In Orbit - On orbital velocity of a satellite around a spherical body

We consider the problem of determining the initial velocity of a satellite required to make it orbit a spherical body of a certain mass, given the distance between the center of the sphere and the center of the satellite. We call this velocity the Orbital Velocity. If the satellite is in orbit, it's flux of energy is zero, assuming that the force of gravity due to the spherical body is the only force working on the satellite. This is because the direction of the force of gravity is always perpendicular to the direction of motion.

The sphere has radius r1 and mass m1. The mass of the satellite is given by m2 and it's radius by r2. The center of the satellite is placed at distance R from the center of the sphere, where R > r1 + r2.

Now the force required by the satellite to stay in orbit is

    Facc = v^2 * R^-1 * m2.

The force giving this acceleration is the force of gravity working on the satellite, given by

    Fgrav = G * m1 * m2 * R^-2.

Solving the equation Facc = Fgrav we get

    v^2 * R^-1 * m2 = G * m1 * m2 * R^-2,

which reduces to

    v^2 = G * m1 * R^-1.

Taking the square root on both sides of the equation gives us the velocity at which to release the satellite in order to go into orbit at the given distance from the center of the sphere:

    v = sqrt(G * m1 * R^-1).

Note that the obtained result is independant of the mass of the satellite.

Using this equation I made a simulation program that displays the orbit of a satellite around a spherical body. The simluation is such that any movement of the satellite can be simulated. So if the initial velocity is set diffent from orbital velocity, the satellite's path will deviate from a circle and the satellite will either separate from it's orbitee or it will fly into it.