Achieving Circular Orbit

## Question:

A satellite is placed into orbit at a velocity of 6km/s parallel
to the Earth's surface. Determine the proper altitude of the
satellite above the Earth such that its orbit remains circular.
What would will happen to the satellite if its velocity is 5km/s
when placed tangentially into orbit.
## Answer:

In a circular orbit Newton's second law leads us to the
relationship

G*M*m/r^{2}=m*v^{2}/r, or
v^{2}=G*M/r for a circular orbit. This yields

r=G*M/v^{2}
In this problem v=6e3m/s so

r=6.673e-11*5.976e24/6e3^{2}=1.108e7m

which corresponds to 4.7e6m orbital altitude.

If the initial velocity were 5km/s instead of 6km/s tangential
at this altitude, the satellite would have only (5/6)^{2}, or 69%
of the kinetic energy of the first case. Due to the mismatch
between the kinetic energy and potential energy at that altitude,
the satellite would slide down the potential energy well, gaining
kinetic energy as it falls. The orbit then would be elliptical
with its aphelion at the insertion altitude. If the perihelion is
below the surface of the Earth it will crash. You should try to
determine if this will happen. Refer to
satellite speed at extremes.

This information is brought to you by M. Casco Associates, a
company dedicated to helping humankind reach the stars through
understanding how the universe works. My name is James D. Jones.
If I can be of more help, please let me know. JDJ