mass of planet?

Question

A satellite orbits Saturn at twice the distance of the Moon from the Earth. Its period is 8 days .Many times more massive than the Earth is Saturn?

Answer 1

Ah...what?

I think you are saying that the satellite is orbiting the Saturn at twice the distance out Moon is from Earth. Is it true?

If so still centripetal force must be equal to the gravitational pull of monstrous Saturn.

Fc=Fg
mV^2/R=GMm/R^2
so
M=RV^2/G
V= 2piR/t= 2 pi R/ (3600 x 24 x 8) I hope you meant Earth days...

Now we have

M=R (2piR/t)^2/G
M= R^3 (2pi/t)^2/G
Let me get the value of R and G

R(mean) = 384,400 km =3.844E+8 m
G=6.673 E-11 m^3 /(kg s^2)
M=(3.844E+8)(2 x pi/(3600 x 24 x 8) )/6.673 E-11=
M=7.033E+25kg
mass of the Earth =5.976e+24
so it appears only to be only 12 times more massive than the Earth.

Actual mass of Saturn is 5.688e+26 (we are off by a factor of 8 as the Earth is almost 95 times lighter)

Answer 2

Orbital angular velocity of the satellite is
(Ω/Ωmoon) = 28days/8days
times greater than that of the moon.

Centriperal acceleration of the satellite is
(A/Amoon)= (Ω/Ωmoon)² R/Rmoon = (28/8)² x 2
times greater than that of the moon.

Mass ratio of Saturn and Earth
Ms/Me = (A/Amoon) (R/Rmoon)² = (Ω/Ωmoon)² (R/Rmoon)³


Answer: Ms/Me = (28/8)² x 2³ = 98
Note that the answer may be derived from astronomamical obervations (of natural satellites of Saturn) only.