Astron. Nachr. / AN 237, No. 4, 304 – 308 (2006) / DOI 10.1002/asna.200510537 The Gyld´ en-type problem revisited: More refined analytical solutions A. Pal 1 , D. S ¸ elaru 2 , V. Mioc 3, , and C. Cucu-Dumitrescu 2 1 Babes ¸-Bolyai University, Faculty of Mathematics and Computer Science, Str. M. Kog˘ alnicean 2 Institute for Space Sciences, Str. Mendeleev 21–25, RO-010362, Bucharest, Romania 3 Astronomical Institute of the Romanian Academy, Str. Cut ¸itul de Argint 5, RO-040557 Bucharest, Romania Received 2005 Aug 11, accepted 2005 Oct 18 Published online 2006 Apr 20 Key words celestial mechanics, stellar dynamics – methods: analytical We resume and consistently extend our previous researches concerning the Gyld´ en-type problem (a two-body problem with time-dependent equivalent gravitational parameter). To approach most of the concrete astronomical situations to be modelled in this way, we consider a periodic small perturbation. For the nonresonant case, we present a second-order analytical solution. For the resonant case, we adopt the most realistic astronomical situation: only one dominant term of the Hamiltonian. In this case we point out a fundamental model of resonance, common to every resonant situation, and, moreover, identical to the first fundamental model of resonance. Considering the simplest model of periodic change of the equivalent gravitational parameter, we find that all possible resonances are confined to the first fundamental model. c  2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction In astronomy and celestial mechanics the Kepler prob- lem with time-dependent gravitational parameter bears the name of the Swedish astronomer Hugo Gyld´ en (e.g., Had- jidemetriou 1963; Deprit 1983). This model of interaction can be described by the Hamiltonian H (q, p,t)= |p| 2 2 − ν (t) |q| , (1) where ν (t) > 0 is the gravitational parameter, while q and p stand for the relative position vector of the two-body problem and the associated momentum vector, respectively. It was introduced by Gyld´ en (1884) in order to explain the secular acceleration observed in the Moon’s longitude. Later on, the applicability area of the model extended to a lot of physical and (especially) astronomical situations. Some integrable cases for a secular time-dependence of ν were pointed out by Messchersky (1902) and discussed by Savedoff and Vila (1964); see also Saari (1977) and Vinti (1977). Within the classical gravitational framework, the secular variation of ν (via change of the gravitational constant or/and masses) models many concrete astronomi- cal situations; we quote arbitrarily Dirac (1938), Brans & Dicke (1961), Hadjidemetriou (1963), Will (1971). Since the changes of ν can also be due to nongravi- tational forces, S ¸ elaru, Cucu-Dumitrescu & Mioc (1992) generalized the terminology by introducing the notion of changing equivalent gravitational parameter, suggested by one of the authors of this paper (A.P.). They ex- tended Gyld´ en’s model by including nongravitational, cen- tral, inverse-square perturbing forces, absorbed in ν . The ⋆ Corresponding author: vmioc@aira.astro.ro source of such a generalization was Saslaw’s (1978) corner- stone paper, which treated the photo-gravitational (gravita- tion plus radiation) Gyld´ en-type problem. In this paper we deal with the most interesting case of the equivalent gravitational parameter variation: the peri- odic one. Here we point out only some astronomical situa- tions approachable in this way: dynamics of particles (from dust to satellites and planets) around pulsating stars, stars with spots, neutron stars, etc.; evolution of a protosolar or protostellar nebula and of planetesimals, planetary nebulae and accretion disks; planetary satellite (artificial or not) dy- namics under the influence of the re-emitted solar radiation pressure, and so forth. Our problem can be transformed into a perturbed two- body problem (Deprit 1983) which will be our kind of ap- proach. In the case of periodic perturbations, it is natural to think about resonances. Let us give some astronomical examples of resonances within our framework. Suppose a “landlord” star with periodic luminosity changes commen- surable with the orbital period of a satellite. The appear- ance of resonances is then clear. It is the same for constant- luminosity “landlords” and spinning satellites with a non- constant albedo over the surface, whose spin period is com- mensurable with the orbital period. Of course, these two cases can be combined in several ways. Mathematically speaking, the Gyld´ en-type problem with periodically changing ν was approached by us from many standpoints: first-order analytical solutions in the case of a periodic variation of the equivalent gravitational pa- rameter (S ¸elaru et al. 1992), the slowly-changing equivalent gravitational parameter (Cucu-Dumitrescu & S ¸ elaru 1997), KAM theory applied to this problem (S ¸ elaru & Mioc 1997), c  2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim