Calculation

Solve the equation of orbital motion (see Exercise 13) using the fourth order Runge-Kutta method.

The 4th order Runge-Kutta Method method was described in Lecture and is summarized below for the first order equation:

The value at the next time step is found from:

where

To solve this equation, we need to supply initial conditions. Let the calculation start at x=1 AU, y=0 AU, and z=0 AU. Let the initial velocity in the x and z directions be 0, and write a script that solves this differential equation for an initial velocity in the y direction that is input by the user.

Please make a subdirectory for this exercise under your directory ~schloerb/ph281/username and then copy the following into that subdirectory:

• a copy of your script (if you call functions, include copies of these as well).
• a plot of the complete orbit of an object with an initial velocity of 8 AU per year.

Extra Credit

Consider the accuracy of the 4th order Runge-Kutta method compared to the 2nd order Runge-Kutta method. Using a circular orbit as a metric, compare the ability of the two methods to complete one orbit and return precisely to the originating point. Make a graph of the error after one orbit versus time step size for the two methods.

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