Simulations of Binary Star System with Earth Mass Planet Craig Swenson

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1 Simulations of Binary Star System with Earth Mass Planet Craig Swenson For this project I wanted to investigate that stability of planetary orbits in binary systems. Logic dictates that for systems where the binary stars are very close together (very little motion in center of mass), planets could be found in stable orbits. Habitable planets would need to be closer to the stars and would likely be more effected by the moving center of mass, which could make it more difficult for stable orbits to form. I will experiment with several different positional and velocity parameters and attempt to find (in the spirit of Problem 5.11) stable planetary orbits that encircle both stars, [create] figure eights, [and]...encircle only one star. Physics For a three body system, Newton's Universal Gravitation Law gives us the following equations, F 1 = Gm 1 m 2 r 21 2 r 21 Gm m 1 3 r 31 2 r 31 F 2 = Gm 1 m 2 r 12 2 r 12 Gm m 2 3 r 32 2 r 32 F 3 = Gm 1 m 3 r 13 2 r 13 Gm m 2 3 r 23 2 For my purposes I will define Gm 1 = Gm 2 = 4π 2 for the binary stars, and let Gm 3 = x4π 2 so as to create an artificial Earth massed planet. Because the step-size necessary may vary during the computation (shorter step-size needed when planet is moving quickly near a star), I will be using the RK45Multistep ODE solver. Simulations In order to make it easier to read the graphs, I have colored the stars and planet, as well as their trails. NOTE: In order to color each mass, as well as their trails, I was required to change the Color variable in the Circle class and the AbstractTrail class to be STATIC variables. Mass1 and Mass2 are the binary stars, and are colored red and green, respectively. Leaving the Earth massed planet as the blue circle and trail. The first stable orbit mentioned by Problem 5.11, and the orbit that I would assume is the most common and most easily achieved, is that of the star orbiting around both stars. Figure 1 shows a stable orbit with the planet at a distance of ~10 AU from the center of mass of the system. Close inspection shows that the orbit is not closed, which is due to the slight movement of the center of mass. Parameters used for the orbit are shown in Table 1. r 23

2 Parameter Value x1 2 vy1 2 x2-2 vy2-2 x3-10 vx3 - vy3-2.8 Table 1. Parameters for Figure 1. Figure 1. An Earth massed planet orbiting around two solar massed stars. Stable orbits are easily achieved at these distances.

3 Although the orbit is stable, the planet would not be habitable. At an average distance of 10 AU, the planet would be far too cold for live. The second case of a stable orbit is that of the planet directly orbiting around one star in the binary system. This will allow the planet to be close to at least one of the stars at all times, possibly making it warm enough to be habitable. This case should be more difficult to find correct parameters for, because the planet is effected more by the stars individually, as compared to the previous case where the planet's distance makes the center of mass more important. Figure 2 shows an example of a stable, but not closed, orbit of the planet primarily orbiting one star. The scale has been changed to make the figure less cluttered. Table 2 provides the parameters used. Minor variations of these parameters, especially x3 and vy3, yield dozens stable orbits all with slightly different ellipticity for the planet, but, surprisingly, this configuration also seems to be able to produce stable orbits relatively easily. Parameter Value x1 2 vy1 2 x2-2 vy2-2 x3-2.5 vx3 0 vy3-12 Table 2. Parameters for Figure 2.

4 Figure 2. Earth massed planet in primary orbit around one of the two solar massed binary stars. Orbit is stable, but is not closed. The average distance between the planet and the star it is primarily orbiting is ~1-2 AU which would definitely make it habitable. This case, though not as trivial as the first case, is still relatively easy to produce. The third case, and what I assume will be the most difficult to model, is that of a planet in a figureeight type model orbiting both stars equally. This is a much more exotic type of orbit and should be difficult to achieve. Because the planet will be alternating back and forth between which star is exerting the greater force, the parameters will be very difficult to set. If the planet ever gets too close to one of the stars, the extra acceleration will eventually through it out of the stable orbit. Figure 3 shows a stable orbit with the planet in a figure-eight type orbit around the stars. Table 3 provides the parameters.

5 Parameters Value x1 1 vy1 3.1 x2-1 vy2-3.1 x3-2.5 vx3 0 vy3 2 Table 3. Parameters for Figure 3. Figure 3. Earth massed planet in stable figure-eight like orbit around two solar massed binary stars.

6 To illustrate how difficult it is to achieve this type of orbit, I ran a second simulation and varied the initial vy1. You can actually see the instability growing on the angular momentum plot (Figure 4). As the angular momentum jumps back and forth, eventually it reaches the point where it can't recover and the planet is lost. Figure 5 shows the results of a.1 difference in vy1 with all other parameters held constant. Table 4 shows the new set of parameters. Parameter Value x1 1 vy1 3 x2-1 vy2-3.1 x3-2.5 vx3 0 vy3 2 Table 4. Modified parameters for Figure 4. Figure 4. Visible spike in angular momentum at approximately t = 6. System never recovers.

7 Figure 5. Moderate deviations from the stable configuration result in catastrophic consequences. It is very likely that a planet in an orbit like that shown in Figure 3 would be habitable. The planet has a minimum distance of ~1 AU from either star and a maximum of 2.5 AU. Given the right types of stars in the binary system, it is possible that the entire planetary orbit would be in the habitable zone. Unfortunately, it would be nearly impossible to achieve this type of orbit, because the amount of tolerable variation is so small that even mild perturbations can cause the planet to fall into one of the stars or be flung into empty space. All of these simulations were performed with only one planet. The introduction of a more planets would only serve to make things increasingly difficult. It is likely that the amount of tolerable variation in parameters would be even less with the introduction of more planets and the possible consequences even more extreme (