of 17
MNRAS
467,
3066–3082 (2017)
doi:10.1093/mnras/stx293
Advance Access publication 2017 February 2
Eccentricity and spin-orbit misalignment in short-period stellar binaries
as a signpost of hidden tertiary companions
Kassandra R. Anderson,
1
‹
Dong Lai
1
,
2
and Natalia I. Storch
3
1
Cornell Center for Astrophysics and Planetary Science, Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
2
Institute for Advanced Study, Princeton, NJ 08540, USA
3
TAPIR, Walter Burke Institute for Theoretical Physics, Mailcode 350-17, Caltech, Pasadena, CA 91125, USA
Accepted 2017 January 31. Received 2017 January 28; in original form 2016 October 8
ABSTRACT
Eclipsing binaries are observed to have a range of eccentricities and spin-orbit misalignments
(stellar obliquities). Whether such properties are primordial or arise from post-formation
dynamical interactions remains uncertain. This paper considers the scenario in which the binary
is the inner component of a hierarchical triple stellar system, and derives the requirements that
the tertiary companion must satisfy in order to raise the eccentricity and obliquity of the inner
binary. Through numerical integrations of the secular octupole-order equations of motion of
stellar triples, coupled with the spin precession of the oblate primary star due to the torque from
the secondary, we obtain a simple, robust condition for producing spin-orbit misalignment in
the inner binary. In order to excite appreciable obliquity, the precession rate of the stellar spin
axis must be smaller than the orbital precession rate due to the tertiary companion. This yields
quantitative requirements on the mass and orbit of the tertiary. We also present new analytic
expressions for the maximum eccentricity and range of inclinations allowing eccentricity
excitation (Lidov–Kozai window), for stellar triples with arbitrary masses and including the
non-Keplerian potentials introduced by general relativity, stellar tides and rotational bulges.
The results of this paper can be used to place constraints on unobserved tertiary companions
in binaries that exhibit high eccentricity and/or spin-orbit misalignment, and will be helpful
in guiding efforts to detect external companions around stellar binaries. As an application,
we consider the eclipsing binary DI Herculis, and identify the requirements that a tertiary
companion must satisfy to produce the observed spin-orbit misalignment.
Key words:
binaries: close – binaries: eclipsing – stars: kinematics and dynamics.
1 INTRODUCTION
Stellar binaries can exhibit a rich variety of dynamical behaviour. In
systems with sufficiently small separations, the orbit can precess due
to non-Keplerian potentials (e.g. general relativistic corrections),
and may also be sculpted by tidal dissipation. If the binary is a
member of a higher multiplicity system, or previously experienced
a close encounter with a neighbouring star, the orbital properties
can be further modified. In many observed binary systems, whether
the orbital elements reflect the properties of the protostellar cloud,
or result from post-formation dynamical evolution, remains an open
question. Distinguishing between the two possibilities can shed light
into star and binary formation processes.
A possible signature of post-formation dynamical evolution is
stellar spin-orbit misalignment (obliquity). One method of probing
stellar obliquities in binaries is by comparing the inclination of the
stellar equator (estimated through measurements of
v
sin
i
and the

E-mail:
kra46@cornell.edu
rotational period) with the orbital inclination. Using this method,
Hale (
1994
) found that solitary binaries tend to have low obliquities
when the separation is less than 30–40 au, but for separations be-
yond 30–40 au, the obliquities are randomly distributed. However,
for binaries residing in hierarchical multi-systems, even those with
small separations can have substantial spin-orbit misalignments, as
a result of post-formation dynamical evolution.
More recently, obliquities have been inferred from mea-
surements of the Rossiter–McLaughlin effect (Rossiter
1924
;
McLaughlin
1924
). A handful of eclipsing binaries have orbital
axes that are misaligned (in projection) with respect to the spin
axis of one or both members. In the ongoing BANANA project
(Binaries Are Not Always Neatly Aligned), an effort to mea-
sure obliquities in comparable-mass eclipsing binaries, Albrecht
et al. (
2007
,
2009
,
2011
,
2013
,
2014
) present Rossiter–McLaughlin
measurements of several systems. Thus far, four systems exhibit
spin-orbit alignment (Albrecht et al.
2007
,
2011
,
2013
), while
two systems contain misaligned components: in DI Herculis, both
the primary and secondary are misaligned, with
λ
pri

72
◦
and
λ
sec
−
84
◦
(Albrecht et al.
2009
); in CV Velorum, the
C

2017 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Eccentricity and obliquity in stellar binaries
3067
primary and secondary have
λ
pri
−
52
◦
and
λ
sec

3
◦
(Albrecht
et al.
2014
). A complementary study of spin-orbit misalignments in
unequal-mass eclipsing binaries (consisting of FGK-M members)
is being undertaken via the EBLM project (Triaud et al.
2013
).
Although the current sample of binaries with Rossiter–Mclaughlin
measurements still consists of only a few members, these efforts,
and others (e.g. eclipsing binaries observed by
Kepler
; see Dong,
Katz & Socrates
2013
), will increase the sample in the coming
years.
In general, it is not clear whether large spin-orbit misalignments
in eclipsing binaries are primordial (reflecting the initial state of the
protostellar cloud), or have been driven to misalignment due to dy-
namical interactions with a perturber. In this paper, we consider the
latter scenario, where the eclipsing binary is the inner component
of a hierarchical triple stellar system, with a tertiary companion
orbiting the centre of mass of the inner binary. If the inclination be-
tween the inner and outer orbits is sufficiently high, the eccentricity
of the inner binary can undergo periodic excursions to large val-
ues, known as Lidov–Kozai (LK) cycles (Lidov
1962
; Kozai
1962
),
see also Harrington (
1968
). It is widely believed that binaries with
P
orb

7 d are not primordial, but have evolved from wider config-
urations via LK cycles with tidal friction (Mazeh & Shaham
1979
;
Eggleton & Kiseleva-Eggleton
2001
; Fabrycky & Tremaine
2007
;
Naoz & Fabrycky
2014
). Indeed, binaries with periods shorter than
this threshold are known to have high tertiary companion fractions
(of up to 96 per cent for periods
<
3 d; see Tokovinin et al.
2006
),
supporting the idea that three-body interactions have played a ma-
jor role in their formation. There should also exist a population
of longer-period, eccentric binaries that are undergoing LK-driven
orbital decay (see Dong et al.
2013
).
It is important to recognize that even a strong perturbation from
a tertiary companion on the inner binary does not guarantee the
production of spin-orbit misalignment in the inner binary. If the in-
ner binary achieves a sufficiently small pericentre distance, a torque
due to the stellar quadrupole (arising from stellar oblateness) may
induce a change in the direction of the spin axis, but the degree
of spin-orbit misalignment depends on several factors. In previous
work (Storch, Anderson & Lai
2014
; Anderson, Storch & Lai
2016
),
we have investigated the spin dynamics of a planet-hosting star, as
a result of the planet undergoing LK oscillations due to a distant
stellar companion (see also Storch & Lai
2015
). The evolution of
the stellar spin axis can be complicated, with several qualitatively
distinct types of possible behaviour, depending on the combination
of planet mass, stellar spin period and the orbital geometries of
the inner and outer binaries. In particular, for increasingly massive
planets (
M
p

5–10
M
J
), the coupling between the star and planet
can be so strong that spin-orbit misalignment cannot be generated,
despite drastic changes in the orbital inclination. As the mass of the
secondary body increases from the planetary to the stellar regime,
the ability to generate spin-orbit misalignment is even further
hindered.
In light of these previous results, the main goal of this paper is
to identify under what circumstances large spin-orbit misalignment
can be generated in stellar binaries, due to secular interactions with
a tertiary companion. Tertiary companions can also excite the bi-
nary eccentricity. Another goal of this paper is thus to identify the
requirements for a tertiary companion to increase the eccentricity
of the inner binary from
e

0 to an observed eccentricity
e
=
e
obs
.
The results of this paper will help interpret current observations of
eclipsing binaries, and guide future efforts to detect tertiary com-
panions in binaries exhibiting large spin-orbit misalignment and/or
high eccentricities.
We do not consider the effects of tidal dissipation in this study. If
tidal dissipation is sufficiently strong to circularize the orbit, it will
almost certainly align the spin axis with the orbital axis on a shorter
time-scale, thereby erasing any obliquity excitation due to the outer
companion. To avoid this complication, we focus here exclusively
on the subset of systems that achieve minimum pericentre distances
that are too large for dissipative tides to act. This is in similar spirit
to the focus of the BANANA project (Albrecht et al.
2011
).
This paper is organized as follows. In Section 2, we review aspects
of LK oscillations in hierarchical triples with comparable masses,
and including the effects of short-range forces (SRFs, due to general
relativity and tidal and rotational distortion). This section also con-
tains new results concerning the ‘LK window’ of inclinations for
eccentricity excitation under general conditions. In Section 3, we
discuss the spin-orbit dynamics of binaries undergoing LK cycles,
and identify a requirement for generating spin-orbit misalignment.
Section 4 presents numerical integrations of the octupole-order sec-
ular equations of motion for a large number of triple systems, and
compares with the analytic results in Sections 2 and 3. In Section 5,
we apply the results to the observed eclipsing binary system DI
Herculis, and conclude in Section 6.
2 LIDOV-KOZAI CYCLES IN TRIPLES WITH
COMPARABLE ANGULAR MOMENTUM
AND SHORT-RANGE FORCES
2.1 Setup and equations
We consider a hierarchical triple stellar system, composed of an
inner binary with masses
m
0
and
m
1
, and outer companion with
mass
m
2
, orbiting the centre of mass of
m
0
and
m
1
. In this notation,
m
0
is the primary body of the inner binary, so that the secondary body
always satisfies
m
1
≤
m
0
. The reduced mass for the inner binary is
μ
in
=
m
0
m
1
/
m
01
, with
m
01
≡
m
0
+
m
1
. Similarly, the outer binary
has reduced mass
μ
out
=
m
01
m
2
/
m
012
, with
m
012
≡
m
0
+
m
1
+
m
2
.
The orbital semi-major axis and eccentricity of the inner and outer
binaries are (
a
in
,
e
in
)and(
a
out
,
e
out
), respectively. For convenience
of notation, we will frequently omit the subscript ‘in’, and define
e
=
e
in
and
j
=

1
−
e
2
in
. The orbital angular momenta of the inner
and outer binaries are denoted by
L
in
and
L
out
, respectively.
When the inclination between the inner and outer binaries is
sufficiently high, the eccentricity and inclination of the inner bi-
nary can undergo large, cyclic excursions, known as Lidov–Kozai
(LK) oscillations (Lidov
1962
; Kozai
1962
). See, for example,
fig. 1 of Holman, Touma & Tremaine (
1997
). These oscillations
are driven by the disturbing potential from the tertiary companion.
To quadrupole order of the potential, the oscillations occur on a
characteristic time-scale
t
k
given by
1
t
k
=
m
2
m
01
a
3
in
a
3
out
,
eff
n,
(1)
where
n
=

Gm
01
/a
3
in
is the orbital mean motion of the inner
binary, and we have introduced an ‘effective outer binary separation’
a
out, eff
,
a
out
,
eff
≡
a
out

1
−
e
2
out
.
(2)
The octupole potential of the outer companion further contributes
to the secular dynamics of the system, introducing under some
conditions even higher maximum eccentricities and orbit flipping
(Ford, Kozinsky & Rasio
2000
; Naoz et al.
2013a
), as well as chaotic
MNRAS
467,
3066–3082 (2017)
3068
K. R. Anderson, D. Lai and N. I. Storch
orbital evolution (Li et al.
2014
). The ‘strength’ of the octupole
potential (relative to the quadrupole) is determined by
ε
oct
=
m
0
−
m
1
m
0
+
m
1
a
in
a
out
e
out
1
−
e
2
out
.
(3)
Thus, for equal-mass inner binaries (
m
0
=
m
1
), or outer binaries
with
e
out
=
0, the octupole contributions vanish.
Additional perturbations on the orbit of the inner binary occur
due to Short-Range Forces, including contributions from general
relativity (GR), and tidal and rotational distortions of the inner
bodies. These non-Keplerian potentials introduce additional peri-
centre precession of the inner orbit that acts to reduce the maxi-
mum achievable eccentricity (e.g. Wu & Murray
2003
; Fabrycky
& Tremaine
2007
), and can suppress the extreme orbital features
introduced by octupole-level terms (Liu, Mu
̃
noz & Lai
2015b
).
In Section 2, for simplicity, we treat the secondary body in the
inner binary (
m
1
) as a point mass (although
m
1
can be comparable
to
m
0
). As a result, we do not consider the SRFs from tidal and
rotational distortion of
m
1
.
1
In order to attain analytical results, for
the rest of this section, we consider the gravitational potential of the
tertiary companion only to quadrupole order (except in Section 2.5,
where we briefly discuss coplanar hierarchical triples). These re-
sults are thus exact for equal-mass inner binaries (
m
0
=
m
1
), or
outer binaries with
e
out
=
0. In Section 4, we perform numerical
integrations with octupole included, and including all SRFs (GR,
and tidal and rotational distortion in both
m
0
and
m
1
).
Here we present key results of LK oscillations with SRFs in sys-
tems where the angular momenta of the inner and outer binaries
are comparable. The results of this section review and generalize
several previous works. For example, Fabrycky & Tremaine (
2007
)
derived the expression for the maximum eccentricity in LK oscilla-
tions (
e
max
) with the effects of GR included, in the limit where the
angular momentum ratio satisfies
L
in
/
L
out
→
0. Liu et al. (
2015b
)
presented results for general SRFs (GR, tides and rotational distor-
tion) and general angular momentum ratios. For
L
in
/
L
out

1, they
identified the existence of a ‘limiting eccentricity’ (see Section 2.3),
but for general
L
in
/
L
out
, Liu et al. (
2015b
) did not fully explore the
behaviour of
e
max
and the boundaries of parameter space that allow
LK oscillations (the ‘LK window’, see Section 2.2). When SRFs
are neglected, the equations for general
L
in
/
L
out
are first given by
Lidov & Ziglin (
1976
, and rederived by Naoz et al.
2013a
), along
with the analytical expression for the LK window. This is further
studied by Martin & Triaud (
2016
) in the context of circumbinary
planets.
The total orbital angular momentum of the system
2
L
tot
=
L
in
+
L
out
is constant, with magnitude
L
2
tot
=
L
2
in
+
L
2
out
+
2
L
in
L
out
cos
I,
(4)
where
I
is the mutual inclination between the two orbits. To
quadrupole order,
e
out
and
L
out
are constant. We can rewrite
equation (4) in terms of the conserved quantity
K
,where
K
≡
j
cos
I
−
η
2
e
2
=
constant
,
(5)
1
For example, the potential energy due to tidal distortion of
m
1
is
W
Tide
,
1
∼
k
2
,
1
Gm
2
0
R
5
1
/r
6
, while the energy due to tidal distortion of
m
0
is
W
Tide
,
0
∼
k
2
,
0
Gm
2
1
R
5
0
/r
6
,where
k
2, 0
and
k
2, 1
are the Love numbers of
m
0
and
m
1
.
For the low-mass main-sequence stars of interest in this paper, with
R
∝
m
0.8
,
we have
W
Tide, 1
/
W
Tide, 0
∼
(
m
1
/
m
0
)
2

1.
2
We have neglected the contribution from the spins of
m
0
and
m
1
, since for
stellar parameters of interest in this paper, the spin angular momentum
S
of
each star satisfies
S
/
L
in

1.
and where we have defined
η
≡

L
in
L
out

e
in
=
0
=
μ
in
μ
out

m
01
a
in
m
012
a
out
(1
−
e
2
out
)

1
/
2
.
(6)
In the limit of
L
in

L
out
(
η
→
0), equation (5) reduces to the
usual ‘Kozai constant’,
√
1
−
e
2
cos
I
=
constant. We will set the
initial eccentricity
e
0

0 for the remainder of this paper, so that
K

cos
I
0
. See Appendix A for a brief consideration of the initial
condition
e
0
=
0.
The total energy per unit mass is conserved, and (to quadrupole
order) given by

=

Quad
+

SRF
.
(7)
The first term in equation (7),

Quad
, is the interaction energy be-
tween the inner and outer binaries,

Quad
=−

0
8

2
+
3
e
2
−
(3
+
12
e
2
−
15
e
2
cos
2
ω
)sin
2
I
=−

0
8
2
+
3
e
2
−
(3
+
12
e
2
−
15
e
2
cos
2
ω
)
×

1
−
1
j
2
K
+
η
2
e
2
2

,
(8)
where
ω
is the argument of pericentre of the inner binary, and

0
=
Gm
2
a
2
in
a
3
out
,
eff
.
(9)
The second term in equation (7),

SRF
, is an energy term due to
SRFs that lead to additional pericentre precession. The contributions
to

SRF
consist of the general relativistic correction, as well as tidal
and rotational distortion of
m
0
,sothat

SRF
=

GR
+

Tide
+

Rot
,
with (e.g. Liu et al.
2015b
)

GR
=−
ε
GR

0
j
,

Tide
=−
ε
Tide

0
15
1
+
3
e
2
+
(3
/
8)
e
4
j
9
,

Rot
=−
ε
Rot

0
2
j
3
,
(10)
where
ε
GR

3
×
10
−
2
̄
m
2
01
̄
a
3
out
,
eff
̄
m
2
̄
a
4
in
,
ε
Tide

9
.
1
×
10
−
7
̄
k
2
,
0
̄
m
1
̄
m
01
̄
R
5
0
̄
a
3
out
,
eff
̄
m
2
̄
m
0
̄
a
8
in
,
ε
Rot

2
.
9
×
10
−
5
P
∗
10 d
−
2
̄
k
q,
0
̄
m
01
̄
R
5
0
̄
a
3
out
,
eff
̄
m
0
̄
m
2
̄
a
5
in
.
(11)
Here,
P

is the spin period of
m
0
. The various dimensionless masses
and radii,
̄
m
i
and
̄
R
i
, are the physical quantities scaled by M

and
R

.
̄
a
in
=
a
in
/
1au, and
̄
a
out
,
eff
=
a
out
,
eff
/
100 au.
̄
k
2
,
0
is the tidal
Love number of
m
0
scaled by its canonical value
k
2, 0
=
0.03. Simi-
larly,
̄
k
q,
0
depends on the interior structure of
m
0
and helps quantify
the degree of rotational distortion, and is scaled by its canonical
value
k
q
,0
=
0.01 (Claret & Gimenez
1992
).
3
Corresponding terms
3
k
q,
0
=
(
I
3
−
I
1
)
/m
0
R
2
0
ˆ

2
0
,where
I
1
and
I
3
are the principal moments of
inertia, and
ˆ

0
is the spin rate of
m
0
in units of the breakup rate.
k
q
,0
is
related to the apsidal motion constant
κ
by
k
q
,0
=
2
κ/
3.
MNRAS
467,
3066–3082 (2017)
Eccentricity and obliquity in stellar binaries
3069
for the tidal and rotational distortions of
m
1
are obtained by switch-
ing the indices 0 and 1 in equations (11) (but are neglected in
Section 2).
In the expression for

Rot
in equation (10), we have assumed
alignment of the spin and orbital axes. When the spin and orbital
axes are not aligned,

Rot
depends on the spin-orbit misalignment
angle. In this situation, the problem is no longer integrable, and
numerical integrations are required (however, see Correia
2015
for
an analytic treatment). In order to attain analytic results, we will
assume that the spin and orbital axes are aligned for the remainder
of Section 2, and consider the spin-orbit dynamics separately, in
Section 4 via numerical integrations.
For the system parameters of interest in this paper, the GR con-
tribution to the SRFs usually dominates over the rotational contri-
bution at low to moderate eccentricities, and the tidal contribution
dominates at very high eccentricities (
e

0.9). As a result,

Rot
can
often be neglected. This approximation requires that
S

L
in
(where
S
is the spin angular momentum of
m
0
), and is always satisfied for
the systems considered in this paper. We also require
ε
Rot
/
2
j
3

1
(so that the rotational contribution does not suppress the LK cycles),
and
ε
Rot
/
2
j
3

ε
GR
/
j
(so that

Rot


GR
, i.e. rotational distortion is
negligible compared to GR). Thus, ignoring the effects of rotational
distortion is justified for eccentricities that satisfy
1
−
e
2

5
.
9
×
10
−
4
̄
k
q,
0
̄
m
01
̄
R
5
0
̄
a
3
out
,
eff
̄
m
0
̄
m
2
̄
a
5
in
2
/
3
P

10 d
−
4
/
3
,
(12)
and
1
−
e
2

4
.
8
×
10
−
4
̄
k
q,
0
̄
R
5
0
̄
m
0
̄
m
01
̄
a
in

P

10 d

−
2
.
(13)
Therefore,

Rot
is often negligible, unless the spin period is excep-
tionally rapid, or if the star has a large radius.
For a given initial condition (
I
0
and
e
0

0), the conservation of

(equation 7) and
K

cos
I
0
(equation 5) yield
e
as a function of
ω
. The maximum eccentricity (where d
e
/
d
ω
=
0) is achieved when
ω
=
π
/
2and3
π
/
2.
2.2 Range of inclinations allowing eccentricity excitation
The ‘window’ of inclinations allowing LK oscillations (starting
from an initial eccentricity
e
0

0) can be determined by enforcing
e
max
>
0. Expanding for
e
2

1, the conservations of energy and
K
=
cos
I
0
[valid to
O
(
e
6
)] reduce to
ae
6
+
be
4
+
ce
2
=
0
,
(14)
where
a
=
η
2
4

4
−
5cos
2
ω

−
ε
GR
6
+
5
ε
Rot
12
+
7
ε
Tide
b
=
η
2
4
+
(4
−
5cos
2
ω
)(1
+
η
cos
I
0
)
−
1
−
ε
GR
3
+
ε
Rot
2
+
10
ε
Tide
3
c
=
5cos
2
ω
sin
2
I
0
+
5cos
2
I
0
+
η
cos
I
0
−
3
+
4
ε
GR
3
+
2
ε
Rot
+
4
ε
Tide
3
.
(15)
For
e
>
0, equation (14) becomes
ae
4
+
be
2
+
c
=
0
.
(16)
This equation determines
e
as a function of
ω
for various parameters
I
0
,
η
,
ε
GR
,
ε
Tide
and
ε
Rot
. The maximum eccentricity occurs at
ω
=
π
/
2and3
π
/
2. In order for this
e
max
=
0 to be reachable from
e
0

0, we require that equation (16) admit
e
=
e
0

0 as a solution for
some value of
ω
0
≡
ω
(
e
0
). Evaluating equation (16) at
e
=
e
0
=
0
yields
cos
2
ω
0
=−
5cos
2
I
0
+
η
cos
I
0
−
3
+
ε
SRF
5sin
2
I
0
,
(17)
wherewehavedefined
ε
SRF
≡
4
3
ε
GR
+
2
ε
Rot
+
4
3
ε
Tide
.
(18)
Requiring that cos
2
ω
0
≥
0 translates into the condition
(cos
I
0
)
−
≤
cos
I
0
≤
(cos
I
0
)
+
,
(19)
where
(cos
I
0
)
±
=
1
10

−
η
±

η
2
+
60
−
20
ε
SRF

.
(20)
In order for (cos
I
0
)
±
to be real,
η
and
ε
SRF
must satisfy
η
2
+
60
−
20
ε
SRF
≥
0
.
(21)
If
ε
SRF
<
3, then equation (21) is satisfied for all values of
η
.If
ε
SRF
>
3 and equation (21) is not satisfied, eccentricity oscillations
cannot be induced for any value of cos
I
0
.
Note that while (cos
I
0
)
+
is less than unity for all values of
η
and
ε
SRF
[provided that equation (21) is satisfied], (cos
I
0
)
−
>
−
1 only
when
η<
2
+
ε
SRF
and
η<
10
.
(22)
On the other hand, requiring that cos
2
ω
0
≤
1 implies that
cos
I
0
≥−
2
η
1
+
1
2
ε
SRF
.
(23)
Thus, if
η>
2
ε
SRF
, then the condition cos
I
0
≥
(cos
I
0
)
−
(in equa-
tion 19) must be replaced by equation (23). If
ε
SRF
=
0, the re-
quirement that cos
I
0
≥−
2
/η
is recovered, as identified by Lidov
& Ziglin (
1976
).
The above conditions (equations 19 and 23) guarantee that energy
conservation equation (14) has a physical solution (
e
,
ω
)
=
(0,
ω
0
).
Requiring
e
2
=
e
2
max
>
0at
ω
=
π
/
2 implies that
c
(cos
ω
=
0)
<
0,
which translates into the condition (19).
Fig.
1
shows the ‘LK window’ of inclinations allowing eccen-
tricity oscillations, determined by equations (20) and (23), as a
function of
η
, for several illustrative values of
ε
GR
(and with
ε
Tide
,
ε
Rot
=
0). At moderate eccentricities, the SRF contribution due to
GR dominates over the tidal contribution (since
ε
Tide

ε
GR
), and
for solar-type stars, GR also dominates over the rotational distortion
(since
ε
Rot

ε
GR
). As a result, adopting the approximation
ε
Tide
,
ε
Rot
=
0 is often a valid approximation, except for eccentricities
near unity, or for large values of the stellar radius and spin rate, see
equations (12) and (13).
Inside the LK window, the maximum eccentricity is also shown,
as calculated in Section 2.3, equation (24). When
ε
GR
=
0and
η
=
0,
the window of inclinations allowing LK oscillations is given by the
well-known form
−
√
3
/
5
≤
cos
I
0
≤
√
3
/
5. For increasing
ε
GR
,the
window narrows for most values of
η
.When
ε
GR
>
2.25, the win-
dow closes and eccentricity oscillations are completely suppressed
for small values of
η
. For larger (

1) values of
η
, LK oscillations
remain possible, but occur only within a very narrow range of in-
clinations, and are limited to retrograde (cos
I
0
<
0) configurations.
MNRAS
467,
3066–3082 (2017)
3070
K. R. Anderson, D. Lai and N. I. Storch
Figure 1.
Left and centre panels: the ‘window’ of inclinations (shaded regions) that allow LK oscillations, versus the angular momentum ratio
η
, for various
values of
ε
GR
(we have set
ε
Tide
=
ε
Rot
=
0). The solid lines are obtained from equation (19), and the dashed line from equation (23). Inside the window, the
LK maximum eccentricity is also shown, as calculated in Section 2.3, equation (24). Combinations of cos
I
0
and
η
below the dashed line allow LK eccentricity
oscillations, but these oscillations are not connected to the
e
0

0 trajectory. This is illustrated in the rightmost panel, where we show example phase-space
trajectories (
ω
,
e
) for energies corresponding to the coloured crosses in the neighbouring uppermost panel (with
ε
GR
=
1.0).
We find that for
ε
GR

5, the LK window is so narrow for all values
of
η
that LK oscillations are for all practical purposes completely
suppressed. The rightmost panel of Fig.
1
shows phase-space trajec-
tories (contours of constant energy) for two representative points.
The trajectory located just inside the LK window shows that the
eccentricity can increase to a large value, starting from
e
0

0. In
contrast, the trajectory just outside of the LK window does not con-
nect to
e
0

0. As a result, for (
η
,cos
I
0
) located below the dashed
curves in Fig.
1
, LK oscillations starting from
e
0

0 are completely
suppressed.
2.3 Maximum and limiting eccentricities
Evaluating the eccentricity at
e
0
=
0 (where
I
=
I
0
)and
e
=
e
max
(where
ω
=
π
/
2) allows energy and angular momentum conserva-
tion to be expressed as
3
8
j
2
min
−
1
j
2
min

5

cos
I
0
+
η
2

2
−
⎛
⎝
3
+
4
η
cos
I
0
+
9
4
η
2
⎞
⎠
j
2
min
+
η
2
j
4
min
+


SRF

0





e
max
0
=
0
,
(24)
where
j
min
≡

1
−
e
2
max
. When the effects of SRFs are negligible,
and in the limit
η
→
0, the solution of equation (24) yields the well-
known relation
e
max
=

1
−
(5
/
3) cos
2
I
0
. Note that the properties
of the tertiary companion (
a
out
,
e
out
,
m
2
) enter equation (24) only
through the combination
a
out
,
eff
/m
1
/
3
2
and
η
.
For general
η
,
ε
GR
,
ε
Tide
and
ε
Rot
, equation (24) must be solved
numerically for
e
max
. Fig.
2
shows an example of
e
max
versus
I
0
,
for an equal-mass inner binary (
m
0
=
m
1
=
1M

) with an orbital
period of 15 d, a low-mass outer companion (
m
2
=
0.1 M

)and
outer binary separations,
a
out
=
10
a
in
,30
a
in
,65
a
in
as labelled.
Figure 2.
The maximum eccentricity of the inner binary, versus the ini-
tial inclination
I
0
.Wehavefixed
m
0
=
m
1
=
1M

,
m
2
=
0.1 M

,
a
in
=
0.17 au (so that the orbital period is
∼
15 d),
e
out
=
0andvarying
a
out
,as
labelled. The solid curves show results with SRFs included, and the dashed
curves show results without SRFs. The dotted curve depicts the standard re-
sult
e
max
=

1
−
(5
/
3) cos
2
I
0
, applicable in the limit
η
→
0and
ε
GR
,
ε
Rot
,
ε
Tide
→
0.
Inspection of Fig.
2
reveals that there is a maximum (limiting)
achievable value of
e
max
, denoted here as
e
lim
, which occurs at a
critical initial inclination
I
0, lim
. This limiting eccentricity
e
lim
occurs
when the initial inclination satisfies the condition d
e
max
/
d
I
0
=
0, or
MNRAS
467,
3066–3082 (2017)
Eccentricity and obliquity in stellar binaries
3071
when d
j
min
/
d
I
0
=
0. Defining
j
lim
≡

1
−
e
2
lim
, and differentiating
equation (24) with respect to
I
0
,wefindthat
I
0, lim
is given by
cos
I
0
,
lim
=
η
2

4
5
j
2
lim
−
1

.
(25)
Obviously, the existence of
I
0, lim
requires
η<
2
/
(1
−
4
j
2
lim
/
5). No-
tice that
I
0, lim
depends on both
η
and on the strength of the SRFs
(through
e
lim
). When
η
→
0,
I
0, lim
→
90
◦
.As
η
increases, the
critical inclination is shifted to progressively retrograde values
(
I
0, lim
>
90
◦
).
Substituting equation (25) into equation (24), we find that the
limiting eccentricity
e
lim
is determined by
3
8
(
j
2
lim
−
1)

−
3
+
η
2
4

4
5
j
2
lim
−
1


+

SRF

0





e
=
e
lim
e
=
0
=
0
.
(26)
Equation (26) may sometimes permit a physical solution for
j
0, lim
,
but imply unphysical values for cos
I
0, lim
. In such cases,
e
lim
cannot
be achieved. As a result, any solution obtained from equation (26)
must also be substituted into equation (25) to ensure that cos
I
0, lim
exists.
Fig.
3
shows
e
lim
and
I
0, lim
as determined from equations (25) and
(26), along with the ranges of inclinations allowing LK oscillations
of any amplitude, from equations (19) and (23), as a function of
a
out
,
eff
/m
1
/
3
2
. In this example, we have set
a
in
=
0.17 au and
e
out
=
0,
and adopted two values of the tertiary mass: a solar-type perturber
(
m
2
=
1M

) and a brown-dwarf perturber (
m
2
=
0.1 M

). Since
equation (26) depends on
η
only through
η
2
,
e
lim
is nearly degenerate
in terms of
a
out
/m
1
/
3
2
for the adopted parameters in Fig.
3
.For
the solar-mass tertiary,
I
0, lim

90
◦
for all values of
a
out, eff
, because
η

1 is always satisfied. For the brown-dwarf tertiary,
I
0, lim
>
90
◦
for small values of
a
out, eff
, because
η
∼
1.
2.4 Constraints on hidden tertiary companions
from inner binary eccentricities
For an observed binary system with eccentricity
e
obs
, we can de-
rive constraints on a possible unseen tertiary companion driving the
eccentricity from
e
0

0to
e
=
e
obs
through LK cycles. The LK max-
imum eccentricity must satisfy
e
max
≥
e
obs
; this places constraints on
the mass of the perturber, and the range of mutual inclinations
I
0
and
effective outer separations
a
out, eff
. In Fig.
4
, we plot curves of con-
stant
e
max
=
0.2, 0.5, 0.8 in (
I
0
,
a
out
) space assuming an equal-mass
inner binary (
m
0
=
m
1
=
1M

) with orbital period
P
orb
=
15 d,
e
out
=
0, and adopting both solar-type and brown-dwarf perturbers.
The curves were obtained by solving equation (24). For a given
e
max
contour, the regions inside the curve indicate the parameter
space able to produce
e
≥
e
max
. For example, if an observed binary
system has
e
obs
=
0.8, a solar-mass perturber must be located within
∼
10 au in order to produce the observed eccentricity, and the neces-
sary inclination is restricted to the range 60
◦

I
0

120
◦
. Similarly,
a brown-dwarf companion must be located within
∼
6 au, most likely
in a retrograde orbit (
I
0

90
◦
).
For
η

1, the properties of the outer perturber required to
produce a given eccentricity can be explicitly calculated, without
having to resort to numerical root finding in equation (24) or (26).
Neglecting the SRF contribution from rotational and tidal distortion
(so that
ε
Rot
=
ε
Tide
=
0), the LK window (equation 20) is
|
cos
I
0
|≤
1
5

15
−
20
3
ε
GR
.
(27)
Figure 3.
Limiting eccentricity
e
lim
and critical inclination
I
0, lim
,asa
function of (
a
out
/a
in
)
̄
m
−
1
/
3
2
. The black curves show
m
2
=
1M

,andthe
red curves show
m
2
=
0.1 M

. The other parameters are
m
0
=
m
1
=
1M

,
a
in
=
0.17 au and
e
out
=
0. In the lower panel, the solid lines indicate
I
0, lim
,
and the dashed lines show the range of inclinations capable of exciting
LK oscillations (
I
0,
±
), as determined from equations (19) and (23). As
L
out
decreases relative to
L
in
(i.e.
η

1),
I
0, lim
is shifted to progressively
retrograde values. For the brown-dwarf tertiary, cos
I
0, lim
does not exist for
small values of
a
out
m
−
1
/
3
2
; as a result,
e
lim
cannot always be achieved. Notice
that
e
lim
is nearly degenerate in terms of (
a
out
)
̄
m
−
1
/
3
2
(thus the red and black
curves nearly coincide in the top panel).
Thus, LK oscillations are completely suppressed (
e
max
=
0) when
ε
GR
satisfies (see also Liu et al.
2015b
)
ε
GR
>
9
4

1
−
5
3
cos
2
I
0

for
η

1
.
(28)
For an inner binary with specified properties, this translates into a
maximum effective perturber distance for LK oscillations (of any
amplitude) to occur:
a
out
,
eff
<
19
.
6au
̄
m
2
̄
m
2
01
1
/
3
a
in
0
.
1au
4
/
3
1
−
5
3
cos
2
I
0
1
/
3
.
(29)
Setting
I
0
=
I
0, lim
=
90
◦
yields the absolute maximum effective
distance
a
out, eff
for LK oscillations to occur (for any inclination).
For
η

1, the limiting perturber distance able to drive the eccen-
tricity to
e
obs
can be solved explicitly by setting
e
max
=
e
obs
=
e
lim
,
MNRAS
467,
3066–3082 (2017)
3072
K. R. Anderson, D. Lai and N. I. Storch
Figure 4.
Curves in (
I
0
,
a
out
) parameter space able to produce a given value
of
e
max
, as labelled. For each contour of
e
max
, the region bounded by the
curve and the
x
-axis indicates combinations of (
I
0
,
a
out
) that will yield even
higher maximum eccentricities. Results are shown for a solar-mass outer
companion (top), and a brown-dwarf outer companion (bottom). The inner
binary properties are fixed at
m
0
=
m
1
=
1M

,
P
orb
=
15d(
a
in
=
0.17 au)
and
e
out
=
0. See also Fig.
13
where we show similar calculations applied
to the eclipsing binary system DI Herculis.
and neglecting the terms in equation (26) proportional to
η
2
,
a
out
,
eff

15
.
5au
a
in
0
.
1au
4
/
3
̄
m
2
̄
m
2
01
1
/
3
×

F
1
+
F
2
̄
m
1
̄
R
5
0
̄
m
0
̄
m
01

a
in
0
.
1au

−
4

−
1
/
3
,
(30)
where we have defined
F
1
=
1
j
lim
(
j
lim
+
1)
(31)
F
2
=
2
.
02
×
10
−
2
1
−
j
2
lim

1
+
3
e
2
lim
+
(3
/
8)
e
4
lim
j
9
lim
−
1

.
(32)
Expanding
F
1
and
F
2
appropriately, and setting
e
lim
=
0, recovers
equation (29) evaluated at
I
0
=
90
◦
.
In Fig.
5
, we plot the maximum effective separation required to
generate an eccentricity
e
obs
=
0.2 and 0.8, by solving equation
(26). We also compare this with the approximate (
η

1 limit) ex-
pression given in equation (30). The exact solution agrees well with
equation (30), because the criterion for determining the limiting
eccentricity (equation 26) depends on the angular momentum ra-
tio only as
η
2
. Therefore, only when
η
→
1 does the approximate
solution deviate from the exact expression.
Figure 5.
Effective perturber distance required to generate a limiting ec-
centricity
e
lim
, as labelled, as a function of the inner binary orbital period.
The solid lines depict a solar-mass outer perturber (
m
2
=
1M

), whereas
the dashed lines depict a low-mass brown-dwarf perturber (
m
2
=
0.05 M

).
The dashed lines correspond to the expression (30), valid in the
η
→
0 limit.
For a given inner binary period
P
in
, in order for an unseen perturber to gen-
erate an eccentricity
e
obs
=
0.2 (0.8), the perturber must have an effective
separation lower than the black (blue) value. Note that the
y
-axis has been
scaled by (
m
2
/
M

)
−
1
/
3
.
2.5 Eccentricity excitation in coplanar systems
If the inner and outer orbits are coplanar, and the octupole con-
tribution is non-vanishing (
ε
oct
=
0), the inner and outer binaries
can exchange angular momentum, thereby periodically exciting the
eccentricity of the inner binary. In the case of exact coplanarity,
the maximum eccentricity can be calculated algebraically (Lee &
Peale
2003
).
The general interaction potential up to octupole order is given in,
e.g., Ford et al. (
2000
), Naoz et al. (
2013a
) and Liu et al. (
2015b
).
If the orbits are exactly coplanar, the interaction energy simplifies
to

Int
=

Quad
+

Oct
=

0
8

−
2
−
3
e
2
+
15
8
e
(3
e
2
+
4)
ε
oct
cos
,
(33)
where
=
in
−
out
, with
the longitude of periapsis.
The total angular momentum
L
tot
=
L
in
+
L
out
is also conserved.
For a given set of orbital geometries (so that both

and
L
tot
are
fully specified),
e
in
and
e
out
as a function of
can be obtained.
The maximum value of
e
in
,
e
max
, occurs at either
=
0or
π
,
depending on the initial value of
, and whether
librates or
circulates.
If either the inner or outer orbit is initially circular, the interac-
tion energy is independent of the initial orientation (
)ofthe
two orbits. The procedure for calculating
e
max
is as follows: we
specify the initial total energy

, including the effects of SRFs
(

=

Int
+

SRF
), and the angular momentum (
L
tot
), calculate
e
as
a function of
and determine the maximum value of
e
(see also
Petrovich
2015a
). As before, we neglect the contribution to

SRF
from rotational distortion (

Rot
=
0).
MNRAS
467,
3066–3082 (2017)
Eccentricity and obliquity in stellar binaries
3073
Figure 6.
Maximum eccentricity
e
max
for coplanar (
I
=
0) hierarchical
triple systems, versus the outer binary semi-major axis. The properties of
the inner binary are fixed, with masses
m
0
=
1M

,
m
1
=
0.5 M

and
P
orb
=
15 d.
In Fig.
6
, we fix the properties of the inner binary (
m
0
=
1M

,
m
1
=
0.5 M

,
P
orb
=
15 d), and plot the maximum eccentric-
ity for the two fiducial masses for the perturber (1 and 0.1 M

),
and varying initial values of
e
out
. The solar-mass perturber must
be sufficiently close (
∼
1 au) and eccentric to excite a substantial
eccentricity in the inner binary. In such configurations, the secular
approximation is in danger of breaking down. The brown-dwarf
perturber is able to excite higher eccentricities, with a sharp peak.
The sharp peak of
e
max
at a specific value of
a
out
coincides when the
angle
changes from circulating to librating. The existence of
librating solutions allows for higher maximum eccentricities (Lee
& Peale
2003
), and can be understood in terms of an ‘apsidal pre-
cession resonance’ (Liu et al.
2015a
). This ‘resonance’ occurs when
the apsidal precession of the inner binary (driven by GR and the
outer binary) matches that of the outer binary (driven by the inner
binary). However, note that this does not qualify as a ‘true reso-
nance’ (see Laskar & Robutel
1995
; Correia et al.
2010
;Laskar,
Bou
́
e & Correia
2012
, for further discussion on the nature of this
‘resonance’).
3 SPIN-ORBIT DYNAMICS IN SYSTEMS
UNDERGOING LK OSCILLATIONS
Due to rotational distortion, each member of the inner binary pos-
sesses a quadrupole moment, causing a torque and mutual preces-
sion of the spin axis
S
and the orbital axis
L
in
. Here we discuss the
precession of the primary member of the inner binary (
m
0
). Similar
results for the spin precession of
m
1
are obtained by switching the
indices 0 and 1 in the following expressions.
The spin axis of
m
0
precesses around
ˆ
L
in
=
ˆ
L
according to
d
ˆ
S
d
t
=

ps
ˆ
L
×
ˆ
S
,
(34)
where the symbolˆdenotes unit vectors, and where the precession
frequency

ps
is given by

ps
=−
3
Gm
1
(
I
3
−
I
1
)cos
θ
sl
2
a
3
in
j
3
S
.
(35)
In equation (35), the spin-orbit angle is defined by cos
θ
sl
=
ˆ
S
·
ˆ
L
,
and
I
3
−
I
1
are the principle moments of inertia of
m
0
.
4
Meanwhile, the orbital axis of the inner binary precesses
and nutates around the total orbital angular momentum axis
J
=
L
in
+
L
out
, with frequency

L
=|
d
ˆ
L
/
d
t
|
. In general,

L
is
a complicated function of eccentricity, but takes the approximate
form (Anderson et al.
2016
)

L

3(1
+
4
e
2
)
8
t
k
√
1
−
e
2
|
sin 2
I
|
.
(36)
Equation (36) is exact at
e
=
0and
e
max
.Both

ps
and

L
are strong
functions of eccentricity, and thus can undergo large variation during
a single LK cycle.
AsdescribedinStorchetal.(
2014
), the dynamical behaviour of
ˆ
S
under the influence of a secondary body undergoing LK oscillations
depends on the ratio
|

ps
/
L
|
. Here we summarize the key aspects
of the dynamics (see also Storch & Lai
2015
; Anderson et al.
2016
).
If
|

ps
||

L
|
throughout the LK cycle, denoted as the ‘non-
adiabatic regime’,
ˆ
S
cannot ‘keep up’ with
ˆ
L
as
ˆ
L
precesses
around
ˆ
J
. As a result,
ˆ
S
effectively precesses around
ˆ
J
,sothat
θ
sj
≡
cos
−
1
(
ˆ
S
·
ˆ
J
)

constant. On the other hand, if
|

ps
||

L
|
throughout the LK cycle, denoted as the ‘adiabatic regime’,
ˆ
S
‘follows’
ˆ
L
, and the spin-orbit angle
θ
sl

constant. Finally, if
|

ps
|∼|

L
|
at some point during the LK cycle, the dynamical
behaviour is complicated due to secular resonances, and chaotic
evolution of
ˆ
S
can ensue (Storch & Lai
2015
). We denote this as
the ‘trans-adiabatic regime’.
In some cases, inclusion of the backreaction torque from the
oblate star on the orbit can considerably complicate this simple clas-
sification. In particular, our previous work, beginning with Storch
et al. (
2014
), focused on systems in which the secondary member
of the inner binary was a planet. In such cases,
L
in
and
S
are often
comparable during the high-eccentricity phases of the LK cycles,
and the backreaction torque from the oblate star on the orbit can be
significant. In contrast, here we consider a stellar mass secondary
body, so that
L
in

S
is well satisfied. As a result, the torque on the
orbital axis from the oblate star is negligible,
5
resulting in simplified
behaviour.
We introduce an ‘adiabaticity parameter’ that characterizes the
degree to which the stellar spin axis
ˆ
S
‘follows’ the precession of
4
There is also a spin–spin interaction, of order
GQ
0
Q
1
/
r
5
,where
Q
0, 1
=
(
J
2
mR
2
)
0, 1
is the rotation-induced quadrupole moment. This is
much smaller than the
S
–
L
terms, of order
GQ
0, 1
m
1, 0
/
r
3
. In addition, spin–
spin resonances may occur when the precession frequencies of the spin
axes (equation 35) become equal (Correia, Bou
́
e&Laskar
2016
). However,
although this latter effect is captured by our numerical integrations in Sec-
tion 4, such spin–spin interactions do not play an important dynamical role
in the systems of interest here.
5
However, note that, although the expression for d
L
in
/
d
t
is negligible here,
the oblate star still causes additional pericentre precession of the orbit.
MNRAS
467,
3066–3082 (2017)