Neptune has a mass of about 1.02x10^26 kg.

Neptune has a mass of about 1.02x10^26 kg. The length of a day on Neptune is 16.11 hrs. Your task is to put a satellite into a circular orbit around Neptune so that it stays above one spot on the surface, orbiting Neptune once each Neptune day. At what distance from the center of the planet should you place the satellite?

The "stays above one spot on the surface" and "orbiting Neptune once each Neptune day" come hand-in-hand. All you want to do is to find the distance from the center such that the satellite orbits Neptune every 16.11 hours = 57996 s.

Using T = 2π√{r^3/[G*m(neptune)]}, we see that:
T^2 = 4π^2*r^3/[G*m(neptune)] ==> r = [T^2*G*m(neptune)/(4π^2)]^(1/3).

Plugging in our given information yields:
r = [T^2*G*m(neptune)/(4π^2)]^(1/3)
= {[(57996 s)^2*(6.67 x 10^-11 N*m^2/kg^2)(1.02 x 10^26 kg)]/(4π^2)}^(1/3)
= 8.34 x 10^7 m.

I hope this helps!

There is only one main force acting on a satellite when it orbits a planet, and that is the gravitational force exerted on the satellite. This force is constantly pulling the satellite towards the centre of that planet which it orbits.

The formula for centripetal force is: F = (mv²)/r
The formula for the gravitational force between two bodies of mass M and m is (GMm)/r²

Since the satellite has to maintain a geostationary orbit around neptune.It moves around Neptune at the same angular speed that the Earth rotates on its axis.

We can use our formulae above to work out characteristics of the orbit.

(mv²/r) = (GMm)/r²
=> v²/r = (GM)/r²

Now, v = (2πr)/T.

=> (((2πr)/T)²)/r = (GM)/r²
=> (4π²r)/T² = (GM)/r²
=> r³ = (GMT²)/4π²

T(time) is one day on Neptune, since this is the period of Neptune. so T is 57.996 x 10^3 seconds.
M is the mass of Neptune, which is1.02x10^26 kg .
Lastly, we know that G (Newton's Gravitational Constant) is 6.67 x 10^-11 m3/kg.s2


So we can work out r.

r = 83,503.9 km. approx

So the satellite will have to orbit Neptune at a height of 83,503.9 km from the centre of the planet.