# Homework Help: Energy of Planetary Motion

1. Feb 23, 2010

### jmtome2

1. The problem statement, all variables and given/known data
OK so here goes.

I'm using an ODEsolver in java to plot the total energy over time of a planetary system. So I've been trying to calculate the rate of energy (per unit mass), $$\frac{E}{m}$$.

2. Relevant equations
The equation for total energy (per unit mass) of a planetary system is:
$$\frac{E}{m}=1/2\cdot v^2-\frac{G\cdot M}{r}$$

G is the gravitational constant
M is the mass of the sun (constant)
v is the velocity of the planet, $$v^2=v^{2}_{x}+v^{2}_{y}$$
r is the distance of the planet from the sun, $$r^2=x^{2}+y^{2}$$

Essentially I need help finding $$\frac{dE}{dt}$$

3. The attempt at a solution
The answer I got for the rate is:

$$\frac{dE}{dt}=v\cdot\left(a+\frac{G\cdot M}{r}\right)$$

where a is the acceleration of the planet, $$a^2=a^{2}_{x}+a^{2}_{y}$$

The problem is that everytime I throw this equation into the ODEsolver, I get a plot of ever-increasing energy as time goes on which I know is not correct.

Help anybody???

2. Feb 23, 2010

### kuruman

Isn't dE/dt = 0 because energy is conserved? Am I missing something here?

3. Feb 23, 2010

### jmtome2

YES!!! Of course it is.... what was I thinking. But this creates a whole problem... I've got to figure out how to plot E now inside the program without using the ODEsolver, which hasn't been mentioned in the book yet. *sigh*

Thanks for clarifying :)