SOLAR OBLIQUITY INDUCED BY PLANET NINE
Elizabeth Bailey, Konstantin Batygin, and Michael E. Brown
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA;
ebailey@gps.caltech.edu
Received 2016 June 24; revised 2016 August 12; accepted 2016 August 23; published 2016 October 21
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 have recently shown that
the physical alignment of distant Kuiper Belt orbits can be explained by a
Å
m
5
20
–
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 speci
fi
c 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.
Key words:
planets and satellites: dynamical evolution and stability
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
rendering 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 system has been recognized as a
long-standing question
(
Kuiper
1951
; Tremaine
1991
; Heller
1993
)
, and remains elusive to this day.
With the advent of extensive exoplanetary observations, it
has become apparent that signi
fi
cant spin
–
orbit misalignments
are common, at least among transiting systems for which the
stellar obliquity can be determined using the Rossiter
–
McLaughlin effect
(
McLaughlin
1924
; Rossiter
1924
)
. Numer-
ous such observations of planetary systems hosting hot Jupiters
have revealed spin
–
orbit misalignments spanning tens of
degrees
(
Hébrard 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
,
2011
; Bayliss et al.
2010
)
.
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 speci
fi
c physical mechanism induces a
misalignment. Furthermore, the signi
fi
cance 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 in
fl
uence capable of
providing an external torque on the solar system suf
fi
cient to
produce a six-degree misalignment over its multi-billion-year
lifetime Heller
1993
)
, virtually all explanations for the solar
obliquity thus far have invoked mechanisms inherent to the
nebular stage of evolution. 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
)
.
However,
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 protostellar
core. Accordingly, asymmetric infall of turbulent protosolar
material has been proposed as a mechanism for the Sun to have
acquired an axial tilt upon formation
(
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 disk
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-disk
misalignment
(
Batygin
2012
; Spalding & Batygin
2014
;Lai
2014
)
. To this end,
ALMA observations of misaligne
d disks in stellar binaries
(
Jensen & Akeson
2014
; Williams et al.
2014
)
have provided
evidence for the feasibility of t
his effect. Although individu-
ally sensible, a general quali
tative drawback of all of the
above mechanisms is that they are only testable when applied
to the extrasolar population of planets, and it is dif
fi
cult to
discern which
(
if any
)
of the aforementioned processes
operated in our solar system.
Recently, Batygin & Brown
(
2016
)
determined that the
spatial clustering o
f the orbits of Kuiper Belt objects with
semimajor axes of
a
250
au can be understood if the solar
system hosts an additional
=
Å
mm
520
9
–
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 parameter space spanning hundreds of astronom-
ical units
in semimajor axis, signi
fi
cant eccentricity, and tens
of degrees of inclination, with a
perihelion distance of roughly
~
q
250
9
au
(
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.
The Astronomical Journal,
152:126
(
8pp
)
, 2016 November
doi:10.3847
/
0004-6256
/
152
/
5
/
126
© 2016. The American Astronomical Society. All rights reserved.
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
)
1
Induction of solar obliquity of some magnitude is an
inescapable consequence of the existence of Planet Nine. That
is, the effect of a distant pertur
ber residing on an inclined orbit
is to exert a mean-
fi
eldtorqueontheremainingplanetsofthe
solar system, over a timespan of
∼
4.5 Gyr. In this manner, the
gravitational in
fl
uence of Planet Nine induces precession of
the angular momentum vectors of the Sun and planets about
thetotalangularmomentumvectorofthesolarsystem.
Provided that angular momentu
m exchange between the solar
spin axis and the planetary orbits occurs on a much longer
timescale, this process leads to
a differential misalignment
of the Sun and planets. Below, we quantify this mechanism
with an eye toward explaining the tilt of the solar spin axis
with respect to the orbital angular momentum vector of the
planets.
The paper is organized as follows. Section
(
2
)
describes the
dynamical model. We report our
fi
ndings in Section
(
3
)
.We
conclude and discuss our results in Section
(
4
)
. Throughout the
manuscript, we adopt the following notation. 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 vector 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
e
.
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 framework of this
approach, Keplerian motion is averaged out, yielding semi-
major axes that are frozen in time. Correspondingly, the
standard
N
-planet problem is replaced with a picture in which
N
massive wires
(
whose line densities are inversely proportional
to the instantaneous orbital velocities
)
interact gravitationally
(
Murray & Dermott
1999
)
. Provided that no low-order
commensurabilities exist among the planets, this method is
well known to reproduce the correct dynamical evolution on
timescales that greatly exceed the orbital period
(
Mard-
ling
2007
; Li et al.
2014
)
.
In choosing which
fl
avor of secular theory to use, we must
identify small parameters inherent to the problem. Constraints
based upon the critical semimajor axis beyond which orbital
alignment ensues in the distant Kuiper Belt
suggest that Planet
Nine has an approximate perihelion distance of
~
q
250
9
au
and an appreciable eccentricity
e
0.
3
9
(
Batygin & Brown
2016
; Brown & Batygin
2016
)
. Therefore, the semimajor axis
ratio
aa
9
(
)
can safely be assumed to be small. Additionally,
because solar obliquity itself is small and the orbits of the giant
planets are nearly circular, here we take
e
=
0 and
i
sin
1
()
.
Under these approximations, we can expand the averaged
planet
–
planet gravitational potential in small powers of
aa
9
(
)
,
and only retain terms of leading order in
i
sin
()
.
In principle, we could self-consistently compute the interac-
tions among all of the planets, including Planet Nine. However,
because the fundamental secular frequencies that characterize
angular momentum exchange among the known giant planets
are much higher than that associated with Planet Nine, the
adiabatic principle
(
Henrard
1982
; Neishtadt
1984
)
ensures that
Jupiter, Saturn, Uranus, and Neptune will remain coplanar with
each-other throughout the evolutionary sequence
(
see, e.g.,
Batygin et al.
2011a
; Batygin
2012
for a related discussion 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 semimajor axis
a
and mass
m
, and possessing equivalent total angular
momentum and moment of inertia
å
å
=
=
ma
m a
ma
m a
,1
j
jj
j
j
j
2
2
()
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 governing the
evolution of two interacting wires is
(
Kaula
1962
; Mard-
ling
2010
)
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
e
=--
+W-W
mm
a
a
a
ii
ii
4
11
4
3cos
1 3cos
1
3
4
sin 2 sin 2 cos
,
2
9
99
2
9
3
22
9
99
( ())( ())
() ( ) (
)
()
where
Ω
is the longitude of the
ascending node and
e
=-
e
1
9
9
2
. 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 planetary eccentri-
cities remain unmodulated, consistent with the numerical
simulations of Batygin & Brown
(
2016
)
and
Brown & Batygin
(
2016
)
, where the giant planets and Planet Nine are observed to
behave in a decoupled manner.
Figure 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, mutually 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, gray, 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 inclination, respectively. Although not shown in the
fi
gure, 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
fi
nal plane of the planets.
2
The Astronomical Journal,
152:126
(
8pp
)
, 2016 November
Bailey, Batygin, & Brown
Although readily interpretable, Keplerian orbital elements do
not constitute a canonically conjugated set of coordinates.
Therefore, to proceed, we introduce Poincaré action-angle
coordinates:
e
G=
G=
=G -
=-W
=G -
=-W
mMa
mMa
Ziz
Ziz
1cos
1cos
.
3
99
99
99
9
9
(())
(())
()
Generally, the action
Z
represents the de
fi
cit of angular
momentum along the
k
axis
ˆ
‐
, and to leading order,
»G
iZ
2
. Accordingly, dropping higher-order corrections
in
i
, expression
(
2
)
takes the form
⎜⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎛
⎝
⎞
⎠
⎛
⎝
⎜
⎜
⎛
⎝
⎜
⎞
⎠
⎟
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎤
⎦
⎥
=-
G
-
G
-
+-
G
-
GGG
-
mm
a
a
a
ZZ
ZZZZ
zz
4
11
4
2
6
31
1
31
1
2
22
cos
.
4
9
99
2
9
3
9
9
2
9
9
9
9
9
9
9
()
()
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 Poincaré coordinates is required to
formulate a practically useful set of equations
(
Morbi-
delli
2002
)
. This transformation is shown explicitly in the
Appendix
.
To complete the speci
fi
cation 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 momentum budget is negligible
compared to that of the planets, 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
secular theory, by considering the response of a test ring with
the
semimajor axis
(
Spalding & Batygin
2014
,
2015
)
:
⎡
⎣
⎢
⎤
⎦
⎥
w
=
a
kR
IM
16
9
,5
2
2
2
6
2
13
̃()
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
pw
=
2
10
days and is taken to decrease as
w
μ
t
1
,in
accordance with the Skumanich relation
(
Gallet &
Bouvier
2013
)
.
De
fi
ning scaled actions
G
=
Ma
̃
̃
and
=
Z
̃
G
-
i
1cos
̃
((
̃
))
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:
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
å
=
G
+
G
G
-
m
a
a
ZZZ
zz
4
33
4
22
cos
.
6
j
j
j
3
2
̃
̃
̃
̃
̃
̃
(
̃
)()
Note that contrary to Equation
(
4
)
, here we have assumed 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 interac-
tion coef
fi
cients have been expanded as hypergeometric series,
to leading order in the
semimajor axis ratio
(
Murray &
Dermott
1999
)
. Although not particularly signi
fi
cant in
magnitude, we follow the evolution of the solar spin axis for
completeness.
Quantitatively speaking, the
re are two primary sources of
uncertainty in our model. The
fi
rst 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 probably attained its
fi
nal orbit
within the
fi
rst
∼
100 Myr of the solar system
’
s lifetime.
Although we recognize the
∼
2% error associated with this
ambiguity, we adopt an integration timescale of
4
.5
Gyr for
de
fi
nitiveness.
A second source of error stems from the fact that the solar
system
’
s orbital architecture almost certainly underwent a
instability-driven transformation sometime early in its history
(
Tsiganis et al.
2005
; Nesvorný & Morbidelli
2012
)
. Although
the timing of the onset of instability remains an open question
(
Levison et al.
2011
; Kaib & Chambers
2016
)
, we recognize
that the
failure of our model to re
fl
ect this change in
a
and
m
(
through Equation
(
1
))
introduces a small degree of inaccuracy
into our calculations. Nevertheless, it is unlikely that these
detailed complications constitute a signi
fi
cant drawback to our
results.
Figure 2.
Time evolution of the solar obliquity
i
e
in the frame of the solar
system, starting with an aligned con
fi
guration of the solar system, and a
Å
m
1
0
Planet Nine with starting parameters in the example range
Î
a
400, 600 au
9
[]
,
Î
e
0.4, 0.6
9
[
]
, and
Î
i
20, 30 de
g
9
[]
.
3
The Astronomical Journal,
152:126
(
8pp
)
, 2016 November
Bailey, Batygin, & Brown
3. RESULTS
As shown in Figure
(
2
)
, the effect of Planet Nine is to induce
a gradual differential precession of the Sun and the solar
system
’
s invariable plane,
2
resulting in a solar obliquity of
several degrees over the lifetime of the solar system. The Sun
’
s
present-day inclination with respect to the solar system
’
s
invariable plane
(
Souami & Souchay
2012
)
is almost exactly
=
i
6
. Using this number as a constraint, we have calculated
the possible combinations of
a
9
,
e
9
,
and
i
9
for a given
m
9
, that
yield the correct spin
–
orbit misalignment after
4
.5
Gyr of
evolution. For this set of calculations, we adopted an initial
condition in which the Sun
’
s spin axis and the solar system
’
s
total angular momentum vector were aligned.
The results are shown in Figure
(
3
)
. For three choices of
=
m
10
9
, 15, and
Å
m
2
0
, the
fi
gure depicts contours of the
required
i
9
in
-
a
e
99
space. Because Planet Nine
’
s perihelion
distance is approximately
~
q
250
9
au, we have only con-
sidered orbital con
fi
gurations with
<<
q
150
350
9
au. More-
over, within the considered locus of solutions, we neglect the
region of parameter space where the required solar obliquity
cannot be achieved within the lifetime of the solar system. This
section of the graph is shown with a light-brown shade in
Figure
(
3
)
.
For the considered range of
m
9
,
a
9
,
and
e
9
, characteristic
inclinations of
~
i
15 30
9
–
are required to produce the
observed spin
–
orbit misalignment. This compares favorably
with the results of Brown & Batygin
(
2016
)
, where a similar
inclination range for Planet Nine is obtained from entirely
different grounds. However, we note that the constraints on
a
9
and
e
9
seen in Figure
(
3
)
are somewhat more restrictive than
those in previous works. In particular, the illustrative
=
Å
mm
10
9
,
=
a
700
9
au,
e
9
=
0.6 perturber considered by
Batygin & Brown
(
2016
)
, as well as virtually all of the
“
high-
probability
”
orbits computed by Brown & Batygin
(
2016
)
fall
short of exciting 6
°
of obliquity from a strictly coplanar initial
con
fi
guration. Instead, slightly smaller spin
–
orbit misalign-
ments of
~
i
35
–
are typically obtained. At the same time,
we note that the lower bound on the semimajor axis of Planet
Nine quoted in Brown & Batygin
(
2016
)
is based primarily on
the comparatively low perihelia of the unaligned objects, rather
than the alignment of distant Kuiper Belt objects, constituting a
weaker constraint.
An equally important quantity as the solar obliquity itself, is
the solar longitude of the
ascending node
3
W
68
. This
quantity represents the azimuthal orientation of the spin axis
and informs the direction of angular momentum transfer within
the system. While the angle itself is measured from an arbitrary
reference point, the difference in longitudes of ascending node
D
W=W -W
9
is physically meaningful, and warrants
examination.
Figure
(
4
)
shows contours of
D
W
within the same parameter
space as Figure
(
3
)
. Evidently, the representative range of the
relative longitude of ascending node is
D
W~-
60
to 40
°
,
with the positive values coinciding with high eccentricities and
low semimajor axes. Therefore, observational discovery of
Planet Nine with a correspondent combination of parameters
a
9
,
e
9
,
i
9
, and
D
W
depicted anywhere on an analog of
Figures
(
3
)
and
(
4
)
constructed for the speci
fi
c value of
m
9
,
would constitute formidable evidence that Planet Nine is solely
responsible for the peculiar spin axis of the Sun. On the
contrary, a mismatch of these parameters relative to the
expected values
would imply that Planet Nine has merely
modi
fi
ed the Sun
’
s spin axis by a signi
fi
cant amount.
Although
W
9
is not known, Planet Nine
’
s orbit is
theoretically inferred to reside in approximately the same plane
as the distant Kuiper Belt objects, whose longitudes of
ascending node cluster around
á
Wñ =
113 13
(
Batygin &
Brown
2016
)
. Therefore, it is likely that
WáWñ
9
, implying
that
D
W
45
. Furthermore, the simulation suite of Brown &
Batygin
(
2016
)
approximately constrains Planet Nine
’
s long-
itude of the
ascending node to the range
of
W
80 120
9
–
,
yielding
<DW<
12
52
as an expected range of solar spin
axis orientations.
If we impose the aforementioned range of
D
W
as a constraint
on our calculations, Figure
(
4
)
suggests that
a
500
9
au and
Figure 3.
Parameters of Planet Nine required to excite a spin
–
orbit misalignment of
=
i
6
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
10,
9
15, and
Å
m
2
0
respectively. Due to independent constraints stemming from the dynamical state of the distant Kuiper Belt, only orbits that fall
in the
<<
q
1
50
350
9
au range are considered. The portion of parameter space where a solar obliquity of
=
i
6
cannot be attained are obscured with a light-brown
shade.
2
Although we refer to the instantaneous plane occupied by the wire with
parameters
a
and
m
as the invariable plane, in our calculations, this plane is not
actually invariable. Instead, it slowly precesses in the inertial frame.
3
The quoted value is measured with respect to the invariable plane, rather
than the ecliptic.
4
The Astronomical Journal,
152:126
(
8pp
)
, 2016 November
Bailey, Batygin, & Brown
e
0.4
9
. Although not strictly ruled out, orbits that fall in this
range are likely to be incompatible with the observed orbital
architecture of the distant Kuiper Belt. As a result, we speculate
that either
(
1
)
Planet Nine does not reside in the same plane as
the distant Kuiper Belt objects it shepherds
or
(
2
)
our adopted
initial condition, where the Sun
’
s primordial angular momen-
tum vector coincides exactly with that of the solar system, is
too restrictive. Of these two possibilities, the latter is somewhat
more likely.
While a null primordial obliquity is a sensible starting
assumption, various theoretical studies have demonstrated that
substantial spin
–
orbit misalignments can be excited in young
planetary systems
(
Lai et al.
2011
; Batygin
2012
; Lai
2014
;
Spalding & Batygin
2014
,
2015
; Fielding et al.
2015
)
, with
substantial support coming from existing exoplanet data
(
Huber
et al.
2013
; Winn & Fabrycky
2015
)
. At the same time, the
recent study of Spalding & Batygin
(
2016
)
has suggested that a
fraction of multi-transiting exoplanet systems would be
rendered unstable if their host stars had obliquities as large as
that of the Sun, and instead inclinations as small as
12
–
are
more typical. Accordingly, it is sensible to suppose that the
initial obliquity of the Sun was not too different from the rms
inclination of the planets
~
i
1
rms
.
To examine this possibility, we considered whether a Planet
Nine with
=
q
250
9
au and
D
W
within the quoted range is
consistent with a primordial solar obliquity of the order
of
~
12
–
. As an illustrative example, we adopted
=
a
500
9
au,
=
e
0.5
9
,
=
mm
15
9
, and evolved the system
backward
in time. Because Hamiltonian
(
4
)
is integrable, a
present-day combination of parameter maps onto a unique
primordial state vector.
The calculations were performed for
=
i
10
9
,20
°
, and
3
0
,
and the results are shown in Figure
(
5
)
. Speci
fi
cally, the panels
depict a polar representation of the Sun
’
s spin axis evolution
tracks measured from the instantaneous invariable plane, such
that the origin represents an exactly aligned con
fi
guration. The
color of each curve corresponds to a current value of
W
9
.
Evidently, for the employed set of parameters, the calculations
yield a primordial inclination range of
i
16
–
. Intriguingly,
the speci
fi
c choice of
=
i
20
9
, and
WáWñ
9
yields the lowest
spin
–
orbit misalignment, that is consistent with
i
rms
. Therefore,
we conclude that the notion of Planet Nine as a dominant driver
of solar obliquity is plausible.
4. DISCUSSION
Applying the well-established analytic methods of secular
theory, we have demonstrated that a solar obliquity of the order
of
several degrees is an expected observable effect of Planet
Nine. Moreover, for a range of masses and orbits of Planet
Nine that are broadly consistent with those predicted by
Batygin & Brown
(
2016
)
and
Brown & Batygin
(
2016
)
, Planet
Nine is capable of reproducing the observed solar obliquity of
6
°
, from a nearly coplanar con
fi
guration. The existence of
Planet Nine, therefore, provides a tangible explanation for the
spin
–
orbit misalignment of the solar system.
Within the context of the Planet Nine hypothesis, a strictly
null tilt of the solar spin
axis is disallowed. However, as
already mentioned above, in addition to the long-term
gravitational torques exerted by Planet Nine, numerous other
physical processes are thought to generate stellar obliquities
(
see, e.g., Crida & Batygin
2014
and the references therein
)
.A
related question then, concerns the role of Planet Nine with
respect to every other plausible misalignment mechanism.
Within the context of our model, this question is informed by
the present-day offset between the longitudes of the ascending
node of Planet Nine and the Sun,
D
W
. Particularly, if we
assume that the solar system formed in a con
fi
guration that was
strictly coplanar with the Sun
’
s equator, the observable
combination of the parameters
maei
,,,
9999
maps onto a
unique value of the observable parameter
D
W
.
Importantly, our calculations suggest that if the orbit of
Planet Nine resides in approximately the same plane as the
orbits of the
a
250
au Kuiper Belt objects
(
which inform the
existence of Planet Nine in the
fi
rst place
)
, then the inferred
range of
D
W
and Planet Nine
’
s expected orbital elements are
incompatible with an exactly co-linear initial state of the solar
spin axis. Instead, backward
integrations of the equations of
motion suggest that a primordial spin
–
orbit misalignment of the
same order as the rms spread of the planetary inclination
(
~
i
1
)
is consistent with the likely orbital con
fi
guration of
Planet Nine. In either case, our results contextualize the
Figure 4.
This set of plots depict the same parameter space as in Figure
(
3
)
, but the contours represent the longitude of ascending node of Planet Nine, relative to that
of the Sun,
D
W
. As before, the values are quoted in degrees.
5
The Astronomical Journal,
152:126
(
8pp
)
, 2016 November
Bailey, Batygin, & Brown
primordial solar obliquity within the emerging extrasolar trend
of small spin
–
orbit misalignments in
fl
at planetary systems
(
Morton & Winn
2014
)
, and bring the computed value closer to
the expectations of the nebular hypothesis. However, we note
that, at present, the range of unconstrained parameters also
allows for an evolutionary sequences in which Planet Nine
’
s
contribution does not play a dominant role in exciting the solar
obliquity.
Gomes et al.
(
2016
)
independently reached similar conclu-
sions. The primary differences between the two studies arise
from the speci
fi
c choice of methodology and the preference of
Gomes et al.
(
2016
)
to consider select inclinations of Planet
Nine, which are signi
fi
cantly higher than the
∼
20
°
inclination
of the distant cluster of Kuiper Belt objects that
fi
rst
engendered the Planet Nine hypothesis
(
Batygin &
Brown
2016
)
.
The integrable nature of the calculations performed in this
work imply that observational characterization of Planet Nine
’
s
orbit will not only verify the expansion of the solar system
’
s
planetary album, but will yield remarkable new insights into
the state of the solar system, at the time of its formation. That
is, if Planet Nine is discovered in a con
fi
guration that
contradicts a strictly aligned initial condition of the solar spin
axis and planetary angular momentum, calculations of the type
performed herein can be used to deduce the true primordial
obliquity of the Sun. In turn, this information can potentially
constrain the mode of magnetospheric interactions between the
young Sun and the solar nebula
(
Konigl
1991
; Lai et al.
2011
;
Spalding & Batygin
2015
)
, as well as place meaningful limits
on the existence of a putative primordial stellar companion of
the Sun
(
Batygin
2012
; Xiang-Gruess & Papaloizou
2014
)
.
Finally, this work provides not only a crude test of the likely
parameters of Planet Nine, but also a test of the viability of the
Planet Nine hypothesis. By de
fi
nition, Planet Nine is
hypothesized to be a planet having parameters suf
fi
cient to
induce the observed orbital clustering of Kuiper Belt objects
with semimajor axis
>
a
250 au
(
Batygin & Brown
2016
)
.
According to this de
fi
nition, Planet Nine must occupy a narrow
swath in
a
−
e
space such that
~
q
250
9
au, and its mass must
reside in the approximate range
=
Å
mm
520
9
–
. If Planet Nine
were found to induce a solar obliquity signi
fi
cantly higher than
the observed value, the Planet Nine hypothesis could be readily
rejected. Instead, here we have demonstrated that, over the
lifetime of the solar system, Planet Nine typically excites a
solar obliquity that is similar to what is observed, giving
additional credence to the Planet Nine hypothesis.
We are grateful to Chris Spalding and Roger Fu for useful
discussions, and to the anonymous reviewer for insightful
comments.
APPENDIX
To octupole order in
aa
9
(
)
, the full Hamiltonian governing
the secular evolution of a hierarchical triple is
(
Kaula
1962
;
Mardling
2010
)
⎜⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎛
⎝
⎞
⎠
⎤
⎦
⎥
m
e
w
=-
+
-
́-+
+W-W
+W-W
m
a
a
a
ei
iei
ii
ii
1
4
1
1
3
2
1
4
3cos
1
3cos
1
15
14
sin cos 2
3
4
sin 2 sin 2 cos
3
4
sin sin cos 2
2
,
9
99
2
9
3
2
9
22
99
22
99
(())
(())
()()
() ( ) (
)
()
( )
(
)
where elements without a subscript refer to the inner body, and
elements with subscript 9 refer to the outer body, in this case,
Planet Nine. Here
m
=+»
Mm M
m
m
()(
)
, and
e
9
is
equal to
-
e
1
9
2
.
To attain integrability, we drop the Kozai harmonic because
comparatively rapid perihelion precession of the known giant
planets
’
orbits ensures that libration of
ω
is not possible
(
Batygin et al.
2011b
)
. Because the eccentricities of the known
Figure 5.
Illustrative evolution tracks of the solar spin axis, measured with respect to the instantaneous invariable plane. The graphs are shown in polar coor
dinates,
where
i
e
and
W
represent the radial and angular variables respectively. The integrations are initialized with the Sun
’
s present-day con
fi
guration
(
=
i
6
,
W=
68
)
,
and are performed backward
in time. For Planet Nine, parameters of
=
Å
mm
15
9
,
=
a
500
9
au, and
e
9
=
0.5 are adopted throughout. Meanwhile, the left, middle,
and right panels show results corresponding to
=
i
10 ,
9
2
0,
and
3
0
respectively. The present-day longitude of the ascending node of Planet Nine is assumed to lie in
the range of
<W <
80
120
9
and is represented by the color of the individual evolution tracks.
6
The Astronomical Journal,
152:126
(
8pp
)
, 2016 November
Bailey, Batygin, & Brown
giant planets are small, we adopt
e
=
0 for the inner orbit.
Additionally, because the inclination of the inner orbit is
presumed to be small throughout the evolutionary sequence, we
neglect the higher-order
W- W
cos 2
2
9
()
harmonic, because it
is proportional to
ii
sin
sin 2
1
2
()
( )
.
Keeping in mind the trigonometric relationship
=
i
sin
-
i
1cos
2
, and adopting canonical Poincaré action-angle
variables given by Equation
(
3
)
, the Hamiltonian takes the
approximate form
⎜⎟
⎜⎟
⎜⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎛
⎝
⎜
⎛
⎝
⎞
⎠
⎞
⎠
⎟
⎛
⎝
⎜
⎜
⎛
⎝
⎜
⎞
⎠
⎟
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎞
⎠
⎟
⎟
e
=-
-
G
-
́-
G
-+ -
G
́--
G
-
G
--
G
́-
mm
a
a
a
Z
ZZ
ZZ Z
zz
1
4
11
4
31
1
31
1
3
4
21
11
21
11
cos
.
9
99
2
3
2
9
9
2
2
9
9
9
9
2
9
()]
Because the inner orbit has a
small inclination, it is suitable to
expand
to leading order in
Z
. This yields the Hamiltonian
given in Equation
(
4
)
.
Since Hamiltonian
(
4
)
possesses only a single degree of
freedom, the Arnold
–
Liouville theorem
(
Arnold
1963
)
ensures
that by application of the Hamilton
–
Jacobi equation,
can be
cast into a form that only depends on the actions. Then, the
entirety of the system
’
s dynamics is encapsulated in the linear
advance of cyclic angles along contours de
fi
ned by the
constants of motion
(
Morbidelli
2002
)
. Here, rather than
carrying out this extra step, we take the more practically simple
approach of numerically integrating the equations of motion,
while keeping in mind that the resulting evolution is strictly
regular.
The numerical evaluation of the system
’
s evolution can be
robustly carried out after transforming the Hamiltonian to
nonsingular Poincaré Cartesian coordinates
==
==
xZzy Zz
xZzy Zz
2cos
2sin
2cos
2sin .
999
9
99
()
()
()
()
Then, the truncated and expanded Hamiltonian
(
4
)
becomes
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎛
⎝
⎜
⎛
⎝
⎜
⎞
⎠
⎟
⎞
⎠
⎟
⎛
⎝
⎜
⎜
⎛
⎝
⎜
⎜
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎤
⎦
⎥
⎥
e
=-
-
G
+
́-
G
+
-
+-
G
+
́-
G
+
GG
+
mm
a
a
a
xy
xy
xy
xy
xx
yy
1
4
11
4
2
6
2
31
1
2
1
31
1
2
1
1
22
1
.
9
99
2
9
3
22
9
9
2
9
2
2
9
9
2
9
2
9
9
2
9
2
9
9
9
()
Explicitly, Hamilton
’
s equations
=-¶ ¶
d
xdt
y
,
=
d
ydt
¶
¶
x
take the form
⎛
⎝
⎜
⎜
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
=
G- -
G
G- -
GG
+
G
-
G- -
G
¶
¶
=
GG
́GG-- +-G
+G+ -G+ + -G
¶
¶
=
G
-+
́
G- -
G
+G--
G- -
G
+
G
G- - + - G
-+
G- -
GG- -
¶
¶
=-
G
-+
́
G- -
G
+G--
́
G- -
G
+
G
G- - + - G
-+
G- -
GG- -
dx
dt
amm
a
yxy
xy
y
xy
y
t
amm
a
xxyxy
xxyyxy
x
t
amm
a
yxx yy
xy
yxy
xy
yxyxy
yxx yy
xy
xy
y
t
amm
a
xxx yy
xy
xxy
xy
xxyxy
xxx yy
xy
xy
4
32
4
4
3
2
1
32
4
3
32
24
2
8312 36 2
3
16
2
4
2
4
1
23 3
2
2
4
3
16
2
4
2
4
1
23 3
2
2
4
.
2
9
9
3
9
3
9
9
9
2
9
2
9
9
9
2
9
2
9
2
9
9
2
9
2
2
9
2
2
9
9
3
9
2
9
3
99
9
2
9
2
9
2
9
2
9
9
2
9
4
9
9
2
9
4
9
2
9
2
9
9
2
9
9
3
9
2
9
3
9
9
9
9
9
2
9
2
9
9
2
9
2
9
9
2
9
2
9
22
9
2
9
2
9
9
9
9
9
9
2
9
2
9
9
2
9
2
9
2
9
9
3
9
2
9
3
99
9
9
9
2
9
2
9
9
2
9
2
9
9
2
9
2
9
22
9
2
9
2
9
99
9
9
9
2
9
2
9
9
2
9
2
()
()
((
)(
)
(
(
)))
((
)
()
()()
()
()
((
)
()
()()
()
()
The evolution of the Sun
’
s axial tilt is computed in the same
manner. The Hamiltonian describing the cumulative effect of
the planetary torques exerted onto the solar spin axis is given
by Equation
(
6
)
.De
fi
ning scaled Cartesian coordinates
==
x
Zz y Zz
2cos
2sin ,
̃
̃
(
̃
) ̃
̃
(
̃
)
we have
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎡
⎣
⎢
⎛
⎝
⎜
⎞
⎠
⎟
⎤
⎦
⎥
å
=
G
+
+
GG
+
m
a
a
xy
xx
yy
4
3
2
3
4
1
.
j
j
j
3
2
22
̃
̃
̃
̃ ̃
̃
( ̃
̃ )
Accordingly, Hamilton
’
s equations are evaluated to character-
ize the dynamics of the Sun
’
s spin pole, under the in
fl
uence of
7
The Astronomical Journal,
152:126
(
8pp
)
, 2016 November
Bailey, Batygin, & Brown
the planets:
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎞
⎠
⎟
å
å
=-
GG
+
G
=
GG
+
G
dx
dt
m
a
ay
y
dy
dt
m
a
ax
x
4
3
4
13
4
3
4
13
j
j
j
j
j
j
3
2
3
2
̃
̃
̃
̃
̃
̃
̃
̃
̃
̃
Note that unlike
Γ
and
G
9
, which are conserved,
G
̃
is an explicit
function of time, and evolves according to the Skumanich
relation. The above set of equations fully speci
fi
es the long-
term evolution of the dynamical system.
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