By F.T., Westerman, H.R. Geyling

Conventional, extremely popular textbook detailing the advent of orbital mechanics. a primary version.

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**Example text**

Some of these series are particularly useful when lies near zero or unity and they will be needed for the perturbation methods of Chapter 8. We therefore consider briefly a few formal manipulations required for these purposes. 37) as the elliptic orbit approaches circularity. The most elementary and strictly formal approach consists of develop- r(M), e e). 36) as a binomial series in powers of and integrating term by term. 3 e3 3 + �32e4 4J+ O(e5 ). 6. 1) vice A more useful result, since it provides position in terms of time rather than is an expansion for E in terms of M.

T. of Time 7] 49 observation to the decimal fraction of a day by using AENA, Table X, Co � version of Hours, Minutes, and Seconds to Decimals of a Day. 000147 In our example, this fraction of a day will not change the Equation of the Equinoxes from its value of - 0�5 1 6. M. T. SLT. of observation = 04h43m41 �424 e) Convert the result of (d) to units of arc (24h Table XI, Conversion of Time to Arc. We have 41� 0�424 h 04 43m41�424 = 3 60°) by means of AENA, = 70° 45 ' 10 ' 1 5" = 6': 36 = 70° 55 ' 21�36 This is the angle between the true line of equinoxes and the Greenwich meridian at the time of observation.

3 e3 3 + �32e4 4J+ O(e5 ). 6. 1) vice A more useful result, since it provides position in terms of time rather than is an expansion for E in terms of M. We let E M u . Then Kepler's equation becomes sin (M U Now we assume that u ) u . 3) where the coefficients Un are expressible in terms of trigonometric functions of M. 3) into . 2), so that Equating powers of are U1 e, we obtain a set of algebraic equations in Un whose solutions z= 1 . 2 3= 3 . 3 -1 . � eZ e4 ) + -eZ �1 - eZ -e4) = + e 1 --+ 24 8 192 2 +...