Fundamentals of astrodynamics and applications 4th Edition (2013)

14 EQUATIONS OF MOTION 1.2
Directrix
l = p/e
•
p
r/e
r
r F F'
•
F'
n
•
F
r p
r p
/e
Figure 1-4. Geometry for Conic Sections. We create conic sections by recognizing that the
sum of the distance from both foci to any point on the orbit is constant. The ratio of
the distance from a focus to the orbit and the distance from that point to the directrix
is also a constant called the eccentricity, e. The closest point in the orbit to the primary
focus, F, is the radius of periapsis, r p
. The distance l is a standard quantity used
to describe conic sections.
simple means of creating an ellipse is to take a fixed-length string, two tacks, and a pencil
to stretch out the string and move on a blank page [See Fig. 1-4]. As the pencil
stretches the string and moves, the length from each focus changes, and the pencil traces
out the ellipse. Prussing and Conway (1993:62) show this mathematically for any point
on the conic section,
r F′ + r F = constant = 2a
(1-1)
r F – r F′ = constant = 2c
Each focus of a conic section has a corresponding stationary line called a directrix.
The ratio of the distance from a focus to a point on the orbit and the distance from that
point to the focus’ corresponding directrix is a constant defined as the eccentricity. The
eccentricity, e, is a fixed constant for each type of conic section; it indicates the orbit’s
shape—its “roundness” or “flatness.” The eccentricity is never negative, and its value
determines the type of conic section for common orbits. It’s unity for parabolic and rectilinear
orbits, less than unity for ellipses, zero for circles, and greater than unity for
hyperbolas. The eccentricity is
c
e = --
(1-2)
a