Frozen orbits of the Hill problem are determined in the double-averaged problem, where short and long-period terms are removed by means of Lie transforms. Due to the perturbation method we use, the initial conditions of corresponding quasi-periodic solutions in the nonaveraged problem are computed straightforwardly. Moreover, the method provides the explicit equations of the transformation that connects the averaged and nonaveraged models. A fourth-order analytical theory is necessary for the accurate computation of quasi-periodic frozen orbits.
Besides its original application to the motion of the Moon [
A classical result shows that low eccentricity orbits around a primary body are unstable for moderate and high inclinations due to third-body perturbations [
The study of the long-term dynamics is usually done in the double-averaged problem. After removing the short- and long-period terms, and truncating higher-order terms, the problem is reduced to one degree of freedom in the eccentricity and the argument of the periapsis. As the double-averaged problem is integrable and the corresponding phase space is a compact manifold, the solutions are closed curves and equilibria. The latter are orbits that, on average, have almost constant eccentricity and fixed argument of the periapsis, and are known as frozen orbits.
To each trajectory of the double-averaged problem it corresponds a torus of quasiperiodic solutions in the nonaveraged problem. The accurate computation of initial conditions on the torus requires the recovery of the short- and long-period effects that were eliminated in the averaging. This is normally done by trial and error, making iterative corrections on the orbital elements, although other procedures can be applied [
Our analytical theory is computed with Deprit's perturbation technique [
While, in general, the third-order theory provides good results in the computation of quasiperiodic, frozen orbits, its solutions are slightly affected by long-period oscillations. This fact may adversely affect the long-term evolution of the frozen orbits and it becomes apparent in the computation of science orbits about planetary satellites, a case in which small perturbations are enough for the unstable dynamics to defrost the argument of the periapsis. Then, the orbit immediately migrates along the unstable manifold with an exponential increase in the eccentricity.
We find that a higher-order truncation is desirable if one wants to use the analytical theory for computing accurate initial conditions of frozen orbits. The computation of the fourth-order truncation removes almost all adverse effects from the quasiperiodic solutions, and shows a high degree of agreement between the averaged and nonaveraged models even in the case of unstable orbits.
Whereas the third-body perturbation is the most important effect in destabilizing science orbits around planetary satellites, the impact of the nonsphericity of the central body may be taken into account. The previous research including both effects has been limited up to third-order theories (see [
The equations of motion of the Hill problem are derived from the Hamiltonian
The problem is of three degrees of freedom, yet admitting the Jacobi constant
To apply perturbation theory, we formulate the problem in Delaunay variables
Our theory is based on the use of Lie transforms as described by Deprit [
The double-averaged Hamiltonian (
The flow can be integrated from the differential equations mentioned previously, (
Flow in the doubly reduced phase space.
Delaunay variables are singular for zero eccentricity orbits, where the argument of the periapsis and the mean anomaly are not defined, and for equatorial orbits, where the argument of the node is not defined. Hence, it is common to study the reduced phase space in the variables introduced by Coffey et al. [
Then, after dropping constant terms and scaling, Hamiltonian (
Equations (
The real roots of (
Regions in the parameters plane with different numbers of equilibria.
Note that the curve given by (
Bifurcation lines in the parameter plane.
A powerful test for estimating the quality of the analytical theory is to check the degree of agreement of the bifurcation lines of the analytical theory with those computed numerically in the nonaveraged problem. To do that we compute several families of three-dimensional, almost circular, periodic orbits of the Hill (nonaveraged) problem that bifurcate from the family of planar retrograde orbits at different resonances. For variations of the Jacobi constant the almost circular periodic orbits evolve from retrograde to direct orbits through the 180 degrees of inclination. At certain critical points, almost circular orbits change from stable to unstable in a bifurcation phenomenon in which two new elliptic periodic orbits appear. The computation of a variety of these critical points helps in determining stability regions for almost circular orbits [
The tests done show that the fourth-order theory gives good results for
Comparison between the bifurcation line of circular, averaged orbits (full line), and the curve of critical periodic orbits (dots).
Hill's case of orbits close to the smaller primary is a simplification of the restricted three-body problem, which in turn is a simplification of real models. Therefore, the final goal of our theory is not the generation of ephemerides but to help in mission designing for artificial satellites about planetary satellites, where frozen orbits are of major interest.
For given values of the parameters
We choose
Initial orbital elements of an elliptic frozen orbit for
Theory | ||||||
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Classical | ||||||
2nd order | ||||||
3rd order | ||||||
4th order |
Long-term evolution of the orbital elements of the elliptic frozen orbit.
When computing a second-order theory with the Lie-Deprit perturbation method we arrive exactly at the classical Hamiltonian obtained by a simple removal of the short-period terms and the classical bifurcation condition that results in the critical inclination of the third-body perturbations
The results of the third- and fourth-order theories are presented in the last two rows of Table
Long-term evolution of the orbital elements of the elliptic frozen orbit.
If we choose the same value for
Long-term evolution of the orbital elements of the circular stable frozen orbit. (a) and (c) third-order theory. (b) and (d) fourth-order theory.
For
Long-term evolution of the orbital elements of the circular, unstable, frozen orbit.
Alternatively to the temporal analysis mentioned previously, a frequency analysis using the Fast Fourier Transform (FFT) shows how initial conditions obtained from different orders of the analytical theory can be affected of undesired frequencies that defrost the orbital elements.
Thus, Figure
(a) FFT analysis of the instantaneous argument of the periapsis of the elliptic solution. (b) Magnification over the low frequencies region.
Figure
(a) FFT analysis of the instantaneous eccentricity of the elliptic solution. (b) Magnification over the low frequencies region.
An FFT analysis of unstable circular orbits has not much sense because of the time scale in which the orbit destabilizes.
Frozen orbits computation is a useful procedure in mission designing for artificial satellites. After locating the frozen orbit of interest in a double-averaged problem, usual procedures for computing initial conditions of frozen orbits resort to trial-and-error interactive corrections, or require involved computations. However, the explicit transformation equations between averaged and nonaveraged models can be obtained with analytical theories based on the Lie-Deprit perturbation method, which makes the frozen orbits computations straightforward.
Accurate computations of the initial conditions of frozen, quasiperiodic orbits can be reached with higher-order analytical theories. This way of proceeding should not be undervalued in the computation of science orbits around planetary satellites, a case in which third-body perturbations induce unstable dynamics.
Higher-order analytical theories are a common tool for computing ephemeris among the celestial mechanics community. They are usually developed with specific purpose, sophisticated algebraic manipulators. However, the impressive performances of modern computers and software allow us to build our analytical theory with commercial, general-purpose manipulators, a fact that may challenge aerospace engineers to use the safe, well-known techniques advocated in this paper.
Let
To average the short-period effects we write Hamiltonian (
Since the radius
After applying the Delaunay normalization [
The Lie transform of generating function
A new application of the recurrence (
The new Lie transform of generating function
This work was supported from Projects ESP 2007-64068 (the first author) and MTM 2008-03818 (the second author) of the Ministry of Science and Innovation of Spain is acknowledged. Part of this work has been presented at 20th International Symposium on Space Flight Dynamics, Annapolis, Maryland, USA, September, 24–28 2007.