The Orbit of Kepler 16b NASA's Kepler space telescope recently made the news by finding a planet that orbits a double-star system, a situation that brought to mind the fictional planet Tatooine of the movie Star Wars. On such a planet, if it had a solid surface, you could see a double sunset. 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This worksheet explores the orbital mechanics of such a system. The orbital properties of the Kepler 16 system are as follows. There are two stars, Kepler 16A with mass 0.6897 times the mass of our Sun, and Kepler 16B with mass 0.20255 times the mass of our Sun. They orbit their common centre of mass in elliptical orbits with a period of 41.079220 days. The semi-major axis is 0.22431 AU (where 1 AU is the semi-major axis of the Earth's orbit around the Sun) and the eccentricity is 0.15944. The planet Kepler 16b has a period of 228.776 days, a semi-major axis 0.7048 AU, and eccentricity 0.0069. The two stars and the planet are almost in the same plane - I will simplify matters slightly by assuming they are exactly in the same plane.

I'll start with the two stars (known as Kepler 16A and Kepler 16B), which orbit their common centre of mass in elliptical orbits. According to NASA, the orbital data are as follows: Mass of Kepler 16A 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 Mass of Kepler 16B 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 Semi-major axis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEoMC4yMjQzMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSNBVUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0Yv Eccentricity 0.15944 Period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEqNDEuMDc5MjIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEifkYnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1JI21pR0YkNiVRJWRheXNGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGLw== Here LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW9HRiQ2LVEsJkNpcmNsZURvdDtGJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdGUUY+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRidGPg== is the mass of our Sun, and 1 AU is approximately the mean distance from the Earth to the Sun. The gravitational attraction of the two stars causes them both to orbit around their common centre of mass in elliptical orbits. The semi-major axis figure above actually refers to the displacement vector from one to the other. Kepler's Third Law (this is the 17th century Kepler, not the space telescope) takes the form 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 T is the period of the orbit, a the semi-major axis, G the gravitational constant, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIkFGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIkJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn the masses of the stars. Let's verify this with Maple. I'll use the subscript s to refer to the stars, as in the next section there will be other data for the planet. 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 QyQ+SSlUaGlyZExhd0c2Ii8qJCZJIlRHRiU2I0kic0dGJSIiIywkKipJI1BpRyUqcHJvdGVjdGVkR0YsSSJHR0YlISIiLCYmSSJtR0YlNiNJIkFHRiUiIiImRjU2I0kiQkdGJUY4RjImSSJhR0YlRioiIiQiIiVGOA== We need to simplify the units. QyQtSSlzaW1wbGlmeUc2IjYjSSIlR0YlIiIi That's reasonably good agreement, given the uncertainties in the measurements. Actually, the period is one of the few parameters that can be observed directly, and the others are estimated using, among other things, Kepler's Third Law. I will redefine LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRicvRjZRJ25vcm1hbEYn so that the Third Law is satisfied exactly (up to roundoff error) with the given value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn. PiZJImFHNiI2I0kic0dGJS1JKGNvbnZlcnRHJSpwcm90ZWN0ZWRHNiUtSSlzaW1wbGlmeUc2JEYqSShfc3lzbGliR0YlNiMqJCoqJkkiVEdGJUYmIiIjSSJHR0YlIiIiLCYmSSJtR0YlNiNJIkFHRiVGNyZGOjYjSSJCR0YlRjdGNy1JIi9HRi42IywkKiRJI1BpR0YqRjUiIiVGNyNGNyIiJEkmdW5pdHNHRiUtSSVVbml0R0YlNiMuSSNBVUdGJQ== A polar equation of an ellipse with semi-major axis a and eccentricity \316\265 and one focus at the origin is 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 Stars A and B will be on opposite sides of their common centre of mass (which we take as the origin), and their distances from the origin are inversely proportional to their masses, so star A's orbital path is PiZJInJHNiI2I0kiQUdGJSosJkkibUdGJTYjSSJCR0YlIiIiLCYmRipGJkYtRilGLSEiIiZJImFHRiU2I0kic0dGJUYtLCZGLUYtKiQmSShlcHNpbG9uR0YlRjMiIiNGMEYtLCZGLUYtKiZGN0YtLUkkY29zR0YlNiNJJnRoZXRhR0YlRi1GMEYw while star B's, in terms of the same parameter \316\270, isLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= PiZJInJHNiI2I0kiQkdGJSwkKiwmSSJtR0YlNiNJIkFHRiUiIiIsJkYqRi4mRitGJkYuISIiJkkiYUdGJTYjSSJzR0YlRi4sJkYuRi4qJCZJKGVwc2lsb25HRiVGNCIiI0YxRi4sJkYuRi4qJkY4Ri4tSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmdGhldGFHRiVGLkYxRjFGMQ== Here, then, are the orbits of the two stars. I'll save the plot for future use. QygtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnBsb3RzR0YlISIiLUkoZGlzcGxheUdGKDYmLUklcGxvdEdGKDYmJkkickdGKDYjSSJBR0YoL0kmdGhldGFHRig7IiIhLCRJI1BpR0YmIiIjL0knY29vcmRzR0YoSSZwb2xhckdGKC9JJ2NvbG91ckdGKFEqR29sZGVucm9kRigtRjA2JiZGMzYjSSJCR0YoRjZGPS9GQUkkcmVkR0YoL0koc2NhbGluZ0dGKEksY29uc3RyYWluZWRHRigvSSdsYWJlbHNHRig3JEkhR0YoRlAiIiI+SStTdGFyT3JiaXRzR0YoSSIlR0YoRis= We'll also want to see how the positions depend on time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. By Kepler's second law (conservation of angular momentum) 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 is constant: the constant turns out to be PiZJImhHNiI2I0kic0dGJS1JJXNxcnRHRiU2IyoqLCYiIiJGLSokJkkoZXBzaWxvbkdGJUYmIiIjISIiRi0mSSJhR0YlRiZGLUkiR0dGJUYtLCYmSSJtR0YlNiNJIkFHRiVGLSZGODYjSSJCR0YlRi1GLQ== A fine stew of units that is! QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjJkkiaEdGKDYjSSJzR0YoIiIi But it would be more convenient to use AU for distance and days for time. LUkoY29udmVydEclKnByb3RlY3RlZEc2JUkiJUc2IkkmdW5pdHNHRictSSVVbml0R0koX3N5c2xpYkdGJzYjKiYuSSNBVUdGJyIiIy5JJGRheUdGJyEiIg== I actually just want the number. QyQ+JkkiaEc2IjYjSSJzR0YmLUkoY29udmVydEclKnByb3RlY3RlZEc2JEkiJUdGJkkqdW5pdF9mcmVlR0YmIiIi While we're at it, I'll also remove the units from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, which is already expressed in AU, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYnJSFH, which is already in days. QyY+JkkiYUc2IjYjSSJzR0YmLUkoY29udmVydEclKnByb3RlY3RlZEc2JEYkSSp1bml0X2ZyZWVHRiYiIiI+JkkiVEdGJkYnLUYqNiRGMEYtRi4= We might try to solve the differential equation for \316\270LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2L1EifkYnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRklGNQ== as a function of t. QyQtSSdkc29sdmVHNiI2JDwkLyomLUkiXkclKnByb3RlY3RlZEc2JComLSZJImFHRiU2I0kic0dGJTYjLCYiIiJGNiokJkkoZXBzaWxvbkdGJUYyIiIjISIiRjYsJkY2RjYqJkY4RjYtSSRjb3NHNiRGLEkoX3N5c2xpYkdGJTYjLUkmdGhldGFHRiU2I0kidEdGJUY2RjtGO0Y6RjYtSSVkaWZmR0YsNiRGQ0ZGRjYmSSJoR0YlRjIvLUZENiMiIiFGT0kpaW1wbGljaXRHRiVGNg== But it is impossible to get a "closed-form" formula to solve this, so instead we use a numerical solution. QyQ+SSVzb2wxRzYiLUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiU8JC8qJi1JIl5HRik2JCooJkkiYUdGJTYjSSJzR0YlIiIiLCZGN0Y3KiQmSShlcHNpbG9uR0YlRjUiIiMhIiJGNywmRjdGNyomRjpGNy1JJGNvc0dGKDYjLUkmdGhldGFHRiU2I0kidEdGJUY3Rj1GPUY8RjctSSVkaWZmR0YpNiRGQ0ZGRjcmSSJoR0YlRjUvLUZENiMiIiFGT0kobnVtZXJpY0dGKS9JJ291dHB1dEdGJUkubGlzdHByb2NlZHVyZUdGJUY3 QyQ+SSZUaGV0YUc2Ii1JJXN1YnNHJSpwcm90ZWN0ZWRHNiRJIiVHRiUtSSZ0aGV0YUdGJTYjSSJ0R0YlIiIi Let's check that after one period we're back at the start. QyQtSSZUaGV0YUc2IjYjJkkiVEdGJTYjSSJzR0YlIiIi This is (up to slight numerical error) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dGOEYvRi8=, i.e. one revolution. QyQtSSZldmFsZkclKnByb3RlY3RlZEc2IywmLUkmVGhldGFHNiI2IyZJIlRHRio2I0kic0dGKiIiIkkjUGlHRiUhIiNGMA== Thus at time t the radial coordinate (in AU) for the two stars are QyY+JkkiUkc2IjYjSSJBR0YmLUkoY29udmVydEclKnByb3RlY3RlZEc2JC1JJWV2YWxHRis2JCZJInJHRiZGJy9JJnRoZXRhR0YmLUkmVGhldGFHRiY2I0kidEdGJkkqdW5pdF9mcmVlR0YmIiIiPiZGJTYjSSJCR0YmLUYqNiQtRi42JCZGMUY8RjJGOEY5 Here is an animation of one orbit of the stars. The relative sizes of the yellow and red dots correspond to those of the two stars (but they are not to scale relative to the orbits). 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

The planet, known as Kepler 16b, has the following orbital data: Period 228.776 days Semi-major axis 0.7048 AU eccentricity 0.0069 This makes it sound like the orbit is elliptical (and quite close to a circle, because the eccentricity is so small). But that can't really be true, because there are two stars pulling the planet in different directions. This is a "three body problem". There are very few exact results for three body problems, so we will have to depend on numerical solutions of the system of differential equations. The deviations from an elliptical orbit turn out to be fairly significant, and these orbital elements are really only referriing to conditions at some particular point in time. Here are those orbital elements for the planet. Again, the units are AU and days. QyQ+JkkoZXBzaWxvbkc2IjYjSSJiR0YmJCIjcCEiJSIiIg==QyQ+JkkiYUc2IjYjSSJiR0YmJCIlW3EhIiUiIiI=QyQ+JkkiVEc2IjYjSSJiR0YmJCIndyhHIyEiJCIiIg==NiI= I'll take an initial condition at distance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRldGRA== from the centre on the positive y axis, but I don't know what initial velocity to take. For the moment I'll just leave it as a parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+. Compared to the stars, the planet's mass is quite small, so I will neglect the planet's effect on the stars: that makes this a "restricted three body problem". I could use the previous section's \316\230 to describe the motion of the stars, but it may be more efficient to include the differential equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg== in the system we solve. Here are the initial conditions, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn are the x and y coordinates of the planet. I'm assuming for simplicity that the three bodies are in the x-y plane: in fact they are not quite coplanar, but that shouldn't make much of a difference. QyQ+SSRpY3NHNiI8Jy8tSSZ0aGV0YUdGJTYjIiIhRisvLSZJInhHRiU2I0kiYkdGJUYqRisvLSZJInlHRiVGMEYqJkkiYUdGJUYwLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGLkYqLCQmSSJ2R0YlRiohIiIvLS1GOzYjRjRGKkYrIiIi I'll need the products of the gravitational constant G with the masses of the two stars, expressed in the appropriate units. QyY+JkkjZ21HNiI2I0kiQUdGJi1JKGNvbnZlcnRHJSpwcm90ZWN0ZWRHNiQtRio2JS1JKXNpbXBsaWZ5RzYkRitJKF9zeXNsaWJHRiY2IyomSSJHR0YmIiIiJkkibUdGJkYnRjZJJnVuaXRzR0YmLUklVW5pdEdGMjYjKiYuSSNBVUdGJiIiJC5JJGRheUdGJiEiI0kqdW5pdF9mcmVlR0YmRjY+JkYlNiNJIkJHRiYtRio2JC1GKjYlLUYwNiMqJkY1RjYmRjhGR0Y2RjlGOkZERjY= Here are the x and y coordinates of the stars, according to the analysis from the previous section. Note that I'll be neglecting the gravitational effect of the planet on the stars, since the planet is much smaller in mass, making this a "restricted three-body problem". The effect is actually quite small but measurable (the planet has approximately 1/3 the mass of Jupiter, or about 0.00046 the mass of Kepler 16A). . Qy4+JkkiUkc2IjYjSSJBR0YmLUkoY29udmVydEclKnByb3RlY3RlZEc2JC1JJWV2YWxHRis2JCZJInJHRiZGJy9JJnRoZXRhR0YmLUYzNiNJInRHRiZJKnVuaXRfZnJlZUdGJiEiIj4mRiU2I0kiQkdGJi1GKjYkLUYuNiQmRjFGO0YyRjdGOD4mSSJ4R0YmRicqJkYkIiIiLUkkY29zRzYkRitJKF9zeXNsaWJHRiY2I0Y0RkZGRj4mSSJ5R0YmRicqJkYkRkYtSSRzaW5HRklGS0ZGRkY+JkZERjsqJkY6RkZGR0ZGRkY+JkZORjsqJkY6RkZGUEZGRkY= The first two differential equations express the components of acceleration of the planet in terms of the gravitational attraction of the stars; the third is the differential equation for \316\270 that we used in the previous section. 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 We can solve the differential equation numerically. In this first attempt, I'll be trying to find a value for the parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+ so that the planet comes back to the y axis at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn. In order to do that, I first use dsolve with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+ as parameter. QyQ+SSRzb2xHNiItSSdkc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JS1JJnVuaW9uR0YpNiRJJGRlc0dGJUkkaWNzR0YlSShudW1lcmljR0YpL0krcGFyYW1ldGVyc0dGJTcjJkkidkdGJTYjIiIhIiIi Now write a procedure that takes a value v, sets that as the parameter, then solves the differential equation to time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and returns the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlLUYvNiVRIlRGJ0YyRjVGOEY9L0Y2USdub3JtYWxGJ0ZKRko=. QyQ+SSJGRzYiZio2I0kidkdGJUYlRiVGJUMkLUkkc29sR0YlNiMvSStwYXJhbWV0ZXJzR0YlNyM5JC1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQtRis2IyZJIlRHRiU2I0kiYkdGJS0mSSJ4R0YlRjk2I0kidEdGJUYlRiVGJSIiIg== This can then be used in fsolve to find the desired parameter value that makes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInZGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZULUZANi1RIj1GJ0Y9RkNGRkZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGZW5GPy1JI21uR0YkNiRRIjBGJ0Y9Rj0=. Unevaluation quotes around LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInZGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== are needed to prevent premature evaluation (we don't want to call F with a symbolic parameter). Since the desired value should be close to 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, which is the speed at which you would go around a circle of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn in time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, I'll give that as an initial value to fsolve. I'll assign the resulting value to the variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RIi5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRldGPg== QyQ+Jkkidkc2IjYjIiIhLUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiQuLUkiRkdGJjYjRiUvRiUsJCooSSNQaUdGLCIiIiZJImFHRiY2I0kiYkdGJkY3JkkiVEdGJkY6ISIiIiIjRjc= Now let's solve the differential equation for up to about 100 orbits ("about" because not all will have the same return time). I'll set the option LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnbWF4ZnVuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== to 0 so Maple won't complain about too many function evaluations. QyQ+SSRzb2xHNiItSSdkc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Ji1JJnVuaW9uR0YpNiRJJGRlc0dGJUkkaWNzR0YlSShudW1lcmljR0YpL0kmcmFuZ2VHRik7IiIhLCQmSSJUR0YlNiNJImJHRiUiJCwiL0knbWF4ZnVuR0YlRjUhIiI= Here is the first orbit of the planet (in black), as produced by odeplot. For comparison, I'll also include the circle of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn (in green). QyQ+SSNiZ0c2Ii1JKGRpc3BsYXlHRiU2JEkrU3Rhck9yYml0c0dGJS1JJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JiZJImFHRiU2I0kiYkdGJS9JJnRoZXRhR0YlOyIiISwkSSNQaUdGLSIiIy9JJ2NvbG91ckdGJUkmZ3JlZW5HRiUvSSdjb29yZHNHRiVJJnBvbGFyR0YuISIi QyYtSShkaXNwbGF5RzYiNiQtSShvZGVwbG90R0YlNidJJHNvbEdGJTckLSZJInhHRiU2I0kiYkdGJTYjSSJ0R0YlLSZJInlHRiVGL0YxL0YyOyIiISZJIlRHRiVGLy9JJ3JlZmluZUdGJSIiJC9JJ2NvbG91ckdGJUkmYmxhY2tHRiVJI2JnR0YlIiIiPkkpT25lT3JiaXRHRiVJIiVHRiUhIiI= In most places the orbit is very close to the green circle. However, the later orbits won't repeat (because the stars will be in different positions). Here is an animation of 100 frames, each one showing the orbit over a time interval of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn. 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 It looks like after somewhere around 79 orbits we're back to the circle, and then the process repeats. We'd like to think that Kepler 16 is a stable system, so this can go on indefinitely. It is not at all obvious that this would be the case. The planet's energy is not conserved; it can either gain energy from the stars or lose it to the stars. If it gained more than it lost over the course of time, it might be pushed into orbits farther and farther away. Or its orbits might become more and more eccentric, eventually perhaps colliding with a star or being expelled from the system. It seems that doesn't happen here. Laurance R. Doyle et al, the authors of the paper in Science about the Kepler 16 system, computed the motion a million years into the future (perhaps with rather more computing power and patience than what I have available here) and report that it does seem to be stable. I'll content myself with following the system up to 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 or about 450 Earth years. I'll use dsolve with the output=Array(...) option to record data at intervals of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, the orbital period of the stars. QyQ+SSRSZXNHNiItSSdkc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Ji1JJnVuaW9uR0YpNiRJJGRlc0dGJUkkaWNzR0YlSShudW1lcmljR0YpL0knb3V0cHV0R0YlLUkmQXJyYXlHRik2IzcjLUkkc2VxR0YpNiQqJkkiaUdGJSIiIiZJIlRHRiU2I0kic0dGJUY9L0Y8OyIiISIlK1MvSSdtYXhmdW5HRiVGREY9 This result has a rather strange format: a 2 \303\227 1 Matrix where the [1,1] entry tells us what each column of the data signifies, and the [2,1] entry is a Matrix with the data itself, one row for each entry of the Array specified in the dsolve command. QyQ+SSVEYXRhRzYiJkkkUmVzR0YlNiQiIiMiIiJGKg== First I'll plot the data in column 5, which gives us LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ0RidGMkY1L0Y2USdub3JtYWxGJ0ZIRkg= after each revolution of the stars. 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 It looks like the upper and lower envelopes of these points follow sinusoidal curves with a period of about 18000 days or about 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. Taking a closer look, let's look for data points very close to the starting point. The distance function I'll use will be the sum of the absolute differences of entries 3 to 6 of the row (giving LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and their derivatives). QyQ+SSVkaXN0RzYiZio2I0kiaUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkYWRkRyUqcHJvdGVjdGVkRzYkLUkkYWJzR0YuNiMsJiZJJURhdGFHRiU2JDkkSSJqR0YlIiIiJkY1NiRGOUY4ISIiL0Y4OyIiJCIiJ0YlRiVGJUY5 How close do any of the later data points get to our starting point? QyQtSSRtaW5HJSpwcm90ZWN0ZWRHNiMtSSRzZXFHRiU2JC1JJWRpc3RHNiI2I0kiaUdGLC9GLjsiJSs/IiUsUyIiIg== And which ones are not too much farther than that from the starting point? QyQtSSdzZWxlY3RHJSpwcm90ZWN0ZWRHNiRmKjYjSSJpRzYiRio2JEkpb3BlcmF0b3JHRipJJmFycm93R0YqRioyLUklZGlzdEdGKjYjOSQkIiM3ISIkRipGKkYqNyMtSSIkR0YlNiM7IiIiIiUsU0Y7 QyQtSSVkaXN0RzYiNiMiJTVSIiIi QyQsJComJkkiVEc2IjYjSSJzR0YnIiIiLUkiL0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjJCImRGwkISIjRioiJTRSRio= So at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRJTM5MTBGJ0Y5Rjk= (i.e. after 3909 revolutions of the stars, or about 439.64 Earth years, both the stars and the planet are in very close to the same configuration I started them in. Here are the two data points: QyQ2JCZJJURhdGFHNiI2JCIiIjtGKCIiJyZGJTYkIiU1UkYpRig= The second entries, the values of \316\270, look different, but the difference should be very close to 3909 times LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dGOEYvRi8=. QyQtSSZldmFsZkclKnByb3RlY3RlZEc2IyomJkklRGF0YUc2IjYkIiU1UiIiIyIiIi1JIi9HNiRGJUkoX3N5c2xpYkdGKjYjLCRJI1BpR0YlRi1GLkYu

It might be interesting to look at other starting positions for the planet. We should expect the orbit to become unstable if it was much closer to the stars. There also could be interesting situations where the planet's orbit is "phase-locked" to the stars, where the ratio of periods of planet and stars is a rational number with small numerator and denominator. Kepler 16b's orbital period is not too far from 5 times the period of the stars: what if it was exactly 5 times? Would this lead to stable or unstable orbits? One could also try to confirm that the motion of Kepler 16b is quasi-periodic rather than chaotic, perhaps by computing Liapunov exponents. These are left as "exercises" for the reader.