EVIDENCE FOR A DISTANT GIANT PLANET IN THE SOLAR SYSTEM
Konstantin Batygin and Michael E. Brown
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA; kbatygin@gps.caltech.edu
Received 2015 November 13; accepted 2016 January 10; published 2016 January 20
ABSTRACT
Recent analyses have shown that distant orbits within the scattered disk population of the Kuiper Belt exhibit an
unexpected clustering in their respective arguments of perihelion. While several hypotheses have been put forward
to explain this alignment, to date, a theoretical model that can successfully account for the observations remains
elusive. In this work we show that the orbits of distant Kuiper Belt objects (KBOs) cluster not only in argument of
perihelion, but also in physical space. We demonstrate that the perihelion positions and orbital planes of the objects
are tightly confined and that such a clustering has only a probability of 0.007% to be due to chance, thus requiring a
dynamical origin. We find that the observed orbital alignment can be maintained by a distant eccentric planet with
mass 10 m
⊕
whose orbit lies in approximately the same plane as those of the distant KBOs, but whose perihelion
is 180° away from the perihelia of the minor bodies. In addition to accounting for the observed orbital alignment,
the existence of such a planet naturally explains the presence of high-perihelion Sedna-like objects, as well as the
known collection of high semimajor axis objects with inclinations between 60° and 150° whose origin was
previously unclear. Continued analysis of both distant and highly inclined outer solar system objects provides the
opportunity for testing our hypothesis as well as further constraining the orbital elements and mass of the distant
planet.
Key words: Kuiper Belt: general – planets and satellites: dynamical evolution and stability
1. INTRODUCTION
The recent discovery of 2012VP113, a Sedna-like body and
a potential additional member of the inner Oort cloud,
prompted Trujillo & Sheppard (2014) to note that a set of
Kuiper Belt objects (KBOs) in the distant solar system exhibits
unexplained clustering in orbital elements. Specifically, objects
with a perihelion distance larger than the orbit of Neptune and
semimajor axis greater than 150 AU—including 2012VP113
and Sedna—have arguments of perihelia, ω, clustered approxi-
mately around zero. A value of ω=0 requires that the object’s
perihelion lies precisely at the ecliptic, and during ecliptic-
crossing the object moves from south to north (i.e., intersects
the ascending node). While observational bias does preferen-
tially select objects with perihelia (where they are closest and
brightest) at the heavily observed ecliptic, no possible bias
could select only for objects moving from south to north.
Recent simulations (de la Fuente Marcos & de la Fuente
Marcos 2014) confirmed this lack of bias in the observational
data. The clustering in ω therefore appears to be real.
Orbital grouping in ω is surprising because gravitational
torques exerted by the giant planets are expected to randomize
this parameter over the multi-Gyr age of the solar system. In
other words, the values of ω will not stay clustered unless some
dynamical mechanism is currently forcing the alignment. To
date, two explanations have been proposed to explain the data.
Trujillo & Sheppard (2014) suggest that an external
perturbing body could allow ω to librate about zero via the
Kozai mechanism.
1
As an example, they demonstrate that a
five-Earth-mass body on a circular orbit at 210 AU can drive
such libration in the orbit of 2012VP113. However, de la
Fuente Marcos & de la Fuente Marcos (2014) note that the
existence of librating trajectories around ω=0 requires the
ratio of the object to perturber semimajor axis to be nearly
unity. This means that trapping all of the distant objects within
the known range of semimajor axes into Kozai resonances
likely requires multiple planets, finely tuned to explain the
particular data set.
Further problems may potentially arise with the Kozai
hypothesis. Trujillo & Sheppard (2014) point out that the Kozai
mechanism allows libration about both ω=0 as well as
ω=180, and the lack of ω∼ 180 objects suggests that some
additional process originally caused the objects to obtain
ω∼0. To this end, they invoke a strong stellar encounter to
generate the desired configuration. Recent work (Jílková
et al. 2015) shows how such an encounter could, in principle,
lead to initial conditions that would be compatible with this
narrative. Perhaps a greater difficulty lies in that the dynamical
effects of such a massive perturber might have already been
visible in the inner solar system. Iorio (2014) analyzed the
effects of a distant perturber on the precession of the apsidal
lines of the inner planets and suggests that, particularly for low-
inclination perturbers, objects more massive than the Earth with
a∼200–300 AU are ruled out from the data (see also
Iorio 2012).
As an alternative explanation, Madigan & McCourt (2015)
have proposed that the observed properties of the distant
Kuiper Belt can be attributed to a so-called inclination
instability. Within the framework of this model, an initially
axisymmetric disk of eccentric planetesimals is reconfigured
into a cone-shaped structure, such that the orbits share an
approximately common value of ω and become uniformly
distributed in the longitude of ascending node, Ω. While an
intriguing proposition, additional calculations are required to
assess how such a self-gravitational instability may proceed
when the (orbit-averaged) quadrupolar potential of the giant
planets, as well as the effects of scattering, are factored into the
simulations. Additionally, in order to operate on the appropriate
timescale, the inclination instability requires 1–10 Earth masses
The Astronomical Journal, 151:22 (12pp), 2016 February doi:10.3847/0004-6256/151/2/22
© 2016. The American Astronomical Society. All rights reserved.
1
Note that the invoked variant of the Kozai mechanism has a different phase-
space structure from the Kozai mechanism typically discussed within the
context of the asteroid belt (e.g., Thomas & Morbidelli 1996).
1
of material to exist between ∼100 and ∼10,000 AU (Madigan
& McCourt 2015).
Such an estimate is at odds with the negligibly small mass of
the present Sedna population (Schwamb et al. 2010). To this
end, it is worth noting that although the primordial
planetesimal disk of the solar system likely comprised tens of
Earth masses (Tsiganis et al. 2005; Levison et al. 2008, 2011;
Batygin et al. 2011), the vast majority of this material was
ejected from the system by close encounters with the giant
planets during, and immediately following, the transient
dynamical instability that shaped the Kuiper Belt in the first
place. The characteristic timescale for depletion of the
primordial disk is likely to be short compared with the
timescale for the onset of the inclination instability (Nes-
vorný 2015), calling into question whether the inclination
instability could have actually proceeded in the outer solar
system.
In light of the above discussion, here we reanalyze the
clustering of the distant objects and propose a different
perturbation mechanism, stemming from a single, long-period
object. Remarkably, our envisioned scenario brings to light a
series of potential explanations for other, seemingly unrelated
dynamical features of the Kuiper Belt, and presents a direct
avenue for falsification of our hypothesis. The paper is
organized as follows. In Section 2, we reexamine the
observational data and identify the relevant trends in the
orbital elements. In Section 3, we motivate the existence of a
distant, eccentric perturber using secular perturbation theory.
Subsequently, we engage in numerical exploration in Section 4.
In Section 5, we perform a series of simulations that generate
synthetic scattered disks. We summarize and discuss the
implications of our results in Sections 6 and 7, respectively.
2. ORBITAL ELEMENT ANALYSIS
In their original analysis, Trujillo & Sheppard (2014)
examined ω as a function of semimajor axis for all objects
with perihelion, q, larger than Neptune’s orbital distance
(Figure 1). They find that all objects with q>30 AU and
a>150 AU are clustered around ω∼0. Excluding objects
with q inside Neptune’ s orbit is sensible, since an object that
crosses Neptune’s orbit will be influenced by recurrent close
encounters. However, many objects with q>30 AU can also
be destabilized as a consequence of Neptune’s overlapped outer
mean-motion resonances (e.g., Morbidelli 2002), and a search
for orbits that are not contaminated by strong interactions with
Neptune should preferably exclude these objects as well.
In order to identify which of the q>30 AU and
a>150 AU KBOs are strongly influenced by Neptune, we
numerically evolved six clones of each member of the clustered
population for 4 Gyr. If more than a single clone in the
calcuations exhibited large-scale semimajor axis variation, we
deemed such an objects dynamically unstable.
2
Indeed, many
of the considered KBOs (generally those with
30<q<36 AU) experience strong encounters with Neptune,
leaving only 6 of the 13 bodies largely unaffected by the
presence of Neptune. The stable objects are shown as dark
blue-green dots in Figure 1, while those residing on unstable
orbits are depicted as green points.
Interestingly, the stable objects cluster not around ω=0 but
rather around ω=318°±8°, grossly inconsistent with the
value predicted from by the Kozai mechanism. Even more
interestingly, a corresponding analysis of longitude of ascend-
ing node, as a function of the semimajor axis reveals a similarly
strong clustering of these angles about Ω=113°±13°
(Figure 1). Analogously, we note that longitude of peri-
helion,
3
vw=+
W
, is grouped around ϖ=71±16 deg.
Essentially the same statistics emerge even if long-term
stability is disregarded but the semimajor axis cut is drawn at
a=250 AU. The clustering of both ϖ and of Ω suggests that
not only do the distant KBOs cross the ecliptic at a similar
phase of their elliptical trajectories, the orbits are physically
aligned. This alignment is evident in the right panel of
Figure 2, which shows a polar view of the Keplerian
trajectories in inertial space.
To gauge the signi ficance of the physical alignment, it is
easier to examine the orbits in inertial space rather than orbital
element space. To do so, we calculate the location of the point
of perihelion for each of the objects and project these locations
into ecliptic coordinates.
4
In addition, we calculate the pole
orientation of each orbit and project it onto the plane of the sky
at the perihelion position. The left panel of Figure 2 shows the
projected perihelion locations and pole positions of all known
Figure 1. Orbits of well-characterized Kuiper-belt objects with perihelion distances in excess of q>30 AU. The left, middle, and right panels depict the longitude of
perihelion, ϖ, longitude of ascending node, Ω, and argument of perihelion ω as functions of semimajor axes. The orbits of objects with a<150 AU are randomly
oriented and are shown as gray points. The argument of perihelion displays clustering beyond a>150 AU, while the longitudes of perihelion and ascending node
exhibit confinement beyond a>250 AU. Within the a>150 AU subset of objects, dynamically stable bodies are shown with blue-green points, whereas their
unstable counterparts are shown as green dots. By and large, the stable objects are clustered in a manner that is consistent with the a>250 AU group of bodies. The
eccentricities, inclinations, and perihelion distances of the stable objects are additionally labeled. The horizontal lines beyond a >250 AU depict the mean values of
the angles and the vertical error bars depict the associated standard deviations of the mean.
2
In practice, large-scale orbital changes almost always result in ejection.
3
Unlike the argument of perihelion, ω, which is measured from the ascending
node of the orbit, the longitude of perihelion, ϖ , is an angle that is defined in
the inertial frame.
4
The vector joining the Sun and the point of perihelion, with a magnitude e is
formally called the Runge–Lenz vector.
2
The Astronomical Journal, 151:22 (12pp), 2016 February Batygin & Brown
outer solar system objects with q>30 AU and a>50 AU.
The six objects with a>250 AU, highlighted in red, all come
to perihelion below the ecliptic and at longitudes between 20°
and 130°.
Discovery of KBOs is strongly biased by observational
selection effects that are poorly calibrated for the complete
heterogeneous Kuiper Belt catalog. A clustering in perihelion
position on the sky could be caused, for example, by
preferential observations in one particular location. The
distribution of perihelion positions across the sky for all
objects with q>30 and a>50 AU appears biased toward the
equator and relatively uniform in longitude. No obvious bias
appears to cause the observed clustering. In addition, each of
our six clustered objects were discovered in a separate survey
with, presumably, uncorrelated biases.
We estimate the statistical significance of the observed
clustering by assuming that the detection biases for our
clustered objects are similar to the detection biases for the
collection of all objects with q>30 AU and a>50 AU. We
then randomly select six objects from the sample 100,000 times
and calculate the root mean square (rms) of the angular distance
between the perihelion position of each object and the average
perihelion position of the selected bodies. Orbits as tightly
clustered in perihelion position as the six observed KBOs occur
only 0.7% of the time. Moreover, the objects with clustered
perihelia also exhibit clustering in orbital pole position, as can
be seen by the nearly identical direction of their projected pole
orientations. We similarly calculated the rms spread of the
polar angles, and find that a cluster as tight as that observed in
the data occurs only 1% of the time. The two measurements are
statistically uncorrelated, and we can safely multiply the
probabilities together to find that the joint probability of
observing both the clustering in perihelion position and in pole
orientation simultaneously is only 0.007%. Even with only six
objects currently in the group, the significance level is about
3.8σ. It is extremely unlikely that the objects are so tightly
confined purely due to chance.
Much like confinement in ω, orbital alignment in physical
space is difficult to explain because of differential precession.
In contrast to clustering in ω, however, orbital confinement in
physical space cannot be maintained by either the Kozai effect
or the inclination instability. This physical alignment requires a
new explanation.
3. ANALYTICAL THEORY
Generally speaking, coherent dynamical structures in particle
disks can either be sustained by self-gravity (Tremaine 1998;
Touma et al. 2009) or by gravitational shepherding facilitated
by an extrinsic perturber (Goldreich & Tremaine 1982; Chiang
et al. 2009). As already argued above, the current mass of the
Kuiper Belt is likely insufficient for self-gravity to play an
appreciable role in its dynamical evolution. This leaves the
latter option as the more feasible alternative. Consequently,
here we hypothesize that the observed structure of the Kuiper
Belt is maintained by a gravitationally bound perturber in the
solar system.
To motivate the plausibility of an unseen body as a means of
explaining the data, consider the following analytic calculation.
In accord with the selection procedure outlined in the preceding
section, envisage a test particle that resides on an orbit whose
perihelion lies well outside Neptune’s orbit, such that close
encounters between the bodies do not occur. Additionally,
assume that the test particle’s orbital period is not commensu-
rate (in any meaningful low-order sense—e.g., Nesvorný &
Roig 2001) with the Keplerian motion of the giant planets.
The long-term dynamical behavior of such an object can be
described within the framework of secular perturbation theory
(Kaula 1964). Employing Gauss’s averaging method (see Ch. 7
of Murray & Dermott 1999; Touma et al. 2009), we can replace
the orbits of the giant planets with massive wires and consider
long-term evolution of the test particle under the associated
torques. To quadrupole order
5
in planet–particle semimajor
axis ratio, the Hamiltonian that governs the planar dynamics of
the test particle is
M
a
e
ma
Ma
1
4
1.1
i
i
i
232
1
4
2
2
() ()
å
=- -
-
=
In the above expression,
is the gravitational constant, M is the
mass of the Sun, m
i
and a
i
are the masses and semimajor axes
of the giant planets, while a and e are the test particle’s
semimajor axis and eccentricity, respectively.
Equation (1) is independent of the orbital angles, and thus
implies (by application of Hamilton’s equations) apsidal
Figure 2. Orbital clustering in physical space. The right panels depicts the side and top views of the Keplerian trajectories of all bodies with a>250 AU as well as
dynamically stable objects with a>150 AU. The adopted color scheme is identical to that employed in Figure 1, and the two thin purple orbits correspond to stable
bodies within the 150<a<250 AU range. For each object, the directions of the angular momentum and Runge–Lenz (eccentricity) vectors are additionally shown.
The left panel shows the location of perihelia of the minor bodies with q>30 AU and a>50 AU on the celestial sphere as points, along with the projection of their
orbit poles with adjacent lines. The orbits with a>250 AU are emphasized in red. The physical confinement of the orbits is clearly evident in both panels.
5
The octupolar correction to Equation (1) is proportional to the minuscule
eccentricities of the giant planet and can safely be neglected.
3
The Astronomical Journal, 151:22 (12pp), 2016 February Batygin & Brown
precession at constant eccentricity with the period
6
e
Ma
ma
4
3
1, 2
i
i
i
22
1
4
2
2
() ()
å
=-
w
=
where
is the orbital period. As already mentioned above, in
absence of additional effects, the observed alignment of the
perihelia could not persist indefinitely, owing to differential
apsidal precession. As a result, additional perturbations (i.e.,
harmonic terms in the Hamiltonian) are required to explain
the data.
Consider the possibility that such perturbations are secular in
nature, and stem from a planet that resides on a planar, exterior
orbit. Retaining terms up to octupole order in the disturbing
potential, the Hamiltonian takes the form (Mardling 2013)
M
a
e
ma
Ma
m
a
a
a
e
e
a
a
ee
e
e
1
4
1
1
4
13 2
1
15
16
13 4
1
cos , 3
i
i
i
232
1
4
2
2
2
2
232
3
2
252
()
(())
(())
()()
⎜⎟
⎜⎟
⎡
⎣
⎢
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎤
⎦
⎥
å
vv
=- -
-
¢
¢¢
+
-¢
-
¢
¢
+
-¢
¢-
-
=
where primed quantities refer to the distant perturber (note that
for planar orbits, longitude and argument of perihelion are
equivalent). Importantly, the strength of the harmonic term in
Equation (3) increases monotonically with e′. This implies that
in order for the perturbations to be consequential, the
companion orbit must be appreciably eccentric.
Assuming that the timescale associated with secular coupling
of the giant planets is short compared with the characteristic
timescale for angular momentum exchange with the distant
perturber (that is, the interactions are adiabatic—see, e.g.,
Neishtadt 1984; Becker & Batygin 2013), we may hold all
planetary eccentricities constant and envision the apse of the
perturber’s orbit to advance linearly in time: ϖ′=ν t, where
the rate ν is obtained from Equation (2).
Transferring to a frame co-precessing with the apsidal line of
the perturbing object through a canonical change of variables
arising from a type-2 generating function of the form
t
2
()
nv=F -
, we obtain an autonomous Hamiltonian
(Goldstein 1950):
M
a
e
ma
Ma
Ma e
m
a
a
a
e
e
a
a
ee
e
e
1
4
1
11
1
4
13 2
1
15
16
13 4
1
cos , 4
i
i
i
232
1
4
2
2
2
2
2
232
3
2
252
()
()
(())
(())
() ()
⎜⎟
⎜⎟
⎡
⎣
⎢
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎤
⎦
⎥
å
n
v
=- -
+--
-
¢
¢¢
+
-¢
-
¢
¢
+
-¢
D
-
=
where
Ma e11
2
(
)
F= - -
is the action conjugate to
the angle
vv v
D
=¢-
. Given the integrable nature of
,
we may inspect its contours as a way to quantify the orbit-
averaged dynamical behavior of the test particle.
Figure 3 shows a series of phase-space portraits
7
of the
Hamiltonian (4) for various test particle semimajor axes and
perturber parameters of m′=10 m
⊕
, a′=700 AU and
e′ = 0.6. Upon examination, an important feature emerges
within the context of our simple model. For test-particle
semimajor axes exceeding a 200 AU, phase-space flow
characterized by libration of Δϖ (shown as red curves)
materializes at high eccentricities.
This is consequential for two (inter-related) reasons. First,
the existence of a libration island
8
demonstrates that an
eccentric perturber can modify the orbital evolution of test
particles in such a way as to dynamically maintain apsidal
alignment. Second, associated with the libration of Δϖ is
modulation of orbital eccentricity. This means that a particle
initially on a Neptune-hugging orbit can become detached by a
secular increase in perihelion distance.
We note that the transition from trivial apsidal circulation to
a picture where librating trajectories occupy a substantial
fraction of the parameter space inherent to the Hamiltonian (4)
depends sensitively on the employed parameters. In particular,
the phase-space portraits exhibit the most dramatic dependence
on a′, since the harmonic term in Equation (4) arises at a higher
order in the expansion of the disturbing potential than the term
responsible for coupling with the giant planets. Meanwhile, the
sensitivity to e′ is somewhat diminished, as it dominantly
regulates the value of e that corresponds to the elliptic
equilibrium points shown in Figure 3. On the other hand, in
a regime where the last two terms in Equation (4) dominate
(i.e., portraits corresponding to a350 AU and companions
with a′∼ 500–1000 AU, e′0.6 and m′a few Earth
masses), m′ only acts to determine the timescale on which
secular evolution proceeds. Here, the choice of parameters has
been made such that the resulting phase-space contours match
the observed behavior, on a qualitative level.
Cumulatively, the presented results offer credence to the
hypothesis that the observed structure of the distant Kuiper Belt
can be explained by invoking perturbations from an unseen
planetary mass companion to the solar system. Simultaneously,
the suggestive nature of the results should be met with a
healthy dose of skepticism, given the numerous assumptions
made in the construction of our simple analytical model. In
particular, we note that a substantial fraction of the dynamical
flow outlined in phase-space portraits ( 3 ) characterizes test-
particle orbits that intersect that of the perturber (or Neptune),
violating a fundamental assumption of the employed secular
theory.
Moreover, even for orbits that do not cross, it is not obvious
that the perturbation parameter (a/a′) is ubiquitously small
enough to warrant the truncation of the expansion at the
utilized order. Finally, the Hamiltonian (4)
does not account for
possibly relevant resonant (and/or short-periodic) interactions
with the perturber. Accordingly, the obtained results beg to be
re-evaluated within the framework of a more comprehensive
model.
6
Accounting for finite inclination of the orbit enhances the precession period
by a factor of
ii
1
cos 1 ...
22
() ++
.
7
Strictly speaking, Figure 3 depicts a projection of the phase-space portraits
in orbital element space, which is not canonical. However, for the purposes of
this work, we shall loosely refer to these plots as phase-space portraits, since
their information content is identical.
8
Note that Figure 3 does not depict any homoclinic curves, so libration of
Δϖ is strictly speaking not a secular resonance (Henrard & Lamaitre 1983).
4
The Astronomical Journal, 151:22 (12pp), 2016 February Batygin & Brown
4. NUMERICAL EXPLORATION
In an effort to alleviate some of the limitations inherent to the
calculation performed above, let us abandon secular theory
altogether and employ direct N-body simulations.
9
For a more
illuminating comparison among analytic and numeric models,
it is sensible to introduce complications sequentially. In
particular, within our first set of numerical simulations, we
accounted for the interactions between the test particle and the
distant companion self-consistently, while treating the gravita-
tional potential of the giant planets in an orbit-averaged
manner.
Practically, this was accomplished by considering a central
object (the Sun) to have a physical radius equal to that of
Uranus’s semimajor axis (
a
U
=
) and assigning a J
2
moment
to its potential, of magnitude (Burns 1976)
J
ma
M
1
2
.5
i
i
i
2
1
4
2
2
()
å
=
=
In doing so, we successfully capture the secular perihelion
advance generated by the giant planets, without contaminating
the results with the effects of close encounters. Any orbit with a
perihelion distance smaller than Uranus’s semimajor axis was
removed from the simulation. Similarly, any particle that came
within one Hill radius of the perturber was also withdrawn. The
integration time spanned 4 Gyr for each calculation.
As in the case of the analytical model, we constructed six
test-particle phase-space portraits
10
in the semimajor axis range
a
50, 550()Î
for each combination of perturber parameters.
Each portrait was composed of 40 test-particle trajectories
whose initial conditions spanned
e 0, 0.95()Î
in increments of
Δe=0.05 and Δϖ=0°, 180°. Mean anomalies of the
particles and the perturber were set to 0° and 180°,
respectively, and mutual inclinations were assumed to be null.
Unlike analytic calculations, here we did not require the
perturber’s semimajor axis to exceed that of the test particles.
Accordingly, we sampled a grid of semimajor axes
a
200, 2000(
)
¢Î
AU and eccentricities
e 0.1, 0.9()¢Î
in
increments of Δa′=100 AU and Δe′ = 0.1, respectively.
Given the qualitatively favorable match to the data that a
m′=10 m
⊕
companion provided in the preceding discussion,
we opted to retain this estimate for our initial suite of
simulations.
Computed portraits employing the same perturber para-
meters as before are shown in Figure 4. Drawing a parallel with
Figure 3, it is clear that trajectories whose secular evolution
drives persistent apsidal alignment with the perturber (depicted
with orange lines) are indeed reproduced within the framework
of direct N-body simulations. However, such orbital states
possess minimal perihelion distances that are substantially
larger than those ever observed in the real scattered Kuiper
Belt. Indeed, apsidally aligned particles that are initialized onto
orbits with eccentricities and perihelia comparable to those of
the distant Kuiper Belt are typically not long-term stable.
Consequently, the process described in the previous section
appears unlikely to provide a suitable explanation for the
physical clustering of orbits in the distant scattered disk.
While the secular confinement mechanism is disfavored by
the simulations, Figure 4 reveals that important new features,
possessing the same qualitative properties, materialize on the
phase-space portrait when the interactions between the test
particle and the perturber are modeled self-consistently.
Specifically, there exist highly eccentric, low-perihelion
apsidally anti-aligned orbits that are dynamically long-lived
(shown with blue dots). Such orbits were not captured by the
analytical model presented in the previous section, yet they
have orbital parameters similar to those of the observed
clustered KBOs.
Figure 3. Phase-space portraits (projected into orbital element space) corresponding to the autonomous Hamiltonian (4). Note that unlike Figure 1, here the longitude
of perihelion (plotted along the x-axis) is measured with respect to the apsidal line of the perturber’s orbit. Red curves represent orbits that exhibit apsidal libration,
whereas blue curves denote apsidal circulation. On each panel, the eccentricity corresponding to a Neptune-hugging orbit is emphasized with a gray line. The unseen
body is assumed to have a mass of m′=10 m
⊕
, and reside on a a′=700 AU, e′=0.6 orbit.
9
For the entirety of the N-body simulation suite, we utilized the mercury6
gravitational dynamics software package (Chambers 1999). The hybrid
symplectic-Bulisch–Stoer algorithm was employed throughout, and the
timestep was set to a twentieth of the shortest dynamical timescale (e.g.,
orbital period of Jupiter).
10
We note that in order to draw a formal parallel between numerically
computed phase-space portraits and their analytic counterparts, a numerical
averaging process must be appropriately carried out over the rapidly varying
angles (Morbidelli 1993). Here, we have not performed any such averaging and
instead opted to simply project the orbital evolution in the
e v-D
plane.
5
The Astronomical Journal, 151:22 (12pp), 2016 February Batygin & Brown
