(1)
Astromechanics
TwoBody Problem (Cont)
5. Orbit Characteristics
We have shown that the in the twobody problem, the orbit of the satellite about the
primary (or viceversa) is a conic section, with the primary located at the focus of the conic
section. Hence the orbit is either an ellipse, parabola, or a hyperbola, depending on the orbit
energy and hence eccentricity. For conic sections we have the following classifications:
e < 1
ellipse
e = 1
parabola
(An exception this relation is that all rectilinear orbits
have e = 1, and angular momentum = 0)
e > 1
hyperbola
Of main interest for Earth centered satellites (Geocentric satellites) and Sun centered satellites
(Heliocentric satellites) are elliptic orbits. However when we go from one regime to another such
as leaving the Earth and entering into an interplanetary orbit then we must deal with hyperbolic
orbits. In a similar manner, if we approach a planet from a heliocentric orbit, hyperbolic orbits are
of interest. Parabolic orbits, on the other hand are more theoretical than practical and simply
define the boundary between those orbits which are periodic and “hang around,” (elliptic orbits),
and those orbits which allow one to escape from the system (hyperbolic orbits). So one might say
that the parabolic orbit is the minimum energy orbit that allows escape. In the following, we will
determine the properties of each of these types of orbits and write some equations that are
applicable only to the type of orbit of interest.
PROPERTIES OF THE ORBITS
Parabolic Orbit (e = 1, En = 0)
The parabolic orbit serves as a boundary between the elliptic (periodic) orbits and the
hyperbolic (escape) orbits. It is the orbit of least energy that allows escape. The orbit equation
becomes,
Further, the periapsis distance = r(0) = p/2 = h
2
/ 2
:
.
The energy equation becomes:
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(2)
(3)
Escape Velocity
Equation (2) defines the
escape velocity
, which is the minimum velocity to escape the two body
system at the given radius r. Note that the speed required for escape is independent of its
direction!
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 Spring '07
 Shoberg
 mechanics, Elliptic orbit, Orbit Energy

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