Fundamentals of astrodynamics and applications 4th Edition (2013)

20 EQUATIONS OF MOTION 1.3
from apoapsis. In addition, f fpa
is zero only at apoapsis and periapsis. The values of
f fpa
for parabolic and hyperbolic orbits may also be positive or negative depending on
whether the satellite is heading away from or towards periapsis, respectively. It’s zero at
periapsis.
Table 1-1 summarizes values for parameters of basic conic sections. I’ll develop
these relations in greater detail in Sec. 1.3.
TABLE 1-1. Values for Conic Sections. This table shows characteristic values for the basic
parameters. Sec. 2.2 discusses the limited values.
a e p n f fpa
Circle a = r e = 0 p = r Undefined f fpa
= 0
Ellipse r p
≤ a ≤ r a
0 < e < 1 r p
< p < 2r p
0°≤ n≤ 360° −90°≤ f fpa
≤ 90°
Parabola a = ∞ e = 1 p = 2r p
Limited Limited
Hyperbola a < 0 e > 1 p > 2r p
Limited Limited
Rectilinear Ellipse r p
≤ a ≤ r a
e = 1 p = 0
Rectilinear Parabola a = ∞ e = 1 p = 0
Rectilinear Hyperbola a < 0 e = 1 p = 0
1.3 Two-body Equation
This section explores the two-body equation of motion and initial analyses. Complete
knowledge of the two-body problem and its assumptions is crucial to astrodynamics.
These techniques are extremely useful for back-of-the-envelope calculations and often
serve as the starting point for more complex study.
Newton’s second law and his universal law of gravitation are the starting points for
virtually any study of orbital motion, especially when combined with Kepler’s Laws.
His first law states that bodies tend to stay at rest or in uniform motion unless they are
acted upon by an external force. Although this idea is easily dismissed in modern times
as trivial, it was revolutionary in 1687. Previously, concepts of motion were based on
Aristotelian philosophy. Most scientists believed that an object’s natural state was at
rest, because they knew that friction causes objects to slow down and eventually stop.
The concept of bodies staying in motion was new.
Most people are familiar with the second law, which states that the time rate of
change of linear momentum is proportional to the force applied. For a fixed-mass system,
F
= --------------- dmv ( ) = ma
dt
(1-11)