Page 44 - Intro to Space Sciences Spacecraft Applications

Orbital Principles
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The final case, then, is when the eccentricity is greater than zero but
still less than one (0 < e e 1). For fixed values of the eccentricity and
semi-major axis, we can see that the orbital radius is a function of the
true anomaly which is a measure of exactly where along the conic sec-
tion the orbiting body is. Substituting different values for the true
anomaly, one would find that the minimum radius occurs when u = 0"
and the maximum radius occurs when u = 180". These distances cor-
respond to the periapsis and upoupsis points in the orbit shown in Fig-
ure 2- 1 and prove that they are associated with the minimum and max-
imum distances between the bodies in an elliptical orbit. Values of
eccentricity within this range, then, correspond to an elliptical conic
section and would indicate an elliptical orbit.
At periapsis (u = 0') and at apoapsis (u = 180°), the conic section equa-
tion (eq. 2-4) simplifies to reveal two useful relationships:

r, = a(l + e) and rp = a(l - e) (2-5)

It is important to realize that orbital radius increases continuously from
periapsis to apoapsis, and decreases continuously from apoapsis to peri-
apsis when moving in an elliptical orbit.

Kepler's Second Law. Kepler's second law reveals that a line drawn
between the two bodies will sweep out the same amount of area during the
same time period anywhere along the orbital path. This characteristic is
illustrated in Figure 2-3.

At2 = At1
Areal =Area2
Figure 2-3. Aredtirne relationship. Kepler's second law reveals that the
velocity of an object changes with orbital radius.