5OrbitCharacteristics - Astromechanics Two-Body...

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(1) Astromechanics Two-Body Problem (Cont) 5. Orbit Characteristics We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) 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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