# Coupled ODE Solver: Plantary Motion

Description| How it works| Planetary Motion

.xls (142 KB)
.zip (61 KB)

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

Y1" = - k(Y1)/R3
Y2" = -k(Y2)/R3

where R2 = Y12 + Y22

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