Formation and Stability of Planetary Orbits in Binary Systems a Case Study of the IM Pegasi Binary

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 s- type 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 3 1.1 Observational evidence ...... 4

2 Theory 5 2.1 Movement equations ...... 5 2.2 Keplerian orbits ...... 5 2.3 Restricted 3-body problem ...... 7 2.3.1 Jacobi constant ...... 7 2.3.2 Lagrange points ...... 8 2.4 Numerical methods ...... 9 2.4.1 Taylor series expansion ...... 10 2.4.2 Runge-Kutta integration scheme ...... 11 2.4.3 Adjustment of the step-size ...... 12 2.4.4 Floating point numbers ...... 13 2.4.5 Local step sizes ...... 15 2.5 Multipole expansion ...... 16 2.5.1 Tidal deformations ...... 18 2.5.2 Calculating the deformation ...... 19 2.6 Computation method ...... 20 2.6.1 Accuracy test ...... 20

3 Results 22 3.1 System parameters ...... 22 3.2 Comparison with previous research ...... 23 3.3 Prograde orbits ...... 24 3.3.1 Metastable orbits ...... 27 3.3.2 Prograde orbits: Conclusion ...... 28 3.4 Retrograde orbits ...... 28 3.4.1 Eccentric retrograde orbits ...... 33 3.4.2 Orbits in an eccentric binary system ...... 34 3.5 Origin of the system ...... 36 3.5.1 Model I: Particle ejection from the primary star ...... 37 3.5.2 Model II: Particle stream from outside the system ...... 37 3.5.3 Model III: Particle disc around the primary star ...... 37 3.5.4 Model IV: Reorientation of the orbital axis ...... 39 3.5.5 Origin of the system: Conclusion ...... 40 3.6 Accounting for quadrupolar distortion ...... 41 3.6.1 Calculating the quadrupole moment ...... 41 3.6.2 Quadrupolar distortion: Orbital stability ...... 42

4 Conclusion 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 × 107y) 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.

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2 Theory

2.1 Movement equations A point-type object with the mass M generates a gravitational field around it. It is given by GM ~g(~x) = − ~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 ] = 9 [AU, M , y], where AU = 149.597 × 10 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: GM ~g(~x) = −∇Φ(~ ~x) = −∇ 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. GMm F~ = − ~x i r3 i 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:

N m ~a = −G X j ~x i r3 ij i6=j ij

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 p r(ϕ) = 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.

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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 (Kep- lerian 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:

2π 2 M M J (r,~v) = r2 + 2G 1 + 2 − ||~v||2 T r1 r2 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 co- rotating 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 deter- mining 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

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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-space- station-astronauts.html

The Lagrange points are located at local extrema of the eﬀective potential of the two- body system. The eﬀective potential is the sum of the gravitational potential and the centrifugal potential. It is given by

L02 Φ = Φ + effective 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.

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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:

N 1 dk ~x (n + 1) = X ~x (n) (3) i k! dtk i k=0 N−1 1 dk ~v (n + 1) = X ~v (n) (4) i k! dtk i k=0

In (4), we summed only to N − 1 instead of N as we did for the calculation of the displacements in (3). The Nth 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 d dk−2 ~x = f ~x, ~x,... ~x ; for 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.

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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 first- order ordinary differential equation of the form:

y˙ = 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 +y ˙n∆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: 1 y = y + (k + 2k + 2k + k ) n+1 n 6 1 2 3 4 where the variables k1 to k4 represent the slope of the function, using different arguments for each: 1 k = f (y , t ) 1 n n 2 1 1 k = f y + k , t + ∆t 2 n 2 1 n 2 1 1 k = f y + k , t + ∆t 3 n 2 2 n 2 1 k = f (y + k , t + ∆t) 3 n 3 n 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

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Now, we calculate the increment for both ~v and ~x: 1 ~v = ~v + (k + 2k + 2k + k ) n+1 n 6 1 2 3 4 1 ~x = ~x + (l + 2l + 2l + l ) n+1 n 6 1 2 3 4 with the parameters defined by: 1 k = ~a ∆t 1 n 2 1 l = ~v ∆t 1 n 2 ~v∆t ∆t k = f ~x + , t + ∆t 2 n 2 2 k ~a ∆t ~a ∆t2 l = g ~v + 1 ∆t = g ~v + n ∆t = ~v ∆t + n 2 n 2 n 2 n 2 ! ~v∆t ~a∆t2 ∆t k = f ~x + + , t + ∆t 3 n 2 4 2 k k l = g ~v + 2 ∆t = ~v ∆t + 2 ∆t 3 n 2 n 2 1 k = f (~x + l , t + ∆t) ∆t 4 n 3 2 1 l = g (~v + k ) ∆t = ~v ∆t + k ∆t 4 n 3 n 3 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 5 A = 5 ~x ∆t 5! dt

4 1 dk B = X ~x ∆tk 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 x e = lim 1 + = 2.718281... x→∞ x If we calculate this expression in MATLAB with varying x, we get drastically different results:

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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:

∆tk dk ~x (n + 1) = ~x (n) + ~x k k k! dtk And the position of the object is obtained with:

N X ~x(n + 1) = ~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 ...   M21 − M23 ... M =   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, ...MNi), 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 n resulting number of steps is a power of two (2 , with n ∈ N), otherwise the individual steps become desynchronized. As an example, if we calculate the system Sun, Earth, Moon 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 2nSM 2nSJ 2nSE − 2nEM 2nEJ  M 0 =    nSM nEM nMJ  2 2 − 2  2nSJ 2nEJ 2nMJ − with nEM > nSE = nSM > nSJ > nEJ = nMJ . 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 3 2 Qij = d r%(~r)(3rirj − δijri ) where % is the mass density and δij the Kronecker-delta. The potential associated with the quadrupole moment is given by: Gm Q x x Φ (~x) = − ij i j Q 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   Q =  0 Q11 0  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 GQ 1 15x2 ~g(~x) = −∇Φ(~x) = − ~x − 11 (3~x + 6x ~e ) − 3 ~x (5) r3 2 r5 3 3 r7

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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 Q Gm0(r) ~g(~x) = −G + 11 ~x = − ~x r3 2 r5 r3 where m0(r) is a distance-dependent "effective mass":

3Q m0(r) = m + 11 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: m Q ~g(~x) = −G − 3 11 ~x r3 r5 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.

19 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. How- ever 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 many- body 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

E0 − En error = max , 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 conﬁgura- tion, as assumed in 2.1, the energy violation amounted to 1.1 × 10−12y−1, and with the secondary star removed, the semi-major axis shift amounted to 2.7×10−12AUy−1, both linearly increasing. This test showed that the error of the code can be assumed to be irrelevant for calculations shorter than approximately 107y, as shifts of the semi-major axis only start to become signiﬁcant then.

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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 103y. In addition, I performed an extended, high accuracy calculation of 105y 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

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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 an- swering 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.

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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:

T ψ = AC 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.

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◦ Table 1: Stability of the prograde circular orbit. Initial angle ϕ0 = 270 . The calculation is run for 103y.

# r0[AU] ψ Stability 1 0.0600 0.169 stable 2 0.0620 0.178 stable 3 0.0640 0.196 stable 4 0.0660 0.206 stable 5 0.0680 0.226 stable 6 0.0720 0.236 stable 7 0.0740 0.247 stable 8 ≥0.0760 – 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 1 previous case, a new small "island of stability" arises at the 3:1 resonance (ψ = 3 ) as well.

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◦ Table 2: Stability of the prograde circular orbit. Initial angle ϕ0 = 180 .

# r0[AU] ψ Stability 1 ≤0.0800 0.237 stable 2 0.0825 0.247 stable 3 0.0850 0.256 stable 4 0.0875 0.265 stable 5 0.0900 – unstable 6 0.0925 – unstable 7 0.0950 – unstable 8 0.0975 – unstable 9 0.1000 – unstable 10 0.1025 – unstable 11 0.1050 – unstable 12 0.1075 – unstable 13 0.1110 0.330 stable 14 0.1125 0.337 stable 15 ≥0.1150 – 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 105y 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.

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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

27 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 1 orbits with ψ > 3 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 phe- nomena 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 103y, then selected configurations at the edge of stable parameter space were computed

28 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 .

# r0[AU] ψ Stability 1 ≤0.1000 0.401 stable 2 0.1015 0.413 stable 3 0.1030 – unstable 4 0.1045 – unstable 5 0.1060 0.456 stable 6 0.1075 0.466 stable 7 0.1090 0.480 stable 8 0.1105 0.494 stable 9 0.1120 0.509 stable 10 0.1135 0.525 stable 11 0.1150 0.543 stable 12 0.1165 0.562 stable 13 0.1175 0.577 stable 14 0.1185 0.596 stable 15 0.1195 0.618 stable 16 ≥0.1205 – 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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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 re- markable. 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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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.

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◦ Table 4: Stability of the retrograde circular orbit. Initial angle ϕ0 = 180 .

# r0[AU] ψ Stability 1 ≤0.1000 0.353 stable 2 0.1025 0.367 stable 3 0.1050 0.381 stable 4 0.1075 0.395 stable 5 0.1100 0.410 stable 6 0.1125 0.424 stable 7 0.1150 0.439 stable 8 0.1175 0.455 stable 9 0.1200 0.470 stable 10 0.1225 0.486 stable 11 0.1250 0.503 stable 12 0.1275 0.519 stable 13 0.1300 0.536 stable 14 0.1325 0.554 stable 15 0.1350 0.572 stable 16 0.1375 0.591 stable 17 0.1400 0.610 stable 18 0.1425 0.632 stable 19 ≥0.1450 – 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 .

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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 re- ◦ spectively, 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.

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Table 5: Stability of the retrograde elliptical orbit. The initial speed of the planet is ◦ increased by 10%. Initial angle ϕ0 = 180 .

# r0[AU] ψ Stability 1 ≤0.0700 0.311 stable 2 0.0725 0.321 stable 3 0.0750 0.340 stable 4 0.0775 0.361 stable 5 0.0800 0.386 stable 6 0.0825 – unstable 7 0.0850 – unstable 8 0.0875 – unstable 9 0.0900 0.468 stable 10 0.0925 0.482 stable 11 0.0950 0.515 stable 12 ≥0.0975 – 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 de- viations 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 .

# r0[AU] ψ Stability # r0[AU] ψ Stability 1 ≤0.1075 0.393 stable 1 ≤0.1075 0.387 stable 2 0.1100 0.407 stable 2 0.1100 0.402 stable 3 0.1125 0.422 stable 3 0.1125 0.416 stable 4 0.1150 – unstable 4 0.1150 – unstable 5 0.1175 0.452 stable 5 0.1175 0.445 stable 6 0.1200 0.468 stable 6 0.1200 0.466 stable 7 0.1225 0.482 stable 7 0.1225 0.479 stable 8 0.1250 0.499 stable 8 0.1250 0.486 stable 9 0.1275 0.516 stable 9 0.1275 0.510 stable 10 0.1300 0.533 stable 10 0.1300 0.525 stable 11 0.1325 0.551 stable 11 0.1325 0.542 stable 12 0.1350 0.569 stable 12 0.1350 0.560 stable 13 0.1375 0.588 stable 13 0.1375 0.581 stable 14 0.1400 0.613 stable 14 ≥0.1400 – unstable 15 ≥0.1425 – unstable

# r0[AU] ψ Stability # r0[AU] ψ Stability 1 ≤0.0875 0.271 stable 1 0.0600 0.155 stable 2 0.0900 0.283 stable 2 0.0625 0.164 stable 3 0.0925 0.294 stable 3 0.0650 0.174 stable 4 0.0950 – unstable 4 0.0675 0.184 stable 5 0.0975 0.323 stable 5 0.0700 0.194 stable 6 0.1000 0.328 stable 6 ≥0.0725 – unstable 7 0.1025 0.343 stable 8 0.1050 0.355 stable 9 0.1075 0.369 stable 10 0.1100 0.382 stable 11 ≥0.1125 – unstable

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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 (superpo- sition). 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 0 of the simulation. Right: End of the simulation for M = 0.5MC (see main text).

0 The simulation was repeated with diﬀerent masses for the planet M , 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 0 primary. For the case M = 1MC , this disk did not remain existent.

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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.

0 M [MC ] Object Impacts A 616 1 B 32 C 215 A 421 0.5 B 35 C 215 A 85 0 B 6 C–

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 co- planar 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.

◦ # r0[AU] i[ ] Stability 1 0.10 10 stable 2 0.10 15 stable 3 0.10 20 stable 4 0.10 25 unstable 5 0.11 25 stable 6 0.11 30 stable 7 0.09 35 unstable 8 0.10 35 unstable 9 0.11 35 unstable 10 0.12 35 unstable

39 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 es- timation, 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 compara- bly 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 corre- −5 −2 sponding quadrupole moment. The acquired value was Q11 = −1.2 × 10 M AU . 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 un- known. 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 or- ◦ bits 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 × 10−8AUy−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 # q r0[AU] SMAS[AUy ] 1 0.5 0.11 < 10−9 2 1.0 0.11 6.0 × 10−8 3 2.0 0.11 7.4 × 10−8 4 10.0 0.11 8.2 × 10−7 5 1.0 0.08 < 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 per- turbing 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.

−2 # r[AU] Q11[M AU ] 1 2.00 −1.50 × 10−7 2 1.00 −5.95 × 10−7 3 0.50 −2.44 × 10−6 4 0.25 −1.06 × 10−5

−2 2 3 −8 By means of minimization, I acquired Q11(r)[M AU ] = (57r + 2.4r ) 10 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.

# MB[M ] tmax[y] 1 0.50 9946 2 0.75 8977 3 1.00 4286 4 1.25 2272 5 1.50 1450

48 5 Addendum

5.2 Numerical code z x z dy x The following is the command loop of the 3-body numerical code. (Ay ) ≡ dty (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 Gmimj between the objects i and j. Only the main routine is shown.

2 2 2 r = Ri − Rj F2 = −µ d e2 + 2d1e1 + d2e 2 2 2 2 v = Vi − Vj F3 = −µ d e3 + 3d1e2 + 3d2e1 + d3e 2 2 −1 P d =√ 1/ ||r|| A2i = mi F2 2 −1 P d = d A3i = mi F3 3 2 d = d d a2 = A2i − A2j 4 3 −1 d = d d d4 = e3v + 3e2a + 3e1a˙ + ea1 5 4 2 −1 3 −1 −1 −1 2 d = d d d4 = −d d4 + 12d d3 d1 + (d2 ) ... 4 −1 2 −1 5 −1 4 e = rd −36d (d1 ) d2 + 24d (d1 ) −1 2 2 d1 = v ◦ e d4 = 8d3d1 + 6(d2) + 2dd4 2 −1 d1 = −d d1 e4 = da2 + 4 (d1a1 + d3v) + 6d2a + d4r 2 2 2 2 2 d1 = 2dd1 F4 = −µ d e4 + 4 d3e1 + d1e3 + 6d2e2 + d4e −1 P e1 = dv + d1r A4i = mi F4 3 a a F = −µd r Ri = Ri + Vi∆t 2 2 b b 2 F1 = −µ d e1 + d1e Ri = Ri + 1/2Ai∆t −1 P c c 3 Ai = mi FRi = Ri + 1/6A1i∆t −1 P d d 4 A1i = mi F1 Ri = Ri + 1/24A2i∆t e e 5 a = Ai − Aj Ri = Ri + 1/120A3i∆t f f 6 a1 = A1i − A1j Ri = Ri + 1/720A4i∆t −1 a a d2 = e1 ◦ v + e ◦ a Vi = Vi + Ai∆t 2 −1 3 −1 2 b b 2 d2 = −d d1 + 2d (d1 ) Vi = Vi + 1/2A1i∆t 2 2 c c 3 d2 = 2 (d1) + dd2 Vi = Vi + 1/6A2i∆t d d 4 e2 = da + 2d1 + d2r Vi = Vi + 1/24A3i∆t −1 e e 5 d3 = e2 ◦ v + 2e1 ◦ a + e ◦ a1 Vi = Vi + 1/120A4i∆t 2 −1 3 −1 −1 4 −1 3 a b c d e f d3 = −d d3 + 6 d d2 d1 − d (d1 ) Ri = Ri + Ri + Ri + Ri + Ri + Ri a b c d e e3 = da1 + 3 (d1a + d2v) + d3r Vi = Vi + Vi + Vi + Vi + Vi

49 6 References

6 References

Alcock C., The effect of the heavy element abundance on the evolution of stars. II - The structure of the models, Acta Astronomica, 1980 Alcock C. & Paczynski B., The effect of the heavy element abundance on the evolution of stars, The Astrophysical Journal, 1978 Bellerose J. & Scheeres D. J., Restricted Full Three-Body Problem: Application to Binary System 1999 KW4, Journal of Guidance, Control, and Dynamics, 2008 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 Eberle J., Cuntz M. & Musielak Z. E., Orbital stability of planets in binary systems: A new look at old results, International Astronomical Union, 2008 Eggenberger A., Udry S., Mayor M., Beuzit J.-L. Lagrange A. M. & Chauvin, G., De- tection and Properties of Extrasolar Planets in Double and Multiple Star Sys- tems, ASP Conference Series, 2004 Kiseleva L. G. & Eggleton P. P., The Effect of Tidal Friction and Quadrupolar Dis- tortion on Orbits of Stars or Planets in Hierarchical Triple systems, ASP Con- ference Series, 1998 Marsden S. V., Berdyugina S. V., Donati J.-F., Eaton J. A., Williamson M. H., Ilyin I., Fischer D. A., Muñoz M., Isaacson H. & Ratner M. I. , A Sun in the Spectroscopic Binary IM Pegasi, the Guide Star for the Gravity Probe B Mission, The Astro- physical Journal, 2005 Musielak Z. E., Cuntz M., Marshall E. A., & Stuit T. D., Stability of planetary orbits 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 Quarles B., Cuntz M. & Musielak Z. E., The stability of the suggested planet in the ν Octantis system: a numerical and statistical study, Monthly Notices of the Royal Astronomical Society, 2012 Quarles B., Musielak Z.E. & Cuntz, M., Study of resonances for the restricted 3-body problem, WILEY-VCH Verlag GmbH & Co., 2012 Runge-Kutta Method for Solving Two Coupled 1st Order Differential Equations or One 2nd Order Differential Equation, phy.davidson.edu/fachome/dmb/py200 /RungeKuttaMethod.htm Nolting W., Grundkurs Theoretische Physik 3, Springer Verlag, 2002 Pozrikidis C., Numerical Computation in Science and Engineering, Oxford University Press, 1998

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!