_{1}

^{*}

We demonstrate in this paper that periodic variations of the
J
_{2}
gravity coefficient of a planet induce small cumulative perturbations on a given family of circular equatorial orbits, and that these perturbations could be measurable with current radiosciences technology. For this purpose, we first consider a Poincaré expansion of the Newtonian equations of motion. Then, by using Floquet’s theory, we demonstrate that, unlike the excitation mechanism, the perturbations are non-periodic, and that the orbit is not “stable” in the long-term, with perturbations growing exponentially. We give the full theory and an application to the case of planet Mars.

Chao and Rubincam [_{2} harmonic moment of Mars is subject to large annual variation as about one quarter of the CO_{2} atmosphere condenses during winters at the poles, and sublimes during summers (see

where_{0} is the Mars

mean polar inertial moment; Ω_{0} is the mean Mars angular rotation velocity; ω is the mean Mars angular orbital velocity; R is the Mars radius; and M is the planet’s mass. Similar variations, but of a lesser amplitude, are also observed on the Earth. In this paper, we show that these small variations can end up in cumulative perturbations on selected families of orbits. This work can be easily extended to semi-annual perturbations and larger degree and order gravity coefficients [

With respect to a given inertial cartesian coordinate frame, the Newtonian equations of motion of a space probe orbiting a planet are

where

maining part of the gravity field (without the J_{2} term), and the _{2} terms. The aims of the notation

We now consider a probe orbiting the planet on a high altitude (i.e. with

where

It is clear that the last equation is decoupled from the first two ones. The solution of the third one corresponds to an oscillation with respect to the mean orbital plane, of no interest for the following discussion.

If we switch to cylindrical coordinates

The constant h_{0} can be identified as an angular momentum.

Jezewski [

where_{c}. One can demonstrate that

ent periods, and define an “ellipsis” with an apse line slowly rotating in the equatorial plane, with a period

The Hamiltonian of the unperturbed motion is given by

If

where

For the first order, after some uninteresting algebra, we arrive at

This system can be rewritten as a first order system by using the usual trick

Similar equations can be obtained from the formalisms of Hill or Lagrange. The approach that we retained is the simplest one. Considering an equatorial circular orbit is a fundamental assumption, as it allows us to write a very simple analytical solution.

System (5) is of Floquet’s type [

The solution of the system with the second member w is given, as

where

with

If

with

This relation shows that the behavior of this system in the “long” term is governed by_{2}. To obtain a common period for N and w, we just have to slightly adjust the altitude of the spacecraft, in order to have an entire number of orbital periods during a Martian year that is then becoming the common period T.

Let us consider the solution for the particular case of the planet Mars and for a circular equatorial orbit. The period T of a circular equatorial orbit of radius d is given from (1’) by

We take the numerical values from [

From these values, we derive^{−6} relative variation with respect to the

We now consider a circular orbit well beyond the atmosphere, at an altitude of 1000.629961 km (semi-major axis 4394.829961 km), in order to have exactly 6716 orbits/Martian year, corresponding to an orbital period of 147.266 min.

This leads to

For the first year, the perturbations (norm of the differences between the perturbed and unperturbed motion) range up to 172.58 m in position, most of it in the along-track direction, and 122.69 mm/s in velocity. They are measurable with state-of-the-art technology, both for laser and Doppler tracking [

a sufficient amount of time, from other Mars satellites, or even from the Earth, if it is equipped with “active” laser receptors [

We believe that this phenomenon is general, and that the theory described in this paper deserves to be generalized to any type of orbit, including polar orbits dedicated to mapping. The analysis will be then complicated by the presence of the secular perturbations caused by the even zonal coefficients of the gravity field, and other long period perturbations. The effect of the perturbations that originate from the triaxiality of Mars is investigated in Appendix B. We plan also to study semiannual variations of the J_{2} gravity coefficient, and to understand how the excitation mechanism described in this paper acts on the orbits of Phobos and Deimos that are near circular equatorial orbits.

This research was funded by the Centre National d’Etudes Spatiales (CNES).

Jean-Pierre Barriot, (2015) Cumulative Perturbations Affecting a Spacecraft on a Mars Equatorial Orbit from the Waxing and Waning of the Polar Caps of the Planet. Open Access Library Journal,02,1-9. doi: 10.4236/oalib.1102272

We have, with

with

Therefore,

Let us verify that this formula defines a continuous mapping of

For

For

thus proving the continuity.

The above analysis supposes that the equatorial moments of Mars are equal. Unfortunately, because of the Tharsis uplift, Mars is the terrestrial planet for which this assumption is the least accurate. If we take into account this triaxiality, the equations of motion (1) become, in an ad’hoc reference frame and up to degree and order two [

where

uation (3)), with respect to both