Planetary Orbits Around Binary Star Systems A. A. Lascelles School of Physics and Astronomy University of Southampton SO17 1BJ United Kingdom E-mail: [email protected] May 3, 2016 Abstract A simulation of planetary orbits around a binary star system using a Runge-Kutta procedure is presented. Through choices of initial velocities and positions, orbit stability is tested for a range of P- and S-type orbits confined to a disc in the x-y plane around the central stars. The boundary conditions and time-evolution of such orbits is explored. Chaos theory, habitable zones, and other such related phenomena are discussed. 1 1 Introduction Binary stars are systems containing two companion stars orbiting each other via gravi- tational attraction with a common centre of mass. It is generally believed that more than half of all star systems we observe contain multiple stars[1]. The search for a habitable exoplanet is currently an active area of research in astronomy, and recent results confirm the existence of exoplanets that orbit binary star systems[2]. Studying such planets is of great importance for exo-biology. For life to exist, the planet on which it lives must keep to a stable orbit over long time-scales. This is needed because the time for protoplanetary formation and for evolution would encompass billions of orbital periods. There are two species of orbit possible for a planet in a binary system, namely P-type and S-type. P-type orbits will orbit the combined centre of mass of the two stars as if they are one single body whereas planets in an S-type orbit only travel around one of the stars with the other acting as a perturbation[3]. To solve the differential equations involved in a 3-body system, the equations of motion must first be derived (as shown in appendix A). This assumes Newtonian physics is appropriate, and combines Newton's 2nd law with the gravitational attraction of the two stars. Satellites do not strictly follow this motion because of perturbations from other planets, comets, asteroids, and even effects such as accretion changing the gravitational field. However, for the scope of this paper, where we neglect such effects, they can be considered valid. Throughout, we assume that masses are in units of solar masses (M ) so that M = 1M , distances are in terms of astronomical units (AU) so that R = 1AU, and time in units of years. It is further assumed that the system is confined to a 2D plane, as we observe in most solar systems. In the following section, the methodology of this paper will be discussed. The two-fold method includes mathematical manipulation of differential equations, derived in appendix A, in order to achieve a form that can be computationally evaluated. The purpose of Sec- tion 3, which forms the main bulk of the scientific findings, is to explain the physics behind the various orbits found. Boundary conditions for each type of orbit will be discussed along with the reasons behind the shape of the S-type, P-type, circular, and elliptical orbits found together with the extension detailing chaotic orbits and habitable zones. Section 4 concludes the findings of this paper, whilst the appendix is where derivations of the equations of motion used herein can be found. 2 2 Methodology 2.1 Manipulation of Differential Equations In order to make the differential equations suitable for the computational Runge-Kutta algorithm, they are first non-dimensionalised and then converted into four first-order equations. These four equations can be sub-divided into two representing the x-velocity and x-acceleration and two representing the y-velocity and y-acceleration as detailed in the following paragraphs. First consider a pair of stars obeying Kepler's laws of motion and rotating with a period T around their common centre of mass on a circle of radius R. Their Cartesian coordinates will be given by, X1 = R cos(Ωt);Y1 = R sin(Ωt); (1) X2 = −R cos(Ωt);Y2 = −R sin(Ωt): (2) Introducing a planet of negligible mass into this scenario creates the 3-body system where the position, r1;2, of the planet from star 1 or star 2 is given by, 2 2 2 r1;2 = (x − X1;2) + (y − Y1;2) : (3) Here, R is the radial distance coordinate, where the stars are set apart at a distance R, r1;2 are the radial distances of the planet from star 1 and 2 respectively, X1;2 and Y1;2 are the distances from the origin of star 1 and 2 respectively, x and y are the positions of 2π the planet with respect to the origin, and angular frequency Ω = T is the inverse of the period of orbit, T, for the two stars. This allows us to write the equations of motion for a 3-body system in the x and y directions (see appendix A), 1 x − X1 1 x − X2 x¨ = −GM 2 + 2 ; (4) r1 r1 r2 r2 1 y − Y1 1 y − Y2 y¨ = −GM 2 + 2 ; (5) r1 r1 r2 r2 where G is the gravitational constant and M is the mass of each star (equal mass stars are assumed). From Eq. (4) and (5) we can define: dx v = ; (6) x dt 3 and dy v = ; (7) y dt for the x and y velocities of the planet respectively - forming two of the four first order equations required. Subsequently, by a reformation of the two equations of motion in terms of the dimen- 0 0 0 0 0 0 0 0 sionless variables t , x , y , vx, vy, r1;2, X1;2, and Y1;2, we can derive the final two first order equations needed as the inputs for the Runge-Kutta algorithm. The replacements are as follows, r t0 = Ωt; r0 = 1;2 ; (8) 1;2 R x y x0 = ; y0 = ; (9) R R X Y X0 = 1;2 ;Y 0 = 1;2 ; (10) 1;2 R 1;2 R 0 0 where vx and vy come from the implementation of these replacements in Eq. (4) and (5). After these substitutions and exploiting the connection between R and Ω { GM = R3Ω2 from Kepler's 3rd law, we arrive at, 0 0 0 0 0 dvx 1 dvx x − X1 x − X2 0 = 2 = − 0 3 + 0 3 ; (11) dt RΩ dt r1 r2 and the equivalent in the y-direction, 0 0 0 0 0 dvy 1 dvy y − Y1 y − Y2 0 = 2 = − 0 3 + 0 3 : (12) dt RΩ dt r1 r2 Hence, we have found the four first-order differential equations which act as the input for the Runge-Kutta algorithm - Eq. (6), (7), (11) and (12). 2.2 Runge-Kutta Algorithm The first-order equations listed in the previous section are input into the Runge-Kutta algorithm and four variables { x-position, y-position, x-velocity, and y-velocity are allowed to be changed manually in order to search for special types of orbit. When investigating certain orbit types, it is more useful to view them in the rotating frame of the stars. For example, if the planet moves in an ellipse around one star faster than the orbital period T, then in the fixed frame we would see a very complex motion opposed to simple 4 ellipses/circles in the rotating frame. The rotating frame is established through a change of variables when plotting the final function of the orbit, namely, ξ = x cos Ωt + y sin Ωt η = −x sin Ωt + y cos Ωt: (13) This is computed and plotted alongside the fixed frame to investigate whether the orbit is stable or not. Orbits can be narrowed down by whether they return to a similar place in the rotating frame whilst precessing or if they form closed loops indicating they had completed an orbit and returned to their initial positions. With this established, the next step is to enter the initial conditions for displacement and velocity and loop through the possibilities in order to search for interesting orbits. A function is defined that contains all of the initial conditions, including the length of time for which to run the program, τ, and finally calls the Runge-Kutta process. A suitable step size, N −1, and value for τ are chosen which limits the error appropriately at each calculated variable whilst keeping the calculation time to a minimum. In order to determine this step size, the process is repeated for varying step sizes until the calculated orbit remained relatively constant regardless of changes in this parameter. Then, using this value for step size, a test is run with varying values of τ and a value is chosen which gave the full orbit with sufficient iterations to check stability but not too many iterations that time is wasted. For the most part, the value of τ chosen is 100,000 and a step size of 0.001. Using error analysis[4], the value for the fourth-order Runge-Kutta error is found to be 1.00x10−8. To investigate orbits around a single combined centre of mass or a single star (so-called `P-type' orbits), the star-planet separation is made to be large. If sufficiently large, no matter what change in displacement along the x or y axis, a circular or elliptical orbit is returned. It is found that starting the planet on the x axis as opposed to on the y axis produces a change in the final orbit due to the stars being initially aligned on the x axis at positions (0,R) and (0,-R), where R is the radius of the circular path that they follow.