By Victor Brumberg

The goal of this ebook is to explain modern analytical and semi analytical recommendations for fixing ordinary celestial-mechanics difficulties. The observe "techniques" is used the following as a time period intermediate among "methods" and "recipes". One usually conceives a few approach to answer of an issue as a normal mathematical instrument, whereas no longer taking a lot care with its computa tional cognizance. nevertheless, the notice "recipes" may perhaps these days be understood within the experience of the well known publication Numerical Recipes (Press et al. , 1992), the place it capacity either algorithms and their particular software realiza tion in Fortran, C or Pascal. Analytical recipes indicate using a few basic or really expert machine algebra method (CAS). The variety of varied CAS at present hired in celestial mechanics is simply too huge to specify quite a few of the main greatest platforms. along with, it kind of feels average to not combine the essence of any set of rules with its specific software implementation. For those purposes, the analytical suggestions of this e-book are to be considered as algorithms to be carried out in numerous methods reckoning on the and software program to be had. The ebook was once preceded by means of Analytical Algorithms of Celestial Mechanics by means of a similar writer, released in Russian in 1980. even with there being a lot universal among those books, the current one is in reality a brand new mono graph.

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1) p+max{O,v-a} (1) p+max{O,a-v} p=O L a=-oo p=max{O,v-a} and, lastly, wrote p + max{O, v - (J} instead of p. 4) u=-oo with _ (_l)V (-x)max{O,v-a}( -y)max{O,a-v} la-vl+n ,a ( n,x,y,v,a ) (1) a x lu-vl x F( -x + max{O, v - (J}, -y + max{O, (J - v}, 1 + I(J - vi, a 2 ). 5) This form enables one to use different transformations of the hypergeometric function. In particular, if a 2 is not so small it may be suitable. 3 General Terms of the Elliptic-Motion Expansions 39 a l<1-vl+n ( ) _ (-1)1' (-X)max{O,v-<1}( -Y)max{O,<1-v} 1<1 n, x, Y, v, a (1)1<1-1'1 (1 _ ( 2 )-x- y-1 x x F(1 + x + max{O, a - v}, 1 + y + max{O, v - a}, 1 + la - vi, ( 2) .

2 . 5 The Keplerian Processor with the Aid of Elliptic Functions 55 The (e, M) expansions are widespread because they enable one to express the coordinates of the two-body problem as explicit functions of time. But this is not of primary importance for constructing analytical theories of motion. As we shall see below, the integration of the equations of perturbations may be performed using the Hansen device to interrelate different angular arguments. If one wants to use instead of M some other anomaly in canonical equations of motion this may be easily achieved by the technique of Bond and Janin (1981).

For example, in the motion of the Earth's artificial satellites the perturbations due to the Earth's oblateness may be found as closed-form expressions in terms of the true anomaly v whereas the main luni-solar perturbations are representable as finite expressions in terms of the eccentric anomaly 9 and the mean longitude of the disturbing body. 15) containing both v and g. 15), presented below, belongs to Jefferys (1971). This algorithm involves several steps. 1. 15). 2. Using the expressions cos v = ~ ( 77 2 ; - 1) , .