To establish a geosynchronous orbit at a distance of 8.6 105 m, the spacecraft must orbit the planet in the same period as the planet's rotation.
This means that the spacecraft must complete one orbit every T = 31 hours. The orbital period is related to the orbital radius by the equation T2 = 4π2R3/GM, where G is the gravitational constant and M is the mass of the planet.
Substituting in the values for T and R into this equation, we can solve for GM: GM = 4π2(8.6 105 m)3/T2 = 2.08 1017 m3/s2.
The spacecraft must then use its thrusters to adjust its speed and position until it enters an orbit with a period of 31 hours and a radius of 8.6 105 m. It can achieve this by using a combination of delta-v maneuvers to adjust its speed and trajectory. The first maneuver will be a prograde burn to increase the spacecraft's speed until it enters an orbit that is slightly more distant than the desired orbit. The spacecraft will then use a series of impulsive maneuvers to adjust its trajectory until it is in an orbit with the desired radius. Finally, the spacecraft will use a retrograde burn to reduce its speed until it reaches the desired orbital period.
Once the spacecraft has completed all of these maneuvers, it will be in a geosynchronous orbit at a distance of 8.6 105 m from the planet's surface.
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