Page 49 - Intro to Space Sciences Spacecraft Applications

Introduction to Space Sciences and Spacecraft Applications
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Orbital Velocities. Another very useful expression for the total energy is:

Relating this to equation 2-8, we can write:
--I--
-cL
V2
-
2 r 2a
and solving for velocity we get:

(2 - 10)

Equation 2- 10 is a general equation for velocity at any point in an orbit
based on its orbital radius r. This equation can be simplified for three spe-
cific cases of particular interest:
In a circular orbit we know that r = a always, so equation 2-10 may
be rewritten as:

(2-11)

This simple relationship shows that circular orbital velocities vCim are
constant and determined by the circular orbital radius rcim only.
From equation 2-5, we know that at apoapsis in an elliptical orbit, r =
ra = a(l + e). Using this in equation 2-10, we can write:

(2 - 12)
va=J= a(l +e)

Since this corresponds to the point of maximum orbital distance,
Kepler and Newton’s laws tell us that this apogee velocity v, is the
slowest during the orbit.

Similarly, using r = rp = a(1 - e) for the orbital radius at periapsis,
equation 2-10 becomes: