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By Jean-Paul Pier

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For an elliptic orbit, one ends up having to solve Kepler's equation, which is csin(E) = E - M, where M = net - T) and c is the eccentricity of the orbit. This equation determines E, the eccentric anomaly, in terms of the time variable t. After finding E, the radial and angular coordinates of the object are calculated using formulas from geometry. Note that in this equation n, T, and c are positive constants. (a) After sketching the functions in Kepler's equation on the same graph, explain why there is at least one solution.

02). Assuming c « 1, find the first three terms in an asymptotic expansion for E. (c) Show that your result agrees, through the third term, with the series solution (Bessel, 1824) E = +2L 00 M n=l 1 -In(nc) sin(nM). 5 Asymptotic Solution of Algebraic and Transcendental Equations 25 It is interesting that Bessel first introduced the functions In(x) when solving Kepler's equation. He found that he could solve the problem using a Fourier series, and this led him to an integral representation of In(x).

1 - 2a = -a =? a = 1 =? 0(c 1 ) and = 0(1). In this case, the conclusions are consistent with the original assumptions, and so this is the balancing we are looking for. This is said to be a distinguished limit for the equation. 9a) 00, o. 10) where A is an arbitrary constant. 3). This is important for the successful completion of Step 3. 7) is supposed to describe the solution in the immediate vicinity of the endpoint x = O. 4) applies over the remainder of the interval (this is assuming there are no other layers).

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