3rd year project report

Cardiff University
School of Physics and Astronomy
Student number: C1005749
Star formation studies with Herschel space
observatory
Author:
Pawala Ariyathilaka
Supervisor:
Dr. Pete Hargrave
May 8, 2015

Contents
1 Abstract 2
2 Introduction 3
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Initial mass function (IMF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2.1 Core mass function (CMF) . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Background theory 5
3.1 Molecular clouds and Dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Jeans Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Pre Stellar Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 From gas cloud to Protostar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.5 The Herschel space observatory . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Data collection and discussion 7
4.1 The images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Tools developed and used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2.1 GAIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2.2 Photometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2.3 Greybody curve ﬁtting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Core Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3.1 Mass estimation function . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3.2 CMF plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.3 Error discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Conclusion 18
1

1 Abstract
Pre stellar cores (PSCs) within the Aquila rift were studied. A PSC is a starless gravitationally
bound core within a cloud of gas and dust. The aim was to measure the mass of the cores and
to construct a core mass function (CMF), to see whether there is a relationship between the
CMF and the initial mass function (IMF). An IMF is the mass distribution of stars at the
time they enter the main sequence, and the CMF is the mass distribution of the cores.
Using a written photometry python script, the flux of selected sources were calculated. The
flux was plotted and a greybody function was fitted to the data. With the greybody function,
the temperatures of the cores were estimated. The temperatures were used in a mass equation
to calculate the core mass.
Comparing the outcome with the Konyves CMF (which was used as a base for comparison),
the calculated CMF was five orders of magnitudes smaller. Reasons for discrepancies are the
error caused by the mass calculation python script, along with the shape of the aperture and
the aperture size used.
2

2 Introduction
2.1 Overview
The main objective of the project was to locate pre stellar cores within the Aquila rift, then
to produce a core mass function (CMF), which demonstrates the distribution of mass of these
dense cores. These dense pre stellar cores are one of the first stages of star formation. Data
collected from the Herschel space observatory was used to look for pre stellar cores within the
Aquila region. Two maps from the PACS and three maps from the SPIRE instrument were
used. The wavelength range was from 70µm to 500µm.
2.2 Initial mass function (IMF)
IMF is the mass distribution of stars, at the time they enter the main sequence. Through
observations, it is concluded that the shape of the IMF is universal, through many observations
of star clusters. Yet it is not understood where the shape comes from. One widely used IMF
is the Salpeter IMF[1] which was the first IMF to be derived in 1955. Salpeter fitted available
observational data at the time with a power law between 0.4M and 10M and found,
φ(M)dM KM−2.35
dM (1)
Where K is a constant of proportionality and φ(M)dM is number fraction of stars between
M and M + dM. The available data was main-sequence star luminosities within the solar
neighbourhood. The formulation of the above function involves looking at the different lumi-
nosities, splitting them in to intervals and counting how many stars fall within each interval
using equation (2),
L
L
=
M
M
ν
(2)
Where M and L are the mass and luminosity of the star being observed, and M and L
are the mass and luminosity of the Sun. The value for ν has a range from 3 to 5 (Prialnik -
Theory of stellar evolution). A value of 3.5 is commonly used for main sequence stars.
The IMFs have developed over the years with more observations been taken such as Chabrier
(2003), Kroupa (2001). However it has not changed much from the Salpeter form. Another
IMF is the Kroupa IMF that was published in 2001. This was derived from a sample of young
star cluster observations[2]. He introduced the following,
φ(M)dM KM−2.3
dM : 0 > M > 0.5M (3a)
φ(M)dM KM−1.3
dM : 0.5M > M > 0.08M (3b)
φ(M)dM KM−0.3
dM : 0.08M > M > 0.01M (3c)
Where M is the solar mass and the other symbols are same as of (1). The Kroupa IMF
is a more universal IMF unlike Salpeter given by (1), which is mainly for much larger stars
than the Sun.
3

2.2.1 Core mass function (CMF)
The CMF was only observed recently and derived, when sub-mm astronomy helped observe
the cold cores hidden with gas and dust clouds. Both the IMF and CMF have a similar shape,
but the CMF peak is shifted to higher mass as evident from figure 1.
Figure 1:
Figure showing CMF plotted alongside the Kroupa and Chabrier IMFs. The CMF is shifted
to the right in comparison with the IMF, but when the cores end up as main sequence stars
on the IMF, the peak moves to the left. This shows the inefficiency when moving from the
CMF to the IMF[3].
This suggests the star forming inefficiency within the pre stellar cores. Normally, only
about 1
3 of mass from CMF makes it to the IMF. The reasons behind this is still not clearly
understood. However, the relationship between the efficiency is thought to be related to the
bipolar flow of a newly formed protostar. The outflow is thought to be limiting the accreted
mass on to the protostar. Theoretical work has shown that outflows can give star forming
efficiencies of about 30% which matches the observations[4].
The core mass function is defined by Ward-Thompson and Whitworth in the textbook, An intro-
duction to star formation as follows,
φ(Mcl)dMcl ∝ M−2.3
cl : Mcl ≥ 2.4M (4a)
cl : 2.4M ≥ Mcl ≥ 1.3M (4b)
cl : 1.3M ≥ Mcl ≥ 0.4M (4c)
Where φ(Mcl)dMcl is the number fraction of cores between (Mcl) and (Mcl) + dMcl[5]. The
similarity between the two curves suggest that the IMF for stars are predetermined by the
CMF. If the CMF follows a universal appearance, this can give clues to why the IMF have
this particular shape.
4

3 Background theory
3.1 Molecular clouds and Dust
Star formation takes place in clouds of gas and dust. The matter in the clouds is constantly
moving, acted on by gravity and being influenced by pressure. These gas clouds are vast, and
even though they have a lower density than the Sun, an average mass of a cloud exceeds the
mass of the Sun by many magnitudes.
The universe is full of dust. When doing astronomy, the effect dust imposes on an obser-
vation needs to be taken in to account. Most the visible light emitted by stars are absorbed
by dust and re-emitted at longer wavelengths, specifically at sub-mm and far infra-red. Dust
radiates away energy like a greybody (an adjusted blackbody), with temperatures between
10K and 100K. Dust particles can capture atoms, molecules to form new molecular species
and grains can lump together to form larger grains.
3.2 Jeans Mass
The densest part of molecular clouds need to be observed when looking for new stars and star
nurseries. These dense areas within the clouds are created through stellar winds, magnetic
fields and supernova shock waves which disrupts the ISM. These create perturbations, which
create areas of high density within parts of the giant cloud. Now the gravitational potential
energy within that region, will increase along with the gas pressure. However, the change
in pressure will not be enough to maintain hydrostatic equilibrium. So the result will be
a gravitational collapse of the cloud. We define Jeans mass as the mass upper limit, that
can be contained in hydrostatic equilibrium within a region of a given volume. If the cloud
mass exceeds the Jeans mass, the cloud will gravitational collapse. Jeans mass is defined by
Prialnik as,
MJ =
3
4π
1
2
3
α
3
2
RT
µG
3
2
1
√
ρ
(5)
Where α is a constant of the order of unity, R is the ideal gas constant, G is the gravitational
constant, T is the temperature, ρ is the density and µ is the sum of the mean atomic mass of
stellar material which has a value of 0.61 for the solar composition.
3.3 Pre Stellar Core
Prior to having a protostar, a pre stellar core(PSC) is formed. The proper definition of
a pre stellar core, as defined by Ward-Thompson and Whitworth (2011) is, a phase in which
a gravitationally bound core has formed in a molecular cloud, and moves towards higher degrees of
central condensation, but no protostar exists within the core. The material in the pre stellar core
is sufficient for an individual star or a small solar system. With temperatures ranging from
7K-15K these are some of the coldest regions in the universe. They emit radiation in the far
infrared and sub-mm bandwidth region. Typical size of PSCs are around 0.05pc in diameter
and a density of 105
cm−3
[7].
5

3.4 From gas cloud to Protostar
Once the pre stellar core becomes gravitationally unstable, it collapses. The initial released
gravitational energy, is radiated away thus keeping the temperature relatively constant. Be-
cause of the large radius, there is high luminosity and a relatively low temperature.
The collapse will create a central matter concentration. This will turn in to a protostar
when hydrostatic equilibrium is reached. As the matter at the center reaches hydrostatic
phase, due to contraction and expansion of the collapsing cloud, an opaque gaseous envelope
will form. As this envelope becomes more dense, it becomes harder for gravitational potential
energy to radiate away. This reduces the luminosity. This evolutionary path is called the
Hayashi track as shown in figure 2. The Hayashi track is followed by cores that have a mass
< 3M .
Figure 2: Different paths protostars take as they join the Main sequence [8]
The object at the center of the protostellar core starts to gather in-falling material from
the accretion disk, that is now formed around the protostar. Most mass of the protostar is
attained at this stage, whilst constantly warming up.
Whilst material is being accreted towards the central mass, there is also an ejection of
material through opposite directions of the protostar, called bipolar flow as shown in figure 3.
It is thought that this process takes place in order to carry away excess angular momentum of
the in falling matter. The protostar is called a pre main sequence star when it has gathered
most of its final main sequence mass.
Eventually the interior of the protostar becomes hot enough for radiative energy to dom-
6

Figure 3: Figure showing the accretion disk and bipolar flow of a protostar [9]
inate. Still supplying most of its luminosity through contracting and releasing gravitational
potential energy. But both temperature and luminosity are now increasing and the protostar
moves down the Henyey track shown in figure 2 until the main sequence phase. When at the
main sequence phase the central density and temperature are high enough for nuclear fusion.
Thus the star stops contracting and reaches its more stable years[10].
3.5 The Herschel space observatory
The five images used in this project, were imaged by the Herschel space observatory. Herschel
is one of European Space Agency’s (ESA) cornerstone missions, launched in 2009 to study
the ’cold’ universe.
Herschel operates at the far infra-red and sub-millimetre (submm) wavelength region. This
was the largest infra-red telescope in space as of the launch date with a mirror diameter of
3.5 meters. Herschel will mainly observe dust and cold objects in the universe, which emits
radiation in the spectrum Herschel operates in.
Herschel has 3 instruments, PACS (a camera and medium resolution spectrometer), SPIRE
(a camera and a spectrometer) and lastly HIFI (high resolution heterodyne spectrometer).
PACS is centred on 70µm and 160µm with the broad bandwidth (∆λ
λ )<0.5 (Poglitsch 2010)[11].
For SPIRE, the central wavelengths are 250µm, 350µm and 500µm. SPIRE has broad band-
width (∆λ
λ ) of ≈ 3 (M.Griffin et al 2010)[12]. The instruments work in different bandwidths
from 60µm all the way to 670µm.
4 Data collection and discussion
4.1 The images
The data used in this report were from the Gould belt survey[13], which allows the study
of different aspects of star formation, from pre stellar cores all the way to protostars. The
images used were of the Aquila rift, part of the Aquila constellation.
7

Each map is 2314 × 2314 pixels, with each pixel representing six arcseconds. In total there
were five maps, two from PACS instrument and three from SPIRE.
Figure 4 represents a multi-wavelength image of the Aquila rift. The Herschel maps are in
units of MJy/Sr (MegaJanskys/steradian) which will need to be converted to Janskys when
the flux values are found.
Figure 4: Figure showing the Aquila rift. This image has all the different wavelengths
super-imposed.[14]
4.2 Tools developed and used
The main aim of the project was to construct a core mass function for the Aquila rift. In order
to achieve this, 250 sources were selected and their flux were measured. This was done using
a photometry script written in python. The flux calculated was plotted against frequency.
Then a greybody function was fitted amongst these points to calculate the temperature of
the source. Then the temperature was used to calculate the dust mass, which allowed the
calculation of the mass of the source. These steps were repeated for all the sources within the
study. The first step was to use GAIA to select sources.
4.2.1 GAIA
The acronym stands for Graphical Astronomy and Image Analysis tool. This is an image
display tool that has the capabilities to carry out photometry[15].
The Herschel images were loaded on to GAIA, and source selection was carried out on
the Herschel maps, one at a time. By changing the ’cut’ level within the maps one can find
relatively faint sources.
To do photometry, select a circular (only available one on GAIA) aperture of any radius
(given in pixels), and place around a source. This then calculates the total flux within that
aperture, given in counts along with the position. The sky background aperture is drawn at
8

the same time the main aperture is defined. When GAIA presents the final counts values, it
has taken to account of the sky background.
Even though it can perform its own photometry, the main purpose of GAIA was to select
the sources. The source positions are selected when the apertures are placed on the maps.
Since the aperture radius was kept constant throughout the project, care was taken to choose
sources that had a diameter similar to that of the aperture. A total of 250 sources were
selected. All the data is saved in to a table that can be loaded on to python. Data includes
the flux, X and the Y positions of the selected sources.
4.2.2 Photometry
A major part of the project was to develop a piece of python code to effectively measure the
flux of the selected sources. This was the first step towards obtaining a CMF.
It simply works the following way. The pre selected X and Y positions are loaded on to
python. Then, a square aperture of diameter 12 pixels (72 arcseconds) is placed around each
selected source to calculate the count rate. The photometry script is written such that, it
measures the flux for each source from all five Herschel maps. This output flux (which is in
MJy/sr) is then converted to Jy. Figure 5 shows the 250 selected sources mapped out on one
of the Herschel maps.
Figure 5: Figure showing the positions of the selected sources next to a multi-wavelength
image of the same area.
Two methods were used to measure the sky background count rate. Again, these count
rates also have to be converted to Janskys.
The first method was to have a sky background aperture, that was 0.5 larger than the
source aperture. This sky background aperture was placed around each source. The area of
the sky background aperture and the main source aperture were the same. When the final
counts were calculated, the sky background will be deducted from the counts within the source
9

aperture. This way, the sky background count varies from one source to another. So the 250
different sources had 250 different sky background apertures.
The second method was to have a sky background aperture in a quiet part of the map
that do not have any sources. Having a background aperture of such, it is possible to get a
true value of the background, one that has not been affected by any filaments. Then using
this, the average sky background Janskys per pixel was calculated. This calculated value, was
deducted from each pixel within the source aperture. This way a constant background value,
that is deducted from each source was obtained.
Figure 6 shows the two different sky background selecting methods along with the GAIAs
default aperture.
Figure 6: Figure showing the two different sky background selecting methods and the GAIA
aperture for comparison. (A)shows the first method and it is similar to the approach GAIA
takes. A source aperture in the center, and a background apertures around it. The only
difference between this method and GAIA is the shape of the aperture. (B) shows the default
GAIA circular aperture. Sub figure (C) shows just the source aperture, the sky background
is obtained from a separate part of the sky, as shown in (D)
Converting the map units of MJy/sr to Janskys is important. Janskys are normally used to
describe point sources, since PSCs are assumed spherical, working with Janskys is required.
The conversion is done as below,
Area in steradians =
2π
360
2
(2r)2
(6)
Where r, the radius is in degrees and is given by,
r =
Pixel radius
3600
(7)
Where Pixel radius is in arc seconds.
Use (6) and (7) to calculate the area of the aperture in steradians. Then multiply the counts
by 106
in order to convert to Jy/sr. Finally multiply by steradians to get an output of counts
in Janskys.
10

The counts from GAIA were used to check the accuracy of the photometry script. Figure
7 shows the difference in counts between the photometry script and GAIA. This is when the
first method for sky background reduction was used.
Figure 7: Figure showing the difference in counts for method one, of sky background reduc-
tion. The average counts for all the sources using the photometry script is 3357.5, the average
with GAIA is 3363.3 counts. The average difference in counts between the two photometry
methods is 5.8 counts, which suggests that the photometry code does a good job of what
GAIA does, since the method of measuring the counts is exactly the same as GAIA. The only
difference is GAIA uses a circular aperture.
Figure 8 shows the same concept as figure 7 but for the second sky background reduction
method.
The first method is exactly what GAIA implements, with a central source aperture and
surrounding larger background aperture. So it is expected that most the data points agree
with the written python script. A potential for errors is having a square aperture whereas
GAIA uses a circle. The area difference between the two mean, more sky background than
the circle. Also the very high flux count values are unlikely to be from a pre stellar core (more
likely to be a protostar).
In the second method, an aperture was placed around a quiet part of the sky. From this,
an average flux value was found per pixel. This was always likely to be a small value, since
the background aperture will be placed in an area with no sources, so little flux. This was
then used as the sky background value. So this value was subtracted from the flux calculated
for all 250 sources on this project. So when this small sky background value deducted from a
bright source (that we will expect from the main sub-field of this map), the expected outcome
11

Figure 8: Figure showing the difference in counts for method two of the sky background
reduction. GAIA and photometry do not agree with each other here. Doing the background
in this method has the advantage that an average background noise from each pixel is taken
out, means the background noise value will be a constant for each source. By using this method
we could eliminate the errors that could arise from having lots of background emission (such
as a filament that do not belong to the source). Also having a sky background aperture in
an area with no sources will give a proper value for the background, one that has not been
affected by emission from a source.
should represent something of the sort shown in figure 8. Again, having the square aperture
might have some implications on the outcome, in terms of more background.
In order to produce the CMF presented in this report, the second method was used as
the sky background reduction method. The greybody fitting function was implemented with
both sky background methods initially. Through this test, the second method came out as
the more reliable one to obtain the core temperatures. Even with the discrepancy between
GAIA and the photometry python script, when the output flux values were used within the
curve fit function code, more than 97 percent of the sources gave out an expected greybody
function fit and a temperature. Out of the 250 selected sources, only on 7 sources the curve
fit function failed. The comparison between the two is shown in figure 9,
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Figure 9: Left hand side image shows method one, and the right side shows method two.
When method one was implemented, there were 20 graphs that didn’t produce a well fit
greybody function. When looked closer, it was not entirely clear the exact cause for the
discrepancies. But the most likely scenario is that the sky background aperture had more flux
than the source aperture, the presence of a filament that goes through the sky background
aperture could cause these discrepancies. Towards the most active part of the cloud, there
were sources that clearly had filaments through the apertures. This is evident if a closer look
is taken at the first image on figure 6. A filament that goes through the sky background
aperture is visible. These are very difficult to avoid. There was also one instance that two
apertures were overlapping which was corrected.
4.2.3 Greybody curve fitting
The final output from the photometry code is an array of 250 × 5 which contains all the flux
values, for all the sources, from the five maps in Janskys. Afterwards, this data was plotted
against frequency, along with a modified greybody function using (8). The curve fit function
on python is used in this instance.
Sν = Ω
c
ν
β
B(ν, T) (8)
Where Sν is the flux density at frequency ν, Ω is the dust constant, β is the dust emissivity
index (fixed at 2 for purposes of this report, Hildebrand 1983[16]) and B(ν, T) is the Planck
blackbody function in terms of frequency given by (9),
B(ν, T) =
2hν3
c2
×
1
e
hν
kBT
− 1
(9)
Where h is the Plancks constant, c is the speed of light, ν is the frequency, T is the
temperature and kB is the Boltzmann constant.
In order to plot equation (8), some initial guess values for Ω, β and T had to be estimated.
These estimates were used within the curve fit function on python. The initial estimates were
0, 2 and 20K in that order. The outcome of this piece of code is shown in figure 10, the five
13

flux values plotted with a modified greybody curve. A graph was produced for each one of
the 250 sources. The greybody functions gives out a calculated temperature of the source and
saves the values to an array.
Figure 10 shows the plotted flux and the modified greybody curve. This is one example
from a list of 250. The calculated temperature from figure 10 was 9.64K which lies in the
range we expect for a pre stellar core.
Figure 10: Figure showing the flux values obtained from GAIA plotted with the modified
greybody function. The second sky background reduction has been used. This greybody
function gives out an accurate temperature of the source its looking at.
The average temperature for all the sources were about 15.049 Kelvins. The expected
temperature range for PSCs are between 7 and 20 Kelvins. Within the sources selected, there
could be protostars. When the flux is plotted against frequency, if the source is a protostar,
the flux is expected to increase at low wavelengths, rather than drop. If such plots were found,
the data was removed from the final set.
4.3 Core Mass Function
4.3.1 Mass estimation function
With the temperatures obtained, the mass of these cores can be determined. Using equation
(10) (Hildebrand 1983) which is shown below, the mass of the dust between Earth and the
pre stellar core can be figured out. Since only 1% of ISM is dust, to find the mass of the core,
the calculated dust mass is multiplied by 100.
Md =
SνD2
kνB(ν, T)
(10)
Here Sν is the adjusted blackbody curve, D is the distance, B(ν, T) is the Planck blackbody
function and kν is the dust absorption coefficient(Konyves 2010[17]) given by,
14

kν = 0.1
ν
1000GHz
β
(11)
where ν is the frequency and β is the dust emissivity index.
There is extreme uncertainty in the value for kν since it is difficult to measure and the value
changes with the physical environment as well. Values between 0.1-1.5 are used. The value
Konyves used which this paper adopted was 0.5 for kν. That value is mostly accepted for 350
and 500µm since these were the wavelengths previous studies took place in.
4.3.2 CMF plot
Using (10), the dust mass was calculated. By multiplying this by 100, the mass of the core
was attained since the core is a gravitationally held gas. The CMF was obtained by plotting
the core mass, against the number of cores in each of these mass intervals. The expected
graph should be similar to the shape of the IMF but shifted to a higher mass.
A python script was used to obtain the mass values of the selected sources. The mass was
calculated using (10), using an initial value of 0.5(Konyves 2010) for kν. However with this
value the CMF obtained was five magnitudes out, when compared with Konyves. This CMF
is presented in figure 11. The error could be credited to a problem in the mass calculation,
thus a problem with the written python code. Attempts should be made to rectify this if time
allows.
Figure 11:
The photometry gives the expected shape of the CMF, even though it is shifted to much
lower mass than expected. A value of 0.5 was used for the kν constant (konyves
2010).Method two was used as the background subtraction method.
If the value of kν is adjusted to 10−5
, a Konyves CMF is obtained. Here, the Konyves
CMF is assumed to be a base CMF. This is not completely incorrect since kν is an unknown
15

constant, however suggesting such a small value is similar to saying there is no dust in space
and implying that dust has a very small effect, which is not the case. With this new value for
kν, a CMF for the Aquilla rift is obtained. This shown in figure12
Figure 12:
The figure shows the obtained CMF for the Aquilla rift. A value of 10−5 for kν was used.
The second method has been used for background reduction.
This CMF looks similar to the Konyves CMF. However due to a smaller sample number
figure 11 has a lower number of cores per bin.
4.3.3 Error discussion
When the Konyves CMF is compared with the calculated CMF, the difference between the
two is evident, with the calculated CMF being 5 magnitudes smaller.
The main source of error in the project was a problem with the code that was used to
calculate the mass. However, there are also a other sources of error that will lead to the
discrepancies between the two CMFs. After adjusting the mass function, these are the other
sources of error that needs to be corrected.
Error arising from the positioning of the apertures. In order to record the X and Y positions,
250 separate sources had to be placed within the Herschel maps using GAIA. If the aperture
is not central on the source initially, it will be reflected in the photometry script. So if the
GAIA apertures were off center, when python places its own square aperture, those apertures
it will be off center from the source too. This mean the counts output is different. The counts
output could be larger, or smaller than the real source counts.
The main error in this project (after the mass calculation) arises from the kν constant. The
value of this is not known well at all, and a slight change in the value can cause big differences
in the outcome. Values from 0.1 to 0.7 have been used previously ( Hildebrand 1983, Kramer
16

et al 2003) and even 1.5. This is a hard constant to validate since the value also depends on
the environment the dust is in.
By changing from the current square aperture to a circular one as shown in figure 13 will
have a big impact on the flux, thus the core mass.
Figure 13: Figure showing the current square aperture and the potential circular one. As
evident, the background that the circular aperture measures will be less. The source is shown
in red.
This way, the source will be covered without having extra background being involved, since
we are assuming a spherical shape for the sources. The reason for not using circular apertures
in this project was because, the difficulty of programming a circular aperture. With more
time available, this should be possible and should yield better results.
Another factor that could change the outcome is having different radius values for different
sources. In this project the radius was fixed at 6 pixels. However, what this means is that
one would get more flux sometimes, and less flux at other times as shown by figure 14.
Figure 14: Figure showing the problems that arise from having a constant aperture. With
a changing aperture size for each source, a more accurate flux value can be calculated. The
source is shown in red
Applying the above method will change the flux obtained from the cores. This also mean
that the list of sources that can be looked at will increase since sources of all sizes can be
17

observed. In this project, on purpose, the sources were chosen such that they were as close
to a diameter of 12 pixels as possible. But when we can work with multi radii levels more
sources can be looked at. This will increase the data sample, which will give more PSCs to
analyse.
Another option to try out would be a follow the Konyves paper as an example and concen-
trate mostly on the main sub-field. This is the brightest part of Aquila rift where most the
activity is happening. The main sub-field is the blue area shown on figure 5.
5 Conclusion
The CMF obtained using a kν value within the accepted range, does not reflect the CMF
obtained by Konyves. Since this project closely followed that paper, it is safe to place the
Konyves CMF as a base to compare the outcome with.
The CMF obtained was five magnitudes smaller than the Konyves CMF. A typical IMF
drops off at small masses. This is because stars with a small mass are less luminous, thus
it is difficult to observe these faint main sequence stars. Since the CMF is presumed as the
progenitor for the IMF, it is expected that the CMF will also drop at lower mass values.
However according to the CMF obtained in this project, most the observed cores were faint.
But this is an extremely unlikely scenario due to observation difficulties, and therefore most
likely false. Bearing this fact in mind it is safe to assume that there is likely to be some error
in the mass calculation that caused this discrepancy.
This would be the first error to be corrected with more time. It is assumed that the error
originated at the mass calculation python script. After this correction, there are some other
error sources to study further and improve on.
One of them is to change the aperture size and and shape. Changing from a square aperture
to a circle will reduce the background signal that will be recorded. This means the flux values
will be different, thus resulting in different mass estimates which will lead to an improved
CMF. Having a variable aperture radius for different sources will allow the use of a larger
data set. Also with this in place, the maximum possible flux can be measured accurately.
The aperture will be as close to the radius of the source as possible, thus most the source flux
can be recorded.
With these errors rectified, a CMF that can be compared with the Konyves CMF could be
achieved.
18

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[17] The Aquila prestellar core population revealed by Herschel, Author: V.Konyves et al,
Publisher: Astronomy and Astrophysics, Date: 02/05/2010
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