October 2011
Intermediate to advanced
749 pages
23h 36m
English
Content preview from Optimal Estimation of Dynamic Systems, 2nd Edition
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620 Optimal Estimation of Dynamic Systems
law in the inertial frame for each body we obtain
M
¨
r
M
=
GMm
||r||
3
r (A.218a)
m
¨
r
m
= −
GMm
||r||
3
r (A.218b)
The negative sign in Equation (A.218a) is due to the opposite direction of the force.
Since, as shown in Figure A.8, r = r
m
−r
M
, then from Equation (A.218) we obtain
¨
r = −
G(M + m)
||r||
3
r (A.219)
If the mass m is much smaller than M (which is a very accurate assumption for
orbiting spacecraft), then we can effectively ignore m so that
¨
r = −
μ
||r||
3
r (A.220)
where
μ
≡ GM is called the gravitational parameter. The gravitational parameter is
more commonly used in orbital mechanics of spacecraft since it can be meas
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