A synchronous orbit is one in which the orbiting satellite has the same period a

Question

A synchronous orbit is one in which the orbiting satellite has the same period as the planet itself, thus allowing the satellite to remain fixed over the same spot on the planets surface. Find the radius of a synchronous orbit about Saturn, and the radius of a synchronous orbit about the sun. Express your answers in terms of Saturn radii R sub sat,  for saturn. And in solar radii for the sun. Some data is provide belowe. Note that 100 AU = 1.5 x 10^8 km.

                    Mean Rotational period             Mass                   Radius

Saturn                    0.44 days                 5.68 x 10^26 kg           6.03 x 10^4 km

Sun                        27 days                   1.99 x 10^30 kg            6.96 x 10^5 km.

Please be thorough in your response. Thanks.

Explanation / Answer

FOR SATURN

R = cuberoot(T^2*G*M/4pi^2) = cuberoot(0.44^2*86400^2*6.6*10^-11*5.68*10^26/(4*3.142^2)) = 1.11*10^8m

R = 1.11*10^8/(6.03*10^7)Rsat = 1.84 * Rsat

FOR SUN

R = cuberoot(T^2*G*M/4pi^2) =

cuberoot(27^2*86400^2*6.6*10^-11*1.99*10^30/(4*3.142^2)) = 2.62*10^10m

R = (2.62*10^10/(6.96*10^8))*Rsun = 37.64*Rsun

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