Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.     

Artificial frozen orbits around Mercury

Orbits around Mercury are influenced by the strong elliptic third-body perturbation, especially for high eccentricity orbits, the periapsis altitude changes dramatically. Frozen orbits whose mean eccentricity and argument of perigee remain constants are obviously a good choice for space missions, but the forming conditions are too harsh to meet practical needs. To deal with this problem, a continuous control method that combines analytical theory and parameter optimization is proposed to build an artificial frozen orbit. The artificial frozen orbits are investigated on the basis of double averaged Hamiltonian, of which the second and third zonal harmonics and the perturbation of elliptic third-body gravity are considered. In this paper, coefficients of perturbations which satisfy the conditions of frozen orbits are involved as control parameters, and the relevant artificial perturbations are compensated by the control strategy. So probes around Mercury can be kept on frozen orbit under the influence of continuous control force. Then complex method of optimization is used to search for the energy optimized artificial frozen orbits. The choosing of optimal parameters, the objective function setting and other issues are also discussed in the study. Evolution of optimal control parameters are given in large ranges of semi-major axis and eccentricity, through the variation of these curves, the fuel efficiency is discussed. The result shows that the control method proposed in this paper can effectively maintain the eccentricity and argument of perigee frozen.     

Analytical investigations of quasi-circular frozen orbits in the Martian gravity field

Frozen orbits are always important foci of orbit design because of their valuable characteristics that their eccentricity and argument of pericentre remain constant on average. This study investigates quasi-circular frozen orbits and examines their basic nature analytically using two different methods. First, an analytical method based on Lagrangian formulations is applied to obtain constraint conditions for Martian frozen orbits. Second, Lie transforms are employed to locate these orbits accurately, and draw the contours of the Hamiltonian to show evolutions of the equilibria. Both methods are verified by numerical integrations in an 80 × 80 Mars gravity field. The simulations demonstrate that these two analytical methods can provide accurate enough results. By comparison, the two methods are found well consistent with each other, and both discover four families of Martian frozen orbits: three families with small eccentricities and one family near the critical inclination. The results also show some valuable conclusions: for the majority of Martian frozen orbits, argument of pericentre is kept at 270° because J ₃ has the same sign as J ₂; while for a minority of ones with low altitude and low inclination, argument of pericentre can be kept at 90° because of the effect of the higher degree odd zonals; for the critical inclination cases, argument of pericentre can also be kept at 90°. It is worthwhile to note that there exist some special frozen orbits with extremely small eccentricity, which could provide much convenience for reconnaissance. Finally, the stability of Martian frozen orbits is estimated based on the trace of the monodromy matrix. The analytical investigations can provide good initial conditions for numerical correction methods in the more complex models.     

A note on lower bounds for relative equilibria in the main problem of artificial satellite theory

In the analytical approach to the main problem in satellite theory, the consideration of the physical parameters imposes a lower bound for normalized Hamiltonian. We show that there is no elliptic frozen orbits, at critical inclination, when we consider small values of H, the third component of the angular momentum. The argument used suggests that it might be applied also to more realistic zonal and tesseral models. Moreover, for almost polar orbits, when H may be taken as another small parameter, a different approach that will simplify the ephemerides generators is proposed.     

Short Term Evolution of Artificial Satellites

When the elimination of the parallax and the elimination of the perigee is applied to the zonal problem of the artificial satellite, a one-degree of freedom Hamiltonian is obtained. The classical way to integrate this Hamiltonian is by applying the Delaunay normalization, however, changing the time to the perturbed true anomaly and the variable to the inverse of the distance, the Hamilton equations become a perturbed harmonic oscillator. In this paper we apply the Krylov—Bogoliubov—Mitropolsky (KBM) method to integrate the perturbed harmonic oscillator as an alternative method to the Delaunay normalization. This method has no problem with small eccentricities and inclinations, and shows good numerical results in the evaluation of ephemeris of satellites.This revised version was published online in October 2005 with corrections to the Cover Date.     

Relativistic and the first sectorial harmonics corrections in the critical inclination

The problem of the critical inclination is treated in the Hamiltonian framework taking into consideration post-Newtonian corrections as well as the main correction term of sectorial harmonics for an earth-like planet. The Hamiltonian is expressed in terms of Delaunay canonical variables. A canonical transformation is applied to eliminate short period terms. A modified critical inclination is obtained due to relativistic and the first sectorial harmonics corrections.     

The Main Problem in Satellite Theory Revisited

Using the elimination of the parallax followed by the Delaunay normalization, we present a procedure for calculating a normal form of the main problem (J ₂ perturbation only) in satellite theory. This procedure is outlined in such a way that an object-oriented automatic symbolic manipulator based on a hierarchy of algebras can perform this computation. The Hamiltonian after the Delaunay normalization is presented to order six explicitly in closed form, that is, in which there is no expansion in the eccentricity. The corresponding generating function and transformation of coordinates, too lengthy to present here to the same order; the generator is given through order four. This revised version was published online in July 2006 with corrections to the Cover Date.     

Averaged model to study long-term dynamics of a probe about Mercury

This paper provides a method for finding initial conditions of frozen orbits for a probe around Mercury. Frozen orbits are those whose orbital elements remain constant on average. Thus, at the same point in each orbit, the satellite always passes at the same altitude. This is very interesting for scientific missions that require close inspection of any celestial body. The orbital dynamics of an artificial satellite about Mercury is governed by the potential attraction of the main body. Besides the Keplerian attraction, we consider the inhomogeneities of the potential of the central body. We include secondary terms of Mercury gravity field from \(J_2\) up to \(J_6\), and the tesseral harmonics \(\overline{C}_{22}\) that is of the same magnitude than zonal \(J_2\). In the case of science missions about Mercury, it is also important to consider third-body perturbation (Sun). Circular restricted three body problem can not be applied to Mercury–Sun system due to its non-negligible orbital eccentricity. Besides the harmonics coefficients of Mercury’s gravitational potential, and the Sun gravitational perturbation, our average model also includes Solar acceleration pressure. This simplified model captures the majority of the dynamics of low and high orbits about Mercury. In order to capture the dominant characteristics of the dynamics, short-period terms of the system are removed applying a double-averaging technique. This algorithm is a two-fold process which firstly averages over the period of the satellite, and secondly averages with respect to the period of the third body. This simplified Hamiltonian model is introduced in the Lagrange Planetary equations. Thus, frozen orbits are characterized by a surface depending on three variables: the orbital semimajor axis, eccentricity and inclination. We find frozen orbits for an average altitude of 400 and 1000 km, which are the predicted values for the BepiColombo mission. Finally, the paper delves into the orbital stability of frozen orbits and the temporal evolution of the eccentricity of these orbits.     

Integrable approximation of J 2-perturbed relative orbits

Most existing satellite relative motion theories utilize mean elements, and therefore cannot be used for calculating long-term bounded perturbed relative orbits. The goal of the current paper is to find an integrable approximation for the relative motion problem under the J ₂ perturbation, which is adequate for long-term prediction of bounded relative orbits with arbitrary inclinations. To that end, a radial intermediary Hamiltonian is utilized. The intermediary Hamiltonian retains the original structure of the full J ₂ Hamiltonian, excluding the latitude dependence. This formalism provides integrability via separation, a fact that is utilized for finding periodic relative orbits in a local-vertical local-horizontal frame and determine an initialization scheme that yields long-term boundedness of the relative distance. Numerical experiments show that the intermediary-based computation of orbits provides long-term bounded orbits in the full J ₂ problem for various inclinations. In addition, a test case is shown in which the radial intermediary-based initial conditions of the chief and deputy satellites yield bounded relative distance in a high-precision orbit propagator.     

Perturbations of the Kepler Problem in Global Coordinates

The usual action-angle variables for the Kepler Problem (the Delaunay variables) are not globally defined, leaving out some orbits (circular orbits or those lying on the xy-plane). Moreover they are trascendental functions of the physical variables, making it quite difficult to write the perturbed Hamiltonian. The way-out proposed here is to pass to a 8-dimensional rank-6 Poisson manifold, that is, to parametrize the state of the Kepler Problem with two 4-dimensional vectors mutually orthogonal and of equal norm. This revised version was published online in July 2006 with corrections to the Cover Date.     

Secular evolution of the orbits of hypothetical satellites of Uranus

The problem of the secular perturbations of the orbit of a test satellite with a negligible mass caused by the joint influence of the oblateness of the central planet and the attraction by its most massive (or main) satellites and the Sun is considered. In contrast to the previous studies of this problem, an analytical expression for the full averaged perturbing function has been derived for an arbitrary orbital inclination of the test satellite. A numerical method has been used to solve the evolution system at arbitrary values of the constant parameters and initial elements. The behavior of some set of orbits in the region of an approximately equal influence of the perturbing factors under consideration has been studied for the satellite system of Uranus on time scales of the order of tens of thousands of years. The key role of the Lidov–Kozai effect for a qualitative explanation of the absence of small bodies in nearly circular equatorial orbits with semimajor axes exceeding ~1.8 million km has been revealed.     

J2 Invariant Relative Orbits for Spacecraft Formations

An analytic method is presented to establish J ₂ invariant relative orbits. Working with mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the J₂ influence. Two first order conditions are established between the differences in momenta elements (semi-major axis, eccentricity and inclination angle) that guarantee that the drift rates of two neighboring orbits are equal on the average. Differences in the longitude of the ascending node, argument of perigee and initial mean anomaly can be set at will, as long as they are setup in mean element space. For near polar orbits, enforcing both momenta element constraints may result in impractically large relative orbits. It this case it is shown that dropping the equal ascending node rate requirement still avoids considerable relative orbit drift and provides substantial fuel savings.This revised version was published online in October 2005 with corrections to the Cover Date.     

The Relegation Algorithm

The relegation algorithm extends the method of normalization by Lie transformations. Given a Hamiltonian that is a power series = ₀+ ₁+ ... of a small parameter , normalization constructs a map which converts the principal part ₀into an integral of the transformed system — relegation does the same for an arbitrary function [G]. If the Lie derivative induced by [G] is semi-simple, a double recursion produces the generator of the relegating transformation. The relegation algorithm is illustrated with an elementary example borrowed from galactic dynamics; the exercise serves as a standard against which to test software implementations. Relegation is also applied to the more substantial example of a Keplerian system perturbed by radiation pressure emanating from a rotating source.This revised version was published online in October 2005 with corrections to the Cover Date.     

Frozen orbits for satellites close to an Earth-like planet

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ ₂ throughJ ₉. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J ₂ only). In particular, we examine the manner in which the odd zonalJ ₃ breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J ₄+J ₂ ² =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.     

Dynamics of a satellite orbiting a planet with an inhomogeneous gravitational field

We study the dynamics of a satellite (artificial or natural) orbiting an Earth-like planet at low altitude from an analytical point of view. The perturbation considered takes into account the gravity attraction of the planet and in particular it is caused by its inhomogeneous potential. We begin by truncating the equations of motion at second order, that is, incorporating the zonal and the tesseral harmonics up to order two. The system is formulated as an autonomous Hamiltonian and has three degrees of freedom. After three successive Lie transformations, the system is normalised with respect to two angular co-ordinates up to order five in a suitable small parameter given by the quotient between the angular velocity of the planet and the mean motion of the satellite. Our treatment is free of power expansions of the eccentricity and of truncated Fourier series in the anomalies. Once these transformations are performed, the truncated Hamiltonian defines a system of one degree of freedom which is rewritten as a function of two variables which generate a phase space which takes into account all of the symmetries of the problem. Next an analysis of the system is achieved obtaining up to six relative equilibria and three types of bifurcations. The connection with the original system is established concluding the existence of various families of invariant 3-tori of it, as well as quasiperiodic and periodic trajectories. This is achieved by using KAM theory techniques.     

Groups of symmetries in the two‐body problem associated to Einstein's PN field

The two‐body problem associated to a spherical post‐Newtonian (PN) field with Einsteinian parameterization is revisited from the single standpoint of symmetries. The corresponding vector fields, in Hamiltonian and standard polar coordinates, or in collision‐blow‐up and infinity‐blow‐upMcGehee‐type coordinates, present symmetries that form diffeomorphic commutative groups endowed with a Boolean structure. The existence of such symmetries is of much help in understanding characteristics of the global flow, or in finding symmetric periodic orbits in more complex problems depending on a small parameter.     

Normalization and the detection of integrability: The generalized Van Der Waals potential

Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled elliptic oscillators are known. In order to test the utility of normalization as a detector of integrability we normalize, to high order, a perturbed Keplerian system known to have several integrable limits; the generalized van der Waals Hamiltonian for a hydrogen atom. While the separable limits give rise to high order degeneracy we find a non-separable, integrable limit for which the normal form does not exhibit degeneracy. We conclude that normalization may, in certain cases, indicate integrability but is not guaranteed to uncover all integrable limits.     

Ptolemaic Transformation in Keplerian Problem

We propose the Ptolemaic transformation: a canonical change of variables reducing the Keplerian motion to the form of a perturbed Hamiltonian problem. As a solution of the unperturbed case, the Ptolemaic variables define an intermediary orbit, accurate up to the first power of eccentricity, like in the kinematic model of Claudius Ptolemy. In order to normalize the perturbed Hamiltonian we modify the recurrent Lie series algorithm of HoriuuMersman. The modified algorithm accounts for the loss of a term's order during the evaluation of a Poisson bracket, and thus can be also applied in resonance problems. The normalized Hamiltonian consists of a single Keplerian term; the mean Ptolemaic variables occur to be trivial, linear functions of the Delaunay actions and angles. The generator of the transformation may serve to expand various functions in Poisson series of eccentricity and mean anomaly.     

Refined model for the evolution of distant satellite orbits

We consider a model that describes the evolution of distant satellite orbits and that refines the solution of the doubly averaged Hill problem. Generally speaking, such a refinement was performed previously by J. Kovalevsky and A.A. Orlov in terms of Zeipel’s method by constructing a solution of the third order with respect to the small parameter m, the ratio of the mean motions of the planet and the satellite. The analytical solution suggested here differs from the solutions obtained by these authors and is closest in form to the general solution of the doubly averaged problem (∼m ²). We have performed a qualitative analysis of the evolutionary equations and conditions for the intersection of satellite orbits with the surface of a spherical planet with a finite radius. Using the suggested solution, we have obtained improved analytical time dependences of the elements of evolving orbits for a number of distant satellites of giant planets compared to the solution of the doubly averaged Hill problem and, thus, achieved their better agreement with the results of our numerical integration of the rigorous equations of perturbed motion for satellites.