Download Formation and stability of planetary orbits in binary systems

Formation and stability of planetary orbits
in binary systems
A case study of the IM Pegasi binary
Martin Schwesinger
August 2013
Bachelor’s thesis
!!!
!!!
!!!
Supervisor
!!!
Prof. Dr. Svetlana Berdyugina
Abstract
Observations done by Berdyugina et al. indicate that a planet is residing in the close
binary system IM Pegasi. The spectroscopic binary star was the guide star for the
Gravity Probe B experiment. In this thesis, I give a basic overview of the stable domains
for s-type (inner) orbits, both prograde and retrograde, for the close binary system IM
Pegasi, utilizing a numerical code I developed based on step-by-step integration of the
Newtonian equations of motions. The research previously done on the stability of stype orbits in binary systems is reproduced. I also simulate a possible scenario for
the creation of the proposed planet, based on the ejection model as suggested by Prof.
Dr. Berdyugina. She proposes that the planet has been formed from material ejected
by the primary red giant star IM Pegasi A; material which was then trapped in the
observed orbit. Finally, I expand the calculations regarding the stability of s-type orbits
by accounting for the quadrupolar distortion of the primary star resulting from the tidal
forces exerted on it by its companion.
1
Contents
Contents
1 Introduction
1.1 Observational evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Theory
2.1 Movement equations . . . . . . . . . . .
2.2 Keplerian orbits . . . . . . . . . . . . . .
2.3 Restricted 3-body problem . . . . . . . .
2.3.1 Jacobi constant . . . . . . . . . .
2.3.2 Lagrange points . . . . . . . . .
2.4 Numerical methods . . . . . . . . . . . .
2.4.1 Taylor series expansion . . . . .
2.4.2 Runge-Kutta integration scheme
2.4.3 Adjustment of the step-size . . .
2.4.4 Floating point numbers . . . . .
2.4.5 Local step sizes . . . . . . . . . .
2.5 Multipole expansion . . . . . . . . . . .
2.5.1 Tidal deformations . . . . . . . .
2.5.2 Calculating the deformation . . .
2.6 Computation method . . . . . . . . . . .
2.6.1 Accuracy test . . . . . . . . . . .
3
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20
3 Results
3.1 System parameters . . . . . . . . . . . . . . . . . . . . . .
3.2 Comparison with previous research . . . . . . . . . . . . .
3.3 Prograde orbits . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Metastable orbits . . . . . . . . . . . . . . . . . . .
3.3.2 Prograde orbits: Conclusion . . . . . . . . . . . . .
3.4 Retrograde orbits . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Eccentric retrograde orbits . . . . . . . . . . . . .
3.4.2 Orbits in an eccentric binary system . . . . . . . .
3.5 Origin of the system . . . . . . . . . . . . . . . . . . . . .
3.5.1 Model I: Particle ejection from the primary star .
3.5.2 Model II: Particle stream from outside the system
3.5.3 Model III: Particle disc around the primary star .
3.5.4 Model IV: Reorientation of the orbital axis . . . .
3.5.5 Origin of the system: Conclusion . . . . . . . . . .
3.6 Accounting for quadrupolar distortion . . . . . . . . . . .
3.6.1 Calculating the quadrupole moment . . . . . . . .
3.6.2 Quadrupolar distortion: Orbital stability . . . . .
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4 Conclusion
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44
5 Addendum
45
5.1 Quadrupolar distortion: Retrospectively falsified results . . . . . . . . . . 45
5.2 Numerical code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 References
50
2
1 Introduction
1 Introduction
IM Pegasi is a close binary star system, located in the constellation Pegasus and approximately 329 light-years away from earth. It was the guide star for the Gravity Probe
B experiment, which aimed to measure effects predicted by general relativity, such as
gravitomagnetism. The system consists of IM Pegasi A, a K-type red giant star with a
mass of 1.8 solar masses (M from hereon) and a sun-like G-type main sequence star
with a mass of 1.0M , IM Pegasi B. Prior to this thesis, research done by Berdyugina et
al. pointed towards the existence of a planet in the system, orbiting IM Pegasi A inside
the binary.
Planetary orbits in binary systems are a well-researched subject, and many works have
been published that calculate the stability of planetary orbits both inside and outside of
a binary system (Musielak et al. 2005, Quarles et al. 2012). Studies include both prograde orbits (the planet is orbiting the main star in the same direction as the companion
star) and retrograde orbits (the planet is orbiting in the opposite direction). These studies, however, do not answer the question about the origin of those constellations. The
occurrence of planet formation in binary systems is well supported (Patience et al 2002,
Eggenberger et al. 2004 & 2007 and others). However, these planets have all formed
in wide binaries, with orbital distances in the range of 20AU . The IM Pegasi system,
however, is a very close binary, with an orbital distance of 0.2327AU , as derived from
the orbital period observed by Berdyugina et al. Berdyugina suggested that the object
in the IM Pegasi system could have been formed from material emitted by the primary
red giant star, which was then caught in the orbit.
For this thesis, I developed a numerical code in order to compute the IM Pegasi system,
I calculated the range of parameters resulting in stable orbits for the proposed planet,
and I tried to answer the question of its possible origin by simulating the scenario as
proposed by Berdyugina.
3
1 Introduction
1.1 Observational evidence
Figure 1: Location of IM Pegasi on the sky.
From: astronomy.net/constellations/pegasus.html (modified)
In a draft of a paper, Berdyugina et al. reported the first polarimetric detection of a
new exoplanet residing in the IM Pegasi system. A power-spectrum analysis of previous
polarimetric measurements revealed a significant period of 4.89 days, which is about
1/5 of the binary period being 24.64877 ± 0.00003 days (Marsden et al. 2005). In
their paper, Berdyugina et al. suggest that "the simplest interpretation of the observed
periodicity is the presence of a third body orbiting the primary and scattering its light
with two polarization maxima for one revolution (near elongations)." It follows that this
third body must reside inside the binary system and orbit the primary red giant star IM
Pegasi A at a very close distance (s-type orbit). Their subsequent numerical simulations
of the system reveal that the proposed planet is indefinitely stable (> 2 × 107 y) if its
orbit is assumed to be retrograde. Their observational data imposes an upper limit mass
of 6.6 Jupiter masses for the proposed planet and shows a remarkably large radius of its
scattering surface. Since the planet is very close to the surface of the star, this is most
likely caused by an extended halo of gas around that planet, similar to those detected
around comets in our solar system as they approach the sun.
4
2 Theory
2 Theory
2.1 Movement equations
A point-type object with the mass M generates a gravitational field around it. It is
given by
~g (~x) = −
GM
~x
r3
where ~x is the position vector, r = ||~x|| the distance from the object and G is the
gravitational constant. For my calculations, I used the system of units [L, M, T ] =
[AU, M , y], where AU = 149.597 × 109 m is the astronomical unit. In this system of
units, the gravitational constant has the value G = 39.447. The gravitational field can
be obtained by means of the negative gradient of the gravitational potential:
~ x) = −∇ GM
~g (~x) = −∇Φ(~
r
The force acting upon a test mass in the gravitational field of that object is equal to the
force field strength at its location multiplied with the mass m of the test mass i.
GM m
F~i = −
~xi
r3
And since the inertial mass is equal to the gravitational mass, the acceleration ~a experienced by the test mass is equal to the gravitational field strength. The gravitational
field has the dimension of an acceleration:
~ai = ~g (~xi )
In a system of N objects, we have to sum over the accelerations caused by each object
acting upon the object i:
~ai = −G
N
X
mj
i6=j
3
rij
~xij
We obtain a second-order differential equation for the trajectories of the objects (with
¨).
~a = ~x
2.2 Keplerian orbits
For a system of two objects, these equations can be solved analytically. If we assume
that one of the objects has a mass m that is negligible in comparison to the mass M of
the second object, the trajectory of the small mass i in the center of mass inertial frame
of reference is given by
r(ϕ) =
p
1 + ε cos ϕ
5
2 Theory
where ϕ is the angle between the periapsis (the point of closest approach) of the orbit
and the current position, p is the semi-latus rectum, and ε the orbital eccentricity.
Figure 2: Sketch of a Keplerian orbit with various orbital elements shown.
From: healthculturesociety.wikispaces.com (modified)
The value of ε determines the shape of the orbit. For e = 0, we obtain a circular
orbit, for 0 < e < 1 an eccentric or elliptical orbit, for e = 1 a parabolic orbit and for
e > 1 a hyperbolic orbit. This representation is called the parameter representation. To
unambiguously define the position of a planetary orbit in space, in general six parameters
are required. These can be the Cartesian position and velocity vectors

 

x1 (t)
v1 (t)

 

~x(t), ~v (t) = x2 (t) , v2 (t)
x3 (t)
v3 (t)
or the six Keplerian orbital elements
e, a, i, Ω, ω, ν
which are the eccentricity, semi-major axis, inclination, longitude of the ascending node,
argument of periapsis and true anomaly respectively. Their definitions are shown in
Figure 3. However, these are just two commonly used options and any set of six pairwise
linearly independent parameters can be used to locate the orbit in space.
6
2 Theory
Figure 3: Left: Diagram, depicting six orbital parameters.
From: en.wikipedia.org/wiki/File:Orbit1.svg Right: Sketch of Keplerian orbits with orbital eccentricities ε = 0, 0.5, 1 and 2.
From: en.wikipedia.org/wiki/File:OrbitalEccentricityDemo.svg (modified)
In case the object i has a non-negligible mass, the Keplerian equations still remain
valid. However, both objects then orbit around their common barycenter.
2.3 Restricted 3-body problem
The n-body problem is the task of predicting the motion of n particles interacting with
each other by gravitation. While the 2-body problem has been completely solved (Keplerian orbits, see 2.2), for n > 2, systems cannot be solved analytically, except for special
cases. Therefore numerical methods have to be used. In the restricted 3-body problem,
a special case of the 3-body problem, it is assumed that the mass of the third object is
negligible and that the objects all move along co-planar paths. By doing so, the third
dimension can be neglected entirely. The circular restricted 3-body problem adds an
additional constraint, namely that the orbit of the binary system is circular.
2.3.1 Jacobi constant
The Jacobi integral or Jacobi constant is a conserved quantity in the circular restricted
3-body problem, while energy and momentum are not conserved separately. The circular
restricted 3-body problem is best described in a rotating coordinate system placed at
the barycenter of the two massive objects M1 and M2 , with an angular velocity equal
to the angular velocity of the binary system. In this co-rotating coordinate system, the
two massive object remain stationary and the third object follows a trajectory in a way
7
2 Theory
that the Jacobi integral remains constant. It is given by:
J (r, ~v ) =
2π
T
2
r2 + 2G
M1 M2
+
r1
r2
− ||~v ||2
where T is the orbital period of the binary, r is the distance from the barycenter and
r1 , r2 the distances to the respective bodies. The first term is the "centrifugal potential
energy", and the second and third terms are twice the negative potential, respectively
kinetic energy per unit mass.
In order to calculate the trajectory of the third object, it is helpful to utilize the corotating coordinate system and perform the calculations in that frame of reference.
Changes in the Jacobi integral can give information about the accuracy of the utilized
integrator. The value of the Jacobi integral for a given configuration is useful for determining the long-term stability of the system: The solution (x, y) for a specific value of
J with ||~v || = 0 is called the zero-velocity surface. It denotes the region in space which
the body cannot surpass and its shape can give information about the conditions under
which the body can escape from the binary.
Figure 4: "Zero-velocity curves in the x–y coordinate frame for an ellipsoid-sphere system
[...] The small darker circle and ellipse represent the bodies themselves." L1
to L5 denote the five Lagrange points (see 2.3.2). From: Bellerose & Scheeres
(2008)
2.3.2 Lagrange points
There are five possible analytical solutions for the circular restricted three-body problem.
These are known as the Lagrange points. For a given two-body system, the Lagrange
points denote positions in space, at which a test mass remains stationary indefinitely, as
observed from a co-rotating coordinate system. The five Lagrange points are denoted as
8
2 Theory
L1 to L5 and their positions are depicted in Figure 5.
Figure 5: The Lagrangian points of the Earth-Moon system with equipotential lines of
the effective potential shown. From: space.com/14518-nasa-moon-deep-spacestation-astronauts.html
The Lagrange points are located at local extrema of the effective potential of the twobody system. The effective potential is the sum of the gravitational potential and the
centrifugal potential. It is given by
Φef f ective = Φ +
L02
2r2
where L0 is the angular momentum per unit mass.
A natural manifestation of the Lagrange points are the "Jupiter Trojans," a group of
small asteroids caught in the Lagrange points L4 and L5 of Jupiter. Unlike the other
Lagrange points, which are only labile configurations, L4 and L5 are stable.
2.4 Numerical methods
In the following chapter, I will introduce various numerical methods and strategies that
I utilized for calculations in this thesis. Numerical integration programs are widely available and could be used instead. However, I was given the explicit task to develop the
numerical code by myself as part of this thesis. Some of the methods I introduce in this
chapter are taken from literature, and some I developed myself.
I used "The Mathworks MATLAB" as the platform to write and execute the code, a
program specifically optimized for numerical simulations and large array operations.
9
2 Theory
2.4.1 Taylor series expansion
In general, the movement equations for N -body systems with N > 2 cannot be solved
analytically due to the seemingly chaotic unpredictability of those systems. In order to
model the development of such a system, numerical methods have to be used.
The simplest approach is to develop a code that integrates the Newtonian equations
step-by-step. If ∆t is the step size and n is the coefficient of the current step, then the
position ~x and velocity ~v of the object i at the next step n + 1 is:
~xi (n + 1) = ~xi (n) + ~vi (n)∆t
(1)
~vi (n + 1) = ~vi (n) + ~ai (n)∆t
(2)
A numerical error accumulates during each step. To rectify this, several advanced integration methods can be used. If we use (1) and (2) in our calculation, the error per
step would be second-order dependent on the chosen step size ∆t, while the total accumulated error would be of first-order dependence. The step size required to achieve an
acceptable accuracy would have to be so small that computation times would become
unreasonably long.
The tolerable step size can be increased by adding additional higher terms to the equation, obtained with the Taylor series expansion:
~xi (n + 1) =
~vi (n + 1) =
N
X
1 dk
k=0
N
−1
X
k=0
~xi (n)
(3)
1 dk
~vi (n)
k! dtk
(4)
k! dtk
In (4), we summed only to N − 1 instead of N as we did for the calculation of the
displacements in (3). The N th term in the sum requires the knowledge of the (N + 1)th
term of the series expansion, which we do not have. Thus, the numerical error caused
by the calculation of the velocities (4) is always the significant error and the resulting
code is of N − 1 order accuracy (the total accumulated error depends on the (N − 1)th
power of the step size).
Each higher order term is a function of the terms preceding it:
!
dk
dk−2
d
~
x
=
f
~
x
,
~
x
,
...
~x ; f or k ≥ 2
dtk
dt
dtk−2
Thus, a code that runs a Taylor series expansion with N > 2 cannot compute all orders
parallel, but must run a loop and compute the lower orders first, then use that data to
calculate the higher orders. A maximum of two orders can be computed during each
loop.
10
2 Theory
2.4.2 Runge-Kutta integration scheme
The Runga-Kutta method is an integration scheme for differential equations and an
improvement over the Taylor series expansion method. Let us assume we have a firstorder ordinary differential equation of the form:
ẏ = f (y, t)
with the starting condition y(0) = y0 . We can integrate the equation step-by-step,
assuming that the next value yn+1 is the present value plus the increment based on the
current slope of the function:
yn+1 = yn + ẏn ∆t
We can increase the accuracy by considering higher derivatives (Taylor series expansion
as discussed in 2.4.1) or by considering increments based on the slope of the function at
different points along the step, calculating a weighted average. One of these methods
is the widely used Runge-Kutta integration scheme. According to the Runge-Kutta
method of fourth order accuracy, commonly referred to as RK4, the next value yn+1 of
the function is given by:
yn+1 = yn +
1
(k1 + 2k2 + 2k3 + k4 )
6
where the variables k1 to k4 represent the slope of the function, using different arguments
for each:
1
2
1
k2 = f yn + k1 , tn +
2
1
k3 = f yn + k2 , tn +
2
k1 = f (yn , tn )
1
∆t
2
1
∆t
2
1
k3 = f (yn + k3 , tn + ∆t)
2
The Newtonian movement equations are second-order differential equations. These can
be integrated numerically by treating them as two coupled first-order differential equations:
~v˙ = ~a = f (~x, ~v , F (t))
~x˙ = ~v = g (~x, ~v , F (t))
Here, F (t) is the time-dependent force field. Since the acceleration ~a experienced by an
object is independent of its current velocity ~v (at least in a non-relativistic case), we can
simplify the dependencies to:
~v˙ = ~a = f (~x, F (t))
g=~v
~x˙ = g (~v ) −−→ ~v
11
2 Theory
Now, we calculate the increment for both ~v and ~x:
1
(k1 + 2k2 + 2k3 + k4 )
6
1
= ~xn + (l1 + 2l2 + 2l3 + l4 )
6
~vn+1 = ~vn +
~xn+1
with the parameters defined by:
1
2
1
l1 = ~vn ∆t
2
k1 = ~an ∆t
k2
l2
k3
l3
k4
l4
∆t
~v ∆t
,t +
∆t
= f ~xn +
2
2
k1
~an ∆t
~an ∆t2
= g ~vn +
∆t = g ~vn +
∆t = ~vn ∆t +
2
2
2
!
2
∆t
~v ∆t ~a∆t
+
,t +
∆t
= f ~xn +
2
4
2
k2
k2
= g ~vn +
∆t = ~vn ∆t + ∆t
2
2
1
= f (~xn + l3 , t + ∆t) ∆t
2
1
= g (~vn + k3 ) ∆t = ~vn ∆t + k3 ∆t
2
Variations of the Runge-Kutta method with different coefficients and orders of accuracy
exist. A notable one is the Runge-Kutta-Fehlberg method, which uses a fourth order
and fifth order numerical method simultaneously and is thus very excellent when used
in codes with automatically adjusting step sizes, as discussed in the following section.
2.4.3 Adjustment of the step-size
To function properly, the integration code must have the ability to vary the step-size
according to the requirements of the current situation. During a close approach of two
objects, the forces acting change rapidly and thus the integration method converges
slower; the step size should be reduced to avoid divergence. Likewise, in case the system
evolves very slowly, the step size should be increased or otherwise the code will get stuck
in that situation.
The criteria used to determine the step size are arbitrary and it should be used whatever
performs best for the integration method at hand. For my calculations, I wrote a routine
that compares the fifth order term of the equation for the displacements as given in (3)
12
2 Theory
with the sum of all terms preceding it. This routine is executed at the end of each loop.
1 d5 A = 5 ~x ∆t5
5! dt 4
X
1 dk k
B=
~x ∆t
k! dtk k=1
If the code finds that A > C × B, where C is a preset constant, then the step size ∆t
is halved and the code returns to the beginning of the loop. Likewise, if the code finds
that A < 25 × C × B, then the step size is doubled and the code returns to the beginning
of the loop. Otherwise, the code continues. The factor 25 was chosen because doubling
the step size would increase A by a factor of 24 and the remaining factor 2 is put in as a
fail-safe measure to avoid the program getting stuck while trying to adjust the step size.
2.4.4 Floating point numbers
All computers operate using a number system called floating point numbers. A floating
point number X is defined as:
X = S × M × BE
where S is the sign, A is called the mantissa, B the base and E the exponent. Internally,
each number is stored with the base B = 2. The sign S is either + or − (1 bit). The
mantissa is a rational number in the range of 1 to 2: M ∈ Q ∧ ∈ [1, 2].
Depending on the type of floating point numbers used, the mantissa is stored in 23 bits
(single precision) or 52 bits (double precision). Each floating point number is internally "normalized" to be stored in that scheme after each mathematical operation. For
example:
1.24 × 27 + 1.56 × 27 = 2.80 × 27 = 1.40 × 28
"The MathWorks MATLAB" solely uses double precision floating point numbers. With
only 52 bits available to store a number, the precision of a calculation is capped at
52 × log10 2 ≈ 16 decimal points.
Normal rules of addition and multiplication do not apply for floating point numbers.
This is a result of the fact that not all rational numbers can be represented as a floating
point number. For a computer, the equation 1 + 10−17 = 1 is true. These errors can add
up significantly during long calculations and generate a threshold under which a smaller
step size cannot reduce the computation error any further. As an example, we can look
at the limit value representation of e:
1
1+
x
e = lim
x→∞
x
= 2.718281...
If we calculate this expression in MATLAB with varying x, we get drastically different
results:
13
2 Theory
1. x = 103 ⇒ e = 2.7169
2. x = 1010 ⇒ e = 2.7183
3. x = 1015 ⇒ e = 3.0350
4. x = 1016 ⇒ e = 1
The closest approximation for e we get with x ≈ 1010 , then the floating point inaccuracies start to become significant.
Figure 6 shows the accumulated error as a function of the the step length for a test
scenario I calculated with my code for different orders of the Taylor expansion. I have
done these tests as I was unable to figure out the cause of these large inaccuracies I
have encountered. In this figure, we see that for the 3rd order and 4th order Taylor
series expansions, the logarithm of the accumulated error is linearly dependent on the
logarithm of the step size, as predicted in 2.4.1. However, the error for the 5th order
series shows deviant behavior. Though the same linear dependence can be seen for large
step sizes, the error cannot be dropped below 10−12 . For smaller step sizes, the error
increases again, until it concurs with the 4th order.
Figure 6: Accumulated error in a test scenario as a function of the number of steps used
on a double logarithmic scale. Blue: 3rd order Taylor series. Green: 4th order
Taylor series. Red: 5th order Taylor series.
The explanation for this behavior is that for very small step sizes the 4th order in the
Taylor series expansion becomes too small to be significant in the process of calculating
the new positions and velocities.
These floating point errors can be alleviated by using a few different techniques. The
14
2 Theory
system of units should be chosen in a way that as many variables as possible are of equal
magnitude. Equations should be restructured to avoid very large numbers being applied
against very small numbers.
I achieved the most significant minimization of the error by storing each order of the
series expansion separately into a different variable. The positions and velocities of the
objects are no longer stored, but must be calculated by summing over the different orders
when the code needs to know them. By doing so, information is lost. However, this loss
of information is not cumulative, as the exact position of the object is still stored. For
the next step n + 1, the "kth-order-position" is given by:
~xk (n + 1) = ~xk (n) +
∆tk dk
~x
k! dtk
And the position of the object is obtained with:
~x(n + 1) =
N
X
~xk (n + 1)
k=1
The same is true for the velocities. Using this method, I was able to significantly reduce
the error for smaller step sizes and it allowed me to add the sixth order to the series
expansion and reduce the error even further, by approximately five magnitudes.
2.4.5 Local step sizes
The performance of my code written for MATLAB is far from optimal. Integrators
written in C++ are significantly faster, as this language is much more machine-oriented.
However, code optimization can come a long way in increasing the performance of any
computation.
For scenarios with N >> 2 bodies, it is optimal to not use a global step size for the
calculation, but an individual (local) step size for each object. For each of the (N − 1)2
interactions, a subroutine should determine the step size required to compute that interaction at sufficient accuracy. Let Mij be the matrix storing the step size required to
compute the interaction between the objects i and j:

− M12 M13
M
− M23

M =  21
M31 M32
−
...
...
...

...
...


...
...
This matrix is symmetric, so Mij = Mji . The code runs at a step size given by min (M ),
but force field values between the objects i and j are renewed only at a step size given
by the corresponding entry Mij . For a specific object i, the displacement and velocity is
calculated only at a step size given by min (M1i , M2i , ...MN i ), the minimum value in the
column and/or row corresponding to the object i. This is the frequency at which new
force field data becomes available; during this period of time, the force fields acting upon
the object i are assumed to be constant and calculating the displacements and velocity
15
2 Theory
of that object at a lower step size would have no effect. The best way to implement
this method is to preset an interval ∆T and divide that interval into multiple steps,
with an independent step size for each interaction. Each must be chosen so that the
resulting number of steps is a power of two (2n , with n ∈ N), otherwise the individual
steps become desynchronized.
As an example, if we calculate the system Sun, Earth, M oon and Jupiter [S, E, M, J],
the matrix M 0 , storing the number of steps required per interval for each interaction
(reciprocal of the step size) is given by:

−
 2nSE

M 0 =  nSM
2
2nSJ
2nSE
−
2nEM
2nEJ
2nSM
2nEM
−
2nM J
2nSJ
2nEJ 


2nM J 
−

with nEM > nSE = nSM > nSJ > nEJ = nM J . The interaction between the earth and
the moon must be computed at the highest accuracy, and the remaining interactions less
so, appropriate to their significance.
This method can reduce the computation time at constant accuracy significantly, but its
initialization requires a considerable amount of time itself. Therefore, it is only effective
for systems with N >> 2 objects.
2.5 Multipole expansion
In 2.1, we assume that each object in our system is point-like. This is arguably not true
for celestial bodies. However, the gravitational field of a spherically symmetric body
is equivalent to that of a point-type object with the entire mass concentrated in the
center of mass. This simplification is valid in most cases, as the deformation of celestial
bodies from a spherical shape is often negligible. A celestial body may be deformed to
a spheroid due to centrifugal forces on the surface (rotational flattening) and/or tidal
effects.
To account for these effects, we can expand the formula for the gravitational field of
each object by using the multipole expansion. The multipole expansion is widely used
in electrostatics to calculate the electric fields of specific charge distributions, but it can
also be used to calculate the gravitational field of mass distributions. It divides the distribution function into a monopole, a dipole, a quadrupole etc. moment at a single point
and approximates the force field by summing over the contributions from the multipole
moments. The monopole moment is equal to the total charge respectively mass of the
distribution. In the case of electrostatics, the dipole moment is equal to the displacement of the electrical charges from the point of origin. This corresponds to the center
of mass for a mass distribution. However, we can choose the inertial frame of reference
to be moving alongside the center of mass, and the dipole moment of the gravitational
field vanishes in that frame of reference. Therefore, gravitational dipoles do not exist.
A gravitational quadrupole, however, does exist. The quadrupole moment results from
the deviation of the mass distribution from a spherical shape. It is a rank 2 tensor and
16
2 Theory
given by:
Z
Qij =
d3 r%(~r)(3ri rj − δij ri2 )
where % is the mass density and δij the Kronecker-delta. The potential associated with
the quadrupole moment is given by:
ΦQ (~x) = −
Gm Qij xi xj
2
r5
The Einstein notation is used. For a spheroid shaped body, with a deformation along
the x3 -axis, the quadrupole tensor has the form:


Q11
0
0


0 
Q =  0 Q11
0
0 −2Q11
The entry Q11 is positive for oblate spheroids and negative for prolate spheroids. The
associated gravitational field can be obtained by calculating the gradient of the potential
given in (4). Using the simplification we made in (5), we get the following term for the
total field of the mass distribution:
Gm
GQ11
~g (~x) = −∇Φ(~x) = − 3 ~x −
r
2
17
1
15x23
(3~
x
+
6x
~
e
)
−
~x
3
3
r5
r7
!
(5)
2 Theory
Figure 7: Left: Shape of the quadrupole potential (far field). The body is located in the center and is elongated along the x3 -axis. Right: Shape of
the potential and force field of an electric quadrupole (near field). From:
sciencewise.blogspot.de/2008/01/exploring-electrostatics.html
2.5.1 Tidal deformations
Figure 8: Artist’s depiction of a star distorted by a black hole’s gravitational field.
Illustration by Mark Garlick, University of Warwick.
As mentioned in 2.5, a star’s shape may be deformed due to rotational flattening and/or
tidal forces acting upon it by an exterior body and its gravitational field may slightly
deviate from its usual 1/r2 -shape due to those deformations. Observational data suggests
18
2 Theory
that IM Pegasi A is tidally locked to its companion star. Therefore, it has a rotational
period of 24.65d (Marsden et al. 2005) and due to its large size should experience a
significant deformation. However, the rotational axis of a tidally locked body always
lies perpendicular to the orbital plane. This axis is identical to the axis pointing in the
direction of the deformation x3 . Using (5), while keeping in mind that in the orbital
plane x3 = 0, we get the following term for the gravitational field in that plane:
m 3 Q11
Gm0 (r)
~g (~x) = −G 3 +
~
x
=
−
~x
r
2 r5
r3
where m0 (r) is a distance-dependent "effective mass":
m0 (r) = m +
3Q11
2r2
A deformation perpendicular to the orbital plane therefore causes a deviation of the
gravitational field from the usual 1/r2 -shape, but the field still points towards the center
of mass at every point. The same is true if the direction of the deformation points
towards ~x, where the gravitational field is measured. In that case, ~x = x3~e3 , x3 = r and
we get:
~g (~x) = −G
m
Q11
− 3 5 ~x
3
r
r
For two tidally locked objects orbiting each other in a circular orbit, the two effects
described above have no effect as long as we replace the gravitational mass with the
effective mass.
However, these effects have a severe effect on a body orbiting either of those two objects
with a different orbital period. IM Pegasi C is such an object. During each orbital
period, the forces acting upon it by the primary star will not always point towards its
barycenter, which can cause significant perturbations of its orbit.
In 3.6, I account for the quadrupole moment of the primary star, where I solely focus on the deformations caused by the tidal forces for the reasons described above.
2.5.2 Calculating the deformation
Assume we have a liquid, incompressible body in the gravitational field of a second body.
The surface of the liquid body must be shaped so that it is an equipotential. Otherwise,
the liquid will consequently flow from regions of higher potential to those of lower potential until the surface is an equipotential. However, in the case of a compressible body,
such as a star, we have to account for the different layers inside it. Each layer inside the
star, wrapped around a surface of constant pressure, must also be an equipotential, otherwise gas from regions of higher potential will again move to regions of lower potential.
Tidal forces acting upon the star rapidly increase with the distance from the barycenter. Therefore, inner layers are less deformed than the outer atmosphere. This makes
calculating the total quadrupole moment of the star complicated. I used a numerical
approach described in 3.6.
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2 Theory
2.6 Computation method
In 2.4, I introduced several different methods to increase the accuracy and performance
of numerical simulations. The decision which one of these I should utilize for the task
at hand was often based on trial and error.
At the time I encountered problems with the accuracy limit as depicted in Figure 6,
I implemented the Runge-Kutta integration scheme to help increase the accuracy at
equal step size. The main contribution to the error was not caused by the integration
scheme but by the innate inaccuracies of floating point numbers. Consequently the error
accumulation was not reduced. After having fixed these problems, the Runge-Kutta
integration method still did not converge as fast as it should. Looking back at it, this
was caused by a bug in the code. When I added the sixth order to the Taylor series
integration scheme, it showed excellent results. As the computation time required was
well within an acceptable range, I sticked with that scheme. However, with more time
at my disposal, I would have liked to fix the Runga-Kutta code, as it should work more
efficiently.
The method of non-global step sizes described in 2.4.5 was not used to compute the
three-body problems as the advantages do not start to kick in for about N < 10. However in 3.5, up to 103 objects will be computed at once, where this method was used to
reduce the computation time significantly.
The following list summarizes all choices made for the integration method used in future
calculations:
• Sixth order Taylor series expansion for monopole fields.
• Sixth order Taylor series expansion for quadrupole fields in 3.6.
• 214 to 216 steps per year, depending on initial distance of the planet (step size ∼ 8
to 32 minutes).
• Global step size for the three-body problems, non-global step size for the manybody calculations done in 3.5.
The structure of the numerical code is presented in the Addendum (5.2).
2.6.1 Accuracy test
To confirm that the code computes the scenarios with sufficient accuracy, I conducted
two separate tests. In the first test, I computed different scenarios and recorded the
violation of conversation of energy. The energy error, defined as
error =
E 0
max − En , n = 1, 2, 3, ...ntotal
E0 where ntotal is the total number of steps in the calculation, is a rough indicator for the
error, but cannot be used solely, as it is possible that several inaccuracies cancel each
20
2 Theory
other out, so that the total energy remains conserved.
To account for this, I conducted a second test, where the secondary star was removed
and the scenario was reduced to a two-body problem. By doing so, the semi-major axis
of the planet should remain constant during the calculation and any change of it can be
attributed to an error of the code.
At a step size of ∼16 minutes (215 steps per year) and using the orbital configuration, as assumed in 2.1, the energy violation amounted to 1.1 × 10−12 y −1 , and with the
secondary star removed, the semi-major axis shift amounted to 2.7 ×10−12 AU y −1 , both
linearly increasing. This test showed that the error of the code can be assumed to be
irrelevant for calculations shorter than approximately 107 y , as shifts of the semi-major
axis only start to become significant then.
21
3 Results
3 Results
Figure 9: Exemplary initial positions of the objects for the retrograde orbit of the proposed planet, henceforth refered to as IM Pegasi C. The arrows denote the
direction of the velocities. The angle BAC is varied (ϕ0 = 90◦ in the case
shown above), as well as the distance AC.
Figure 9 shows the initial configuration used for the calculations. I have simulated the
system for different configurations and investigated their stability. Both the velocities
of IM Pegasi A & B, as well as the velocity of C is chosen so that the resulting orbit
would be a circular orbit if it was not for the perturbations caused by the third object.
The scenarios were computed for varying periods of time. To get a basic overview of
the stabilities of the different orbits, I calculated the scenarios for 103 y . In addition,
I performed an extended, high accuracy calculation of 105 y for a few explicit scenarios
that were shown to be stable. These calculations changed none of the results. Ejection
times for unstable configurations typically ranged from 1y − 100y and all orbits that
remained stable in that time frame also remained stable in the extended calculation.
3.1 System parameters
For the simulations of the IM Pegasi system, the binary separation distance was chosen
as R = 0.2327AU . This value is derived from the binary orbital period as observed by
Berdyugina et al. and the masses of the stars MA = 1.8M & MB = 1M (Marsden
et al. 2005, Berdyugina et al. 2000). The upper mass limit of the planet was given by
Berdyugina et al. as MC < 6.6MJ , where MJ is the Jupiter mass. The value chosen
for the mass in these simulations is MC = 0.005M (5.24MJ ). I repeated some of the
simulations with different masses, using MC = 1MJ and MC = 10MJ . The results for
these simulations are omitted, as varying the planet’s mass in that range did not change
22
3 Results
any of the outcomes. Finally, the radius of the primary was set to 0.06AU (12.9R )
and all other objects were assumed to be point-like.
3.2 Comparison with previous research
To verify my research, I tried to reproduce the results of previous corresponding numerical calculations. Musielak et al. (2004) investigated the stability of both p-type and
s-type prograde orbits for varying mass ratios MA /MB of the binary system as well as
varying distance ratios RAC /RBC . I have reproduced their calculations for the s-type
orbits, for which their results are shown in Figure 10 (left). The initial conditions they
used are identical to those described in Figure 9 with ϕ0 = 90◦ . Musielak et al. give
definitions for their terms stable, marginally stable and unstable orbits in Figure 10 (left):
"Stable (S) ≤ 5%, marginally stable (MS), given as 5% < (MS) < 35%, and unstable (U) ≥ 35%, where the percentage refers to the orbital variability with respect to
the initial distance between the primary star and the giant planet specified at the begin
of the calculations. The variability is averaged over the number of computed planetary
orbits and thereafter the stability criterion is applied."
As the definitive method Musielak et. al used to calculate the orbital variability does
not arise from the text, I continued to use my standard method of investigating orbital
stability: I calculated the scenarios for 1000y , which is equivalent to approximately 105
orbits, depending on the initial distance, and classified all scenarios in which the planet
remained bound to the primary as stable. Musielak et al. stated that their classification
of stable orbits were motivated by the fact that for the earth to remain inside the habitable zone, its orbital variability must be smaller than 5%. In the case of IM Pegasi,
considerations about the habitable zone are pointless. We are more interested in answering the question whether a planet can exist at the proposed location, and therefore
the criteria I used are more suited for this special task. Musielak et al. also stated that
their classification of unstable orbits as (U) ≥ 35% are motivated by their own research,
which showed that orbits outside that limit typically become unstable eventually. Such
an orbit should also be classified as unstable by my method, as I have calculated the
orbits for a much longer period of time. Thus, although my method does not allow
the classification of "marginally stable" orbits, the regions of unstable orbits should be
identical.
Figure 10 (right) shows my results. The divides between unstable and stable respectively marginally stable orbits are more or less identical. It was possible to reproduce
the results from Musielak et al. The deviation of the stable-unstable divide between
both diagrams is well within the standard deviation, given by the step size at which I
scanned the parameter space. The results verify the validity of my code.
23
3 Results
Figure 10: Left: "Range of mass ratios and separation ratios corresponding to stable,
marginally stable and unstable inner (S-type) planetary orbits in binary systems. The simulations are based on 1000 orbits." From: Musielak et al. (2004)
Right: Comparison with my results obtained for the same scenarios. The
simulations are run for 1000y.
3.3 Prograde orbits
By the current state of knowledge, the occurrence of retrograde orbits is very rare and
we have not ruled out the possibily of a prograde orbit yet. Therefore, the most obvious
conclusion from the observational data is that a planet resides in the IM Pegasi system,
orbiting IM Pegasi A in a prograde orbit. Only if we can rule out the possibility of
a prograde orbit being stable in the proposed 5:2 resonance, or alternatively in a 5:1
resonance, as this orbit could also explain the observed 5:1 period, we should investigate
the possibilities of a retrograde orbit.
Table 1 shows the results of the calculations for varying r0 , the initial distance AC,
while the angle ϕ0 = 270◦ remains constant. ψ denotes the ratio of orbital periods
between B and C:
ψ=
TAC
TAB
The value ψ is the best way to indicate the orbital distance of C as, due to the perturbations caused by the secondary star, its orbit is significantly altered from an ellipsis,
and therefore Keplerian parameters are not fit to define the orbit.
24
3 Results
Table 1: Stability of the prograde circular orbit. Initial angle ϕ0 = 270◦ . The calculation
is run for 103 y.
#
1
2
3
4
5
6
7
8
r0 [AU ]
0.0600
0.0620
0.0640
0.0660
0.0680
0.0720
0.0740
≥0.0760
ψ
0.169
0.178
0.196
0.206
0.226
0.236
0.247
–
Stability
stable
stable
stable
stable
stable
stable
stable
unstable
We can see that there is a potential range of stable orbits within the range 0.06AU <
r0 < 0.076AU , with the 4:1 resonance (ψ = 0.25) being the outermost stable orbit.
The 5:1 resonance (ψ = 0.2) also appears to be stable, a resonance that is also able to
explain the observed periodicity of 4.89 days. However, for the classification of stable
orbits, the finite radius of the primary star has not yet been taken into account, and a
closer look at the data shows that the planet would approach the primary star as close
as that star’s radius for all initial distances in the stable range. Therefore, no stable
orbit is possible for these initial conditions.
Table 2 show the results for the initial angle ϕ0 = 180◦ . For these configurations,
the resulting orbits are far less eccentric and as a result stable for larger ψ . Again,
the orbits are stable up to the 4:1 resonance and slightly above, but in contrast to the
previous case, a new small "island of stability" arises at the 3:1 resonance (ψ = 31 ) as well.
25
3 Results
Table 2: Stability of the prograde circular orbit. Initial angle ϕ0 = 180◦ .
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
r0 [AU ]
≤0.0800
0.0825
0.0850
0.0875
0.0900
0.0925
0.0950
0.0975
0.1000
0.1025
0.1050
0.1075
0.1110
0.1125
≥0.1150
ψ
0.237
0.247
0.256
0.265
–
–
–
–
–
–
–
–
0.330
0.337
–
Stability
stable
stable
stable
stable
unstable
unstable
unstable
unstable
unstable
unstable
unstable
unstable
stable
stable
unstable
Again, all orbits with r0 < 0.0875AU are impossible due to the orbit’s path passing
through the primary star, but the 3:1 resonant orbit is barely stable, with the closest
approach of the planet being 0.065AU . I have calculated this configuration for an extended period of 105 y and it remained stable indefinitely. It is, however, questionable
if such a close approach to the primary would leave the planet unharmed and its orbit
intact. Friction, tidal forces and radiation are not considered in this calculation and may
either disintegrate the planet entirely or cause it to plummet into the primary star.
I considered only configurations with initial angles ϕ0 = 180◦ or ϕ0 = 270◦ . These
are two exemplary configurations, with a very high, respectively a very low eccentricity.
It is unnecessary to consider configurations with intermediate initial angles.
26
3 Results
Figure 11: Left: Probability density (arbitrary units) of IM Pegasi C for the 3:1 resonant prograde orbit. A and B are locked at the marked locations (rotating
coordinate system). Right: Path of the planet in the same coordinate system
(black line) and surface of IM Pegasi A (red line).
Figure 12: True to scale sketch of the stable configurations for the prograde orbit. The
gray area denotes the range of initial distances r0 that result in a stable orbit.
3.3.1 Metastable orbits
In the simulations for prograde orbits, there appeared a range of parameters where the
orbit of the planet is metastable, jumping back and forth between being bound to IM
Pegasi A & B. The mechanism behind this process is the following: The eccentricity
vector of C points at a constant direction, as seen from an outside observer. At each
pass-by of the secondary star, the eccentricity of the planet is increased, until the planet
surpasses the Lagrangian point L1 , at which point it will be captured by the secondary
star. This process repeats itself, now with the primary causing the perturbations. These
kinds of orbits are not indefinitely stable; the planet will be kicked out of the system or
crash into one of the two stars in approximately 1y − 100y . It also should be noted that
in these scenarios the planet repeatedly approaches the stars up to a proximity at which
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3 Results
the tidal forces would either rip it apart or it would crash into the star, depending on
that star’s radius. Figure 13 shows the probability density of the planet for one of these
scenarios. The probability of finding the planet bound to A instead of B is roughly equal
to their mass ratios. One observes that the planet is located at the Lagrangian point
L1 very frequently. This phenomenon of "metastable orbits" did not arise for the case
of retrograde orbits.
Figure 13: Left: Probability density (arbitrary units) of IM Pegasi C for a metastable
orbit. A and B are locked at the marked locations (rotating coordinate system). Right: Path of the planet in the same coordinate system (black line).
Red dot: IM Pegasi A. Green dot: IM Pegasi B.
3.3.2 Prograde orbits: Conclusion
Although I have found a small range of stable configurations at the 3:1 resonance, the
observed 5:2 ratio of the orbital periods cannot be explained with a prograde orbit. All
orbits with ψ > 13 are unstable. If the conclusions from the observational data are
correct, then the planet must reside in a retrograde orbit around the primary.
The occurrence of this "island of stability" at the 3:1 resonance is peculiar. Such phenomena did not appear in the calculations done in chapter 3.2, nor were they discussed
by Musielak et al. The effects of resonances on the stability of planetary orbits was investigated by Quarles et al. (2012). Their results show that similar "islands of stability"
exist for outer (p-type) orbits.
3.4 Retrograde orbits
I have amply calculated different configurations for retrograde orbits. The procedure was
similar as for the case of prograde orbits. First, different parameters were computed for
103 y , then selected configurations at the edge of stable parameter space were computed
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3 Results
for an extended period of time to see if the results change. Table 3 shows the results for
different initial distances r0 with ϕ0 = 90◦ being held constant. We can easily see that
orbits remain stable for much greater ψ than in the prograde case. All configurations up
to the 5:2 resonance are stable. If the initial distance is increased slightly further, orbits
become unstable for a small range. An analogous "island of instability" at that period
ratio will continue to show up on subsequent sets of parameters in this chapter. Then,
for even greater initial distances, orbits remain stable up to the 5:3 resonance.
Table 3: Stability of the retrograde circular orbit. Initial angle ϕ0 = 90◦ .
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
r0 [AU ]
≤0.1000
0.1015
0.1030
0.1045
0.1060
0.1075
0.1090
0.1105
0.1120
0.1135
0.1150
0.1165
0.1175
0.1185
0.1195
≥0.1205
ψ
0.401
0.413
–
–
0.456
0.466
0.480
0.494
0.509
0.525
0.543
0.562
0.577
0.596
0.618
–
Stability
stable
stable
unstable
unstable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
unstable
In contrast to the prograde case, the retrograde 5:2 resonant orbit is indefinitely stable
with the closest approach of the planet to the primary being 0.093AU . Thus, it remains
at a sufficient distance to the primary’s surface (as close as 500, 000km).
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3 Results
Figure 14: Left: Probability density (arbitrary units) of IM Pegasi C for the 5:2 resonant
retrograde orbit with ϕ0 = 90◦ . A and B are both locked at the marked
locations (rotating coordinate system). Right: Path of the planet in the
same coordinate system (black line) and surface of IM Pegasi A (red line).
The distance limit up to which the planetary orbits remain stable is also very remarkable. Figure 15 shows the probability density for the outermost stable orbit with
r0 = 0.1425AU . We can see that the planet routinely surpasses the Lagrangian L1 of
the binary system; the planet’s trajectory is curved into the opposite direction at that
point. Still, the planet remains bound to the primary.
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3 Results
Figure 15: Left: Probability density of IM Pegasi C for the outermost stable retrograde
orbit. A and B are both locked at the marked locations (rotating coordinate
system). L1 denotes the Lagrangian where the forces of A and B cancel each
other out. Right: Path of the planet in the same coordinate system (black
line) and surface of IM Pegasi A (red line).
Having found a stable orbit that can explain the observational data, we can now look
at different variations of the parameters to see for which range the proposed resonant
orbit remains possible. Table 4 shows the results for different initial distances r0 , now
with the initial angle being held constant at ϕ0 = 180◦ . In contrast to all other calculations performed for retrograde orbits, the "island of instability", in distance slightly
above the 5:2 resonance, disappeared. The most reasonable explanation for this would
be that the effects that destabilize the orbits at that period ratio are insufficient to kick
the planet out of the system in this specific case. As already shown for prograde orbits,
configurations with ϕ0 = 180◦ are more stable in general.
31
3 Results
Table 4: Stability of the retrograde circular orbit. Initial angle ϕ0 = 180◦ .
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
r0 [AU ]
≤0.1000
0.1025
0.1050
0.1075
0.1100
0.1125
0.1150
0.1175
0.1200
0.1225
0.1250
0.1275
0.1300
0.1325
0.1350
0.1375
0.1400
0.1425
≥0.1450
ψ
0.353
0.367
0.381
0.395
0.410
0.424
0.439
0.455
0.470
0.486
0.503
0.519
0.536
0.554
0.572
0.591
0.610
0.632
–
Stability
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
unstable
Figure 16: True to scale sketch of the stable configurations for retrograde orbits. The
gray area denotes the range of initial distances r0 that result in a stable orbit.
ϕ0 = 180◦ .
32
3 Results
Figure 17: Exemplary paths of the planet in a co-rotating coordinate system (black line)
and surface of IM Pegasi A (red line). Left: 5:2 resonant retrograde orbit
with ϕ0 = 180◦ . Right: 2:1 resonant retrograde orbit with ϕ0 = 180◦ .
Figure 17 shows the planet’s path for the 5:2 and 2:1 resonant retrograde orbits respectively, with ϕ0 = 180◦ . Notably, this is the first shown orbit, where the planet
follows an exact closed path, as observed from a co-rotating coordinate system. In all
scenarios shown previously, the orbital parameters periodically shifted, but they always
returned to the starting point, so that the orbit still remained stable indefinitely.
That such configurations exist is particularly relevant. In order to explain the observations by Berdyugina et al. coherently, the proposed planet must reside in an orbit with a
low orbital variability, so she stated. The orbital variability is far too high for previously
shown configurations, such as the 5:2 resonant orbit with ϕ0 = 90◦ (Figure 14).
3.4.1 Eccentric retrograde orbits
For the next calculation, the same parameters as in the previous were used, except that
the initial velocity of the planet was increased by 10% to generate an elliptical orbit.
For an undistorted two-body system, this would correspond to an orbital eccentricity of
e ≈ 0.2. We would expect the range of stable configurations to be reduced as circular
orbits tend to be more resistant to perturbations. Table 5 shows the results: The "island
of instability" at ψ > 0.4 has increased in size and the 5:2 resonant orbit is in fact no
longer stable. In addition, the outermost stable orbit is located further inward, only
slightly above the 2:1 resonance.
33
3 Results
Table 5: Stability of the retrograde elliptical orbit. The initial speed of the planet is
increased by 10%. Initial angle ϕ0 = 180◦ .
#
1
2
3
4
5
6
7
8
9
10
11
12
r0 [AU ]
≤0.0700
0.0725
0.0750
0.0775
0.0800
0.0825
0.0850
0.0875
0.0900
0.0925
0.0950
≥0.0975
ψ
0.311
0.321
0.340
0.361
0.386
–
–
–
0.468
0.482
0.515
–
Stability
stable
stable
stable
stable
stable
unstable
unstable
unstable
stable
stable
stable
unstable
Figure 18: Probability densities (arbitrary units). A and B are both locked at the marked
locations (rotating coordinate system). Left: Exemplary elliptical retrograde
orbit. Right: Comparison with an undistorted elliptical orbit (IM Pegasi B
removed).
3.4.2 Orbits in an eccentric binary system
The following calculations investigate the effects which an imposed eccentricity of the
binary system has on the stability of the orbits. Previously, it was assumed that the
binary’s orbit is perfectly circular. I repeated those these calculations with orbital eccentricities of e = 0.045, e = 0.096, e = 0.2 and e = 0.43 (the eccentric orbits were
generated by increasing the initial velocities by 2.5%, 5%, 10% and 20% respectively),
34
3 Results
while keeping the orbital period constant. The results are shown in table 6. Small deviations from a circular orbit do not have a large impact on the stability of the system,
but the irregular perturbations caused by highly eccentric orbits drastically reduces the
range of possible parameters. The 5:2 resonant orbit becomes unstable for e > 0.2.
Table 6: Stability of the retrograde circular orbit in elliptical binaries with varying eccentricities. Top left: e = 0.045, Top right: e = 0.096, Bottom left: e = 0.2,
Bottom right: e = 0.43. Initial angle ϕ0 = 180◦ .
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
r0 [AU ]
≤0.1075
0.1100
0.1125
0.1150
0.1175
0.1200
0.1225
0.1250
0.1275
0.1300
0.1325
0.1350
0.1375
0.1400
≥0.1425
ψ
0.393
0.407
0.422
–
0.452
0.468
0.482
0.499
0.516
0.533
0.551
0.569
0.588
0.613
–
Stability
stable
stable
stable
unstable
stable
stable
stable
stable
stable
stable
stable
stable
stable
stable
unstable
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
r0 [AU ]
≤0.1075
0.1100
0.1125
0.1150
0.1175
0.1200
0.1225
0.1250
0.1275
0.1300
0.1325
0.1350
0.1375
≥0.1400
ψ
0.387
0.402
0.416
–
0.445
0.466
0.479
0.486
0.510
0.525
0.542
0.560
0.581
–
Stability
stable
stable
stable
unstable
stable
stable
stable
stable
stable
stable
stable
stable
stable
unstable
#
1
2
3
4
5
6
7
8
9
10
11
r0 [AU ]
≤0.0875
0.0900
0.0925
0.0950
0.0975
0.1000
0.1025
0.1050
0.1075
0.1100
≥0.1125
ψ
0.271
0.283
0.294
–
0.323
0.328
0.343
0.355
0.369
0.382
–
Stability
stable
stable
stable
unstable
stable
stable
stable
stable
stable
stable
unstable
#
1
2
3
4
5
6
r0 [AU ]
0.0600
0.0625
0.0650
0.0675
0.0700
≥0.0725
ψ
0.155
0.164
0.174
0.184
0.194
–
Stability
stable
stable
stable
stable
stable
unstable
35
3 Results
3.5 Origin of the system
The observational data and the simulations done by Berdyugina et al. suggest that a
third body is located within the IM Pegasi binary in an inner (s-type) orbit, and my
simulations suggest that this orbit is retrograde. If this conclusion is correct, it severely
challenges planet formation theory. While the formation of planets in wide binaries is
well supported (Eggenberger et al. 2004, 2007), planet formation in such close binaries
is unprecedented, even more so for retrograde orbits. This led Berdyugina to propose
a deviant mechanism for the formation of the third object: Red giant stars such as IM
Pegasi A generally have very dense stellar winds, losing a large part of their atmosphere
over the course of their lifespan. This gas eventually forms a planetary nebula. Such a
nebula is shown in Figure 19.
Figure 19: NGC7293, the helix nebula, a planetary nebula as observed by the Hubble
Space Telescope and the Cerro Tololo Inter-American Observatory (superposition). From: de.academic.ru/dic.nsf/dewiki/993838
It is possible, due to the composition of the IM Pegasi system, that some or all of
the material ejected by IM Pegasi A is caught within an s-type orbit. The occurrence
of the retrograde orbit can then be explained either directly with this theory or by a
subsequent "natural selection" of a large range of possible orbits.
I upgraded my numerical code as described in 2.4.5 to simulate this scenario. The
primary star constantly ejects small test particles with random velocities and directions
of motion from the surface. These particles only interact via gravitation with the components of the system and do not interact with each other. Once a specific particle
plummets into a star or recedes too far from the system, it is removed and a new particle is generated. A total of up to 1000 particles are active at a time. This way, if there
is a configuration that results in a particle being caught in a stable orbit, particles will
36
3 Results
slowly accumulate in that orbit due to "natural selection."
3.5.1 Model I: Particle ejection from the primary star
Figure 20: Snapshots of two scenarios calculated with the particle model. The small
black dots represent particles. The objects are not to scale. Left: Particle
ejection from the primary. Right: Particle stream from outside the system
(see 3.5.2).
The model described in 3.5, was run for an extended period of time, with and without the
planet being present. Out of 800,000 particles, no particle was trapped in a long-term
stable orbit. Though some particles were temporarily caught around the primary, their
orbits were all far too eccentric to last longer than up to 5 revolutions. On the contrary,
a cloud of highly eccentric particles orbiting the binary system with semi-major axes of
1 − 2AU slowly accumulated.
3.5.2 Model II: Particle stream from outside the system
In the next approach, I investigated whether it is possible that a rogue planet has been
trapped inside the binary system and is now residing in an orbit around the primary star.
The particle stream was changed to originate from outside the system, as depicted in
20 (right). The particles spawn at a distance of 1AU and move with random velocities
and displacements perpendicular to the direction of motion, and fall onto the binary
system. With this model, again no particles were observed to be caught in long-term
stable orbits. More particles were caught in short-term stable orbits than with model I.
3.5.3 Model III: Particle disc around the primary star
The results of the previous two simulations showed no positive results. We did not
observe the formation of a particle disk around the primary as we would have hoped. I
37
3 Results
changed the approach and investigated whether such a particle disk would be stable in
that system once it had formed. The particles were all spawned in a disk around the
primary star at the same time and the calculation was run for 1000y . Snapshots of the
scenario are shown in Figure 21.
Figure 21: Snapshots of the particle disk scenario. The small black dots represent particles. The objects other than IM Pegasi A are not to scale. Left: Beginning
of the simulation. Right: End of the simulation for M 0 = 0.5MC (see main
text).
The simulation was repeated with different masses for the planet M 0 , namely 0MC
(the planet was removed), 0.5MC and 1MC , where MC is the mass previously used in
all simulations. During the calculation, many particles collided with the primary star.
Others collided with the planet or remained stable inside the disk, either in the two
Lagrange points L4 and L5 of the planet or in a disk very close to the surface of the
primary. For the case M 0 = 1MC , this disk did not remain existent.
38
3 Results
Table 7: Number of particle collisions with the bodies IM Pegasi A, B & C during the
particle disk simulation for different masses of the planet M 0 .
M 0 [MC ]
1
0.5
0
Object
A
B
C
A
B
C
A
B
C
Impacts
616
32
215
421
35
215
85
6
–
3.5.4 Model IV: Reorientation of the orbital axis
It is possible that the planet in the IM Pegasi system did not begin to exist in a coplanar orbit. It is a common occurrence that the orbital axes of orbiting bodies align
very quickly due to the influences those bodies have on each other. The same could have
happened to the proposed planet. It may have started in an entirely different orbital
plane and was forced into a co-planar orbit by the companion star.
For this calculation, I returned to the 3-body numerical code and added the previously neglected third dimension. I investigated the orbital stability of retrograde orbits
for varying inclinations i (see Figure 3). The results are shown in Table 8:
Table 8: Stability of the retrograde orbit with ϕ0 = 180◦ as a function of the inclination
i and the initial distance r0 . The simulations are run for 100y.
#
1
2
3
4
5
6
7
8
9
10
r0 [AU ]
0.10
0.10
0.10
0.10
0.11
0.11
0.09
0.10
0.11
0.12
i[◦ ]
10
15
20
25
25
30
35
35
35
35
39
Stability
stable
stable
stable
unstable
stable
stable
unstable
unstable
unstable
unstable
3 Results
Orbits are stable up to an inclination of i = 30◦ . Therefore, the possibility that
the planet started in an entirely different orbital plane can be ruled out. If the orbital
axis indeed drifted and forced the planet into a retrograde orbit, it must have happened
before IM Pegasi A became a red giant star.
3.5.5 Origin of the system: Conclusion
The scenarios I to III described above were not investigated thoroughly as the utilized
model quickly proved to be insufficient to be applied to this problem. The model has
many shortcomings. It is an improper simplification to assume that the movement of gas
ejected from the star is accurately described by a model of non-interacting particles. A
more accurate model must include friction fields and consider temperature and density
of the particles. I discussed ways to expand the model with Prof. Dr. Berdyugina, but
ultimately, we decided that the focus for the remaining time for this thesis should lie
elsewhere. I left the question for origin of the system unanswered, and expanded on the
orbital stability calculations by accounting for quadrupolar distortions (3.6).
Having only utilized a very simplified model, conclusive answers cannot be drawn from
these simulations. The ejection model seems unlikely, but cannot be definitively falsified. However, it was shown that even in this very close binary with the secondary star
causing heavy perturbations, a particle disk around the primary star can remain stable
and be collected by the planet, once it has accumulated a non-negligible mass.
40
3 Results
3.6 Accounting for quadrupolar distortion
In the previous calculations, it was assumed that the objects are point-type masses,
moving exactly as predicted by Newtonian mechanics. This is only a rough approximation. In reality, there are several effects that can perturb the ideal trajectories and
add up significantly over longer periods of time. For example, general relativity has a
significant impact on the shape of the Mercury orbit, accounting for a bit less than 10%
of its perihelion precession. This effect should be even more dominant for IM Pegasi
C, as it is even closer to its parent star than Mercury, and relativistic effects quickly
decrease with distance. The planet’s interaction with the surrounding gas (for example
the red giant’s stellar wind) may also be significant, as could be orbital perturbations
caused by tidal friction. Numerous works have been published on this subject (Kiseleva
et al. 1998, and others).
Accounting for all these effects would be beyond the scope of this thesis. Therefore,
I focused solely on an effect that I estimated to be the most significant: The tidal
deformation of the primary star by its companion star. The consequences of this deformation are described in 2.5. The goal is to calculate the extent of this deformation and
then check if the orbits that were previously deemed stable still remain stable with this
deformation taken into account.
Figure 22: Initial positions of the objects for the retrograde orbit of IM Pegasi C with
the deformation of A taken into account. The deformation is exaggerated.
3.6.1 Calculating the quadrupole moment
Integrating the quadrupole moment of the deformed star, with the contribution of each
layer, requires knowledge about the density distribution within the star. To get an estimation, I used the density values obtained by Alcock & Paczynski 1979 for a 2M ,
41
3 Results
13.3R (0.062AU ) helium burning red giant star, which describes IM Pegasi A comparably well. I interpolated the points given in the paper to obtain a basic density function. I
computed the tidal deformation for a set of layers, by calculating the shape of the equipotential for that layer. Then, I integrated numerically over the layers to gain the corresponding quadrupole moment. The acquired value was Q11 = −1.2 × 10−5 M AU −2 .
To get a visualization of this value: It corresponds approximately to the quadrupole
moment of an incompressible, spheroidal body with the mass and radius of IM Pegasi
A and the axis dimensions a = b and c = 1.004a. If we replace IM Pegasi A with an
incompressible body of similar mass and radius, the resulting value would be increased
by a factor of 10. It has to be noted that several systematic errors render the confidence
in the obtained value very low, but at the same time, an uncertainty analysis cannot
be performed. Also, the implications of several simplifications I have made remain unknown. Nonetheless, given the results I obtained with different approaches or density
functions, I estimate the accumulated error to be not greater than a factor of two.
3.6.2 Quadrupolar distortion: Orbital stability
With the quadrupolar distortion taken into account, the prograde and retrograde orbits were computed anew, where ϕ0 = 180◦ was used in all calculations. The results
show that the already shaky prograde orbits are no longer stable in the modified field.
Retrograde orbits remained stable throughout the simulations. However, a closer look
at the data shows that the planet moves along a spiral-shaped path due to the perturbation, coming closer to the primary with each revolution. The simulation began with
the planet in the 5:2 resonant orbit. The semi-major axis of the orbit decreased at a
rate of 6 × 10−8 AU y −1 . This is a very small shift and it is possible that it is caused
entirely by numerical errors, but the tests performed in 2.6.1 show that the semi-major
axis shift caused by numerical errors is another four magnitudes smaller. In addition, I
ran a second parallel simulation with the original 3-body numerical code for point-type
objects to verify that the spiral form was indeed caused by the quadrupole fields. The
simulations show no detectable shift within the resolution limit.
Due to time constraints, the simulation could not be finished and the ultimate fate
of the planet in that simulation remains unkown. However, we can fast-forward and
measure the annual semi-major axis shift for orbits closer to the surface. Due to the
low confidence on the estimated quadrupole moment, I also repeated the simulations for
different values. These results are all shown in Table 9:
42
3 Results
Table 9: Semi-major axis shift (SMAS) as a function of the quadrupole moment qQ11 ,
where Q11 is the value obtained in 3.6.1.
#
1
2
3
4
5
q
0.5
1.0
2.0
10.0
1.0
r0 [AU ]
0.11
0.11
0.11
0.11
0.08
SMAS[AU y −1 ]
< 10−9
6.0 × 10−8
7.4 × 10−8
8.2 × 10−7
< 10−9
We can observe a dependence of the semi-major axis shift on the quadrupole moment
used, but some of the results are incoherent. For q = 0.5 no shift was detected within
the resolution limit. The shift also seems to halt at a closer proximity to the primary
star, as shown in #5. This conclusion is consolidated by the simulation with q = 10,
which was run long enough to observe this behavior (see Figure 23).
Figure 23: Averaged distance AC as a function of time for the simulation #4 in Table
9. The fluctuating peaks are artifacts (the distance is recorded at a finite
step-length, thus causing fluctuations).
By linear extrapolation of the semi-major axis shift as observed in #2, we can calculate
the time at which the planet plummets into the primary star to approximately 500,000y .
However, the simulations showed that the shift will halt at some point and the planet
will remain stable indefinitely. We can therefore conclude that the tidal deformation of
the primary star has a significant impact on the evolution of the proposed planet, but
the perturbation caused is not strong enough to render its orbit unstable.
43
4 Conclusion
In the Addendum (5.1), I present previous results on this subject that were proven
to be false due to numerical inaccuracies in the simulation. Although I have improved
the code significantly, it is still possible that the observed effects are caused by similar
inaccuracies. Unfortunately, there was not enough time left to investigate these issues
in depth. Therefore, the results are to be taken with a grain of salt.
4 Conclusion
I was able to reproduce the findings of Musielak et al. (2004) and therefore validate the
numerical code I developed. With this code, I was able to confirm that the proposed
planet is in fact indefinitely stable if its orbit is retrograde. Accounting for the tidal
deformations experienced by the primary star did not change these results. Other perturbing effects, such as effects of general relativity or tidal friction, were not accounted
for. Therefore, all simulations performed for this thesis are only an approximation.
The proposed ejection model that should explain the origin of the proposed planet could
not be confirmed. The method utilized to simulate this scenario was insufficient to be
applied to this problem. It would have been interesting to know what would be the
outcome of simulations performed with an improved model that included friction fields
and other effects beyond what was accounted for in this thesis.
44
5 Addendum
5 Addendum
Previous calculations I did for 3.6 seemingly show that both prograde and retrograde orbits become unstable in short time frames, when the quadrupolar distortion is accounted
for. Fortunately, I was able to detect in time that the shifts of orbital parameters I
observed had to be attributed entirely to numerical inaccuracies. The attractive and
repellent components of the quadrupole field did not compensate each other correctly
and the semi-major axis shift was greatly amplified as a result. Such an error is difficult
to identify, and therefore it was detected only at the very end of the available time for
this thesis. For the sake of completeness and to provide an accurate protocol of this
thesis, the results previously obtained are presented in this addendum.
5.1 Quadrupolar distortion: Retrospectively falsified results
The results show that the modified field has a severe impact on the long-term stability
of the system. Prograde orbits all became unstable after less than 1 year (≈100 orbits).
Retrograde orbits did not appear to be much more stable, although the ejection times
ranged slightly higher, up to 100 years at least. The observed behavior of the planet
depends on its initial distance. For the case of a retrograde orbit and for an initial
distance smaller than 0.110 ± 0.005AU , the planet slowly approaches the primary, until
it plummets into it. For distances greater than that limit, the planet’s orbit becomes
more and more eccentric over time, until it is either kicked out of the system or collides
with the primary. These behaviors are exemplified in Figure 24.
Figure 24: Distance of the planet to the primary as a function of time for retrograde
orbits with quadrupolar distortion taken into account (black line) and surface
of IM Pegasi A (red line). Left: Initial distance r0 = 0.14AU . Right: Initial
distance r0 = 0.09AU .
45
5 Addendum
In the final part of this thesis, I look at planetary orbits in distorted binaries in a
more general case and attempt to map the ejection times of s-type orbits, both prograde
and retrograde as a function of the binary separation distance. These results can give
a rough overview over the stability of planets in close binaries, such as the ν Octanis
system (Quarles et al. 2012). Assuming that the primary star is always the distorted
one is motivated by the fact that in a binary system, the more massive star is always
the first to become a red giant, and is also much more likely to host planets.
Assuming that the primary has the properties of IM Pegasi A, I calculated the quadrupole
moment of the primary star for different separation distances. The results are shown in
Table 10.
Table 10: Quadrupole moment of IM Pegasi A as a function of the separation distance
AB.
#
1
2
3
4
r[AU ]
2.00
1.00
0.50
0.25
Q11 [M AU −2 ]
−1.50 × 10−7
−5.95 × 10−7
−2.44 × 10−6
−1.06 × 10−5
By means of minimization, I acquired Q11 (r) [M AU −2 ] = (57r2 + 2.4r3 ) 10−8 as
the function for the quadrupole moment. Using this function, I mapped the ejection
times for different binary separation ratios and initial distances r0 . The results are
shown in Figures 25 and 26.
46
5 Addendum
Figure 25: Planet ejection time as a function of the binary separation distance R and
the initial distance r0 . The parameters used are given in the main text. Due
to constraints on computation time, the large stable areas (green) have been
scanned less accurately and values were interpolated. The simulations were
stopped after 50, 000y if no ejection was observed. Left: Retrograde orbits.
Right: Prograde orbits.
Figure 26: Left: Maximum ejection time as a function of the binary separation distance R for retrograde orbits (dashed line) and prograde orbits (dotted line).
Right: Ejection time as a function of the initial distance r0 with the binary
separation distance kept constant at 0.9AU for retrograde orbits (dashed line)
and prograde orbits (dotted line).
47
5 Addendum
The results show that for small binary separation distances (inducing high quadrupole
moments), all prograde orbits are extremely unstable and do not last longer than a
few revolutions. However, for larger separations (low quadrupole moments), prograde
orbits surpass the retrograde orbits in stability. Though the tidal deformation has a
destabilizing effect on all otherwise stable orbits, the outermost retrograde orbits are
actually stabilized to some degree by this deformation. For r0 ≈ 0.8R, where R is the
binary separation distance, the planet sustains longer if the quadrupolar distortion is
taken into account.
Figure 27: Probability density (arbitrary units) of IM Pegasi C for the retrograde orbit
with R = 0.45AU and r0 = 0.34AU . A and B are locked at the marked
locations (rotating coordinate system). This configuration becomes unstable
after 150 years.
Table 11: Maximum ejection time tmax as a function of the secondary star’s mass MB
for a constant binary separation distance R = 0.5AU . r0 has been scanned
with a step-size of 0.01AU to find the maximum.
#
1
2
3
4
5
MB [M ]
0.50
0.75
1.00
1.25
1.50
48
tmax [y]
9946
8977
4286
2272
1450
5 Addendum
5.2 Numerical code
y
z
d
x
.
The following is the command loop of the 3-body numerical code. (Axy )z ≡ dt
y (A )
The index y is omitted for y = 0. i and j index the objects. For variables with two
indexes ij , those indexes have been omitted (for example r = rij is the position vector
from object i to object j ). ◦ is the dot product, µij is the interaction constant Gmi mj
between the objects i and j . Only the main routine is shown.
r = Ri − Rj
v = Vi − Vj
d2 =√1/ ||r||2
d = d2
d3 = d2 d
d4 = d3 d
d5 = d4 d
e = rd
d−1
1 =v◦e
d1 = −d2 d−1
1
d21 = 2dd1
e1 = dv + d1 r
F = −µd3 r
F1 = −µ d2 e1 + d21 e
P
F
Ai = m−1
i
P
F1
A1i = m−1
i
a = Ai − Aj
a1 = A1i − A1j
d−1
2 = e1 ◦ v + e ◦ a
3 (d−1 )2
d2 = −d2 d−1
1 + 2d 1
d22 = 2 (d1 )2 + dd2
e2 = da + 2d1 + d2 r
d−1
3 = e2 ◦ v + 2e1 ◦ a + e ◦ a1
3 −1 −1
4 −1 3
d3 = −d2 d−1
3 + 6 d d2 d1 − d (d1 )
e3 = da1 + 3 (d1 a + d2 v) + d3 r
F2 = −µ d2 e2 + 2d21 e1 + d22 e
F3 = −µ d2 e3 + 3d21 e2 + 3d22 e1 + d23 e
P
A2i = m−1
i P F2
A3i = m−1
F3
i
a2 = A2i − A2j
d−1
4 = e3 v + 3e2 a + 3e
1 ȧ + ea1
−1
−1
−1 2
2
3
d4 = −d d4 + 12d d−1
d
+
(d
)
...
3
1
2
−1 2 −1
−1 4
4
5
−36d (d1 ) d2 + 24d (d1 )
d24 = 8d3 d1 + 6(d2 )2 + 2dd4
e4 = da2 + 4 (d1 a1 + d3 v) + 6d2a + d4 r
F4 = −µ d2 e4 + 4 d23 e1 + d21 e3 + 6d22 e2 + d4 e
P
F4
A4i = m−1
i
Ria = Ria + Vi ∆t
Rib = Rib + 1/2Ai ∆t2
Ric = Ric + 1/6A1i ∆t3
Rid = Rid + 1/24A2i ∆t4
Rie = Rie + 1/120A3i ∆t5
Rif = Rif + 1/720A4i ∆t6
Via = Via + Ai ∆t
Vib = Vib + 1/2A1i ∆t2
Vic = Vic + 1/6A2i ∆t3
Vid = Vid + 1/24A3i ∆t4
Vie = Vie + 1/120A4i ∆t5
Ri = Ria + Rib + Ric + Rid + Rie + Rif
Vi = Via + Vib + Vic + Vid + Vie
49
6 References
6 References
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Bellerose J. & Scheeres D. J., Restricted Full Three-Body Problem: Application to
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Berdyugina S. V., Berdyugin A. V., Hackman T., Strassmeier K. G. & Tuominen I.,
The long-period RS CVn binary IM Pegasi - II. First surface images, Astronomy
and Astrophysics, 2000
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tection and Properties of Extrasolar Planets in Double and Multiple Star Systems, ASP Conference Series, 2004
Kiseleva L. G. & Eggleton P. P., The Effect of Tidal Friction and Quadrupolar Distortion on Orbits of Stars or Planets in Hierarchical Triple systems, ASP Conference Series, 1998
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in binary systems, Astronomy & Astrophysics, 2004
Patience J., White R. J., Ghez A. M., McCabe C., McLean I. S., Larkin J. E., Prato L.,
Kim S. Sungsoo, Lloyd J. P. & Liu M. C., Stellar companions to stars with planets, The Astrophysical Journal, 2002
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ν Octantis system: a numerical and statistical study, Monthly Notices of the
Royal Astronomical Society, 2012
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problem, WILEY-VCH Verlag GmbH & Co., 2012
Runge-Kutta Method for Solving Two Coupled 1st Order Differential Equations
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50
6 References
List of Figures
astronomy.net/constellations/pegasus.html, The Constellation Pegasus
healthculturesociety.wikispaces.com, Eccentric rotation of the Earth
en.wikipedia.org/wiki/File:Orbit1.svg, Keplerian orbital elements
Bellerose J. & Scheeres D. J., Restricted Full Three-Body Problem: Application to
Binary System 1999 KW4, Journal of Guidance, Control, and Dynamics, 2008
space.com/14518-nasa-moon-deep-space-station-astronauts.html, The Lagrange points
for the Earth-moon system
sciencewise.blogspot.de/2008/01/exploring-electrostatics.html, A 2-dimensional Elec-
trostatics Applet
Musielak Z. E., Cuntz M., Marshall E. A., & Stuit T. D., Stability of planetary orbits
in binary systems, Astronomy & Astrophysics 2004
de.academic.ru/dic.nsf/dewiki/993838, NGC7293
en.wikipedia.org/wiki/File:OrbitalEccentricityDemo.svg, A diagram of the various forms
of the Kepler Orbit and their eccentricities
51
Acknowledgments
I’d like to thank Prof. Svetlana Berdyugina for her great supervision and for giving me
the chance to write my thesis on this subject. I could not have asked for a subject more
tailored towards my interests. I’d also like to thank the entire Kiepenheuer-Institut for
their warm welcome.
Thanks to Lino Burgold for proof-reading my thesis.
Without the constant support of my family, this thesis wouldn’t have been possible.
Thank you for everything!