Computational Celestial Mechanics means purely analytical, computer-assisted and numerical methods with the typical feature of the necessity of carrying out a vast amount of calculations, aimed to determine dynamical features of bodies of planetary systems.

Table of Contents Introduction Keplerian motion and Kepler's equation Perturbation theory : The precession of the perihelion as an example Computer-assisted stability results Computer-assisted proofs KAM theory Nekhoroshev theory Long-time integrations Stability of the solar system Chaotic obliquity of the planets Synthetic theories Ephemerides computation Choreographies of the N-body problem References See also

An accurate prediction of the dynamics of the objects of the solar system often requires very long computations. Even the easiest problem provided by the two-body model deserves computational skill in solving Kepler's equation, which allows to derive the Keplerian elements of the orbit as a function of time. In passing from the two to the three body problem, an increasing complexity due to nonintegrability and chaos is a trademark of Celestial Mechanics; the three body problem motivated the development of perturbation theories aimed to find approximate solutions of the equations of motion. Refined analytical perturbative techniques, such as KAM or Nekhoroshev theory, can be applied to some problems of Celestial Mechanics under suitable assumptions; most likely, effective results often require very lengthy computations which can be implemented through computer-assisted techniques. Models of Celestial Mechanics can be studied also by numerical integrations, eventually using frequency map analysis or synthetic theories; however, due to the numerical errors introduced by the machine particular care must be taken when running over long time scales, as over the age of the solar system. Both analytical and numerical techniques are worldwide used to compute accurate ephemerides.

The two-body problem is the study of the motion of two material points, e.g. a planet and a satellite, subject to the mutual gravitational attraction. Its solution is provided by Kepler's laws, according to which the motion takes place on an ellipse. Assume that the planet is located at one focus, which is taken as the origin of a reference frame whose abscissa coincides with the perihelion line; in this frame let <math>r</math> (the orbital radius) and <math>f</math> (the true anomaly) be the polar coordinates of the satellite. Let <math>a</math> and <math>e</math> be the semimajor axis and eccentricity of the ellipse. The solution of the two-body problem (see, e.g., [Roy]) is provided by the set of formulae

<math>r=a(1-e\ cos\ E)</math>

<math>f=2 \arctan\Big(\sqrt\ \tan{E\over 2}\Big)

</math>

<math>\ell=E-e\sin E\ ,</math>

where <math>\ell=\ell(t)</math> is the mean anomaly related to the time by <math>\ell(t)=n t +\ell(0)\ ,</math> <math>n</math> denoting the frequency of revolution. Last formula, known as Kepler's equation, must be solved to provide the eccentric anomaly <math>E</math> as a function of <math>\ell\ ,</math> and therefore of the time. Inserting such solution in the first two equations, one obtains the variation with time of the orbital radius and of the true anomaly. Its solution can be found numerically, up to a given precision, through an iterative algorithm e.g. using a Newton's method, or it can be expressed analytically by means of the Bessel's functions <math>J_k(x)</math> as

<math>

E=\ell+\sum_{k=1}^\infty {1\over k}\Big[J_{k-1}(ke)+J_{k+1}(ke)\Big]\sin(k\ell)\ . </math>

Canonical perturbation theory ([FM]) is related with nearly integrable Hamiltonian systems and provides efficient tools for Celestial Mechanics. A typical application is the computation of the precession of the perihelion of Mercury.

Consider a nearly-integrable Hamiltonian function of the form

<math>

H({\underline I},{\underline \varphi})=h({\underline I})+\varepsilon f({\underline I},{\underline \varphi})\ , </math> where <math>h</math> and <math>f</math> are analytic functions depending on the actions <math>{\underline I}</math> (varying on an open set of <math> R^n</math>) and on the angles <math>{\underline\varphi}</math> (belonging to the standard <math>n</math>-dimensional torus); <math>\varepsilon>0</math> is a small parameter which measures the strength of the perturbation. The aim of classical perturbation theory ([FM]) is to construct a near-to-identity canonical transformation, say <math>C:({\underline I},{\underline \varphi})\rightarrow ({\underline I'},{\underline \varphi'})\ ,</math> which allows to remove the perturbation to higher orders in the perturbing parameter; the transformed Hamiltonian takes the form

<math>

H_1({\underline I'},{\underline \varphi'})\ =\ h_1({\underline I'})+\varepsilon^2 f_1({\underline I'},{\underline\varphi'})\ , </math> where <math>h_1</math> and <math>f_1</math> denote the new unperturbed Hamiltonian and the new perturbing function. The integrable part of the transformed Hamiltonian is simply given by the sum of the old integrable Hamiltonian and the average of the perturbation over the angle variables. The explicit derivation of the expressions for <math> h_1</math> and <math> f_1</math> often presents a considerable computational complexity.

An example of the implementation of classical perturbation theory is the computation of the precession of the perihelion of Mercury in the Mercury-Sun-Jupiter system (within the framework of the restricted, planar, circular, three-body model). Delaunay action-angle variables are introduced, where the actions are related to the osculating semimajor axis and eccentricity of the Keplerian orbit, while the angles are the mean anomaly and the difference between the argument of perihelion and the time. Denoting by <math>\varepsilon</math> the mass-ratio of the primaries, the problem is described by the two degrees-of-freedom Hamiltonian

<math>

H(L,G,\ell,g)=-{1\over {2L^2}}-G+\varepsilon R(L,G,\ell,g)\ , </math> where the perturbing function <math>R=R(L,G,\ell,g)</math> represents the interaction between Mercury and Jupiter. Long computations finally lead to a development of the perturbing function; the first few terms are given by

<math>

R(L,G,\ell,g)=R_{00}(L,G)+R_{10}(L,G)\cos\ell+R_{11}(L,G)\cos(\ell+g)+R_{12}(L,G)\cos(\ell+2g))+...\ , </math> where, denoting by <math>e=\sqrt{1-{G^2\over L^2}}\ ,</math> the coefficients <math>R_{ij}</math> are given by the expressions

<math>

R_{00}=-{L^4\over 4}(1+{9\over {16}}L^4 +{3\over 2}e^2)+...\ ,\quad R_{10}=L^4)+...\ , </math>

<math></math>

R_{11}=-{3\over 8}L^6(1+{5\over {8}}L^4)+...\ ,\quad R_{12}=2101.

Celestial Mechanics, Hamiltonian Normal Forms, Hamiltonian Systems, Kolmogorov-Arnold-Moser Theory, N-body simulations (gravitational), Stability of the solar system, Three Body Problem

Category:Astrophysics Category:Celestial mechanics Category:Dynamical Systems Category:Planetary Physics