T br

This equation verifes Kepler's Third law.

From the conic section equation, Eq. (2.2.14), if the angle f is known, the distance r can be determined. However, for most applications, the time of a particular event is the known quantity, rather than the true anomaly, f. A simple relation between time and true anomaly does not exist. The transformation between time and true anomaly uses an alternate angle, E, the eccentric anomaly, and is expressed by Kepler's Equation, which relates time and E as

where n is the mean motion (vm/®3) , t is time, and tp is the time when M2 is located at perifocus. The time tp is a constant of integration. Since Kepler's Equation is a transcendental equation, its solution can be obtained by an iterative method, such as the Newton-Raphson method. If g = E - e sin E — M, where M = n(t —tp), the mean anomaly, then the iterative solution gives

\gVfc where k represents the iteration number and

From known values of a and e and a specifi d t, the preceding iterative method will provide the eccentric anomaly, E. A typical value of E to start the iteration is Eo = M. Some useful relations between the true anomaly and the eccentric anomaly are sinf =

dM f

It can be shown that

(Kepler's Equation)

but in some cases it is useful to express r as a function of time, or mean anomaly. This relation for elliptic orbits, however, involves an infnite series because of the transcendental nature of Kepler's Equation; namely, oo a = 1 + Jrn(TOe) cos mM (2.2.25)

where Jm is the Bessel Function of the fist kind of order m, with argument me (see Smart, 1961; Brouwer and Clemence, 1961). If the eccentricity is small, the following approximate equations may be useful:

sin E = sinM + 2 sin 2M + ... E = M + esinM + ... .

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