Celest Mech Dyn Astr (2011) 109:333–366 DOI 10.1007/s10569-010-9331-1 ORIGINAL ARTICLE Complete closed-form solutions of the Stark problem Gregory Lantoine · Ryan P. Russell Received: 16 April 2010 / Revised: 25 August 2010 / Accepted: 16 December 2010 / Published online: 5 February 2011 © Springer Science+Business Media B.V. 2011 Abstract Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direc- tion. For both the two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions. Keywords Stark problem · Kepler problem · Perturbation · Integrability · Analytical solution · Low-thrust 1 Introduction 1.1 Motivations The purpose of this paper is to describe the so-called Stark problem, the perturbed two-body problem where the perturbation is a constant force in magnitude and direction. This problem is interesting because it is one of the few exactly solvable problems (i.e. systems of governing ordinary differential equations) in Celestial Mechanics. G. Lantoine (B) · R. P. Russell School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected] R. P. Russell e-mail: [email protected] 123 334 G. Lantoine, R. P. Russell This problem is named after the German physicist Johannes Stark who discovered in 1913 the Stark effect (Stark 1914), i.e. the shifting and splitting of spectral lines of atoms and molecules in the presence of an external static electric field. Understanding the Stark effect is important for the analysis of atomic and molecular rotational spectra (Chattopadhyay and Boxer 1995; Pierce and Boxer 1995). Thus, apart from the intrinsic interest of its own, the Stark problem can play the role of a model problem, from which one can obtain information about the properties of atoms when strong electric fields are present. Exploiting this effect can also greatly enhance the utilities of some molecules (Ishigami et al. 2005). By analogy between atoms and planets, systems described by the Stark problem are also common in orbital mechanics. We can mention the case of low-thrust trajectories where a spacecraft engine can generate a continuous acceleration of constant magnitude and direc- tion. The Stark problem can therefore model a low-thrust trajectory discretized by segments of constant thrust (Lantoine and Russell 2009). Another important potential application of this problem is the study of the influence of the solar pressure on the orbit of a satellite, since the corresponding perturbation force can be approximated as constant over a short time inter- val. This is a particularly valid approximation when the satellite does not enter often into the shadow of the Earth, for instance in the cases of sun-synchronous orbits above the terminator. However, in the present work we stop short of these considerations; our aim is confined to the derivation of explicit representations for the solutions of the Stark problem. Next, we give a thorough review of previous work regarding the Stark model over the past three centuries. 1.2 Historical survey The main integrable problems of Celestial Mechanics are easily counted, namely, the Kepler problem, the Euler problem (two center Newtonian gravitational motion), and the Stark problem. This small number of integrable problems along with the interest from the physics community explains why the Stark problem has received special attention over almost two and one-half centuries, with occasionally periods of intense studies. The Stark problem was shown to be integrable first by Lagrange who reduced it to quadr- atures at the end of the eighteenth century (Lagrange 1788). Although elliptic functions were not known at his time, Lagrange’s analysis and reduction is very elegant, and he demon- strates the intuition that the solution can be expressed with some transcendental functions (“rectification of conic sections”). Lagrange also points out that the Stark problem differs significantly from the Euler problem and that a dedicated analysis is necessary. In the middle of the nineteenth century, two mathematicians, Jacobi and Liouville, give crucial contributions towards a more a rigorous treatment of the Stark problem. Their work represents the mathematical foundation that all later studies build upon, including this paper. Following the work of Hamilton on his ‘General Method in Dynamics’ (Hamilton 1835), Jacobi derives a general procedure for the study of dynamical systems through the Hamilton-Jacobi equation. Jacobi also found that the Stark system admits the separation of its variables in the parabolic coordinates and formulates the Hamilton-Jacobi equation for the problem in these coordinates. Complementing the ideas of Jacobi, Liouville derives sufficient conditions for separability of dynamical systems (at the origin of the notion of Liouville integrability), and noted that most of the known integrable problems, including the Stark problem, met his conditions for separability (Liouville 1849). At the beginning of the twentieth century, the Stark problem received other attention due to the first observations of the Stark effect. Within a decade of the appearance of Bohr’s quantum theory, this effect was first explored to explain some characteristics of the 123 Complete closed-form solutions of the Stark problem 335 hydrogen atom in a state excited by a homogeneous electric field (Banks and Leopold 1978a,b; Murray-Krezan 2008; Hezel et al. 1992; Byrd and Mitchell 2004). Most authors relied on the same parabolic coordinates and separation of variables to characterize the motion of the electron in a hydrogen atom, whether from a classical or quantum mechanics perspective. On one hand, Born stays in a classical mechanics context to show that previous theories of the Stark problem are particularly powerful to find transition frequencies of excited systems, but he only finds approximate solutions using Fourier series (Born 1960). Slightly later Epstein treated the same expressions by successive expansions and obtained results up to second order in the electric field strength (Epstein 1916). On the other hand, Froman provides a comprehensive review of all major studies of the Stark effect from a quantum mechanics point of view (Froman 2008). In connection with the development of wave mechanics, it is shown that the Schrodinger equation for a one-electron system in a homogeneous electric field is also separable in parabolic coordinates. This analogy between the treatment of the Stark problem in quantum and classical mechanics is remarkable. More recently, the Stark problem is extended by considering a charged particle moving in the field of one localized dyon inside a homogeneous electric field (referred to as the MICZ-Kepler-Stark system) (Nersessian and Ohanyan 2008). As in the nominal case, this also is an integrable system, which allows separation of variables in parabolic coordinates. Furthermore, the advent of the space age in the 1950s following the launch of Sputnik led to an increase of interest in the Stark problem. There were investigations in two areas: a theo- retical examination of the solutions of the Stark problem; and an investigation of the potential use of solutions of the Stark problem as a basis of approximation for specific solutions in orbital dynamics. In the former, we mention especially the work of Isayev who derived an analytical theory of motion for a balloon-satellite perturbed by a constant solar pressure (Isayev and Kunitsyn 1972); and in the latter we note the interesting work of Beletskii about the planar Stark problem (Beletsky 2001). Following his discussion of the accessible/non- accessible regions, the solution forms are presented in terms of Jacobian elliptic functions based on the analysis of the integrals of motion. The author also characterizes (incompletely) the different types of orbits that can be encountered in the two-dimensional Stark problem and provides representative planar trajectories. However, one negative feature of the form of the Beletskii’s solutions is that some of the parameters in the solutions tend to infinity in the Kepler limit (as the perturbation approaches zero). At about the same time, Vinti investigated the effect of a constant force on a Keplerian orbit with the introduction of Delaunay variables (a common set of variables for perturbed Kepler problems) (Vinti 1964). Note that his result is approximate as he eliminates short periodic terms to focus on secular terms only. Another important milestone in the history of the Stark problem is the work of Kirchgraber in the seventies. He is concerned (like us) with handling perturbations of the Kepler prob- lem and contrary to previous authors who mainly use parabolic coordinates, he uses the so-called KS-variables to show that the Stark problem is separable and to describe it ana- lytically (Kirchgraber 1971). However, the required change of variables is a complicated nonlinear transformation which is likely to be computationally inefficient and presents less physical insight than previous transformations. Also, the final closed-form expressions of the solutions are unfortunately not given. Having arrived at the quadrature, he states that this last integral can be solved by invoking the Jacobi elliptical functions. But there is no indication of how this last step is accomplished. Using Kirchgraber’s method, Rufer applies the Stark formulation to low-thrust trajectory optimization (Rufer 1976).