arXiv subjects

Martin Lara

Publications and source records attributed to Martin Lara.

At least 19 recordsLinked to original sources

arXiv2025

Using the Translation Theorem for the Automated Stationkeeping of Extremely-Low Lunar Missions

Extremely-Low Lunar Orbits (eLLOs) (altitudes $\leq 50$ km) exhibit severe perturbations due to the highly non-spherical lunar gravitational field, presenting unique challenges to orbit maintenance. These altitudes are too low for the existence of stable `frozen' orbits, and naive stationkeeping methods, such as circularization, perform poorly. However, mission designers have noticed a particular characteristic of low lunar orbits, which they have found useful for stationkeeping and dubbed the "translation theorem", wherein the eccentricity vector follows a predictable monthly pattern that is independent of its starting value. We demonstrate this feature results from the low orbital eccentricity combined with the dominant effect of a particular subset of sectoral and tesseral harmonics. Subsequently, automated stationkeeping strategies for eLLOs are presented, utilizing this theorem for eccentricity vector control. Several constraints within the eccentricity vector plane are explored, including circular, annular, and elevation-model derived regions, each forming distinct stationkeeping strategies for varying orbital configurations. Subsequently, the optimal control profiles for these maneuvers within the eccentricity plane are obtained using Sequential Convex Programming (SCP). The proposed strategies offer computational simplicity and clear advantages when compared to traditional methods and are comparable to full trajectory optimization.

astro-ph.IM↗

arXiv2024

Orbital perturbation coupling of primary oblateness and solar radiation pressure

Solar radiation pressure can have a substantial long-term effect on the orbits of high area-to-mass ratio spacecraft, such as solar sails. We present a study of the coupling between radiation pressure and the gravitational perturbation due to polar flattening. Removing the short-period terms via perturbation theory yields a time-dependent two-degree-of-freedom Hamiltonian, depending on one physical and one dynamical parameter. While the reduced model is non-integrable in general, assuming coplanar orbits (i.e., both Spacecraft and Sun on the equator) results in an integrable invariant manifold. We discuss the qualitative features of the coplanar dynamics, and find three regions of the parameters space characterized by different regimes of the reduced flow. For each regime, we identify the fixed points and their character. The fixed points represent frozen orbits, configurations for which the long-term perturbations cancel out to the order of the theory. They are advantageous from the point of view of station keeping, allowing the orbit to be maintained with minimal propellant consumption. We complement existing studies of the coplanar dynamics with a more rigorous treatment, deriving the generating function of the canonical transformation that underpins the use of averaged equations. Furthermore, we obtain an analytical expression for the bifurcation lines that separate the regions with different qualitative flow.

astro-ph.EP↗

arXiv2023

Higher-order composition of short- and long-period effects for improving analytical ephemeris computation

The construction of an analytic orbit theory that takes into account the main effects of the Geopotential is notably simplified when splitting the removal of periodic effects in several stages. Conversely, this splitting of the analytical solution into several transformations reduces the evaluation efficiency for dense ephemeris output. However, the advantage is twofold when the different parts of the mean-to-osculating transformation are composed into a single transformation. To show that, Brouwer's solution is extended to the second order of the zonal harmonic of the second degree by the sequential elimination of short- and long-period terms. Then, the generating functions of the different transformations are composed into a single one, from which a single mean-to-osculating transformation is derived. The new, unique transformation notably speeds up the evaluation process, commonly improving evaluation efficiency by at least one third with respect to the customary decomposition of the analytical solution into three different parts.

physics.class-ph↗

arXiv2023

On mean elements in artificial satellite theory

The merits of a perturbation theory based on a mean to osculating transformation that is pure periodic in the fast angle are investigated. The exact separation of the purely short-period effects of the perturbed Keplerian dynamics from the long-period mean frequencies is achieved by a non-canonical transformation, which, therefore, cannot be computed by Hamiltonian methods. For this case, the evolution of the mean elements strictly adheres to the average behavior of the osculating orbit. However, due to the inescapable truncation of perturbation solutions, the fact that this theory confines the long-period variations of the semimajor axis into the mean variation equations, how tiny they may be, can have adverse effects in the accuracy of long-term semi-analytic propagations based on it

math.DS↗

arXiv2022

A torsion-based solution to the hyperbolic regime of the J2-problem

A popular intermediary in the theory of artificial satellites is obtained after the elimination of parallactic terms from the J2-problem Hamiltonian. The resulting quasi-Keplerian system is in turn converted into the Kepler problem by a torsion. When this reduction process is applied to unbounded orbits the solution is made of Keplerian hyperbolae. For this last case, we show that the torsion-based solution provides an effective alternative to the Keplerian approximation customarily used in flyby computations. Also, we check that the extension of the torsion-based solution to higher orders of the oblateness coefficient yields the expected convergence of asymptotic solutions to the true orbit.

math.DS↗

arXiv2022

Earth satellite dynamics by Picard iterations

The main effects of the Earth's oblateness on the motion of artificial satellites are usually derived from the variation of parameters equations of an average representation of the oblateness disturbing function. Rather, we approach their solution under the strict mathematical assumptions of Picard's iterative method. Our approach recovers the known linear trends of the right ascension of the ascending node and the argument of the perigee, but differs from the accepted solution in the value of the mean motion. This amended rate radically improves the in-track errors of typical orbit propagations. In addition, our truncation of the Picard iterations solution to its secular terms includes the corrections that must be applied to the osculating initial conditions in the right propagation of the mean dynamics.

physics.class-ph↗

arXiv2021

Note on the analytical integration of circumterrestrial orbits

The acclaimed merits of analytical solutions based on a fictitious time developed in the 1970's were partially overvalued due to a common misuse of classical analytical solutions based on the physical time that were taken as reference. With the main problem of the artificial satellite theory as a model, we carry out a more objective comparison of both kinds of theories. We find that the proper initialization of classical solutions notably balances the performance of the two distinct approaches in what respects to accuracy. Besides, extension of both kinds of satellite theories to higher orders show additional pros and cons of each different perturbation approach, thus providing complementary information to prospective users on which kind of analytical solution may better support their needs.

math.DS↗

arXiv2021

A new radial, natural, higher order intermediary of the main problem four decades after the elimination of the parallax

Simplifications in dealing with the equation of the center when the short-period effects are removed from a zonal Hamiltonian are commonly attributed to the elimination of parallactic terms. But this interpretation is incorrect, and the simplifications rather stem from the removal of concomitant long-period terms, an outcome that can also be achieved without need of eliminating the parallax. To show that, a Lie transforms simplification is invented that augments the exponents of the inverse of the radius and still achieves analogous simplifications in handling the equation of the center to those provided by the classical elimination of the parallax simplification. The particular case in which the new transformation does not modify the exponents of the parallactic terms of the original problem, leads to a new intermediary of the main problem that, while keeping higher order effects of J2, is formally analogous to Cid's first order radial intermediary.

math.DS↗

arXiv2021

Design of distant retrograde orbits based on a higher order analytical solution

Because the distant retrograde orbits dynamics inherently depends on special functions, approximate analytical solutions in the literature are commonly constrained to providing rough approximations of the qualitative behavior. We rely on perturbation methods and succeed in computing a higher order analytical solution that extends the range of applicability of previous solutions to the problem of relative, quasi-satellite orbits with large librations. Besides, the analytical solution provides two design parameters that are effectively used in the computation of periodic, distant retrograde orbits without constraining to the typical 1:1 resonance.

math.DS↗

arXiv2020

Brouwer's satellite solution redux

Brouwer's solution to the artificial satellite problem is revisited to show that the complete Hamiltonian reduction is rather achieved in the plain Poincar\'e's style, through a single canonical transformation, than using a sequence of partial reductions based on von Zeipel's alternative for dealing with perturbed degenerate Hamiltonian systems.

math.DS↗

arXiv2020

Solution to the main problem of the artificial satellite by reverse normalization

The non-linearities of the dynamics of Earth artificial satellites are encapsulated by two formal integrals that are customarily computed by perturbation methods. Standard procedures begin with a Hamiltonian simplification that removes non-essential short-period terms from the Geopotential, and follow with the removal of both short- and long-period terms by means of two different canonical transformations that can be carried out in either order. We depart from the tradition and proceed by standard normalization to show that the Hamiltonian simplification part is dispensable. Decoupling first the motion of the orbital plane from the in-plane motion reveals as a feasible strategy to reach higher orders of the perturbation solution, which, besides, permits an efficient evaluation of the long series that comprise the analytical solution.

math.DS↗

arXiv2019

Exploring the long-term dynamics of perturbed Keplerian motion in high degree potential fields

The long-term dynamics of perturbed Keplerian motion is usually analyzed in simplified models as part of the preliminary design of artificial satellites missions. It is commonly approached by averaging procedures that deal with literal expressions in expanded form. However, there are cases in which the correct description of the dynamics may require full, contrary to simplified, potential models, as is, for instance, the case of low-altitude, high-inclination lunar orbits. In these cases, dealing with literal expressions is yet possible with the help of modern symbolic algebra systems, for which memory handling is no longer an issue. Still, the efficient evaluation of the averaged expressions related to a high fidelity potential is often jeopardized for the expanded character of the output of the automatic algebraic process, which unavoidably provides huge expressions that commonly comprise tens of thousands of literal terms. Rearrangement of the output to generate an efficient numerical code may solve the problem, but automatization of this kind of post-processing is a non trivial task due to the ad-hoc heuristic simplification procedures involved in the optimization process. However, in those cases in which the coupling of different perturbations is not of relevance for the analysis, the averaging procedure may preserve the main features of the structure of the potential model, thus avoiding the need of the typical blind computer-based brut force perturbation approach. Indeed, we show how standard recursions in the literature may be used to efficiently replace the brut force approach, in this way avoiding the need of further simplification to improve performance evaluation. In particular, Kaula's seminal recursion formulas for the gravity potential reveal clearly superior to the use of both expanded expressions and other recursions more recently proposed in the literature.

math.DS↗

arXiv2018

HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits

The algorithms used in the construction of a semi-analytical propagator for the long-term propagation of Highly Elliptical Orbits (HEO) are described. The software propagates mean elements and include the main gravitational and non-gravitational effects that may affect common HEO orbits, as, for instance, geostationary transfer orbits or Molniya orbits. Comparisons with numerical integration show that it provides good results even in extreme orbital configurations, as the case of SymbolX.

astro-ph.EP↗

arXiv2018

Nonlinear librations of distant retrograde orbits: a perturbative approach -- The Hill problem case

The non-integrability of the Hill problem makes that its global dynamics must be necessarily approached numerically. However, the analytical approach is feasible in the computation of relevant solutions. In particular, the nonlinear dynamics of the Hill problem close to the origin, and the libration point dynamics have been thoroughly investigated by perturbation methods. Out of the Hill sphere, the analytical approach is also feasible, at least in the case of distant retrograde orbits. Previous analytical investigations of this last case succeeded in the qualitative description of the dynamics, but they commonly failed in providing accurate results. This is a consequence of the essential dependance of the dynamics on elliptic functions, a fact that makes to progress in the perturbation approach beyond the lower orders of the solution really difficult. We propose an alternative perturbation approach that allows us to provide a very simple low order analytical solution in trigonometric functions, on the one hand, and, while still depending on special functions, to compute higher orders of the solution, on the other.

math.DS↗

arXiv2018

Higher Order Approximation to the Hill Problem Dynamics about the Libration Points

An analytical solution to the Hill problem Hamiltonian expanded about the libration points has been obtained by means of perturbation techniques. In order to compute the higher orders of the perturbation solution that are needed to capture all the relevant periodic orbits originated from the libration points within a reasonable accuracy, the normalization is approached in complex variables. The validity of the solution extends to energy values considerably far away from that of the libration points and, therefore, can be used in the computation of Halo orbits as an alternative to the classical Lindstedt-Poincaré approach. Furthermore, the theory correctly predicts the existence of the two-lane bridge of periodic orbits linking the families of planar and vertical Lyapunov orbits.

math.DS↗

arXiv2018

Short-axis-mode rotation in complex variables

Decomposition of the free rigid body Hamiltonian into a "main problem" and a perturbation term provides an efficient integration scheme that avoids the use of elliptic functions and integrals. In the case of short-axis-mode rotation, it is shown that the use of complex variables converts the integration of the torque-free motion by perturbations into a simple exercise of polynomial algebra that can also accommodate the gravity-gradient perturbation when the rigid body rotation is close enough to the axis of maximum inertia.

math.DS↗

arXiv2017

A Hopf variables view on the libration points dynamics

The dynamics about the libration points of the Hill problem is investigated analytically. In particular, the use of Lissajous variables and perturbation theory allows to reduce the problem to a one degree of freedom Hamiltonian depending on two physical parameters. The invariant manifolds structure of the Hill problem is then disclosed, yet accurate computations are limited to energy values close to that of the libration points.

math.DS↗

arXiv2016

Note on the ideal frame formulation

An implementation of the ideal frame formulation of perturbed Keplerian motion is presented which only requires the integration of a differential system of dimension 7, contrary to the 8 variables traditionally integrated with this approach. The new formulation is based on the integration of a scaled version of the Eulerian set of redundant parameters, and slightly improves runtime performance with respect to the 8 dimensional case while retaining comparable accuracy.

math.DS↗