Investigation of the Tidal Migration of 'Hot' Jupiters
1. TH
E
U
N I V E R
S
ITY
OF
E
D I N B U
R
G
H
The University of Edinburgh
School of Physics and Astronomy
MPhys Project Thesis
Simulating Tidal Migration of ‘Hot’
Jupiters
Author:
Calum Hervieu
Supervisor:
Professor Ken Rice
October 11, 2016
Abstract:
Papers published as recently as 2010 have identified that a significant fraction
of ‘hot’ Jupiters orbit in a way that is unexpected from traditional formation
theories. This paper presents an alternative formation theory, namely through
three-body Kozai-Lidov cycles. The theory is then evaluated through the extensive
development and testing of an N-body code, referred to as hermite. The results
agree well with hypotheses, implying that the code has been developed successfully,
and the underlying physical processes are well-understood.
4. Theory and Literature Review
1 Motivation
A significant problem in astronomy is that astrophysical systems, in general, evolve on vast
timescales. This means that direct observational verification of theories on human timescales
can often be very difficult. Instead, other methods must be implemented. One option is to study
several similar systems and piece together the precise manner of evolution using tried and tested
physical laws - e.g. gravitation. However, when observation does not match the theory, as is
the case with the ‘hot’ Jupiters observed on inclined orbits, the next best option is to utilise
computational simulation. The scientific focus of this paper lies with these ‘hot’ Jupiter planets,
a significant proportion of which have been observed to orbit on an incline with respect to their
host star (Triaud et al., 2010); this, according to the traditional formation theory of gravitational
accretion and migration through the protoplanetary disc, should not be the case. Specifically,
this paper will investigate the role of 3-body Kozai-Lidov cycles in perturbing the proto-Jupiter
onto an eccentric orbit, and the subsequent tidally driven migration. This will be done mainly
through N-body simulations, which are a widely used tool in astronomy and astrophysics to
simulate dynamical systems of particles and have vast uses ranging from investigating dark
matter halos in large scale structure down to stellar and planetary models.
Briefly, the paper is structured as follows: a short history of ‘hot’ Jupiters and their formation
will be presented first, followed by how they are detected and the work that led to the discovery
of the inclined systems. Next, an alternative formation theory through Kozai-Lidov cycles will
be proposed before the exact computational methods and theory, tests and simulations are
demonstrated. Finally, an analysis of the project and conclusions will be presented.
2 ‘Hot’ Jupiters
‘Hot’ Jupiters are extrasolar planets of similar size and mass, and therefore density, to that of
Jupiter, but that orbit much closer to their parent star. Typically they are located at 0.015
- 0.2au (astronomical units) (Raymond et al., 2005), and being so close, exhibit much higher
surface temperatures. They follow very circular orbits (e < 0.2) because they have been, or are
being, circularized by the gravitational tidal forces between the planet and host star (Fabrycky
and Tremaine, 2007).
The standard scenario is that all planets form in protostellar discs on prograde, circular orbits
as expected from angular momentum conservation. There are subtle differences between the
formation of the rocky terrestrial planets, like Earth, and gas giants, like Jupiter and Saturn.
A significant constraint on ‘hot’ Jupiter formation is that it must happen relatively fast, before
the gas in the protostellar disc is dissipated.
‘Hot’ Jupiters are thought to form at large radii (beyond the frost line), before undergoing Type II
disc migration (Masset and Papaloizou, 2003) and settling on their observed orbits at r < 0.2au.
2
5. Chapter 1 Theory and Literature Review 3
Briefly, Type II migration occurs when a planetesimal of significant mass (e.g Mp ≥ 0.5MJupiter)
has opened a tidally produced gap in the uniform gas of a protoplanetary disc. This movement
of material causes spiral density waves to propagate radially outward from the planet’s orbit and
will interact with the gas in the vicinity causing a torque. In this regime the subsequent motion
of the planet is defined by the evolution of the disc. The direction of migration is dependent on
the net torque across the planet; overtaking interactions with the gas inside the orbit tend to add
angular momentum, while overtaking interactions outside tend to remove angular momentum.
In practice the net torque is almost always negative; causing energy dissipation, a loss of orbital
angular momentum, and thus migration toward the parent star (Lubow and D’Angelo, 2006).
Planetesimals that have formed a gap do continue to accrete some gas via small streams of
material that cross the orbit. However, the rate of gas accretion is inversely proportional to the
mass of the planetesimal; as a larger planetesimal causes the gap to become deeper and hence
reduces the rate of transfer of gas across the orbit.
3 Detection
3.1 Radial Velocity
Due to their size ‘hot’ Jupiters were the first exoplanets to be discovered, by Mayor and Didier
in 1995 (Mayor and Didier, 1995). The main method of detection is to infer their presence via
radial velocity measurements of the host star. Radial velocity astronomy relies on the ability to
observe the ‘wobble’ of a star as it, and its companion planets, orbit their common barycentre.
As the star orbits away from or toward the observer, its spectral absorption lines are Doppler
shifted from their known laboratory values. Due to the enormous mass difference this barycentre
is very often inside the star itself and so the wobbles are very small, requiring extreme precision
and very sensitive equipment to detect.
The observed periodic shifting of the star’s spectral lines gives vital information about the radial
velocity of the target, and this velocity curve in turn provides one with a direct measurement
of the period of the orbit. From stellar evolution theory the mass of the star is often already
determined, and assuming mp M , Kepler’s third law takes the form
P2
=
4π2
a3
GM
, (1)
where a is the semi-major axis (radius, in the circular case) of the orbit, P is the period and G,
the gravitational constant.
For the planet, under the assumption of a circular orbit, it is also clear that
vp =
2πa
P
. (2)
With the stellar mass known, and the velocity of both planet and star measured, it is trivial to
calculate the mass of the orbiting body via momentum conservation.
6. Chapter 1 Theory and Literature Review 4
The above methodology is simple and has far reaching consequences, but the radial velocity
method also has a distinct disadvantage. It is rare to find a system that is exactly aligned with
the line of sight to Earth - the probability of this occurring is approximately;
Pr =
R∗
a
, (3)
which assumes both a circular orbit of semi-major axis (a) and that the radius of the orbiting
planet (Rp) is negligible compared to the radius of the star (R∗). The fact that this probability
is small (≈ 0.0046 for the Sun-Earth system) means that one is often observing the projection of
radial velocity, at some unknown angle (Φ), so the true velocity is not known. This uncertainty
in the projection angle means the radial velocity method can only place a minimum limit on the
mass that must be in orbit around the target star in order to cause the observed oscillations. It
cannot, however, tell the observer much of how the mass is distributed - i.e how many planets
there may be, or how far out they are orbiting. To clarify, the radial velocity method can
only constrain Msin(Φ). To find the true mass of exoplanets, one must combine radial velocity
measurements with transit photometry.
3.2 Transit Photometry
Transit photometry is a very simple concept. If a planet passes between the star and Earth
down the line of sight (transits) then it will block some light. The change in flux (transit depth)
depends on the observed relative angular sizes of the planet (θp) and star (θ∗),
∆F =
θp
θ∗
2
≡
Rp
R∗
2
. (4)
Clearly angular size is related to physical size, hence the equivalency holds. The condition for
transit requires that the orbit must be almost exactly edge-on; if this is true, the minimum mass
provided by radial velocity measurements is, in fact, the planet’s true mass.
The combination of these two methods can provide the observer with a wealth of information
about the system. From radial velocity the minimum mass can be constrained and from transit
the planet’s radius can be determined; these can obviously be combined to investigate the planet’s
density, which in turn gives clues as to its composition - see Figure 1, (F. Motalebi and Gillon,
2015). Further, when a planet transits, under certain circumstances, it is possible to isolate
the planet’s spectrum from that of the host star and so information about the atmospheric
abundances can also be measured.
7. Chapter 1 Theory and Literature Review 5
Figure 1: Real observed exoplanet density data showing various expected chemical composi-
tion limits. (F. Motalebi and Gillon, 2015)
3.3 Evidence of Alternative Formation Mechanisms
In addition, these measurements can provide an insight into the formation of planets. This was
well demonstrated in 2010, in a paper analysing the radial velocity curves of several ‘hot’ Jupiter
systems observed under the Wide Angle Search for Planets (WASP) consortium (Triaud et al.,
2010). Figure 2 shows the best fit line published for planet WASP-5b, which is sinusoidal,
implying a system with a planet on a stable, circular orbit. Figure 3 shows a zoomed-in view
around v = 0, the point at which the planet is transiting the host star’s meridian, which exhibits
a symmetrical ‘kink’.
This kink is known as the ’Rossiter–McLaughlin effect’, simultaneously observed in different
systems, by Rossiter and McLaughlin in 1924 (see Bibliography). To explain this effect one
must consider a star rotating on it’s axis aligned perpendicularly to the observer. One half of
the star rotates toward the observer, thus being slightly blue-shifted, while the other half rotates
away, being slightly red-shifted. This shift is detected as a uniform broadening in the star’s
spectral lines. As a planet transits it will sequentially block light from each half of the star,
altering the perceived redshift over time, which manifests as an anomaly in the radial velocity
curve. As the planet crosses the meridian the anomaly will switch from positive to negative, or
vice versa depending on stellar spin orientation. Figure 3 shows a highly symmetrical kink across
v = 0, which implies the transiting planet blocked each half of the star for the same length of
time. This suggests either that the planet’s orbit passes over the centre of the star as observed
from Earth or that the stellar spin is orientated exactly perpendicular to, and the orbit is exactly
aligned across, the line of sight. For example, the planet transits the star horizontally as viewed
from Earth, at some angle above or below the plane of the line of sight.
The radial velocity data published for WASP-5b is in good agreement with the idea of ‘hot
Jupiter’ formation via disc migration. However, in the same paper Triaud analysed the radial
velocity curve for WASP-15b, shown in figures 4 and 5, which is similar, however, the ‘kink’ is
extremely asymmetrical and inverted with respect to WASP-5b. This asymmetry across v = 0
indicates the planet spent more time transiting across one side of the star than the other; this
is consistent with the planet transiting on an orbit that is inclined to the line of sight. The
8. Chapter 1 Theory and Literature Review 6
Figure 2: The radial velocity curve
for WASP-5b. (Triaud et al., 2010)
Figure 3: The Rossiter – McLaugh-
lin effect clearly exhibited by WASP-
5b.(Triaud et al., 2010)
inversion of the kink also suggests that this system is in a retrograde orbit with respect to the
parent star’s spin.
Figure 4: The radial velocity curve
for WASP-15b. (Triaud et al., 2010)
Figure 5: A highly asymmetrical
kink around v = 0 exhibited by the
WASP-15b system. (Triaud et al.,
2010)
Unfortunately it is often not possible, at the distances involved, for an observer on Earth to
determine the exact orientation of the stellar spin axis (and by extension the stellar equatorial
plane) relative to the line of sight. This means that is not possible to identify specific systems
which exhibit ‘hot’ Jupiter planets on orbits that are inclined relative to the host star’s equator.
What can be determined, however, is the sky-projected spin-orbit angle (β) of the planet and as
before, the projection of the radial velocity down the line of sight, vsin(I) - where I is the stellar
spin axis inclination relative to the observer. To clarify, due to the challenges of uncoupling
vsin(I) measurements, it is difficult to know whether the target star is spinning slowly, inclined
toward the observer or spinning much faster, angled away from the observer, or indeed aligned
perpendicularly to the line of sight in which case vsin(I) = vtrue.
9. Chapter 1 Theory and Literature Review 7
Using the following geometrical argument taken from Fabrycky & Winn (2009);
cosi = cosIcosΦ + sinIsinΦcosβ, (5)
and assuming the stellar spin axis angle was distributed isotropically across a sample, Triaud
was able to run Monte-Carlo simulations and fit a distribution of likely orbital inclinations (i)
to the WASP data.
It was found that between 45 and 85% of the WASP planets studied were misaligned with
i > 30◦
. Their paper concluded with the following;
“Aligned systems are no longer the norm, radically altering our view on how these hot Jupiters
formed.”
- Triaud et al. (2010)
It now seems unlikely that disc migration can explain the significant fraction of ‘hot’ Jupiters ob-
served on inclined orbits. To account for these systems, one must consider alternative formation
mechanisms.
4 Kozai-Lidov Cycles
As mentioned in section 2, ‘hot’ Jupiters are expected to form beyond the frost line on stable,
circular orbits. However, it is possible for a planet to undergo Kozai-Lidov cycles driven by a
stellar, or sub-stellar, companion on a highly inclined orbit (Kozai, 1962; Lidov, 1962). The
mathematics presented by Kozai and Lidov is very involved, but demonstrates that the com-
ponent of the planet’s orbital angular momentum parallel (which will be denoted Lz ≡ |Lz|ˆL
here) to the angular momentum of the primary and companion’s angular momentum vector is
conserved under the hierarchical (aouter ainner) restricted three-body problem assumptions.
Utilising Delauney variables (Laskar, 2014; Naoz et al., 2013) it is possible to show,
Lz = (1 − e2) cos i, (6)
where the orbital inclination (i) is measured relative to the plane of the outer binary (star and
perturbing companion).
If Lz is conserved, then through equation (6) it is seen clearly that orbital inclination can be
exchanged for eccentricity. Kozai found that there is a critical angle (ic) such that if ic ≤ iinitial ≤
180 − ic, then the planet’s orbit cannot remain circular; both the eccentricity of the planet and
the mutual inclination between inner and outer binaries will exhibit periodic oscillations - this
is demonstrated in figure 6.
Kozai also found that the amplitude of these oscillations is independent of any physical variables
of the three-body system other than the initial inclination of inner binary relative to the outer.
In fact, the only property of these oscillations that does depend on more than the inclination is
the period of the oscillations, which is given by (Kiseleva et al., 1998);
10. Chapter 1 Theory and Literature Review 8
Figure 6: Example simulation showing Kozai-Lidov oscillations of an inner planet and outer
perturber. The blue and red lines represent two types of integration, which is unimportant
here. (Naoz et al., 2013).
τ =
2
3π
P2
outer
Pinner
M1 + M2 + M3
M3
(1 − e2
outer)
3
2 . (7)
The location of the frost line, beyond which the ‘hot’ Jupiters are thought to form, is highly
dependent on which assumptions one makes about the system, but it is certain that it is far
outwith the distance over which tidal forces can be considered plausible as a mechanism to sig-
nificantly shrink the planet’s orbit within a reasonable timescale - typically r ≤ 0.2au (Fabrycky
and Tremaine, 2007). However if the initial inclination between the planet and perturber is
sufficiently large that the eccentricity becomes significant, the pericentre of the orbit may fall
extremely close to the central star. This then allows the host star to exhibit tidal drag on the
planet, which may shrink and circularise its orbit within a timescale akin to that of the system’s
lifetime.
4.1 Gravitational tides
Consider a planet on an elliptical orbit around a host star. The planet is not entirely rigid,
and so will bulge along its orbital axis - the line drawn between the two bodies, perpendicular
to the transverse orbital velocity vector. This bulge is a result of both its own rotation and
the differential of the gravitational force across the planet. As can be seen by Kepler’s laws
the velocity of an elliptical orbit is not uniform - the planet travels faster at distance of closest
approach, or pericentre, than at distance of furthest approach, or apocentre. This means that
the bulge will lag with respect to the orbital axis, with varying degrees of severity at different
points along the orbit and so the planet experiences a torque. This tidal torque couples the
spin of the star to the orbit of the planet; it acts to equalise the spin (Ω1) and orbital angular
velocity via the transfer of angular momentum and dissipation of energy.
11. Chapter 1 Theory and Literature Review 9
In the case of a circular orbit this tidal torque results in spin-orbit synchronisation and alignment
of the spin axis with the orbit normal, where the final angular velocity is determined by the total
angular momentum left to the orbit. If the orbit is eccentric, however, as is the case for a Jupiter
planet undergoing Kozai-Lidov oscillations, the system exhibits ‘pseudosynchronisation’; i.e.,
the star spins at a rate slightly less than the maximum orbital angular velocity (experienced at
pericentre), where the tidal interaction is strongest (Hut, 1981). Circularisation of the planet’s
orbit and a change in obliquity, which is the angle between stellar spin and orbital angular
momentum vector, are regarded as the normal outcomes of tidal dissipation.
Neglecting other processes, such as stellar evolution and winds, and assuming there are no
internal processes such as convection taking place, tidal evolution will cease, at least theoretically,
when both bodies are in aligned, synchronous rotation on a circular orbit - i.e. when the planet
becomes tidally locked to the star.
When in tidal equilibrium, the total angular momentum L(Ω) is the sum of the spin Ls(Ω) and
orbital Lo(Ω) components, which are increasing and decreasing functions respectively. Hut thus
found that there is a critical value of total angular momentum, given by
Lc = 4IG(M1 + M2)
1
2
µ
3I
3
4
, (8)
where µ = M1M2/(M1 + M2), is the reduced mass and I = I1 + I2, is the sum of the moments
of intertia of bodies 1 and 2. If L(Ω) ≡ Ls(Ω) + Lo(Ω) > Lc then it is possible for the system
to reach tidal equilibrium. However, in systems of extreme mass ratio, such as those of a ‘hot’
Jupiter orbiting an approximately solar mass star, there may be no stable end point to tidal
evolution, and the planet can continue to migrate until it is destroyed (Hut, 1980; Ogilvie, 2014).
In reality, most stars exhibit magneto-centrifugal driven winds, which carries particles away
from the upper atmosphere and further drains the star’s angular momentum. The winds create
a constant imbalance in angular momentum between spin and orbit; thus the planet will continue
to migrate until it is engulfed by the host star (Lovelace et al., 2008). This process is constant
and means in reality there may be no stable end point to ‘hot’ Jupiter migration once the planet
is on a very small orbit. This is something that this project hopes to investigate.
4.2 Plausibility
Multiplicity in solar-type stars is quite common, with 30-40% believed to have stellar, or sub-
stellar companions within a distance of 20-20,000 au (Duquennoy and Mayor, 1991; Raghavan
et al., 2010). Simulations ran by Rice et al. (2015) showed that out of their initial population of
5,000 systems, ∼ 5% of planets were scattered onto closer orbits through Kozai-Lidov oscillations
driven by a stellar companion and survived inside 1au. This would imply that ∼ 1 − 2% of all
stellar systems have the ability to produce ‘hot’ Jupiters via this mechanism. It seems plausible
that with this many binary systems, companion driven Kozai-Lidov cycles could be relatively
frequent in the Galaxy and thus play a significant role in the formation of ‘hot’ Jupiters on
inclined orbits. However, it is clear that it is extremely unlikely that the Kozai-Lidov mecha-
nism is the only method of forming ‘hot’ Jupiters, as disc migration, without the need for any
companion, is still prominent.
12. Chapter 1 Theory and Literature Review 10
It is worth noting that if the companion was a brown dwarf, it would be difficult to detect via
direct imaging as they are extremely faint and cool. It should also be noted that the system
could have evolved further so that the companion may no longer even be present.
Figure 7, taken from Rice et al. (2015), shows a population of simulated (diamonds) and known
(circles) exoplanets, and confines the region of e − a space in which planets will form (along the
dashed line) before migrating to the ‘hot’ Jupiter region at a < 0.1au as e → 0.
Figure 7: Eccentricity vs semimajor axis for a population of simulated (diamonds) and known
(circles) exoplanets. (Rice et al., 2015).
And so, to clarify, this paper aims to develop an N-body code to the point where it can accurately
and reliably simulate the evolution of a 3-body system, consisting of an inner star-Jupiter binary
and an external sub-stellar companion, through Kozai-Lidov cycles that leads to the formation
of a ‘hot’ Jupiter.
13. Computational Methods
5 Description of an Orbit
Figure 8: The set up of two orbits crossing paths,
and the orbital parameters used to describe their
relative orientation in 3D space.
Two codes were inherited and used exten-
sively throughout this project, which will
be discussed below. The initial task was
to set up identical hierarchical three-body
systems in each to allow for comparison.
The set up is illustrated in figure 8 to the
right.
One can imagine the plane of reference to
be the x − y plane, with the inner most
binary (star and ‘hot’ Jupiter) situated at
the origin, co-rotating in this plane. The
position of the outer companion and rela-
tive orientation of its orbit can now be de-
scribed in general by seven parameters; the
time (t), semi-major axis (a), eccentricity
(e), inclination (i), argument of periapsis
(ω), longitude of the ascending node (Ω)
and the true anomaly (ν), all of which may
change with time.
The inclination is the angle between the
specific angular momentum vectors of the inner (h1) and outer (h2) orbits. The ascending node
is the point at which the outer companion passes upwards (defined to be the positive ˆz direction)
through the reference plane. This then defines the argument of periapsis, the angle between the
vector n = ˆz×h2 = (−hy, hx, 0), which points from origin to ascending node, and the eccentricity
vector, which points from origin towards the periapsis and has a magnitude equal to the orbit’s
eccentricity - this is also referred to as the Runge-Lenz vector (denoted e). The longitude of the
ascending node measures the angle between the reference direction (positive ˆx) and n. Finally,
the true anomaly is the angle that depicts precisely where the companion is in its orbit at any
time, and is measured relative to the Runge-Lenz vector.
It is worth emphasising that because ‘hot’ Jupiters form on circular, aligned orbits via accretion
as discussed previously, i = 0, meaning ω and Ω are undefined for the inner binary system. For
computation then, they are both set to zero by convention; this is equivalent to letting n point
down the positive x-axis.
11
14. Chapter 2 Computational Methods 12
6 The Secular Code
The secular code, also referred to here as 3DTidal, was written by Professor Ken Rice from the
School of Physics and Astronomy at the University of Edinburgh. Briefly, 3DTidal makes several
assumptions about the three body system; most importantly of these is that the ratio of semi-
major axes of the inner and outer binaries is small enough that it can be used as the expansion
parameter in perturbation analysis and that the ratio of orbital periods is small enough to justify
orbit averaging (Mardling and Lin, 2002). Specifically, it relies on knowing the specific orbital
angular momentum vector of the inner orbit as well as its Runge-Lenz vector, which are given
by;
h = r × ˙r, and e =
˙r × h
G(M1 + M2)
− ˆr, (9)
where r is the position vector and ˙r is the velocity of the planet measured relative to the primary
body. Since for an unperturbed, stable orbit the magnitudes of these vectors are constant at all
times, it is justified to assume that they vary only slowly under small external perturbations.
These assumptions save a lot on computing power, as there is no longer any need to know the
exact position of the planet; its slowly varying orbital parameters are enough to describe its
evolution over long timescales. This rids the description of the true anomaly.
A full description of this approach is given in Mardling and Lin (2002), see also Pollard (1966,
p.31), but the main equations to note are the time-averaged rate of change of these two vectors,
given by;
dh
dt
=
i
r × fi , (10)
de
dt
= i[2(fi · ˙r − (r · ˙r)fi − (fi · r)˙r]
G(M1 + M2)
. (11)
These expressions can be expanded up to octupole order and their averaged values are enough to
uniquely define the orbit. It is then possible to calculate the orbital parameters e, a, i, ω, and Ω
via;
e = |e|, (12)
a =
h2
G(M1 + M2)(1 − e2)
, (13)
i = arccos
h1 · h2
h1h2
, (14)
ω = arccos
n · e
ne
, (15)
Ω = arccos
nx
n
(if ny ≥ 0); Ω = 2π − arccos
nx
n
(if ny < 0). (16)
As previously described, with the slow varying assumptions these five parameters can completely
describe the evolution of the orbit of the planet over long timescales.
15. Chapter 2 Computational Methods 13
7 The N-body Code
The N-body code, referred to here also as hermite, is a 4th-order direct integration code written
by Sverre Aarseth, from the Institute of Astronomy at the University of Cambridge - for further
information on 4th-order integration methods see Kokubo et al. (1998). The details are beyond
the scope of this project; it will suffice to know that it is an explicit integrator that works in
Cartesians, relying on knowing x, y, z, vx, vy and vz of every particle at each time step.
7.1 Additions and Modifications
This project was very much developmental, and required several additions and modifications
to the hermite code, in order to try and be able to accurately simulate the evolution of ‘hot’
Jupiter systems. The main additions and the theory behind them are presented in the following
subsections.
7.1.1 Setting up Identical Systems
The first task was to create the ability to set up identical systems in both codes. For 3DTidal this
was easy, simply requiring the orbital parameters be specified. However as the hermite code is an
explicit integrator, it works in Cartesians and so it was necessary to write a FORTRAN code that
could convert all orbital parameters into Cartesian co-ordinates, and then rotate these initial
co-ordinates by the three orbital angles (i, ω, Ω) to create any completely general 3-dimensional
configuration.
Because ‘hot’ Jupiters are formed via accretion in the protostellar disc, the inner binary was
constrained to co-rotate in the x-y plane, with no possibility of initial inclination between them.
Putting the centre of mass at the origin meant their initial positions can be calculated simply
by;
x1 = −
µ
M1
a(1 − e) and x2 =
µ
M2
a(1 − e), where µ =
M1M2
(M1 + M2)
, (17)
Next the outer companion was added, and the new centre of mass was shifted to the origin, by
treating the inner orbit as a point particle of mass M12 = M1 + M2, since aout ainner . Then
a series of rotations for each i, ω, and Ω were performed using a matrix of the form Λ(θ), shown
below;
Λ(θ) =
cosθ + ux
2
(1 − cosθ) uxuy(1 − cosθ) − uzsinθ uxuz(1 − cosθ) + uysinθ
uyux(1 − cosθ) + uzsinθ cosθ + uy
2
(1 − cosθ) uyuz(1 − cosθ) − uxsinθ
uzux(1 − cosθ) − uysinθ uzuy(1 − cosθ) + uxsinθ cosθ + uz
2
(1 − cosθ)
where ˆu = uxˆux + uyˆuy + uz ˆuz is an arbitrary unit vector around which the rotations of θ take
place. After each rotation checks were put in place to make sure that the magnitude of all orbital
parameters, and physical variables (e.g. angular momentum, velocity) are conserved.
16. Chapter 2 Computational Methods 14
7.1.2 Tidal Force
As is hopefully clear, gravitational tidal forces form the basis in physics of this project. How-
ever, as inherited, the hermite code was not capable of simulating these forces and so this was
implemented using the following vector expression, taken from Mardling and Lin (2002), for the
tidal acceleration on body 2, as a result of body 1 given by
f1
T = −n
6k1
Q1
M2
M1
S1
a
5
a
r
8
[3(ˆr · ˙r)ˆr + (ˆr × ˙r − rΩ1) × ˆr]. (18)
As before, a is the semi-major axis of the orbit, r is the magnitude of the distance between star
and planet, and newly introduced is n; the mean motion of the orbit, with units of s−1
and
defined by
n =
G(M1 + M2)
a3
. (19)
These terms were allowed to vary with time and so were calculated each time step. The Love
number, denoted k1, is used to describe the deformation of a body in response to a tidal potential.
As described by Murray and Dermott (1998) Love numbers are “mostly used as a convenient
way of cloaking our ignorance of a body’s internal structure”. For the case of computation
then, k1 = 0.1, from Rice et al. (2015). The tidal quality factor (Q-value) of body 1, Q1, is a
dimensionless measure of the energy in a stable orbit divided by the amount of energy dissipated
per orbit by tides. Terrestial planets tend to have small Q-values near 100 while the giant gaseous
planets like ‘hot’ Jupiters and stars have Q-values nearer 106
(Barnes, 2011). Finally, S1 is the
radius of the star and was set to 0.005au, which is roughly the radius of the Sun. Written in this
form, it is easy to see that equation (18) represents an acceleration, as the first four bracketed
terms are dimensionless.
Expanded into Cartesian co-ordinates equation (18) has the form
f1
Tx ∝ −
1
r2
z[zvx − xvz − r2
Ωy] − y[xvy − yvx − r2
Ωz] + 3x[xvx + yvy + zvz] , (20)
and similarly for f1
Ty, f1
Tz over two cyclical permutations over x, y, z.
To be explicit, the components of this acceleration multiplied by the timestep (dt) were added
to the net acceleration array of body 2 at each timestep in the main hermite integrator nbint.f.
7.1.3 Stellar Spin
As explained in section 4.1, the star will apply a torque to the planet, simply given by
τ = r × f1
T . (21)
By definition, this represents a change in angular momentum of the orbit, and so through
conservation laws, an inverse change in angular momentum of the spin of body 1. Which way
17. Chapter 2 Computational Methods 15
the angular momentum is exchanged depends on the relative momenta of the planet’s orbit and
stellar spin. If the planet has a shorter orbital period than the star’s rotational period, it will
‘spin up’ the star by transferring its angular momentum, which will in turn cause the orbit to
shrink. Hence the spin evolution of the star is described by
˙Ω1 = −
µ12
I1
dL
dt
, (22)
where µ12 is the reduced mass of the inner binary. Computationally this was implemented by
updating the spin matrix Ω each time step via
Ωi = Ωi −
µ12
I1
τidt for i = x, y, z.
7.1.4 Stellar Wind
To investigate the effect of stellar winds on the migration of the ‘hot’ Jupiter on a small orbit
a simplistic magnetic braking form for the change in stellar wind was implemented (Rice et al.,
2012). The rate of change of the stellar spin, due to angular momentum loss through winds
( ˙Ω1,wind) in the unsaturated regime takes the form
˙Ω1,wind = −κwΩ3
1, (23)
and in the saturated regime, when Ω1 > ˜Ω;
˙Ω1,wind = −κwΩ2
1
˜Ω, (24)
where the saturation spin rate, ˜Ω, is set to 14Ω . The braking efficiency coefficient, κw, is
somewhat empirical, and is set so that the star spins down on a timescale similar to that of the
Sun - i.e. so that the young star’s spin rate changes by 5 - 10 days over a timescale of roughly
a few billion years.
Explicitly, κw is given by
κw =
2
3
B2 τc
τc, Ω
βmp
2kBTw
1
2 S2
1
k2M1
. (25)
Values taken from Collier Cameron and Jianke (1994); Rice et al. (2012) are: a magnetic flux
density, B = 3G, a convective turnover time of τc, = 8.9 × 105
s, an angular velocity of the
Sun of Ω = 4.0 × 10−6
rads−1
, a wind temperature of Tw ≈ 104
K and from Mestel and Spruit
(1987), β = 0.16. As before, M1 = M and S1 ≡ R ≈ 0.005AU. Finally, the effective radius of
gyration for star is taken to be k2
= 0.1 and the constants, mp and kB, are the proton mass and
Boltzmann factor respectively. In cgs units, κw/I1 ≈ 10−6
.
18. Simulations
8 Tests
The first, most basic test to do was to check that the inner binary was being set up in the desired
configuration and that the orbit was stable over long timescales. This was done for randomly
selected parameters, including those stated above and it was checked that the orbit remained
stable for at least 100 million years using the secular code.
Less trivial was creating the ability to set up a stable three body system in any general 3-
dimensional configuration.
Due to the fact that the three body systems experimented with throughout the project were
chosen essentially at random, it was necessary to impose two stability criteria (Naoz et al., 2013;
Rice, 2015) on the initial conditions generator code;
a
a3
e3
1 − e2
3
< 0.1, (26)
and
a3
a
> 2.8 1 +
M3
M1 + M2
2/5
(1 + e3)2/5
(1 − e3)6/5
1 −
0.3i
180
. (27)
The first of these equations ensures the octupole and quadrupole terms dominate, which is
necessary for the secular code to run accurately, while the second equation ensures the triple
system is long-term stable.
Figures 9 and 10 show two randomly selected stable three body systems that meet these criteria.
The models used, though essentially unimportant, are given in caption.
Figure 9: A stable orbit of the brown
dwarf companion (brown) orbiting the in-
ner binary (green and orange) with pa-
rameters a = 0.05au, e = 0.2, a3 =
20.0au, e3 = 0.8, i = 85.0◦, ω = 28.0◦
and Ω = 10.0◦
Figure 10: Another stable orbit of a
Jupiter mass companion (brown) orbiting
the inner binary (green and orange) with
parameters a = 0.05au, e = 0.2, a3 =
10.0au, e3 = 0.8, i = 185.0◦, ω = 75.0◦
and Ω = 65.0◦
16
19. Chapter 3 Simulations 17
The ratio of orbits in figure 9 is a/a3 = 0.0025 so it makes sense at this resolution the inner
orbit appear as just a couple of centralised data points. To better illustrate the inner orbit, the
companion was brought closer and placed at a high inclination in figure 10.
9 Results
9.1 Kozai-Lidov Simulation
There is a useful test, found in Naoz et al. (2013), and repeated by Rice (2015) using the
previously mentioned secular code, that was previously shown in figure 6, section 4. This test
was used as a reference to check that both hermite and 3DTidal were simulating the Kozai-Lidov
cycles correctly. Here the tidal and spin evolution, and wind terms are ignored. The system
consists of the previously mentioned model: M1 = 1M , M2 = 1MJ , M3 = 40MJ ; with an inner
orbit with a = 6 au, e = 0.001 and an outer orbit described by a3 = 100 au, e3 = 0.6, i = 65◦
and ω = 45◦
.
Figure 11: Inclination vs time plot for
the N-body (green) and secular (red) sim-
ulation, showing an oscillating inclination
with a ‘flip’ at i = 90◦ around t = 4×106.
Figure 12: A log10(1 − e) vs t plot
for the N-body (green) and secular (red)
simulation showing how the third body
drives oscillations in the planet’s eccen-
tricity.
As can be seen both hermite and 3DTidal are in very good agreement with figure 6, section
4, with the ‘flip’ around i = 90◦
happening just before t = 4 × 106
years in all three cases.
Originally, this simulation was ran for a full 25 × 106
years as Naoz did, however toward the end
of the project as the code became vastly more complex, N-body simulations on this timescale
were found to be no longer possible with the time constraints faced, and so 10 × 106
years was
deemed a suitable length of time over which to observe numerical agreement. This was in part
due to only having a standard workstation to use but mainly because of the need to run many
simulations at once. It was found that running hermite simulations for 10-20 million years was
the longest that was realistically possible and that this was, in most cases, long enough for the
system to demonstrate the desired or expected behavior.
20. Chapter 3 Simulations 18
The vast range over which the eccentricity may oscillate through Kozai-Lidov cycles is shown
well in figure 12; recall as log10(1 − e) → −∞, then e → 1! Over a mere 4 million years, the
eccentricity has been driven from e0 = 0.001 to e ≈ 1. This provides further support to the
idea that these cycles could cause the pericentre of a ‘hot Jupiters’ orbit to fall in a region close
enough to the host star that the tidal drag becomes significant.
However, as is seen more clearly by Naoz’s figure (fig. 6), these oscillations can, and will, continue
for many millions of years if the tidal forces are neglected. This is because gravity is a central
force, so no torque is being applied to the planet in its orbit and no energy is being lost so
the orbit remains in this oscillatory state. As an aside, in reality, due to numerical errors in
computing there are extremely small amounts of energy lost per timestep but the accuracy of
hermite far outreaches the timescale over which these would be noticeable in these simulations.
9.2 Tidal Force
From Naoz’s model it is clear that a third body can drive the pericentre of a ‘hot’ Jupiter’s orbit
extremely close to the host star. However, as mentioned in the previous section, Kozai-Lidov
oscillations will continue indefinitely if the tidal forces are neglected. So using the secular code,
the tidal effects were turned on and a simulation was ran for a billion years to see if it was
possible for a planet, starting out at a large radius where ‘hot’ Jupiters are expected to form,
would become tidally trapped and begin to migrate.
Figure 13: A figure showing the evolution of the semi-major axis (grey) and pericentre (blue)
against time using initial conditions described in text.
The initial conditions, taken from Rice (2015) consist of a 1.1M central star, with a 7.8MJ
planet in orbit at a = 5au initially with a modest eccentricity of e = 0.1. The companion is
large, with M3 = 1.1M but situated far away on an eccentric orbit described by a3 = 1, 000au
and e3 = 0.5. The tidal love numbers are ks = 0.028, kp = 0.51 with the tidal dissipation quality
factors of Qs = 5.35×107
and Qp = 5.88×105
for star and planet respectively. Finally the outer
companion is inclined at i = 85.6◦
to the inner orbit and the initial spin periods of bodies 1 and
2 are 20 days and 10 hours respectively. Clearly, the secular code is far more advanced in its
21. Chapter 3 Simulations 19
treatment of these systems than the N-body code, which even after the extensive development
of this project is still unable to consider the love number, spin and Q-value of the planet.
The secular code is very sophisticated, and produces a very promising result. Figure 13 shows
the evolution of the semi-major axis and periastron (a(1 − e)) with time for this simulation.
Specifically, it shows the pericentre oscillating close to the star, in the region where tidal forces
become significant. This causes a tidal torque on the planet that acts to shrink and circularise
the orbit, thus the semi-major axis tends to < 0.1au; the region in which ‘hot’ Jupiter planets
are found observationally. The reason that the semi-major axis remains constant right up to
around t = 1 × 108
years is because energy is conserved during Kozai-Lidov interaction. So,
since orbital energy goes as
E = −
GMtot
2a
, (28)
then the semi-major axis remains constant until the tidal forces become significant. In the regime
where tidal forces begin to dominate, energy begins to be dissipated by frictional forces created
due to the bulge lag and thus the semi-major axis decays. The periastron, however, depends on
the specific orbital angular momentum, given by,
h = G(M1 + M2)a(1 − e2), (29)
which as explained in section 6 is one of the vectors that 3DTidal relies on knowing at each
timestep, and so this oscillates rather significantly throughout the simulation. This simula-
tion further supports the theory that Kozai-Lidov oscillations can drive a large gaseous planet,
forming out at r ≈ RJup into a region of e − a space where ‘hot’ Jupiters are expected.
Now that the full ‘hot’ Jupiter formation mechanism through Kozai-Lidov cycles has been demon-
strated, this project set out to develop the hermite code to a point where a comparison could be
made between the N-body and secular methods. Although this turned out not to be an achiev-
able goal, the main developments and results will be presented forthwith. From here, this paper
considers just the inner binary to investigate the effect of tides on the resulting migration in this
regime. For computational purposes this meant setting M3 = 1 × 10−10
M and a3 = 10, 000
au; ensuring the effect of the third body was negligible.
Throughout this section an ideal model will be used, with slight variations depending on the
simulation. The basic set up is an inner binary consisting of a solar mass star (M1 = 1M ) and
a Jupiter mass planet (M2 = 1MJ ) with semi-major axis and eccentricity defined by a and e
respectively.
Starting with the most simplistic system, the effect of pure gravitational tidal interactions on
a ‘hot’ Jupiter following a circular orbit was explored. Below, figure 14 shows the evolution in
time of the semi-major axis and period of the planet due to tides for the circular (e = 0) case,
while spin and wind effects are ignored.
It is clear to see that the tides have caused a drag on the ‘hot’ Jupiter which has caused it to
migrate inward as expected. This supports the idea that if the ‘hot’ Jupiter was perturbed,
through Kozai-Lidov interactions, onto an orbit that brought it to within r ≤ 0.05au, then tidal
forces could play an important role in the resulting migration.
22. Chapter 3 Simulations 20
Figure 14: Semi-major axis and period of the ‘hot’ Jupiter plotted against time, showing a
steadily and slowly shrinking orbit as a result of tidal drag.
However, from figure 14 it is obvious that the timescale of this evolution is very long, with a
fractional change in semi-major axis of just ∆a ≡ (ainitial −afinal)/ainitial = 0.024 over 10 million
years; thus it is necessary to present a short aside.
9.2.1 Q-Values
Briefly mentioned in section 7.1.2 was the Q-value of the star, which is a dimensionless factor
that describes a bodies’ response to tidal forces. Specifically, it is defined as the ratio of the
energy in a stable orbit to the energy dissipated by tides per period1
, and is usually taken to be
around 106
for main sequence stars. Therefore, it should be clear that changing the Q-value of
a simulation will have no ‘physical’ effect on the system other than to increase or decrease the
rate of energy loss per orbit, which will, by extension, alter the evolution timescale of the system
and thus the real simulation time. N-body simulation is very computationally expensive, and
with only workstation computers available, the decision was made to alter the Q-value to make
the simulations more computationally accessible. And so, the timescales presented here are all
unrealistically fast, but all simulation of the underlying physical interactions taking place are
still accurate.
1
Most easily understood mathematically by;
Q =
2πE0
−(dE/dt)dt
, (Wu, 2005)
23. Chapter 3 Simulations 21
For clarity, consider the following equations for rate of change of a and e of the inner binary,
which neglect the effect of the planet’s Q value, taken from Jackson et al. (2009);
1
a
da
dt
= −
9
2
G
M1
1/2
S5
1M2
Q1
1 +
57
4
e2
a−13/2
, (30)
and
1
e
de
dt
= −
225
16
G
M1
1/2
S5
1M2
Q1
a−13/2
. (31)
From here it is easy to see that for a fixed system, the rate of change of the semi-major axis
and eccentricity are proportional to powers of 1/Q1. Thus, by decreasing Q1, one increases the
rate of change of a and e, leading to a faster evolution but leaving the systems’ other physical
descriptions unaltered.
9.3 Tides with Reduced Q-Values
To demonstrate this, the previous simulation from section 9.2 is repeated with a reduced Q-
value, in order to further the evolution of the system on a reasonable computational timescale.
Figure 15 shows the effect that reducing the Q-value from 106
to 103
can have on the evolution
timescale of the semi-major axis for the circular system. In the Q1 = 106
case the fractional
change in semi-major axis was ∆a = 0.024, and in the Q1 = 103
case, ∆a = 0.08 - which is
almost three times as big over the same timescale.
Figure 15: Semi-major axis against time for the e = 0 case with Q1 = 106 (purple) and
Q1 = 103 (blue), showing the effect of purely reducing the Q value.
24. Chapter 3 Simulations 22
However, one needs to take care as there is a lower limit to Q-value stability, under which
the equations of motion will cease to work correctly. This also places a limit on the minimum
simulation time that is required to accurately map the evolution of these systems. Figure 16
shows the semi-major axis of the same circular system, with the Q-value reduced further to just
Q1 = 100. The evolution is extremely rapid, with a large fractional change of ∆a = 0.2 over
just 6.5 × 106
million years before a near-instantaneous cut off leading to planetary destruction.
Clearly, the observed oscillation at t ≈ 8 × 106
years is unphysical as there should be no source
of angular momentum for the planet yet included in hermite and is likely a result of the Q-value
being too small for the equations to remain stable. Again, it is worth emphasising that this
timescale is very unrealistic due to the altered Q-value.
Figure 16: Semi-major axis against time for the previous circular case, showing an unrealistic
evolution in the semi-major axis due to numerical instability caused by the Q-value.
So, the Q-value was set to Q = 104
and another simulation was ran, with the planet slightly
closer, at a = 0.04au in the hope of seeing viable tidal destruction. It was not clear when
beginning these tidal simulations what the end point would be, but it was hypothesised that if
the spin of the star was neglected there would be no stable end point to tidal migration. This is
because if the spin of the star cannot gain angular momentum and if the period of the planet is
initially shorter than the period of the star’s rotation, then their angular momenta will always be
imbalanced, as the planet’s angular frequency will always increase away from the star’s constant
value. This would cause the planet’s orbit to shrink indefinitely and spiral into the star, ending
in destruction.
This effect is seen quite clearly in figure 17; the stellar spin (orange) has been set to a constant
value of T1,s ≈ 10days (corresponding to an angular frequency of Ω1 ≈ 36 rad timestep−1
) so
as the planet continually transfers its orbital angular momentum, its angular frequency (green)
increases further from the spin value. Clearly as the spin value is constant, this imbalance is
continually amplified and equilibrium can never be reached.
25. Chapter 3 Simulations 23
Figure 17: Angular frequency of the stellar spin and planet’s orbit vs time for the circular
case presented in text, with a reduced Q value of Q = 104.
Figure 18 shows the evolution in time of the semi-major (grey) and period (blue) of the ‘hot’
Jupiter in this regime. This shows exactly what is expected, with both lines tracing an ever
sharper gradient, as the planet spirals further in and is eventually engulfed by the host star. In
actual fact, hermite will ‘jam’ long before this happens, as the forces and accelerations tend to
infinite as r → 0 so it was found that getting data in the region a < 0.01 was very difficult.
Figure 18: Semi-major axis, period vs time for the circular case presented in text, with a
reduced Q value of Q = 104.
To further emphasise this migration, figure 19 shows the x − y positions of the Jupiter and
star, given in au. Because this is an N-body simulation, and data points were binned every set
26. Chapter 3 Simulations 24
number of timesteps, one would expect there to be many more points on a smaller orbit than
on a larger one and thus the inside of figure 29 to be vastly more populated than the outside.
The reason this isn’t the case here is because the planet spent many hundreds of thousands of
years more on the larger, outer orbits than the inner ones as the migration became very rapid
at a ≈ 0.03au. Again, because hermite will jam as the planets approach collision distance scales
there is a complete lack of data points at r ≤ 0.01au, but the eventual fate of the ‘hot’ Jupiter
should be abundantly clear.
Figure 19: Positions in the x − y plane, given in au, for the circular case presented in text,
with a reduced Q value of Q = 104.
9.4 Spin
The hope here was that by adding in a mechanism to allow the transfer of angular momentum
to the star, the angular frequency of the spin would eventually increase enough to balance that
of the planet’s orbit, which is also increasing and spin-orbit equilibrium could be reached. This
may seem counter-intuitive, but the reader should be reminded that the relation between stellar
spin period (T1,s) and its angular momentum is the inverse to that of the planet’s orbital period
(T2) and its angular momentum. When the star’s period decreases, it is rotating faster therefore
it has gained angular momentum - e.g. it has been spun up by the planet - but when the planet’s
orbital period has decreased, its orbit has shrunk, thus it has lost angular momentum. Further,
the below graph shows orbital angular frequency, which scales with the inverse of the period.
Figures 20 and 21 show the evolution of the same circular system, but with the spin evolution of
the star enabled - being updated as described by equation 22. The results are encouraging, with
the star and planet becoming tidally locked2
in a reasonable, but probably over-estimated, time
2
‘Tidally locked’ here is a bit of a misnomer as hermite is actually incapable of considering the planet’s spin,
but refers to just the matching of spin-orbit angular momenta and by extension their angular frequency.
27. Chapter 3 Simulations 25
of 4×106
years, again due to the reduced Q-value. As can be seen clearly in figure 20, spin-orbit
equilibrium is reached when Ω1,s = Ω2,o ≈ 115 rad timestep−1
. When the angular frequencies
match, there is no longer a transfer of momentum between the bodies meaning that no torque
is experienced by the planet and hence migration ceases and a corresponding stabilisation is
observed in the semi-major axis (fig. 21).
Figure 20: Figure showing the angular frequency against time of the star’s spin and planet’s
orbit, matching just before t = 4 × 106 years.
The following figure further emphasises the end point of migration, showing a steady decline
to a ≈ 0.04au before levelling out smoothly and remaining very nearly constant for another 8
million years.
Figure 21: Figure showing the semi-major axis vs time, stabilising at a ≈ 0.04au when the
spin-orbit angular momenta match at t = 4 × 106 years.
28. Chapter 3 Simulations 26
9.4.1 Spin and an Eccentric Orbit
Up until this point all these simulations have been carried out on a circular (e = 0) system. This
is because ‘Hot’ Jupiters are observed on very circular orbits (e 0.2), which suggests that
either the high eccentricity caused by Kozai-Lidov cycles is damped during migration or that
protoplanets exhibiting eccentric orbits are very quickly tidally destroyed. It would make sense
that if Kozai-Lidov cycles drive inner binaries into highly eccentric orbits, then tidal damping
of eccentricity is the most likely scenario; so this regime was investigated next. Figures 22 and
23 show the evolution of the semi-major axis, period and eccentricity of a ‘hot’ Jupiter system
with the same parameters as presented previously, except with a fairly modest eccentricity of
e = 0.2.
Figure 22: Semi-major axis and period of the ‘hot’ Jupiter for the e = 0.2 case, showing
rapid tidal destruction.
Clearly, these figures show rapid destruction of the planet within 800,000 years as a → 0. This
is a limitation of the code, and should not be interpreted as a realistic evolution of this regime.
It is likely that because the only momenta being considered here are in the star’s spin and the
planet’s orbit that there is no ability to dampen the eccentricity of the orbit. This means that as
the ‘hot’ Jupiter migrates, its eccentricity is oscillating wildly (fig. 23), and hence its pericentre
falls progressively closer to the star until eventually it is tidally destroyed at r S1 = 0.005au.
Note, as before, the N-body code will jam before this destruction criteria is actually reached,
hence why the lower limit of figure 22 is a = 0.01.
29. Chapter 3 Simulations 27
Figure 23: Eccentricity evolution for the e = 0.2 case, showing vast oscillations and tidal
destruction within 800,000 years.
To rectify this, both the spin of the planet on its axis and the tidal ‘back effect’ of the planet
on the star would need to be considered. This was originally a goal of the project, but was
unfortunately not attained due to time constraints and coding errors. However, it is included
here for completeness and a full discussion of this will be covered in section 10 below.
9.5 Stellar Wind
As discussed in section 4.1, stellar winds carry material away along magnetic field lines, which
dissipates angular momentum and causes the star to naturally spin down over its lifetime. This
wind will increase the torque acting on nearby planets, by increasing the gradient dL/dt, resulting
in further migration and likely eventual destruction of any ‘hot’ Jupiters. However, if the planet
is orbiting far enough out, say at the radius of Jupiter (r ≈ 5.0au), then one would expect the
winds to have very little effect on the period of the orbit over the course of even the stars lifetime
- if it did, then Jupiter would not be in its observed orbit in the Solar System. This suggests
that the natural first test to check the winds are operating correctly would be to ensure the
period of a planet orbiting at this radius remains (relatively) unchanged, while the stellar spin
rate decreases over a reasonable timescale - ‘reasonable’ here being dictated by observation of
main sequence stars.
30. Chapter 3 Simulations 28
Figure 24: The spin rate of the star, spinning itself down due to magnetic wind braking on
an over-estimated timescale.
Figure 24 shows the spin period of the star (T1,s) increasing from T1,s = 10.14 to ≈ 10.24 in
days, over a period of 10 million years. This corresponds to roughly two hours, which again
is a large overestimate; it is expected from stellar evolution theory, and observations of the
sun, that a solar mass protostar will spin down from an initial spin period of around 10 days
to approximately 25 days (≈ T ) within roughly 5 billion years. The wind braking efficiency
coefficient, κw, is not yet very well known by the scientific community and variation in its value
by a few orders of magnitude could be justified to make the simulation more realistic and the
computation more accessible. However, due to already existing ambiguity in the timescales due
to the altered Q-value the efficiency of the winds here are a far smaller source of uncertainty.
What is most important is that the semi-major axis and period of the planet’s orbit (fig. 25)
remains approximately constant, while the star is spinning itself down (fig. 24); thus showing that
the winds are acting correctly as an extra source of depletion of the spin momentum. Although
the effect of the tidal force is essentially negligible over these distances, there is still a small but
finite torque being applied and so the orbit does migrate inward a small fraction. Again, over
these timescales, even this extremely small migration (∆a ≈ 0.00011 au) is an over-estimation
due to the decreased Q-value and the over-efficient winds.
31. Chapter 3 Simulations 29
Figure 25: Semi-major axis and period of
the planet’s orbit, showing a nearly constant
orbit, with only a minuscule fractional change
of (ainitial − afinal)/ainitial = 0.0000278.
Figure 26: x vs y positions, given in
au, to further illustrate that this is a
large, stable circular orbit.
9.6 Winds and the End Point of Migration
Now that the tidal force, stellar spin and winds have been implemented and tested to satisfaction
the combined effects on the end point of ‘hot’ Jupiter migration can be investigated. For the
following simulation, the ‘hot’ Jupiter was placed at r = 0.04au, which is within the typical radius
where tidal forces become significant. It is assumed that had the Jupiter already undergone
Kozai-Lidov cycles it would be on a circular orbit so e = 0 here, and all other parameters remain
as previous. Here it was expected that at these small distances the stellar wind would continually
strip the star of its angular momentum, thus creating a constant imbalance in angular momenta
and resulting in the planet’s eventual destruction.
The results are encouraging. The specific scalar angular momentum stored in the inner orbit is
given by
h = G(M1 + M2)a(1 − e2),
which makes it clear that
dh
dt
∝
da
dt
, (32)
when one remembers that the hermite code is incapable of damping the eccentricity through tidal
interactions. From this then it is expected that as a r → 0, the tidal force f1
T → ∞, so the tidal
torque τ ≡ r × f1
T → ∞ and thus dL/dt → −∞ from the definition of this torque. Ultimately,
this means that the gradient of a also tends to negative infinity. Figure 27 shows exactly this
behaviour; both the period and semi-major axis of the orbit declining first gradually then much
more rapidly as a tends to zero over a reasonable, but as ever over-estimated, timescale.
32. Chapter 3 Simulations 30
Figure 27: Semi-major axis and period of the ‘hot’ Jupiter against time, showing a steady
decrease until a rapid migration around a ≈ 0.03au leading to planetary destruction.
The spin rate of the star throughout the 8 million year simulation is shown in figure 28. Initially
the star was given a spin period of 10 days, but over the course of the evolution the planet has
spun up the star (period goes down) by a significant fraction. As the evolution begins, the star’s
wind is spinning it down slowly which battles the effect of the planet’s migration, acting to spin
it up. However, from the above argument, as the rate of change of angular momentum from the
orbit to the spin (dL/dt) gets faster and faster, the effect of the wind becomes inconsequential
and the star’s spin period rapidly increases as the planet spirals in until it is engulfed. Once the
planet is destroyed one would expect the star to settle on its new spin rate and begin to slow
down again as the winds become the only term affecting the spin period. The reason this is not
seen is again due to the ‘jamming’ effect of N-body simulations when two particles approach
r → 0 and the gravitational accelerations tend toward infinity.
Figure 28: Spin period of the star vs time, showing the star getting spun up massively as the
planet loses (almost) all it’s orbital angular momentum to the star’s spin.
33. Chapter 3 Simulations 31
To further emphasise that this is indeed a migration, a plot of the x vs y positions in the centre of
mass frame of the ‘hot’ Jupiter and host star is shown in figure 29. The rapidity of the migration
is again emphasised by the lack of data points in the central region of this plot.
Figure 29: A plot of x vs y, given in au to further emphasise that this is a rapid migration.
As was explained in section 9.2, the reason fewer points are seen on the smaller orbits is because
of both the rapidity of the migration and the jamming of the N-body code.
9.7 A Boundary Region
With the stellar winds successfully implemented, it is interesting to probe the boundary region
between where winds are the dominant effect leading to a net loss of angular momentum in the
star’s spin and where tidal dissipation causes migration of the planet leading to the star being
rapidly spun back up. This simulation was ran with the previous starting parameters, but with a
slightly more emphasised wind - e.g. a higher value of κw. Consider figure 30, it looks similar to
those presented previous, showing the semi-major axis and period of the ‘hot’ Jupiter declining
as it undergoes a migration inward from a = 0.05aau.
34. Chapter 3 Simulations 32
Figure 30: The semi-major axis (purple) and period (blue) showing migration of the ‘hot’
Jupiter.
However, if one considers also figure 31 a new picture is formed. The period has a very prominent
turning point, at roughly 6 million years.
Figure 31: A figure showing the boundary between where the winds cause a net loss of
angular momentum and the planet’s migration causes a net increase in angular momentum in
the star’s spin.
This is the point at which the winds, acting to slow the star down lose dominance to the migration
of the orbit which is acting to spin it up as before. From here the end point is the same as in
section 9.6; endless migration due to winds leading to planetary destruction. Unfortunately this
end point cannot be shown due to a problem with the Q-value stability. Due to time constraints
faced at the end of the project the Q-value here was reduced to Q = 103
which means that the
simulation became unphysical as t → 25 × 106
years.
35. Discussion
10 Analysis
This paper has presented an alternative theory of formation for ‘hot’ Jupiters, one that can
deal with the findings of papers like Triaud et al. (2010), and is backed up by computational
simulations that utilise both secular and N-body methods.
The project was very much developmental, and although not all the initial goals were accom-
plished, a lot of progress has been made. Full testing and implementation of a tidal acceleration
on body 2 from body 1, a method to update the spin of body 1 and a simple magnetic braking
form of stellar wind has been completed. This has made the N-body code vastly more useful in
simulating these systems.
The simulations have been in good agreement with the hypotheses throughout, which shows that
although these three body systems exist unimaginably far away, there is still a fundamentally
good grasp on the physics which governs their interaction.
This project has shown that over the period of a billion years, it is possible for a giant gaseous
planet which has formed at large radii to become tidally trapped through three-body Kozai-
Lidov cycles, and for its orbit to shrink and circularise to the region of e − a space that ‘hot’
Jupiters are known to occupy. From this region at a ≈ 0.05au, it has also been shown through
N-body simulation that the tidal forces play a prominent, and probably dominant, role in the
subsequent migration of the planet. Under the assumption of no wind, the stabilising effects of
spin interactions was demonstrated before the winds were turned on and their destructive effect
was fully revealed.
As is always the case, this project had its limitations, which will be discussed briefly here. Simu-
lation time turned out to be the main constraint throughout this project, with early simulations
taking up to 5 days to complete. This meant that if mistakes were made (and yes, many were)
that often they would not be discovered until several days had been wasted. If there had been
more time or a faster means of equally accurate N-body simulation this would perhaps have
helped the project develop further - it was possible to turn down the energy tolerance, lending
itself to much faster simulation but resulting in an unacceptable loss of accuracy. Any project
can be made infinitely better with infinitely more time, but if it had been available it would have
been beneficial to probe more different systems to investigate the generality of the theory and
perhaps even simulate the expected evolution of some directly-imaged exoplanet systems, such
as HR8799.
Moving away from computational issues, some of the underlying physics of this project is not as
accurately understood as could be hoped. For example, the literature Q-value of a main sequence
star is often given as “a few times 106
.” Similarly, the tidal love numbers are not particularly
well known and it is easy to justify varying the value of the magnetic braking efficiency coefficient
by several orders of magnitude! Remember, this coefficient is applied every time step and so
varying it by an order of magnitude has a huge effect on the timescales of the evolution of the
33
36. Chapter 4Discussion 34
system. If these values were better constrained then this project could place better limits on the
expected timescale of migration for these ‘hot’ Jupiter systems.
Despite these limitations the project has taken a basic N-body code and developed it to a point
where it is capable of accurately simulating the key aspects of these migrating systems. However,
as was alluded too in section 9.4.1, the code still has some severe limitations that need addressed.
11 Further Work
The ideal end goal here would be to develop hermite to the point where it is capable of a full
tidal simulation like that presented from the secular code in section 9.2, figure 13. To reach this
goal could easily take another year of work, but there are a few, more basic, abilities which could
be developed to vastly improve the quality of the N-body simulation.
The most obvious, which was mentioned in section 9.4.1, would be to include a spin of the
planet and then the tidal ‘back effect’ of the planet on the star. This is exactly as it sounds;
the spin of the planet causes a tidal torque, which acts against the migration, to be experienced
by the star. Implementing this would be the same as it was when adding the effect of body 1
on 2, and so, hopefully, would not be too difficult to construct. This may, or may not, allow
for hermite to exhibit tidal circularisation, which is by far the biggest piece of the puzzle still
missing from this paper. Until this work is completed, this project is restricted to assuming
that tidal circularisation has already occurred during the Kozai-Lidov phase of the evolution
and starting from e = 0.
37. Conclusions
In summary, this project has developed an N-body code to the point where it can accurately
simulate the evolution of three-body systems undergoing gravitational interaction, Kozai-Lidov
cycles and circular tidal migration. Although the ideal end goal of a full tidal simulation, and
comparison with the secular code was not achieved, it should still be considered a success as all of
the simulations agreed with pre-existing hypotheses, which is a credit to both our understanding
of the underlying physical processes of these systems and our ability to successfully simulate
millions of years on real timescales as short as a couple of days.
Acknowledgments
The author would primarily like to thank this project’s supervisor, Professor Ken Rice, for his
guidance, infinite wisdom and constant open door policy.
Also deserving of credit are fellow MPhys students Balint Borgulya and Andrew Mackie, for
always being willing to lend their time to help debug endless lines of dysfunctional code, and
Keith Hardie for his willingness to act as primary grammar checker throughout the writing of
this thesis.
35
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