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Snippedforlif Snippedforlif
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10 years ago
A 19880 kg satellite is in a synchronous orbit around Neptune. Find the radius of the orbit
NOTE: One astronomical unit (AU) is the average distance between the sun and the earth
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10 years ago
Due to the relatively small mass of the satellite, ignoring it wont alter the result very much, and will make the calculation shorter. Strictly speaking though, its a 2 body sysyem, so procedure 1 is the full spec on single body calculation (no mass satellite) . Procedure 2 is the final equation for a synchronous radius for a known mass satellite as part of a 2 body system.
Neptune mass (m1) = 1.0243 e26 kg
Neptunes (sidereal) rotation time = 16.1111 h = 58,000 seconds
Rotation rate of Neptune (RR) = ( 2 * pi ) / 58,000 = 1.0833 e-4 rad / sec  ( satellite to match this )
Satellite mass (m2) = 1.988 e4 kg
G = newtons gravity constant = 6.673 e-11
r = orbital radius

Procedure 1 for no mass satellite in synchronous orbit (equation build) :
Standard equation, rad / sec = v / r , you know rad / sec, find v to find r.
To find v
For stable orbit, gravitational acceleration = centripetal acceleration
 ( G * m) / r ² = v ² / r
So : isolate v,
v = square root ( ( G * m1 ) / r )
Then :
rad / sec = v / r = ( square root ( ( G * m1 ) / r ) ) / r
Isolate r
r = cube root ( ( G * m1 ) / ( rad / sec ² ) )
r = 8.3512 e7 meters

Procedure 2 ( two body system )  :
(satellite used as basis for calcs)
d = center to center distance
d = cube root ( ( G * m1 ) / ( ( RR ² ) * ( m1 / ( m1 + m2 ) ) ) )
r  = ( m1 / ( m1 + m2 ) ) * d

If you need the complete procedure 2 build, let me know.

PS the radius r in procedure 2, is the distance from the 2 body center of mass to the satellite,
in this case r and d are virtually identical, but the point should be noted.
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