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 approx- imately 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 pro- grade 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 stud- ies, 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. 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 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 expe- rienced 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. 6 2 Theory Figure 3: Left: Diagram, depicting six orbital parameters.