Extreme trans neptunian objects and the kozai mechanism

arXiv:1406.0715v2[astro-ph.EP]16Jun2014
Mon. Not. R. Astron. Soc. 000, 000–000 (2014) Printed 17 June 2014 (MN LATEX style file v2.2)
Extreme trans-Neptunian objects and the Kozai mechanism:
signalling the presence of trans-Plutonian planets
C. de la Fuente Marcos⋆
and R. de la Fuente Marcos
Universidad Complutense de Madrid, Ciudad Universitaria, E-28040 Madrid, Spain
Accepted 2014 June 3. Received 2014 June 3; in original form 2014 April 23
ABSTRACT
The existence of an outer planet beyond Pluto has been a matter of debate for decades and the
recent discovery of 2012 VP113 has just revived the interest for this controversial topic. This
Sedna-like object has the most distant perihelion of any known minor planet and the value
of its argument of perihelion is close to 0◦
. This property appears to be shared by almost all
known asteroids with semimajor axis greater than 150 au and perihelion greater than 30 au (the
extreme trans-Neptunian objects or ETNOs), and this fact has been interpreted as evidence for
the existence of a super-Earth at 250 au. In this scenario, a population of stable asteroids may
be shepherded by a distant, undiscovered planet larger than the Earth that keeps the value of
their argument of perihelion librating around 0◦
as a result of the Kozai mechanism. Here, we
study the visibility of these ETNOs and confirm that the observed excess of objects reaching
perihelion near the ascending node cannot be explained in terms of any observational biases.
This excess must be a true feature of this population and its possible origin is explored in the
framework of the Kozai effect. The analysis of several possible scenarios strongly suggest that
at least two trans-Plutonian planets must exist.
Key words: celestial mechanics – minor planets, asteroids: general – minor planets, asteroids:
individual: 2012 VP113 – planets and satellites: individual: Neptune.
1 INTRODUCTION
Are there any undiscovered planets left in the Solar system? The an-
swer to this question is no and perhaps yes! If we are talking about
planets as large as Jupiter or Saturn moving in nearly circular orbits
with semimajor axes smaller than a few dozen thousand astronom-
ical units, the answer is almost certainly negative (Luhman 2014).
However, smaller planets orbiting the Sun well beyond Neptune
may exist and still avoid detection by current all-sky surveys (see
e.g. Sheppard et al. 2011). Nevertheless, the answer to the ques-
tion is far from settled and the existence of an outer planet located
beyond Pluto has received renewed attention in recent years (see
e.g. Gomes, Matese & Lissauer 2006; Lykawka & Mukai 2008;
Fernández 2011; Iorio 2011, 2012; Matese & Whitmire 2011). So
far, the hunt for a massive trans-Plutonian planet has been fruitless.
The recent discovery of 2012 VP113 (Sheppard & Trujillo
2014), a probable dwarf planet that orbits the Sun far beyond Pluto,
has further revived the interest in this controversial subject. Tru-
jillo & Sheppard (2014) have suggested that nearly 250 au from
the Sun lies an undiscovered massive body, probably a super-Earth
with up to 10 times the mass of our planet. This claim is based
on circumstantial evidence linked to the discovery of 2012 VP113
that has the most distant perihelion of any known object. The value
of the argument of perihelion of this Sedna-like object is close to
⋆ E-mail: nbplanet@fis.ucm.es
0◦
. This property appears to be shared by almost all known aster-
oids with semimajor axis greater than 150 au and perihelion greater
than 30 au (the extreme trans-Neptunian objects or ETNOs), and
this has been interpreted as evidence for the existence of a hid-
den massive perturber. In this scenario, a population of asteroids
could be shepherded by a distant, undiscovered planet larger than
the Earth that keeps the value of their argument of perihelion li-
brating around 0◦
as a result of the Kozai mechanism. The Kozai
mechanism (Kozai 1962) protects the orbits of high inclination as-
teroids from close encounters with Jupiter but it also plays a role
on the dynamics of near-Earth asteroids (Michel & Thomas 1996).
Trujillo & Sheppard (2014) claim that the observed excess of
objects reaching perihelion near the ascending node cannot be the
result of observational bias. In this Letter, we study the visibility of
extreme trans-Neptunian objects and the details of possible obser-
vational biases. Our analysis confirms the interpretation presented
in Trujillo & Sheppard (2014) and uncovers a range of additional
unexpected patterns in the distribution of the orbital parameters
of the ETNOs. The overall visibility is studied in Section 2 using
Monte Carlo techniques and an obvious intrinsic bias in declina-
tion is identified. The impact of the bias in declination is analysed
in Section 3. The distribution in orbital parameter space of real ob-
jects is shown in Section 4. In Section 5, our findings are analysed
within the context of the Kozai resonance. Results are discussed in
Section 6 and conclusions are summarized in Section 7.

2 C. de la Fuente Marcos and R. de la Fuente Marcos
2 VISIBILITY OF TRANS-NEPTUNIAN OBJECTS: A
MONTE CARLO APPROACH
Trujillo & Sheppard (2014) focused their discussion on asteroids
with perihelion greater than 30 au and semimajor axis in the range
150–600 au. Here, we study the visibility of these ETNOs as seen
from our planet. The actual distribution of the orbital elements of
objects in this population is unknown. In the following, we will as-
sume that the orbits of these objects are uniformly distributed in or-
bital parameter space. This is the most simple choice and, by com-
paring with real data, it allows the identification of observational
biases and actual trends easily. Using a Monte Carlo approach, we
create a synthetic population of ETNOs with semimajor axis, a ∈
(150, 600) au, eccentricity, e ∈ (0, 0.99), inclination, i ∈ (0, 90)◦
,
longitude of the ascending node, Ω ∈ (0, 360)◦
, and argument of
perihelion, ω ∈ (0, 360)◦
. We restrict the analysis to objects with
perigee < 90 au. Out of this synthetic population we single out ob-
jects with perihelion, q = a(1 − e) > 30 au, nearly 20 per cent
of the sample. Both, the test and Earth’s orbit are sampled in phase
space until the minimal distance or perigee is found (that is some-
times but not always near perihelion due to the relative geometry
of the orbits). Twenty million test orbits were studied. We do not
assume any specific size (absolute magnitude) distribution because
its nature is also unknown and we are not interested in evaluating
any detection efficiency. In principle, our results are valid for both
large primordial objects and those (likely smaller) from collisional
origin. Further details of our technique can be found in de la Fuente
Marcos & de la Fuente Marcos (2014a, b).
Fig. 1 shows the distribution in equatorial coordinates of the
studied test orbits at perigee. In this figure, the value of the pa-
rameter in the appropriate units is colour coded following the scale
printed on the associated colour box. Both perigees (panel A) and
semimajor axes (panel B) are uniformly distributed. The eccentric-
ity (panel C) is always in the range 0.4–0.95. Inclination (panel D),
longitude of the ascending node (panel E) and argument of perihe-
lion (panel F) exhibit regular patterns. The frequency distribution
in equatorial coordinates (see Fig. 2) shows a rather uniform be-
haviour in right ascension, α; in contrast, most of the objects reach
perigee at declination, δ, in the range -24◦
to 24◦
. Therefore, and as-
suming a uniform distribution for the orbital parameters of objects
in this population, there is an intrinsic bias in declination induced
by our observing point on Earth. Under the assumptions made here,
this intrinsic bias and any secondary biases induced by it are inde-
pendent on how far from the ecliptic the observations are made. In
other words, if the objects do exist and their orbits are uniformly
distributed in orbital parameter space, most objects will be discov-
ered at |δ| <24◦
no matter how complete and extensive the surveys
are. But, what is the expected impact of this intrinsic bias on the
observed orbital elements?
3 THE IMPACT OF THE DECLINATION BIAS
So far, all known trans-Neptunian objects with q > 30 au had
|δ| <24◦
at discovery which supports our previous result (see Table
1 and Fig. 2). But this intrinsic bias may induce secondary biases
on the observed orbital elements. If we represent the frequency dis-
tribution in right ascension and the orbital parameters a, e, i, Ω and
ω for test orbits with |δ| <24◦
, we get Fig. 3. Out of an initially
almost uniform distribution in α, i, Ω, and ω (see Figs 2 and A1),
we obtain biased distributions in all these four parameters. In con-
trast, the distributions in a and e are rather unaffected (other than
Figure 1. Distribution in equatorial coordinates of the studied test orbits at
perigee as a function of various orbital elements and parameters. As a func-
tion of the perigee of the candidate (panel A); as a function of a (panel B);
as a function of e (panel C); as a function of i (panel D); as a function of Ω
(panel E); as a function of ω (panel F). The associated frequency distribu-
tions are plotted in Fig. A1. The green circles in panel D give the location
at discovery of all the known objects (see Table 1).
scale factors) by the intrinsic bias in δ. Therefore, most objects in
this population should be discovered with semimajor axes near the
low end of the distribution, eccentricities in the range 0.8–0.9, in-
clinations under 40◦
, longitude of the ascending node near 180◦
and
argument of perihelion preferentially near 0◦
and 180◦
. Any devia-
tion from these expected secondary biases induced by the intrinsic
bias in declination will signal true characteristic features of this
population. Therefore, we further confirm that the clustering in ω
pointed out by Trujillo & Sheppard (2014) is real, not the result of
observational bias. Unfortunately, the number of known objects is
small (13), see Table 1, and any conclusions obtained from them
will be statistically fragile.
4 DISTRIBUTION IN ORBITAL PARAMETER SPACE OF
REAL OBJECTS
The distribution in orbital parameter space of the objects in Ta-
ble 1 shows a number of puzzling features (see Fig. 3). In addi-

Signalling the trans-Plutonian planets 3
Figure 2. Frequency distribution in equatorial coordinates (right ascension,
top panel, and declination, bottom panel) of the studied test orbits at perigee.
The distribution is rather uniform in right ascension and shows a maximum
for declinations in the range -24◦to 24◦. The bin sizes are 0.024 h in right
ascension and 0.◦18 in declination, error bars are too small to be seen. The
black circles correspond to objects in Table 1.
tion to the clustering of ω values around 0◦
(but not 180◦
) already
documented by Trujillo & Sheppard (2014), we observe clustering
around 20◦
in inclination and, perhaps, around 120◦
in longitude of
the ascending node. The distributions in right ascension, semima-
jor axis and eccentricity of known objects appear to be compatible
with the expectations. However, (90377) Sedna and 2007 TG422
are very clear outliers in semimajor axis. Their presence may sig-
nal the existence of a very large population of similar objects, the
inner Oort cloud (Brown, Trujillo & Rabinowitz 2004). The distri-
bution in inclination is also particularly revealing. Such a clustering
in inclination closely resembles the one observed in the inner edge
of the main asteroid belt for the Hungaria family (see e.g. Milani et
al. 2010). Consistently, some of these objects could be submitted to
an approximate mean motion resonance with an unseen planet. In
particular, the orbital elements of 82158 and 2002 GB32 are very
similar. On the other hand, asteroids 2003 HB57, 2005 RH52 and
2010 VZ98 all have similar a, e and i, and their mean longitudes, λ,
differ by almost 120◦
(see Table A1). This feature reminds us of the
Hildas, a dynamical family of asteroids trapped in a 3:2 mean mo-
tion resonance with Jupiter (see e.g. Broˇz & Vokrouhlick´y 2008).
If the three objects are indeed trapped in a 3:2 resonance with an
unseen perturber, it must be moving in an orbit with semimajor axis
in the range 195–215 au. This automatically puts the other two ob-
jects, with semimajor axis close to 200 au, near the 1:1 resonance
with the hypothetical planet. Their difference in λ is also small (see
Table A1), typical of Trojans or quasi-satellites. Almost the same
can be said about 2003 SS422 and 2007 VJ305. The difference in
λ between these two pairs is nearly 180◦
. On the other hand, the
clustering of ω values around 0◦
could be the result of a Kozai res-
onance (Kozai 1962). An argument of perihelion librating around
0◦
means that these objects reach perihelion at approximately the
same time they cross the ecliptic from South to North (librating
around 180◦
implies that the perihelion is close to the descending
node). When the Kozai resonance occurs at low inclinations, the
argument of perihelion librates around 0◦
or 180◦
(see e.g. Milani et
Figure 3. Frequency distribution in right ascension (bottom panel) and the
orbital elements of test orbits with |δ| <24◦. The bin sizes are 0.45 au in
semimajor axis, 0.001 in eccentricity, 0.◦09 in inclination, 0.◦36 in longitude
of the node, 0.◦36 in argument of perihelion and 0.024 h in right ascension,
error bars are too small to be seen. The black circles correspond to objects
in Table 1. Data from Table A1 have been used.
al. 1989). At the Kozai resonance, the precession rate of its argu-
ment of perihelion is nearly zero. This resonance provides a tem-
porary protection mechanism against close encounters with plan-
ets. An object locked in a Kozai resonance is in a metastable state,
where it can remain for a relatively long amount of time before a
close encounter with a planet drastically changes its orbit.
5 DIFFERENT KOZAI SCENARIOS
The most typical Kozai scenario is characterized by the presence of
a primary (the Sun in our case), the perturbed body (a massless test
particle, an asteroid), and a massive outer or inner perturber such as
the ratio of semimajor axes (perturbed versus perturber) tending to
zero (for an outer perturber) or inﬁnity (for an inner perturber). In
the case of an outer perturber, the critical inclination angle separat-
ing the circulation and libration regimes is ∼39◦
; for an inner per-
turber it is ∼63◦
(see e.g. Gallardo, Hugo & Pais 2012). Here, the
libration occurs at ω = 90◦
and 270◦
. Under these circumstances,
aphelion (for the outer perturber) or perihelion (for the inner per-
turber) always occur away from the orbital plane of the perturber.
This lack of encounters greatly reduces or completely halts any dif-
fusion in semimajor axis. A classical example of an object submit-
ted to the Kozai effect induced by an outer perturber is the asteroid

Table 1. Equatorial coordinates, apparent magnitudes (with filter if known)
at discovery time, absolute magnitude, and ω for the 13 objects discussed
in this Letter. (J2000.0 ecliptic and equinox. Source: MPC Database.)
Object α (h:m:s) δ (◦:′:′′) m (mag) H (mag) ω (◦)
(82158) 2001 FP185 11:57:50.69 +00:21:42.7 22.2 (R) 6.0 6.77
(90377) Sedna 03:15:10.09 +05:38:16.5 20.8 (R) 1.5 311.19
(148209) 2000 CR105 09:14:02.39 +19:05:58.7 22.5 (R) 6.3 317.09
2002 GB32 12:28:25.94 -00:17:28.4 21.9 (R) 7.7 36.89
2003 HB57 13:00:30.58 -06:43:05.4 23.1 (R) 7.4 10.64
2003 SS422 23:27:48.15 -09:28:43.4 22.9 (R) 7.1 209.98
2004 VN112 02:08:41.12 -04:33:02.1 22.7 (R) 6.4 327.23
2005 RH52 22:31:51.90 +04:08:06.1 23.8 (g) 7.8 32.59
2007 TG422 03:11:29.90 -00:40:26.9 22.2 6.2 285.84
2007 VJ305 00:29:31.74 -00:45:45.0 22.4 6.6 338.53
2010 GB174 12:38:29.365 +15:02:45.54 25.09 (g) 6.5 347.53
2010 VZ98 02:08:43.575 +08:06:50.90 20.3 (R) 5.0 313.80
2012 VP113 03:23:47.159 +01:12:01.65 23.1 (r) 4.1 293.97
(3040) Kozai that is perturbed by Jupiter. Another possible Kozai
scenario is found when the ratio of semimajor axes (perturbed ver-
sus perturber) is close to one. In that case, the libration occurs at
ω = 0◦
and 180◦
; therefore, the nodes are located at perihelion and
at aphelion, i.e. away from the massive perturber (see e.g. Milani
et al. 1989). Most studies of the Kozai mechanism assume that the
perturber follows an almost circular orbit but the effect is also pos-
sible for eccentric orbits, creating a very rich dynamics (see e.g.
Lithwick & Naoz 2011). Trujillo & Sheppard (2014) favour a sce-
nario in which the perturber responsible for the possible Kozai li-
bration experimented by 2012 VP113 has a semimajor axis close
to 250 au. This puts 2012 VP113 near or within the co-orbital re-
gion of the hypothetical perturber, i.e. the Kozai scenario in which
the ratio of semimajor axes is almost 1. The Kozai mechanism in-
duces oscillations in both eccentricity and inclination (because for
them
√
1 − e2 cos i = const) and the objects affected will exhibit
clustering in both parameters. This is observed in Fig. 3 but the
clustering in e could be the result of observational bias (see above).
6 DISCUSSION
Our analysis of the trends observed in Fig. 3 suggests that a massive
perturber may be present at nearly 200 au, in addition to the body
proposed by Trujillo & Sheppard (2014). The hypothetical object
at nearly 200 au could also be in near resonance (3:2) with the one
at nearly 250 au (e.g. if one is at 202 au and the other at 265 au, it is
almost exactly 3:2). Any unseen planets present in that region must
affect the dynamics of TNOs and comets alike. In this scenario, the
aphelia, Q = a(1 + e), of TNOs and comets (moving in eccentric
orbits) may serve as tracers of the architecture of the entire trans-
Plutonian region. In particular, objects with ω ∼ 0◦
or 180◦
can give
us information on the possible presence of massive perturbers in the
area because they only experience close encounters near perihe-
lion or aphelion (if the assumed perturbers have their orbital planes
close to the Fundamental Plane of the Solar system). Their perihe-
lia are less useful because so far they are < 100 au. However, the
presence of gaps in the distribution of aphelia may be a signature
of perturbational effects due to unseen planets. Fig. 4 shows the
distribution of aphelia for TNOs and comets with semimajor axis
greater than 50 au. The top two panels show the entire sample. The
two panels at the bottom show the distribution for objects with ω <
35◦
or ω > 325◦
or ω ∈ (145, 215)◦
. These objects have their nodes
close to perihelion and aphelion and their distribution in aphelion
shows an unusual feature in the range 200–260 au. The number
of objects with nodes close to aphelion/perihelion is just four. The
total number of objects with aphelion in that range is 18. Immedi-
ately outside that range, the number of objects is larger. Although
we may think that the difference is significant, it is unreliable sta-
tistically speaking because it could be a random fluctuation due
to small number statistics. However, a more quantitative approach
suggests that the scarcity is indeed statistically significant. If 18
objects have been found in ω ∈ (0, 360)◦
, 4 within an interval of
140◦
, and assuming a uniform distribution, we expect 7 objects not
4. The difference is just 0.8σ. Here, we use the approximation given
by Gehrels (1986) when N < 21: σ ∼ 1 +
√
0.75 + N. But Fig.
3 indicates that, because of the bias, objects with ω close to 0◦
or
180◦
are nearly four times more likely to be identified than those
with ω close to 90◦
or 270◦
. So, instead of 7 objects we should
have observed 14 but only 4 are found, a difference of 2σ, that
is marginally significant. Therefore, if they are not observed some
mechanism must have removed them.
Fig. 4 (top panels) shows an apparent overall decrease in the
number of objects with aphelion in the range 200–300 au. The (e, a)
plane plotted in Fig. 5 confirms that the architecture of that region
is unlikely to be the result of a gravitationally unperturbed envi-
ronment. If there are two planets, one at nearly 200 au and another
one at approximately 250 au, their combined resonances may clear
the area of objects in a fashion similar to what is observed between
the orbits of Jupiter and Saturn but see Hees et al. (2014) and Iorio
(2014). On the other hand, it can be argued that ETNOs could be the
result of close and distant encounters between the proto-Sun and
other members of its parent star cluster early in the history of the
Solar system (see e.g. Ida, Larwood & Burkert 2000; de la Fuente
Marcos & de la Fuente Marcos 2001). However, these encounters
are expected to pump up only the eccentricity not to imprint a per-
manent signature on the distribution of the argument of perihelion
in the form of a clustering of values around ω=0◦
. In addition, they
are not expected to induce clustering in inclination.
7 CONCLUSIONS
In this Letter, we have re-examined the clustering in ω found by
Trujillo & Sheppard (2014) for ETNOs using a Monte Carlo ap-
proach. We confirm that their finding is not a statistical coincidence
and it cannot be explained as a result of observational bias. Besides,
(90377) Sedna and 2007 TG422 are very clear outliers in semima-
jor axis. We confirm that their presence may signal the existence of
a very large population of similar objects. A number of additional
trends have been identified here for the first time:
• Observing from the Earth, only ETNOs reaching perihelion at
|δ| <24◦
are accessible.
• Besides clustering around ω = 0◦
, additional clustering in in-
clination around 20◦
is observed.
• Asteroids 2003 HB57, 2005 RH52 and 2010 VZ98 all have
similar orbits, and their mean longitudes differ by almost 120◦
.
They may be trapped in a 3:2 resonance with an unseen perturber
with semimajor axis in the range 195–215 au.
• The orbits of 82158 and 2002 GB32 are very similar. They
could be co-orbital to the putative massive object at 195–215 au.
• The study of the distribution in aphelia of TNOs and comets
shows a relative deficiency of objects with ω close to 0◦
or

Figure 4. Distribution of aphelia for TNOs and comets with semimajor axis
greater than 50 au: all objects (top panels) and only those with ω < 35◦or
ω > 325◦or ω ∈ (145, 215)◦(bottom panels).
180◦
among those with aphelia in the range 200-260 au. The differ-
ence is only marginally significant (2σ), though. Gaps are observed
at ∼205 au and ∼260 au.
We must stress that our results are based on small number statistics.
However, the same trends are found for asteroids and comets, and
the apparent gaps in the distribution of aphelia are very unlikely to
be the result of Neptune’s perturbations or observational bias. Per-
turbations from trans-Plutonian objects of moderate planetary size
may be detectable by the New Horizons spacecraft (Iorio 2013).
ACKNOWLEDGEMENTS
We thank the anonymous referee for her/his helpful and quick re-
port. This work was partially supported by the Spanish ‘Comunidad
de Madrid’ under grant CAM S2009/ESP-1496. We thank M. J.
Fernández-Figueroa, M. Rego Fernández and the Department of
Astrophysics of the Universidad Complutense de Madrid (UCM)
for providing computing facilities. Most of the calculations and
part of the data analysis were completed on the ‘Servidor Central
Figure 5. Centaurs, TNOs, ETNOs and comets in the (e, a) plane. The
dark gray area represents the eccentricity/semimajor axis combination with
periapsis between the perihelion and aphelion of Jupiter, the light gray area
shows the equivalent parameter domain if Neptune is considered instead of
Jupiter. The brown area corresponds to the (e, a) combination with apoapsis
between 190 and 210 au and the orange area shows its counterpart for the
range 250–280 au.
de Cálculo’ of the UCM and we thank S. Cano Alsúa for his help
during this stage. In preparation of this Letter, we made use of the
NASA Astrophysics Data System, the ASTRO-PH e-print server
and the MPC data server.
REFERENCES
Brown M. E., Trujillo C., Rabinowitz D., 2004, ApJ, 617, 645
Broˇz M., Vokrouhlický D., 2008, MNRAS, 390, 715
de la Fuente Marcos C., de la Fuente Marcos R., 2001, A&A, 371, 1097
de la Fuente Marcos C., de la Fuente Marcos R., 2014a, MNRAS, 439, 2970
de la Fuente Marcos C., de la Fuente Marcos R., 2014b, MNRAS, 441, 3,
2280
Fernández J. A., 2011, ApJ, 726, 33
Gallardo T., Hugo G., Pais P., 2012, Icarus, 220, 392
Gehrels N., 1986, ApJ, 303, 336
Gomes R. S., Matese J. J., Lissauer J. J., 2006, Icarus, 184, 589
Hees A., Folkner W. M., Jacobson R. A., Park R. S., 2014, Phys. Rev. D,
89, 102002
Ida S., Larwood J., Burkert A., 2000, ApJ, 528, 351
Iorio L., 2011, MNRAS, 415, 1266
Iorio L., 2012, Celest. Mech. Dyn. Astron., 112, 117
Iorio L., 2013, Celest. Mech. Dyn. Astron., 116, 357
Iorio L., 2014, preprint (arXiv:1404.0258)
Kozai Y., 1962, AJ, 67, 591
Lithwick Y., Naoz S., 2011, ApJ, 742, 94
Luhman K. L., 2014, ApJ, 781, 4
Lykawka P. S., Mukai T., 2008, AJ, 135, 1161
Matese J. J., Whitmire D. P., 2011, Icarus, 211, 926
Michel P., Thomas F. C., 1996, A&A, 307, 310
Milani A., Carpino M., Hahn G., Nobili A. M., 1989, Icarus, 78, 212
Milani A., Knezevic Z., Novakovic B., Cellino A., 2010, Icarus, 207, 769
Sheppard S. S., Trujillo C., 2014, MPEC Circ., MPEC 2014-F40
Sheppard S. S. et al., 2011, AJ, 142, 98
Trujillo C. A., Sheppard S. S., 2014, Nature, 507, 471

Figure A1. Frequency distributions for the orbital elements of test orbits in
Fig. 1. Bin sizes are as in Fig. 3, error bars are too small to be seen.
APPENDIX A: ADDITIONAL FIGURES AND TABLES

Table A1. Various orbital parameters (̟ = Ω + ω, λ = ̟ + M) for the 13 objects discussed in this Letter (Epoch: 2456800.5, 2014-May-23.0 00:00:00.0
UT. J2000.0 ecliptic and equinox. Source: JPL Small-Body Database.)
Object a (au) e i (◦) Ω (◦) ω (◦) ̟ (◦) λ (◦)
(82158) 2001 FP185 220.7545067 0.84492276 30.77926 179.32889 6.76597 186.09486 187.24430
(90377) Sedna 532.2664228 0.85696250 11.92861 144.52976 311.18801 95.71777 93.91037
(148209) 2000 CR105 229.9196589 0.80773939 22.70769 128.23495 317.09262 85.32757 90.37358
2002 GB32 209.4649254 0.83128842 14.18242 177.01044 36.88563 213.89607 213.92324
2003 HB57 161.1315216 0.76362930 15.49540 197.85952 10.63985 208.49937 209.44502
2003 SS422 197.4196450 0.80023290 16.80405 151.10109 209.98241 1.08350 1.72635
2004 VN112 333.5527773 0.85809672 25.52708 66.04930 327.23428 33.28358 33.55408
2005 RH52 152.6816879 0.74449569 20.46892 306.19829 32.59337 338.79166 340.86704
2007 TG422 531.9002265 0.93310126 18.57950 112.98155 285.83713 38.81868 39.06830
2007 VJ305 192.3878720 0.81702908 11.98914 24.38420 338.53140 2.91560 4.04033
2010 GB174 368.2345380 0.86809908 21.53344 130.59114 347.52989 118.12103 121.29941
2010 VZ98 156.4583186 0.78062638 4.50909 117.47040 313.79473 71.26513 68.78231
2012 VP113 264.9446814 0.69599853 24.01737 90.88555 293.97160 24.85715 27.78384

Extreme trans neptunian objects and the kozai mechanism

More Related Content

What's hot (20)

Similar to Extreme trans neptunian objects and the kozai mechanism (20)

More from Georgi Daskalov (20)

Recently uploaded (20)

Extreme trans neptunian objects and the kozai mechanism