Page 43 - Intro to Space Sciences Spacecraft Applications

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Introduction to Space Sciences and Spacecraft Applications
In an orbit, r represents the radial distance (orbital radius) between the
bodies’ mass centers. The true anomaly 2) is the angle measured from the
major axis line (in the direction pointing toward periapsis) to the radial
line between the two bodies. This is measured in the same direction as the
motion of the orbiting body (refer to Figure 2-1).

Example Problem:

With the knowledge (from geometry) that for a circle the dis-
tances a and b are equal, find the eccentricity of a circular orbit and
give the relationship between the periapsis, apoapsis, and semi-
major axis distances.

Solution:

From equation 2-2 with a = b, it is clearly seen that the eccentric-
ity of any circular orbit is zero.
Relating this finding to equation 2-3, for this expression to be
zero, ra must be equal to rp.
In fact, inserting a value of zero for eccentricity into the conic sec-
tion equation (eq. 2-4) we find that r = a for any true anomaly 2).
This result tells us that the orbital radius for any circular orbit
remains constant throughout the orbit.

Orbital Types. A closer look at the conic section equation (eq. 2-4) reveals
that the eccentricity can tell us immediately what type of orbit we are in:
If we only consider positive eccentricities, the first limiting case is
when e = 0, which we already know indicates a circular conic section
and thus, a circular orbit.
The next limiting case occurs when e = 1. For this value the conic
equation gives a value of infinity for the radius at the point where the
true anomaly approaches 180”. This corresponds to a parabolic conic
section. Values of eccentricity greater than 1 indicate a hyperbolic
conic section. Parabolic and hyperbolic orbits represent “open” or non-
repeating orbits. Such orbits are used by deep space probes to escape
the earth or even the solar system as in the case of the Voyager probes.