Bate, Mueller, and White - Fundamentals of Astrodynamics ... - UL FGG

8 TWO-BODY ORBI TAL MECHANICS Ch.1
F TOTAL = F 9 + F OTHER '
(l.2-6)
We are now ready to apply Newton's second law of motion. Thus,
d (mj v j) = FTOTAL. (1.2-7)
The time derivative may be expanded to
m · dVj dmj_ F
I--+V'--dt
I dt TOTAL
(1.2-8)
It was previously mentioned that the body may be expelling some mass
to produce thrust in which case the second term of equation 1.2-8
would not be zero. Certain relativistic effects would also give rise to
changes in the mass m j as a function of time. In other words, it is not
always true-especially in space dynamics-that F = ma. Dividing
throur by the mass m j gives the most general equation of motion for
the it body
r· =
I
FTOTAL
mj
0.2-9)
whereL-.. ________ .... .. ..I
fj is the vector acceleration of the jth body relative to the x, y, z
coordinate system.
mj is the mass of the jth body.
FTOTAL
F 9 = -
is the vector sum of all gravitational forces
Gmj
n
2:
j = 1
j =1= j
and all other external forces
(r .. )
J I