G = gravitational constant
M = mass of the earth
R = radius of orbit of a satellite
r = radius of orbit of a second satellite
v = speed of the satellite
P = period of a satellite
p = period of a second satellite
Equate gravitational acceleration with centripetal acceleration
g = GM/R^2 = v^2/R
Express the orbital speed in terms of the orbit circumference and period
v = 2piR/P
And insert the expression for v into the first equation
GM/R = 4PI^2R^2/P^2
GM/R^3 = 4pi^2/P^2
R^3/P^2 = 4pi^2/(GM) = constant = C
We can do the above since G and M are constants for all earth orbits
So we can write a second equation of the same form for another satellite and equate to get:
R^3/P^2 = r^3/p^2
r^3 = R^3p^2/P^2
r = R(p^2/P^2)^(1/3)
For the second satellite we have p = 8P
r = R(8^2)^(1/3) = R(64)^(1/3) = 4R
