How can you determine if something is in stable orbit

Question

A space probe has a period of 100 minutes around the earth at an altitude of 83 km about the earth's surface?

 

Can we tell if a stable orbit is present?

Answer 1

I assume you are asking if we can tell if it is in a stable circular orbit.

 

T = period of orbit

r = radius of orbit

R_E = radius of earth

h = height of orbit above surface of the earth

M = mass of earth

 

r = R_E + h

 

For a given height,

 

(E1)   v = 2πr/T = 2π(R_E + h)/T

 

In a stable circular orbit, 

 

(E2)   v = √(GM/r) = √[GM/(R_E + h)]

 

Compute v from equations E1 and E2 and compare values.  If E1 gives a higher v than E2, the object is moving too fast to be in a circular orbit, and will rise in altitude.  If E1 gives a lower v than E2, the object is moving too slow, and will lose altitude.