Doubt in the concept of a question on gravitation and satellites

Question

Two satellites of same mass m are revolving around earth(M) in same orbit of radius r. Rotational directions of the 2 satellites are opposite, therefore they can collide. Total mechanical energy of the system (both satellites and earth) is (m<<M) __?
Ans)-GMm/r

My doubts: 1)Does the rotation of the satellites need to be opposite?
2)What happened to the effect of one satellite on other? Won't that affect the total energy?

Answer 1

1)  No, energy is a scalar quantity, not a vector, so the orbital direction does not matter.  The two satellites would have the same total energy if orbiting in the same direction

 

2)  The gravitational attraction between the two satellites does contribute to the total energy, but the contribution is negligibly small since m<<M.

 

For a satellite of mass m orbiting in a circular orbit at a distance r from the Earth's center of mass:

 

mv²/r = GMm/r²

v² = GM/r

 

KE = (1/2)mv² = GMm/2r

 

Total Energy (E) = Potential Energy + Kinetic Energy

E = -GMm/r + GMm/2r = -GMm/2r

 

For two satellites with the same mass (m) and orbital radius (r):

 

E = 2(-GMm/2r) = -GMm/r