The Kepler Problem | Astronomy Computational Problem Set 1, Exercises of Astronomy

Download The Kepler Problem | Astronomy Computational Problem Set 1 and more Exercises Astronomy in PDF only on Docsity! MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Earth, Atmospheric, and Planetary Sciences Department Astronomy 8.282J­12.402J February 28, 2006 Computational Problem Set 1 The Kepler Problem Optional Suggested Procedure: The differential equation of motion for the Kepler problem, in vector form, is: d2�r dt2 = − GM ̂r r2 = − GM�r r3 (1) In Cartesian coordinates we have: d2x dt2 = − GM x r3 (2) d2y dt2 = − GM y r3 (3) As four, first­order, coupled equations: dx dt = vx (4) dy dt = vy (5) dvx dt = − GM x r3 (6) dvy dt = − GM y r3 (7) Equations (4) through (7) can be solved by a simple 4th­order Runge­Kutta integration scheme. A few pages of text describing this method, taken from the Numerical Recipes book, are attached. In the language of the RK4 program (Numerical Recipes’ Runge­Kutta program): Y = the four independent variables (x, y, vx, vy ) X = the time t N = 4 (8 for the case of the two­body problem) H = the time step Δt DERIVS = user supplied analytic derivatives, [right hand sides of (4) through (7)] YOUT = are the incremented Y variables after time step H Or, you can use the canned integrators in mathematical packages such as Matlab, Mathematica, IDL, etc. 1