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

20 TWO·BODY ORBITAL MECHAN ICS Ch.1
Integrating both sides
i X h =
M + B. (1.5-2)
Where B is the vector constant of integration. If we now dot multiply
this equation by r we get a scalar equation:
r 0 i X h = r o M !... + r o B
r
.
Since, in general, a 0 b X c =
a X b . c and a. a = a2
h2 = W + r8 cos v
where V (nu) is the angle between the constant vector B and the radius
vector r. Solving for r, we obtain
r = h2/g
.
1 +(81 J.9 cos V
(1.5-3)
1.5.2 The PolarEquation of a Conic Section. Equation (1.5-3) is the
trajectory equation expressed in polar coordinates where the polar
angle, V, is measured from the ftxed vector B to r. To determine what
kind of a curve it represents we need only compare it to the general
equation of a conic section written in polar coordinates with the origin
located at a focus and where the polar angle, v, is the angle between r
and the point on the conic nearest the focus:
I r-
1 + e cos V
p
(1.5-4)
In this equation, which is mathematically identical in form to the
trajectory equation, p is a geometrical constant of the conic called the
"parameter" or "semi-latus rectum." The constant e is called the
"eccentricity" and it determines the type of conic section represented
by equation (1.5-4) .