# Coupled ODE Solver: Plantary Motion

Description| How it works| Planetary Motion
To download the file used in this example with the equations set-up,
use these links:

.xls (142 KB)

.zip (61 KB)

The ODE solver tool is used to solve the following Differential
Algebraic Equation:

Y1" = - k(Y1)/R^{3}

Y2" = -k(Y2)/R^{3}

where R^{2} = Y1^{2} + Y2^{2}

We take k = 1 for convinience - the idea is to get a feel for the
solution and not get bogged down with the numbers.

The motion of planets is governed by such equations (with a
different k). Y1 and Y2 are the co-ordinates of the planet and R is
the distance of the planet from the star. The star is located at the
origin, and it is assumed to be much heavier than the planet (so its
motion is negligible).

To use the ODE solver tool for this system, we have to break up the
system into a set of first order equations. We do this by
introducing 2 variables Y3 and Y4, defined as follows:

Y3 = Y1', and

Y4 = Y2'

These new variables are actually the velocities in the horizontal
and perpendicular directions.

Our system of equations is then:

Y1' = Y3

Y2' = -Y4

Y3' = -Y1/R3

Y4' = -Y2/R3

We solve this to get 1000 points between t=0 and t=10.

We can now solve this system for different initial conditions.

Setting the initial positions Y1 = -1 and Y2 = 0 and initial
velocities Y3 = 0 and Y4 = 1, we get a circle as the solution.
Interesting results are obtained when the system is solved for lower
Y4. When Y4 = 0.8, we get an ellipse with the star at the focus, but
if Y4 is lowered to 0.35, we get a decaying elliptical orbit.

Y4 = 1

Y4 = 0.7

Y4 = 0.35