Draft version July 21, 2016
Preprint typeset using L
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SOLAR OBLIQUITY INDUCED BY PLANET NINE
Elizabeth Bailey, Konstantin Batygin, Michael E. Brown
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125
Draft version July 21, 2016
ABSTRACT
The six-degree obliquity of the sun suggests that either an asymmetry was present in the solar
system’s formation environment, or an external torque has misaligned the angular momentum vectors
of the sun and the planets. However, the exact origin of this obliquity remains an open question.
Batygin & Brown (2016) have recently shown that the physical alignment of distant Kuiper Belt
orbits can be explained by a 5
−
20
m
⊕
planet on a distant, eccentric, and inclined orbit, with an
approximate perihelion distance of
∼
250 AU. Using an analytic model for secular interactions between
Planet Nine and the remaining giant planets, here we show that a planet with similar parameters can
naturally generate the observed obliquity as well as the specific pole position of the sun’s spin axis,
from a nearly aligned initial state. Thus, Planet Nine offers a testable explanation for the otherwise
mysterious spin-orbit misalignment of the solar system.
1.
INTRODUCTION
The axis of rotation of the sun is offset by six degrees
from the invariable plane of the solar system (Souami
& Souchay 2012). In contrast, planetary orbits have an
RMS inclination slightly smaller than one degree
1
, ren-
dering the solar obliquity a considerable outlier. The
origin of this misalignment between the sun’s rotation
axis and the angular momentum vector of the solar sys-
tem has been recognized as a a longstanding question
(Kuiper 1951; Tremaine 1991; Heller 1993), and remains
elusive to this day.
With the advent of extensive exoplanetary observa-
tions, it has become apparent that significant spin-orbit
misalignments are common, at least among transiting
systems for which the stellar obliquity can be determined
using the Rossiter-McLaughlin effect (Rossiter 1924;
McLaughlin 1924). Numerous such observations of plan-
etary systems hosting hot Jupiters have revealed spin-
orbit misalignments spanning tens of degrees (H ́ebrard
et al. 2008; Winn et al. 2010; Albrecht et al. 2012), even
including observations of retrograde planets (Narita et al.
2009; Winn et al. 2009; Bayliss et al. 2010; Winn et al.
2011). Thus, when viewed in the extrasolar context, the
solar system seems hardly misaligned. However, within
the framework of the nebular hypothesis, the expectation
for the offset between the angular momentum vectors of
the planets and sun is to be negligible, unless a specific
physical mechanism induces a misalignment. Further-
more, the significance of the solar obliquity is supported
by the contrasting relative coplanarity of the planets.
Because there is no directly observed stellar companion
to the sun (or any other known gravitational influence ca-
pable of providing an external torque on the solar system
sufficient to produce a six-degree misalignment over its
multi-billion-year lifetime Heller 1993), virtually all ex-
planations for the solar obliquity thus far have invoked
mechanisms inherent to the nebular stage of evolution.
ebailey@gps.caltech.edu
1
An exception to the observed orbital coplanarity of the planets
is Mercury, whose inclination is subject to chaotic evolution (Laskar
1994; Batygin et al. 2015)
In particular, interactions between the magnetosphere of
a young star and its protostellar disk can potentially lead
to a wide range of stellar obliquities while leaving the
coplanarity of the tilted disk intact (Lai et al. 2011). Yet
another possible mechanism by which the solar obliquity
could be attained in the absence of external torque is an
initial asymmetry in the mass distribution of the proto-
stellar core. Accordingly, asymmetric infall of turbulent
protosolar material has been proposed as a mechanism
for the sun to have acquired an axial tilt upon forma-
tion (Bate et al. 2010; Fielding et al. 2015). However,
the capacity of these mechanisms to overcome the re-
aligning effects of accretion, as well as gravitational and
magnetic coupling, remains an open question (Lai et al.
2011; Spalding & Batygin 2014, 2015).
In principle, solar obliquity could have been excited
through a temporary, extrinsic gravitational torque early
in the solar system’s lifetime. That is, an encounter
with a passing star or molecular cloud could have tilted
the disc or planets with respect to the sun (Heller 1993;
Adams 2010). Alternatively, the sun may have had a
primordial stellar companion, capable of early star-disc
misalignment (Batygin 2012; Spalding & Batygin 2014;
Lai 2014). To this end, ALMA observations of misaligned
disks in stellar binaries (Jensen & Akeson 2014; Williams
et al. 2014) have provided evidence for the feasibility
of this effect. Although individually sensible, a general
qualitative drawback of all of the above mechanisms is
that they are only testable when applied to the extra-
solar population of planets, and it is difficult to discern
which (if any) of the aforementioned processes operated
in our solar system.
Recently, Batygin & Brown (2016) determined that the
spatial clustering of the orbits of Kuiper Belt objects
with semi-major axis
a
&
250 AU can be understood if
the solar system hosts an additional
m
9
= 5
−
20
m
⊕
planet on a distant, eccentric orbit. Here, we refer to
this object as Planet Nine. The orbital parameters of
this planet reside somewhere along a swath of parame-
ter space spanning hundreds of AU in semi-major axis,
significant eccentricity, and tens of degrees of inclina-
tion, with a perihelion distance of roughly
q
9
∼
250 AU
arXiv:1607.03963v2 [astro-ph.EP] 20 Jul 2016
2
i
9
L
ʘ
L
in
L
9
inertial reference plane
i
ʘ
i
in
o
r
b
i
t
9
ΔΩ
i
n
n
e
r
o
r
b
i
t
L
total
Fig. 1.—
Geometric setup of the dynamical model. The orbits
of the planets are treated as gravitationally interacting rings. All
planets except Planet Nine are assumed to have circular, mutu-
ally coplanar orbits, and are represented as a single inner massive
wire. The sun is shown as a yellow sphere, and elements are not to
scale. Black, grey, and dotted lines are respectively above, on, and
below the inertial reference plane. The pink arrows demonstrate
the precession direction of the angular momentum vector of the
inner orbit,
L
in
, around the total angular momentum vector of the
solar system
L
total
. Red and blue arrows represent the differential
change in longitudes of ascending node of the orbits and inclina-
tion, respectively. Although not shown in the figure, the tilting of
the oblate sun is modeled as the tilting of an inner test ring. Over
the course of 4.5 billion years, differential precession of the orbits
induces a several-degree solar obliquity with respect to the final
plane of the planets.
(Brown & Batygin 2016). In this work, we explore the
possibility that this distant, planetary-mass body is fully
or partially responsible for the peculiar spin axis of the
sun.
Induction of solar obliquity of some magnitude is an
inescapable consequence of the existence of Planet Nine.
That is, the effect of a distant perturber residing on an
inclined orbit is to exert a mean-field torque on the re-
maining planets of the solar system, over a timespan of
∼
4
.
5 Gyr. In this manner, the gravitational influence
of Planet Nine induces precession of the angular momen-
tum vectors of the sun and planets about the total an-
gular momentum vector of the solar system. Provided
that angular momentum exchange between the solar spin
axis and the planetary orbits occurs on a much longer
timescale, this process leads to a differential misalign-
ment of the sun and planets. Below, we quantify this
mechanism with an eye towards explaining the tilt of the
solar spin axis with respect to the orbital angular mo-
mentum vector of the planets.
The paper is organized as follows. Section (2) describes
the dynamical model. We report our findings in section
(3). We conclude and discuss our results in section (4).
Throughout the manuscript, we adopt the following no-
tation. Similarly named quantities (e.g.
a
,
e
,
i
) related
to Planet Nine are denoted with a subscript “9”, whereas
those corresponding to the Sun’s angular momentum vec-
tor in the inertial frame are denoted with a tilde. Solar
quantities measured with respect to the solar system’s
invariable plane are given the subscript
.
2.
DYNAMICAL MODEL
To model the long-term angular momentum exchange
between the known giant planets and Planet Nine, we
employ secular perturbation theory. Within the frame-
work of this approach, Keplerian motion is averaged out,
yielding semi-major axes that are frozen in time. Corre-
spondingly, the standard
N
−
planet problem is replaced
with a picture in which
N
massive wires (whose line den-
sities are inversely proportional to the instantaneous or-
bital velocities) interact gravitationally (Murray & Der-
mott 1999). Provided that no low-order commensurabili-
ties exist among the planets, this method is well known to
reproduce the correct dynamical evolution on timescales
that greatly exceed the orbital period (Mardling 2007; Li
et al. 2014).
In choosing which flavor of secular theory to use, we
must identify small parameters inherent to the prob-
lem. Constraints based upon the critical semi-major
axis beyond which orbital alignment ensues in the dis-
tant Kuiper belt, suggest that Planet Nine has an ap-
proximate perihelion distance of
q
9
∼
250 AU and an ap-
preciable eccentricity
e
9
&
0
.
3 (Batygin & Brown 2016;
Brown & Batygin 2016). Therefore, the semi-major axis
ratio (
a/a
9
) can safely be assumed to be small. Addi-
tionally, because solar obliquity itself is small and the
orbits of the giant planets are nearly circular, here we
take
e
= 0 and sin(
i
)
1. Under these approximations,
we can expand the averaged planet-planet gravitational
potential in small powers of (
a/a
9
), and only retain terms
of leading order in sin(
i
).
In principle, we could self-consistently compute the
interactions among all of the planets, including Planet
Nine. However, because the fundamental secular fre-
quencies that characterize angular momentum exchange
among the known giant planets are much higher than
that associated with Planet Nine, the adiabatic princi-
ple (Henrard 1982; Neishtadt 1984) ensures that Jupiter,
Saturn, Uranus and Neptune will remain co-planar with
each-other throughout the evolutionary sequence (see
e.g. Batygin et al. 2011; Batygin 2012 for a related dis-
cussion on perturbed self-gravitating disks). As a result,
rather than modeling four massive rings individually, we
may collectively replace them with a single circular wire
having semi-major axis
a
and mass
m
, and possessing
equivalent total angular momentum and moment of in-
ertia:
m
√
a
=
∑
j
m
j
√
a
j
ma
2
=
∑
j
m
j
a
2
j
,
(1)
where the index
j
runs over all planets. The geometric
setup of the problem is shown in Figure (1).
To quadrupole order, the secular Hamiltonian govern-
ing the evolution of two interacting wires is (Kaula 1962;
Mardling 2010):
H
=
G
4
mm
9
a
9
(
a
a
9
)
2
1
ε
3
9
[
1
4
(
3 cos
2
(
i
)
−
1
)(
3 cos
2
(
i
9
)
−
1
)
+
3
4
sin(2
i
) sin(2
i
9
) cos(Ω
−
Ω
9
)
]
,
(2)
where Ω is the longitude of ascending node and
ε
9
=
√
1
−
e
2
9
. Note that while the eccentricities and inclina-
tions of the known giant planets are assumed to be small,
no limit is placed on the orbital parameters of Planet
Nine. Moreover, at this level of expansion, the plane-
3
q
9
=350AU
q
9
= 150 AU
q
9
=250AU
22
23
25
27
29
12
13
14
16
18
20
25
16
17
19
21
23
25
a
9
(AU)
a
9
(AU)
a
9
(AU)
contours: inclination (deg) of Planet Nine
insufficient
obliquity
distant KBO orbits
do not cluster
Kuiper Belt
destroyed
initial conditions: exact spin-orbit alignment
e
9
e
9
e
9
m
9
= 10
m
m
9
= 15
m
m
9
=20
m
Fig. 2.—
Parameters of Planet Nine required to excite a spin-orbit misalignment of
i
= 6 deg over the lifetime of the solar system,
from an initially aligned state. Contours in
a
9
-
e
9
space denote
i
9
, required to match the present-day solar obliquity. Contour labels are
quoted in degrees. The left, middle, and right panels correspond to
m
9
= 10
,
15, and 20
m
⊕
respectively. Due to independent constraints
stemming from the dynamical state of the distant Kuiper belt, only orbits that fall in the 150
< q
9
<
350 AU range are considered. The
portion of parameter space where a solar obliquity of
i
= 6 deg cannot be attained are obscured with a light-brown shade.
tary eccentricities remain unmodulated, consistent with
the numerical simulations of Batygin & Brown (2016);
Brown & Batygin (2016), where the giant planets and
Planet Nine are observed to behave in a decoupled man-
ner.
Although readily interpretable, Keplerian orbital el-
ements do not constitute a canonically conjugated set
of coordinates.
Therefore, to proceed, we introduce
Poincar ́e action-angle coordinates:
Γ =
m
√
G
M
a
Γ
9
=
m
9
√
G
M
a
9
ε
9
Z
= Γ
(
1
−
cos(
i
)
)
z
=
−
Ω
Z
9
= Γ
9
(
1
−
cos(
i
9
)
)
z
=
−
Ω
9
.
(3)
Generally, the action
Z
represents the deficit of angular
momentum along the
ˆ
k
−
axis, and to leading order,
i
≈
√
2
Z/
Γ. Accordingly, dropping higher-order corrections
in
i
, expression (2) takes the form:
H
=
G
4
mm
9
a
9
(
a
a
9
)
2
1
3
9
[
1
4
(
2
−
6
Z
Γ
)(
3
(
1
−
Z
9
Γ
9
)
2
−
1
)
+ 3
(
1
−
Z
9
Γ
9
)
√
1
−
Z
9
2Γ
9
√
2
Z
Γ
2
Z
9
Γ
9
cos(
z
−
z
9
)
]
.
(4)
Application of Hamilton’s equations to this expression
yields the equations of motion governing the evolution of
the two-ring system. However, we note that action-angle
variables (3) are singular at the origin, so an additional,
trivial change to Cartesian counterparts of Poincar ́e co-
ordinates is required to formulate a practically useful set
of equations (Morbidelli 2002). This transformation is
shown explicitly in the Appendix.
To complete the specification of the problem, we also
consider the torque exerted on the sun’s spin axis by a
tilting solar system. Because the sun’s angular momen-
tum budget is negligible compared to that of the plan-
ets, its back-reaction on the orbits can be safely ignored.
Then, the dynamical evolution of its angular momentum
vector can be treated within the same framework of sec-
ular theory, by considering the response of a test ring
with semi-major axis (Spalding & Batygin 2014, 2015):
̃
a
=
[
16
ω
2
k
2
2
R
6
9
I
2
G
M
]
1
/
3
,
(5)
where
ω
is the rotation frequency,
k
2
is the Love number,
R
is the solar radius, and
I
is the moment of inertia.
Because we are primarily concerned with main-
sequence evolution, here we adopt
R
=
R
and model
the interior structure of the sun as a
n
= 3 polytrope,
appropriate for a fully radiative body (Chandrasekhar
1939). Corresponding values of moment of inertia and
Love number are
I
= 0
.
08 and
k
2
= 0
.
01 respectively
(Batygin & Adams 2013). The initial rotation frequency
is assumed to correspond to a period of 2
π/ω
= 10 days
and is taken to decrease as
ω
∝
1
/
√
t
, in accord with the
Skumanich relation (Gallet & Bouvier 2013).
Defining scaled actions
̃
Γ =
√
G
M
̃
a
and
̃
Z
=
̃
Γ(1
−
cos(
̃
i
)) and scaling the Hamiltonian itself in the same
way, we can write down a Hamiltonian that is essentially
analogous to equation (4), which governs the long-term
spin axis evolution of the Sun:
̃
H
=
∑
j
(
G
m
j
4
a
3
j
)
̃
a
2
[
3
̃
Z
̃
Γ
+
3
4
√
2
̃
Z
̃
Γ
2
Z
Γ
cos( ̃
z
−
z
)
]
.
(6)
Note that contrary to equation (4), here we have as-
sumed small inclinations for both the solar spin axis
and the planetary orbits. This assumption transforms
the Hamiltonian into a form equivalent to the Lagrange-
Laplace theory, where the interaction coefficients have
been expanded as hypergeometric series, to leading or-
der in semi-major axis ratio (Murray & Dermott 1999).
Although not particularly significant in magnitude, we
follow the evolution of the solar spin axis for complete-
ness.
Quantitatively speaking, there are two primary sources
of uncertainty in our model. The first is the integration
timescale. Although the origin of Planet Nine is not well
understood, its early evolution was likely affected by the
presence of the solar system’s birth cluster (Izidoro et al.
2015; Li & Adams 2016), meaning that Planet Nine prob-