Numerical efficiency of the elliptic function expansions of the first-order intermediary for general planetary theory

Abstract

Abstract.We compare numerical efficiency of the two kinds of series for the first-order intermediate orbit for general planetary theory: (1) the classical expansion involving mean longitudes of the planets; (2) an expansion resulting from the theory of elliptic functions. We conclude that mutual perturbations of close couples of planets (the ratio of major semi-axes ~ 1) can be represented in more compact form with the aid of the second kind of series. Two kinds of series for the first-order intermediaryIn spite of significant progress of numerical approaches to construct the theories of motion of the major planets, analytical theory of planetary motion remains to be a challenging and interesting scientific task. Recently the interest in constructing analytical planetary theories has been revived by the idea to use an elliptic function of time (instead of time itself) as independent variable in order to make the resulting series more compact (see, Brumberg (1996) and references therein). Our aim is to compare the numerical efficiency of the classical series involving mean longitudes of planets and the series resulting from the application of elliptic functions for the particular case of the first-order intermediate orbit for general planetary theory (Brumberg, 1994

On the expansions of intermediate orbit for general planetary theory

“…In Section 2 we describe a numerical algorithm allowing one to compute both p-and r-series for higher-order GPT. In Section 3 we extend the results of Brumberg and Klioner (1996) for mutual perturbations of a pair of planets up to the fourth order with respect to the ratio # of the planetary masses to the mass of the central body. Section 4 is devoted to a comparison of the efficiency of the p-series and the ~--series representing mutual perturbations of a triplet of planets of the second and third orders with respect to p. Principal conclusions and some additional remarks are given in Section 5.…”

“…In Brumberg and Klioner (1996) the p-and the r-series have been already computed for the first-order intermediary. The method of calculation used in Bmmberg and Klioner (1996) is quite different from the algorithm we use in the present paper, and involves the explicit expression for IzT) i)-derived in Brumberg (1994) as well as a numerical integration cf some auxiliary function appearing in that explicit expression.…”

“…The method of calculation used in Bmmberg and Klioner (1996) is quite different from the algorithm we use in the present paper, and involves the explicit expression for IzT) i)-derived in Brumberg (1994) as well as a numerical integration cf some auxiliary function appearing in that explicit expression. We have repeated the calculations using the method described in Section 2 The principal conclusions are that the "r-series are more efficient than the p-series when we consider mutual perturbations of close couples of planets (when the ratio of the semi-major axes ai/aj is close to 1) and/or when a relatively high precision is required.…”

“…Following these lines, several steps toward constructing the GPT in the form of the r-series have been undertaken in Brumberg (1992), Klioner (1992), and Brumberg and Klioner (1996). The principal aims of this work are to demonstrate practical feasibility of constructing both the p-series and the T-ones for the intermediate orbit for the GPT as well as to compare numerical efficiency of the two kind of series.…”

“…The principal aims of this work are to demonstrate practical feasibility of constructing both the p-series and the T-ones for the intermediate orbit for the GPT as well as to compare numerical efficiency of the two kind of series. The numerical efficiency has been already investigated in Brumberg and Klioner (1996) for the first-order intermediary. The conclusions were rather promising: the r-series were shown to be more compact than the corresponding p-series.…”