first thin trickle of green into the cracked ice, I was preternaturally alive in hand and brain. The rigorous restriction of my painting to these brief hours of physical and spiritual exaltation must share with my colours the credit for the fact that I had already done work that was to win me a niche distinctively my own as a painter of tropical marines. How much absinthe—or the reaction from absinthe—had to do with my earlier successes was conclusively proven by the way my work at first fell off when those colourful years I was later to spend with the incomparable Huntley Rivers in the Samoas and Marquesas began to bring me back manhood of mind and body and to rid me—I trust for good and all—of the curse saddled upon me in my student days in Paris. But that is neither here nor there as regards the present story. I had ascertained that Allen's train was to arrive from Brisbane at ten in the morning, and that he was to be taken directly from the station to the Town Hall to receive the Freedom of the City. Then, out of consideration for the fact that the hero (as the Herald had it) was still far from recovered from the terrible hardships he had endured as a consequence of his unparalleled self-sacrifice, the remainder of the day was to be left at his disposal to rest in. The further program—in which His Excellency the Governor-General himself was to take part—would be arranged only after the personal desires of the modest hero had been consulted. A 'phone to the gallery where my Exhibition was on—or an inquiry of almost anyone connected with the show at the Town Hall, for that matter—would apprise Allen that I was staying at the Australia, and there I knew he would come direct the moment he could shake himself free from his entertainers. Someone was to take him off to lunch, to be sure, but—especially as it was reported that he was already dieting to get back to riding weight—I felt sure this would not detain him long. It will be about three, I told myself, and left
- 70. word at the office that any man asking for me around that hour should be brought straight to my rooms without further question. I also 'phoned Lady X—— and begged off from showing her and a party of friends from Government House my pictures at four, as I had promised a couple of days previously. Being borne off to the inevitable and interminable Australian afternoon teas—or to anything else I could not easily shake myself free from very shortly after five —was one of the worst ordeals incident to the spell of lionizing that had set in for me from the day of my arrival in Sydney. What did I care for Sydney, anyhow? Paris was my goal—gay, cynical, heartless Paris, who took or rejected what her lovers laid at her feet only as it stirred, or failed to stir, her jaded pulses, asking not how it was made or what it had cost. Paris! To bring that languid beauty fawning to my own feet for a day—even for an hour, my hour—that would be something worth living—or dying—for. For many years I had been telling myself that (between three and five in the afternoon, of course) and now—quite aside from my nocturnal flights there on the wings of the Green Lady—it seemed that the end so long striven for was almost in sight. I lunched lightly—a planked red snapper and a couple of alligator pears—in my room, and toward two o'clock (to be well on the safe side) slipped into the old hunting jacket I have mentioned, and was ready; just that—ready. My nerves were absolutely steady. The hand holding the palette knife with which (to kill the passing minutes) I began daubing pigments upon a rough rectangle of blotched canvas on an easel in the embrasure of the windows, might have adjusted the hair-spring of my wrist-watch, and the beat of my heart was slow and strong and steady like the throb of the engines of a liner in mid-ocean. If either hand or nerve inclined more one way than the other, it was toward relaxation rather than tenseness. Tenseness— with a man who has himself in hand—is for the moment of action, not for the interval of waiting which precedes it. My whole feeling was that of complete adequacy; but then, the sensation was no new one to me—at that time of day.
- 71. Exhausting the gobs of variegated colour on my palette, I went to a table in the bathroom and started chipping the delicately tinted linings from the contents of a packing case of assorted sea shells, confining my attentions for the moment to a species of bivalve whose refulgent inner surface had caught and held the lambent liquid gold of sunshine that had filtered through five fathoms of limpid sea-water to reach the coral caverns where it had grown. Powdering the coruscant scalings in a mortar, I screened them from time to time, carefully noting the gradations of colour—ranging from soft fawn to scintillant saffron—as the more indurated particles stood out the longer against the friction of the pestle. At this time, I might explain, I was in the tentative stage of my experimentation to evolve and perfect a greater variety of media than had hitherto been available with which to express in colour the interminable moods of sea and sky and sunshine. The value of my contribution to art—not yet complete after five years—will have to be judged when I pass it on to my contemporaries and posterity. Of the part these colours played in my later and more permanent success (to differentiate it from the spectacular but transient spell of fame upon the threshold of which I stood at the moment of which I write), I can only say that had I been confined to the pigments with which my predecessors had been forced to express themselves, I should never have risen above the rating of a second or third class dauber of sea-scapes.
- 72. CHAPTER II HARD-BIT DERELICTS With Allen and his coming in the back of my brain, it was only natural that my thoughts, as I ground and sifted and sorted the golden powders, should turn to Kai and the train of events leading up to the ghastly tragedy of the Cora Andrews, so distorted a version of which had gone abroad as a consequence of the fact that Allen was alive and Bell was dead, and that I, so far, had not told what I knew of the circumstances under which the one and the other had been induced to board the stricken black-birder. It must have been, I reflected, its comparative remoteness from all of even the least-sailed of the South Pacific trade routes that was responsible for making Kai Atoll, a barely perceptible smudge on the chart of the Louisiades, the unofficial rendezvous for the most picturesque lot of cut-throats, blackguards and beachcombers that The Islands had known since the days of Bully Hayes and his care-free contemporaries. Like had attracted like after the original nucleus gathered, safety had come with numbers, and at the time of my arrival no man whose misdeeds had not made him important enough to send a gunboat after needed to depart from that secure haven except of his own free will. Among a score of hard-bit derelicts whose grinning or scowling phizzes flashed up in memory at the thought of that sun-baked loop of coral, with its rag-tag of wind-whipped coco palms and its crescent of zinc and thatch-roofed shacks, only three—or four including myself—occupied my mind for the moment. Allen—reckless daredevil that he was—had come to Kai from somewhere in the Solomons for the very good and sufficient reason that it was the only island south of the Line at the time where his welcome would not
- 73. have been either too hot or too cold to suit his fastidious taste. Bell had come, in a stove-in whaleboat, because Kai was the nearest settlement to the point where he put the Flying Scud—the trading schooner that was his last command, if we except the Cora Andrews —aground on Tuka-tuva Reef. The girl, who arrived with Bell in the whaleboat, came because he brought her. The tide-rips of Kai passage and the Devil's own toboggan were all the same to Rona— at this stage of the game, at least—so long as the big, quiet, masterful Yankee was bumping-the-bumps with her. And even afterwards—but let that transpire. I, Roger Whitney, artist, formerly of New York and Paris, and, latterly, man-about-the French-colonies, with no fixed abode, had been landed at Kai by a French gunboat from the Noumea station. I packed myself off from that accursed hole because the suicide of a couple of officers in whose company I had been drinking absinthe at the Cercle Militaire for some weeks had reminded me altogether too poignantly of what I might, in the ordinary course of things, expect to be doing myself before long. A change of scene and, if possible, a modification of habits was the only hope. I would never have had the initiative to tackle even the first had not the feeling persisted that I was on the verge of doing something worth while with my painting. I went to Kai because the archipelago thereabouts was reputed to have the most gorgeous sky and water colouring in Polynesia. Neither the promised beauties nor the reputed badness of Kai stirred me greatly in anticipation. With a bitter smile I told myself that every night I was seeing sights more lovely than anything my eyes were likely to rest on short of Paradise, while the Chamber of Horrors in which I awoke every morning was a veritable annex to the Inferno itself. No, it was out of the question that Kai could unfold in realities, whether to delight or shock, things to outdo those that were already mine in dreams that had themselves become more real than realities. Well, it turned out that I was only half right, or wrong, whichever way you want to put it. While, on the one hand, I found
- 74. the bluff, open badness of Kai rather more refreshing than shocking; on the other hand, it was hardly more than a week before I was ready to swear that not the most ethereal houri that ever laid her cool green hand upon my fevered brow was of a class to run one- two-three with a flame-quivering slip of a nymph whom I had surprised at her bath in a beryline pool inside the windward reef. I began to pull myself together from that hour. Rona, the very sight of whom threw most men out of hand, had quite the opposite effect upon me. I knew she was not for me, and the thought that the world actually held such loveliness in the form of flesh and blood had a sort of reassurance about it, like the knowledge that one has an ample income from government bonds. Because I had landed from the Zelee, and also, perhaps on account of my rig-out (especially the brimless Algerian sun-helmet), the beach of Kai put me down at once as a We-we, and, therefore, a creature quite apart. The only Frenchmen on the island were a couple of escapes from the convict settlement of New Caledonia, and because neither of them could ride or shoot or fight with their fists, they had no standing with the predominant Australian push, most of whom were more or less handy at all three. It was, indeed, the fact that, in spite of all my years in Paris and the French colonies had done to make a physical wreck of me, I still retained something of the quickness of eye and hand and foot which had conspired to make my Harvard record as an all-round- athlete one that only two or three men have equalled even down to the present day, that gave me such easy sledding in making my way with the best people of Kai. It took just three minutes—the length of the first round of the friendly bout I fought with Heifer Halligan, ex-welter-weight champion of Victoria, at Jackson's pub one afternoon—to change Kai's openly expressed contempt for me to something very near respect. I thoroughly appreciated the attitude of that breezy lot of sport-loving rascals toward a Frenchified Yankee artist, especially one that did not appear to be a fugitive from justice, and so took the
- 75. first opportunity to win a standing with them which would at least incline them to let me go my own way when I wanted to. Notwithstanding my wretched condition, I outpointed my chunky opponent a good three to one in that opening round; indeed, the Heifer's excuse for the foul which put me to sleep in the Second was that both his bloomin' peepers were so nearly swelled shut he couldn't see stryght. But it was my swelling groin and battered hands, rather than Heifer's bruised optics, that came in for first attention from deft-fingered Doc Wyndham—once of Guy's, on his own admission. The next day I was waited upon by a delegation sent from Jackson's Sporting Club to urge me to put myself in training for a go-to-the-finish with Shark-mouth Kelly of Suva, the Fiji open champ. My speed would dazzle a cow-footed dolt like Shark-mouth was, they said, and he would be easy picking for me. They further urged that we could clean up all the loose money west of the Hundred and Eightieth—what odds would Fiji not give in backing a fourteen-stone stoker against an artist that only weighed ten stone and looked half dished with the green besides? Moreover, I could keep the whole purse for myself; all they wanted out of it was the sport. God bless the scalawags, it was more than half true, that last. The funny thing about it was that the project actually tempted me at the time, principally, I think, because there seemed a chance that the hard exercise of training—the very thing, indeed, that helped work the miracle a few years later—might effect me at least a temporary separation, if not a permanent divorce, from the Green Lady. I was still temporizing with delegations when the Cora Andrews dropped her hook in Kai Lagoon and gave us something else to think about. If the little cunning I had left with my fists won me the respect of the beach, it remained for my proficiency with the revolver— something which I had never allowed myself to grow rusty in—to give me real prestige. My father had been only less famous as a pistol shot than as a builder of steel bridges, and from my birth it
- 76. had been his dream that I should carry on the tradition in both lines. If it had broken the old boy's heart when I turned my back on engineering for art—insisting on going from Harvard to Beaux Arts instead of to Boston Tec as he had planned—he at least had nothing to complain of on the score of my aptitude for the revolver. He admitted that I had bred true in hand and eye, even on the day that he called my art tomfoolery a throwback from my French grandmother. I have always thought that the one circumstance which prevented the Governor from cutting me off in his will when he finally had definite proofs of the depths to which I had sunk in Paris, was the fact that, on my last visit to the old home on the Hudson, I had beaten him, shot for shot, with his own pistols, and at his favourite distance. They were rather free with their gun play during my first fortnight at Kai, each little affair having been followed by one or two more or less ceremonious burials in the coral-walled cemetery on the south lip of the windward passage. It was merely as a precautionary measure—on the off chance that they should be tempted to draw me into something of the kind at a time when I might not be quite on edge for it—that I took early opportunity to uncover a trifle of what I had crooked in my trigger-finger. A casually winged gull or two, and a few plugged pennies (not a miss at the latter, luckily, even when they tried to spin them edge on to my line of fire) effected all that was necessary. After that, though they were continually sending for me to come down to Jackson's and shoot the wire off champagne corks (fizz, loot of some kind, was the freest flowing drink on the island at the time), or perform some other equally useful and spectacular gun stunt, not the roughest of the gang but took the most meticulous care not to press his invitation the instant it sank home to him that my mood of the moment wasn't of a kind calculated to blend smoothly with the free and easy spirit of a beach-combers' carousal. It was hardly to be expected that they would ever quite understand why a man who could blot out a cove's blinker as easy
- 77. wiv his fist as wiv his gun (as I was told that Reefer Ogiston, penal absentee and pearler, put it one day) and who 'peared mo' than comfitabl' heeled fo' coin, should be light an' looney enuf tu go roun' smearin' smashed barnculs on sail cloth; and yet it was on that very score—or at least to their quick comprehension of what I was driving at in my pictures—that the beach of Kai rendered me a priceless service. Almost from the outset they began to twig my marines, to feel the living atmosphere I was striving to paint into them. They were all men who had lived by the sea, on the sea; yes, and not a few of them had worked under the sea. Well, when I began to see those deep-set, wrinkle-clutched eyes squint to a focus of concentration, and, presently, the quick heave of a hairy chest as the message of the canvas flashed home, I knew that I was on the right track. Nothing less than that would have given me the courage to go on working, as I had set myself to do, on a steadily decreasing allowance of absinthe, a certain supply of which, of course, I had brought with me from Noumea. So much for me and my relations to Kai at the time of which I am writing. Now as to Bell.... Who is that tall, square-jawed chap who looks as though he was not quite sober? I had asked a day or two after I landed. Yank—calls himself Bell, Jackson replied laconically; adding that he was not quite sober when he tried to take a cross-cut over Tuka-tuva Reef with the Flying Scud, that he was not quite sober when he hit the beach in a busted whaleboat, that he had been not quite sober all the time since, and that there was no doubt that he would still be not quite sober when the time came for him to leave the island, whether he went out with the tide in an outrigger canoe or shuffled off up the Golden Stairs. Allus been pickled and allus goin' to be pickled, Jackson continued; then, qualifyingly: Course I don't know he was pickled when he kum int' the world, but I'm willin' to lay any odds that he'll be pickled when he shuffles out of it.
- 78. Just about all of which was, or proved to be, stryght dope. After quoting this terse summing of Jackson's, it may sound a little strange when I say that Bell was a gentleman—not had been, understand (that could have been said with some truth about a dozen or more of us at Kai), but was a gentleman. Though undeniably never quite sober, the fact remained that no one on the island had ever seen him quite drunk. And no matter how much liquor he had stowed under hatches, no one could say that it interfered either with his trim or his navigation. His even rolling gait was always the same, whether it was the glow of his eye-opening plunge at dawn that lighted his face, or the flush of twelve hours of steady tippling that darkened it at twilight. Nor was he ever known to omit that gravely courteous, almost old-fashioned, bow which, with the flicker of smile that was more of his eyes than his mouth, was the invariable greeting he bestowed upon friend and stranger alike. The mellow drawl of his It's suah goin' to be a fine mawnin', had made it easier for me to weather dawns that—in my inflamed imagination—menaced monstrously in jagged lines like a cubist's nightmare. If drink had any effect on his speech, it was to incline him to reserve rather than garrulity. His temper appeared to be under quite as perfect control as his legs. Even when he broke Red Logan's jaw with a swift short-arm jolt the time that sanguine Lochinvar tried to nip Rona off his arm as they passed on the beach in the twilight, they said that Bell hardly raised his voice as he guessed that'd hold the varmit fo' a while. And when, a few days later, Doc Wyndham told him with a grin that Red wouldn't be screwing a diving helmet on his block for some weeks to come, it was said there was real regret in the Yankee's voice as he hoped that the injury wouldn't be pumanant. Yes, before I had been a week at Kai I felt that there was a little addition I could safely make to Jackson's comprehensive estimate. I knew that Bell had been born a gentleman, and—whatever lapses there may have been, or might be—I knew he was going to die a
- 79. gentleman. And that also (had I put it on record) would have proved pretty nearly stryght dope. What stumped me at first was trying to reconcile the remarkable control Bell maintained over all his faculties in spite of his hard drinking with the fact (apparently fully authenticated) that he had run aground—through drunkenness—every ship he had ever commanded, beginning with a U. S. gunboat. He cleared up that matter for me himself one afternoon, however, by casually observing —at the moment he chanced to be watching me trying to transfer to canvas the riot of opalescence between the lapis lazuli of the barely submerged reef and the deep indigo where a hundred fathoms of brine threw back the reflection of the sinister core of cumulo-nimbus in the heart of a menacing squall—that the sea had always acted as a tremendous stimulant to him, especially when he trod a deck. If I could just have managed to cut out the whisky at sea, all would have been smooth sailin', he said in his deep rich Southern drawl. On land—heah ... anywheah—kawn jooce is lak food to me; mah body convuts it into ene'gy just lak an engine does coal. But with a schoonah kickin' undah me—we'ell, I guess theah's just one kick too many, something lak mixin' drinks p'raps. It suah elevates me good an' plenty ... and when I come down theah's natchaly some crash. My ship an' I gen'aly strike bottom at about the same time. But, s'elp me Gawd (a tensing timbre in his voice) on mah next command— It was the one sure sign that Bell was beginning to feel the kick of his kawn jooce when he spoke of his next command. Unless that kick was beginning to carry a pretty weighty jolt behind it he knew just as well as everyone else on the beach did that he would never get his Master's Certificate back again, and that even if he did there was no house from Honolulu to Hobart that would trust a ship to a man who had already beached a half-dozen.
- 80. Kai was glib to the last detail—rig, tonnage, cargo, insurance, owner and the like—respecting the several merchant craft Bell had piled up in the course of his downward career; but the extent of local dope in the matter of the gunboat episode was to the effect that it happened up Manila-way, and that that was the bally smash that started him goin'. Personally, I took little stock in the naval part of the yarn—that is, at first. Then, one morning—it was the day after the tail of a typhoon had sucked up the end of Ah Yung's laundry shack and left everyone on the beach short of clothes—Bell came out in a suit of immaculate starched whites. It was the cut of the jacket and the way he wore it that drew and held my puzzled gaze; that its shoulders were drilled for epaulettes and that its thin pearl buttons barely held in buttonholes that had been worked for something thicker and wider I did not notice till later. Steady-eyed, lean-jawed, square-shouldered, ready-poised—not even a flapping Payta sombrero could quite disguise, nor five years of heavy tippling quite obliterate, the marks of type. Then I understood why it was that Bell, all but down and out though he might be, was, and would remain to the last, a gentleman. There are things the Navy puts into a man that not even a court-martial can take away. The only allusion Bell ever made to his remoter past was drawn from him a few days later, when—he was watching me paint again— I chanced to mention that I had spent a fortnight in the Philippines on my way south from Saigon to Australia. Glancing up at the sound of his sharp intake of breath, I saw his jaw set over the questions that leapt to the tip of his tongue, to relax gradually as a faraway look came into his wide-set grey eyes and a wistful smile of reminiscence parted his lips. Did you heah the band play on the Luneta in the evenin'? he asked eagerly, while the spiggoties in their calesas wuh racin' round the circle, an' the kiddies an' theyah nusses wuh rompin' on the grass, an' the big red sun was goin' down behind Mariveles beyond
- 81. the bay? An' did you know the Ahmy an' Navy Club—not the new one ... the ol' one ovah cross the moat inside the wall? Put up there all my time in Manila, I replied. A very comfy old hangout, especially considering what the hotels were. An'—did you— (he gulped once or twice as though the question came hard) did you evah heah them speak at the Club of a chap called Blake ... Lootenant-Commandah Blake? He was a son of Captain Blake, who helped Sampson polish off Cervera, an' a gran'son of Adm'al Blake. Ol' naval fam'ly. You mean the man who pulled off that coup when Wood was cleaning up the crater of Bud Dajo? Some kind of a bluff on his own with one of the little old gunboats Dewey captured after the Battle of Manila Bay, wasn't it? Scared some Jolo Dato into giving up a bunch of our men he already had lined up against a wall to bolo, didn't he? Of course, I remember perfectly now. General X—— (mentioning the Military Governor of Mindanao by name) told me the yarn himself the night I dined with him in Zamboanga. He said no one but an old poker shark would ever have thought of the stunt, much less had the nerve to bluff it out. Incidentally he mentioned that the chap was the best poker player in the Navy, as he was also the speediest baseball pitcher ever graduated from Annapolis; that he had been missed almost as much for the one as the other since he dropped out of sight several years before. Some difficulty about— Tryin' to push Corregidor out of the entrance to Manila Bay with the nose of his gunboat, Bell cut in harshly, the hell in his soul glowing through his eyes as the glare of the coal-bed welters beyond a stoker's lifted furnace flap. That, and a single sob sucked through his contracted throat as the vacuum in his chest called for air, were the only outward signs of the intensest spasm of throttled emotion I ever saw assail a human being. Then the square jaw tightened, the cords of the muscular neck drew taut, and what would have been another body and soul racking sob was noiselessly absorbed in the
- 82. buffer of a flexed diaphragm. The fires of agony behind the eyes paled and died down like an expiring coal. The corrugations of the brow smoothed out as a smile—half amused, half wistful—relaxed the set lips. The old controlled Bell (I shall continue to call him so) was in the saddle again. So they still remembah mah ball-playin', he drawled musingly, his left hand digits gently massaging the bulbous swelling remaining after some red-hot drive had telescoped the middle finger of his right. Ye'es, of co'se they'd miss mah wing in the Ahmy-Navy game at Ca'nival time. But mah pokah—we'ell I reckon a few of 'em did find mah pokah hand about as bafflin' as mah baseball ahm. But it was straight deliv'ry, tho'—both of 'em. An' they wouldn't be callin' me a fo'-flushah, etha. No, you didn't heah any of 'em say that, I'm right suah. A smile more whimsical than bitter twitched his lips twice or thrice in the minute or two he stood alone with his thoughts. So I've sort o' dropped out o' sight to 'em? he said finally. We'ell, I guess that was about the best thing to happen for all consuned. But, just the same, if you evah go back Manila-way I won't be mindin' it if you tell 'em that, tho' the ol' wing's tuhn'd to glass from long lack o' limberin', an' tho' I don't play pokah down heah fo' feah o' bein' knifed fo' mah luck, I'm still hittin' true to fohm in mah own lil' game of alterin' the sea map with the noses of ships. I reckon they'll know the reason why. There was another interval of silence, but, unlike the other, not charged, electric. Bell's blow-off through the safety-valve of frank speech had taken the peak off the pent-up pressure within, and when he spoke again it was merely to quote what the Governor of North Carolina had said about its having been a long time between drinks. Great thust aggravateh, the Sou'east Trade. Would I mind— ahem—hiking home with him and lubricating my tonsils with a drop of J. Walkah? That was simply his delicate way of pretending to ignore my slavery to absinthe, a habit which not even the most
- 83. whisky-saturated sot of an Anglo-Saxon can ever quite forgive one of his race for falling a victim to. I wouldn't? We'ell, hasta manyanah. With a crunch of coral clinkers under his feet and a stave of Carry Me Back to Ol' Virginny on his lips, Bell, disdaining the smooth path by the beach, swung off through the pandanus scrub on what he called a bee-line for home! He had a weakness for taking short-cuts on land as well as at sea. Never again—not even in the moment of his great decision—did he lift for me or any other man the furnace flap of iron reserve that masked the fires of his innermost soul. Their saving sense of sport, which was the golden vein in the rough iron of the beach push of Kai, made it inevitable that they should have a substantial sense of respect for a man of Bell's stamp, and this might easily have ripened to an active popularity had not the American's quiet but inflexible reserve prevented their knowing him better. They suspected that he was no novice in handling the big Colt's that was flopping on his hip when he landed, they knew that there was a weighty punch behind his long arm, and they were frankly outspoken in their admiration of the manner in which he stowed and carried his booze. But what had impressed them more than anything else was the way in which he had taken the devil out of a vicious imp of a Solomon Island pony on the beach one morning. Hellish hard-handed, Slant Allen had said, as his steel- blue eyes narrowed down to slits in the intensity of his interest and admiration; but a seat like he was screwed to the brute's backbone. Old cross-country rider—hundred to one on it. Man in a million in a steeplechase on a horse strong enough to carry the weight. Gawd, what a seat! All in all, indeed, there was only one thing the beach held against Bell, and that was Rona, or rather his possession of her. There was nothing personal in this, of course. They merely regarded the big American in the same light they had always regarded a man with a chest of pearls or anything else of value that their simple,
- 84. direct natures made them yearn for the possession of. There was this difference, however. Where the push of Kai would have combined to a man to get away with a box of pearls or a cargo of shell, the annexing of a woman was essentially a lone-hand game, and—well, Bell was hardly the kind of a one-man job any of them cared to tackle. I feel practically certain that, but for the disturbance of the even tenor of Kai's way incident to the Cora Andrews affair, his rights in Rona would never have been challenged.
- 85. CHAPTER III THE GIRL HERSELF As for the girl herself, words fail me in trying to picture her, just as my brush and pencil (save perhaps for that one rough memory sketch, done at white heat while still gripped in the exaltation that first glimpse of her splashing inside the reef had thrown me into) have always failed. This is, I fancy, because, unbelievably beautiful though she was, there was still so much of her appeal that was of the spirit rather than the flesh—something intangible which had to be sensed rather than seen. She was compact of contradictions, physical as well as mental. So slender as almost to suggest fragility at a first glance, there was still not a straight line, nor an angle, nor a hint of boniness, from the arch of her instep to the tips of her ears. Again, pixie-like as she was in the dainty perfection of her modelling, there was yet a fairly feral suggestion of suppleness and strength underrunning the soft fluency of contour. The strength was there, too, held in reserve in the flexible frame like the power of a coiled spring. I saw her unleash it one morning when, impatient of the slowness of a clumsy Fijian who was launching a very sizable dugout for her, she yanked him aside by the hair of his fuzzy head and did the job herself. I can still see the run of muscles under the olive-silk skin of arm and ankle, and the bent-bow arch of her slender back, as she gave a last push to the cranky outrigger. Indeed, my mind is full of pictures like that—paddling, swimming, leaning hard against the buffets of a passing squall, with a lock of wet hair streaking across her glowing face and her drenched garments clinging to her lithe limbs; and yet, as I have said, the buoyant, flaming spirit of her always escaped my brush and pencil as it now eludes portrayal by my pen.
- 86. But the most baffling, as it was also the most fascinating, of Rona's contradictions was the combination she presented of inward intensity and outward calm. The fire of her was, perhaps, the first thing one was conscious of. Even I, with my blood thinned and cooled with the ice of absinthe, could never watch her movements without a quickening of my jaded pulses; to the sanguine combers of Kai the sight of her (whether the rippling undulations of arms and shoulders as she drove a canoe through the water, or the hawk-like immobility of her as she poised on a pinnacle of reef waiting for a chance to cast her little Dyak purse-net) was palpably maddening. So much for the flaming appeal of the girl in action, or suspended action, which was, of course, about the only way in which she was ever revealed to the beach. Now picture the same creature (as Bell —and occasionally myself, his only intimate friend on the island—so often saw her) seated cross-legged on a mat, her sloe-eyes, set slightly slant, fixed dreamily on nothingness, like a sort of reincarnated girl-Buddha. The sight of her thus never failed to awaken in my nostrils the smell of smouldering yakka sticks, and to set my ears ringing with the throb of temple bells. To my hyper-sophisticated (I will not say degenerate) senses this Oriental side of the girl made a subtle appeal that was like an enchantment. The passion to paint her—always burning within me when I saw her in action—never assailed me when she fell into one of those contemplative calms. Rather the peace of her soothed me like an opiate and made me content to sit and dream myself. It was the one thing (until I got the habit by the throat years afterward) that ever held my nerves steady when the absinthe hour drew near at the end of the afternoon. As long as Rona would continue to sit Buddha I had myself completely in hand, even till well on after sunset. But if she moved, or spoke, or even showed by her eyes that she was following Bell's words (it was he—less sensitive to this phase of her than I—who did most of the talking at these times), the spell was broken. The haste of my bolt for home was almost indecent. I have sometimes thought that a few months alone with
- 87. Rona at this time might have effected very near to a complete cure in me—by a sort of involuntary mental therapeutic treatment on her part, I mean. But perhaps the other side of her—the unreposeful one—might have complicated the case. Both the fire and the repose of Rona—the passion and the peace of her—were reflected in the olive oval of her face, the one by the full, sensuous lips and the sensitive nostrils, and the other by the smooth, low brow. The low-lidded blue-black eyes were debatable territory, now in the hands of one, now the other. So, too, that infallible gauge of temperament, whose dial is the pucker between the eyebrows. With Rona, this passion-pressure index was a corrugated knot of intensity or an olive blank according as to whether her inner fires were flaming or banked. Bell knew little of the girl's origin and said less. Rona's trousseau consisted of huh peacock sca'f an' this heah baby bolo, he said in his slow drawl one afternoon when he had borrowed the exquisite little dagger to show me how the Jolo juramentado executed his favourite belly-ripping stroke; an' I reckon they'll comprise 'bout the sum total of huh mo'nin' at mah fun'ral. That, and I guess Rona knows no mo' 'bout mah past reco'd than I do 'bout huhs, was all I recollect his ever having said on the subject. He was content to let it rest at that. It was old Jackson who told me that he had seen the girl at Ponape, where she had been brought by an owl-eyed (referring to horn-spectacles rather than to the almond orbs themselves, I took it) chink when he came back to the Carolines after buying bird-of- paradise skins down New Guinea-way. She was dressed Java-style at the time, and was said to have been picked up at Ternate or Ambon in the Moluccas. Although the wily old Celestial kept the girl practically under lock and key from the first, customers of his shop occasionally glimpsed her, and she them, it would seem. Among these was the Yankee skipper of the trading schooner, Flying Scud. The coming together of those two must have been like the touching
- 88. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com