Surface of Equal Gravitational Potential

[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.

 

[quantumdude]

Looks good to me.

 

[amcavoy]

Great, thank you for the help.