# Surface of Equal Gravitational Potential

1. Oct 24, 2005

### amcavoy

In my introductory physics class, we were given a problem to find the equation of the surface between the Earth and the Sun where the gravitational potential is equal. At first (without working anything out), it seems that it might be some sort of ellipsoid or paraboloid. For now, I'm just going to look at a level curve and try to work it out in two dimensions.

Let M be the mass of the sun, and m be the mass of the earth. Let r1 be the distance from the earth to a point (x,y) on the graph and r2 be the distance from the sun to the same point (x,y):

$$-\frac{GM}{r_2}=-\frac{Gm}{r_1}\implies r_1M=r_2m$$

Now I am going to set the center of the sun as point (0,0) and the earth as (0,k). Doing so gives:

$$M\sqrt{x^2+\left(y-k\right)^2}=m\sqrt{x^2+y^2}$$

I also know that when x=0, r1+r2 is the distance from the earth to the sun.

I am not finished yet. I would just like to know if my approach is valid. Could someone let me know?

Thank you.

2. Oct 24, 2005

### Tom Mattson

Staff Emeritus
Looks good to me.

3. Oct 24, 2005

### amcavoy

Great, thank you for the help.