Fundamentals of astrodynamics and applications 4th Edition (2013)

16 EQUATIONS OF MOTION 1.2
19134
v
r
e = 0.0
r = 12,756 km
0.0
F, F′
a, r, p
Figure 1-6.
–19134
–19134
0.0 19134 km
Circular Orbits. For circular orbits, the semimajor axis, semiparameter, and
radius are always equal and constant. This is a special case of the ellipse, in which
both foci coincide.
If we solve Eq. (1-4) for the square of the eccentricity,
b 2
a 2
e 2 = 1 – ---- =
a 2 – b
----------------
2
a 2
we get an alternate equation for the eccentricity:
e
=
a 2 – b
--------------------
2
a
(1-6)
We take only the positive value for eccentricity because eccentricity is defined as a positive
value. Continuing to develop an expression for flattening, use Eq. (1-3) and form
(2f − f 2 ):
2f – f 2
2a – 2b a
-----------------
2 – 2ab + b
= – --------------------------------
2
=
a
a 2
a 2 – b
----------------
2
a 2
Therefore, by Eq. (1-6),
e 2 = 2f – f 2
(1-7)
Another expression, derived from Eq. (1-7), can also relate the two notations: