CEAS Space Journal manuscript No. (will be inserted by the editor) HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits Martin Lara · Juan F. San-Juan · Denis Hautesserres CEAS Space Journal (ISSN: 1868-2502, ESSN: 1868-2510) (2018) doi:10.1007/s12567-017-0152-x (Pre-print version) Abstract The algorithms used in the construction of a semi- intermediary solutions to the J2 problem [34,11], the contin- analytical propagator for the long-term propagation of High- uous increase in the accuracy of observations demanded the ly Elliptical Orbits (HEO) are described. The software prop- use of more complex dynamical models to achieve a sim- agates mean elements and include the main gravitational and ilar precision in the orbit predictions. In particular higher non-gravitational effects that may affect common HEO or- degrees in the Legendre polynomials expansion of the third- bits, as, for instance, geostationary transfer orbits or Mol- body disturbing function are commonly required (see [17, niya orbits. Comparisons with numerical integration show 24], for instance). Useful analytical theories needed to deal that it provides good results even in extreme orbital config- with a growing number of effects, a fact that made that the urations, as the case of SymbolX. trigonometric series evaluated by the theory comprised tens of thousands of terms [5]. Keywords HEO · Geopotential · third-body perturbation · In an epoch of computational plenty, the vast possibili- tesseral resonances · SRP · atmospheric drag · mean ties offered by special perturbation methods clearly surpass elements · semi-analytic propagation those of general perturbation methods in their traditional application to orbit propagation. Apparently by this reason analytical perturbations have these days been cornered to a 1 Introduction downgraded role of providing some insight into the problem under investigation, a task for which a first order averag- The subject of analytical or semi-analytical propagation is ing is usually considered to be enough, yet the computations very old. Since the first analytical orbit propagators based on of higher orders may provide important details on the dy- namics [33,21]. However, analytical theories like the pop- Funded by CNES contract Ref. DAJ-AR-EO-2015-8181. Preliminary ular SGP4 [16] still enjoy a wide number of users mainly results were presented at the 6th ICATT, Darmstadt, Germany, March involved in catalog propagation duties, a role in which other 14 – 17, 2016 tools like the Draper Semi-Analytic Satellite Theory [29,6] M. Lara or the numeric-analytic theory THEONA [12] can compete GRUCACI – University of La Rioja to numerical integration up to a limited accuracy.1 C/ Madre de Dios 53, Edificio CCT, 26006 Logrono,˜ Spain On the other hand, new needs in satellite propagation, arXiv:1808.05462v1 [astro-ph.EP] 18 Jul 2018 Tel.: +34-941-299440 like the challenges derived of compliance with Space Law, Fax: +34-941-299460 motivate the development of software tools based on analyti- E-mail: [email protected] cal or semi-analytical methods, as, for instance, STELA2. In J.F. San-Juan addition, design of end of life disposal strategies may require GRUCACI – University of La Rioja E-mail: [email protected] the long term propagation of thousands of trajectories to find an optimal solution; accurate ephemeris are not needed in D. Hautesserres CCT – Centre National d’Etudes´ Spatiales 18 Av. Edouard Belin, 1 A partial list of orbit propagators can be found in http://facul- 31401 Toulouse CEDEX 4, France ty.nps.edu/bneta/papers/list.pdf, accessed September 29, 2016. E-mail: [email protected] 2 https://logiciels.cnes.fr/content/stela 2 Martin Lara et al. the preliminary design and using semi-analytic propagation be important because they give rise to long-period effects, makes the approach quite feasible [1]. whereas the effects of tesseral harmonics in general aver- Current needs for long-term propagation at the Centre age out to zero. The latter, however, can have an important National d’Etudes´ Spatiales motivate the present research. impact on resonant orbits, as in the case of geostationary HEOSAT, a semi-analytical orbit propagator to study the satellites (1 to 1 resonance) or GPS and Molniya orbits (2 to long-term evolution of spacecraft in Highly Elliptical Orbits 1 resonance). (HEO) is presented. The perturbation model used includes The importance of each perturbation acting on an earth’s the gravitational effects produced by the more relevant zonal satellite depends on the orbit characteristics, and fundamen- harmonics as well as the main tesseral harmonics affecting tally on the altitude of the satellite. Thus, for instance the to the 2:1 resonance of earth’s gravitational potential, which atmospheric drag, which can have an important impact in has an impact on Molniya-type orbits; the third body per- the lower orbits, may be taken as a higher order effect for turbations in the mass-point approximation, which only in- altitudes above, say, 800 km over the earth’s surface, and is clude the Legendre polynomial of second order for the sun almost negligible above 2000 km. and the polynomials from second order to sixth order in the A sketch of the order of different perturbations when case of the moon; solar radiation pressure, in the cannonball compared to the Keplerian attraction is presented in Fig. 1 approximation, and atmospheric drag. based on approximate formulas borrowed from [32, p. 114]. The forces of gravitational origin are modeled taking ad- As illustrated in the figure, the non-centralities of the Geopo- vantage of the Hamiltonian formalism. Besides, the prob- tential have the most important effect in those parts of the or- lem is formulated in the extended phase space in order to bit that are below the geosynchronous distance, where the J2 avoid time-dependence issues. The solar radiation pressure contribution is a first order perturbation and other harmonics and the atmospheric drag are added as generalized forces. cause second order effects. To the contrary, in those parts of The semi-analytical theory is developed using perturbation the orbit that are farther than the geosynchronous distance techniques based on Lie transforms. Deprit’s perturbation the gravitational pull of the moon is the most important per- algorithm [7] is applied up to the second order of the second turbation, whereas that of the sun is of second order when zonal harmonics, J2. In order to avoid as far as possible the compared to the disturbing effect of the moon, and pertur- lost of long-period effects from the mean elements Hamilto- bations due to J2 and solar radiation pressure (SRP) are of nian, the theory is corrected by the inclusion of long-period third order. terms of the Kozai-type [20,10]. The transformation is de- veloped in closed-form of the eccentricity except for tesseral 0.01 Moon resonances, and the coupling between J2 and the moon’s dis- turbing effects are neglected. J2 10-4 Sun This paper describes the semi-analytical theory and pres- ents relevant examples of the numerical validation. An ex- magnitude -6 tensive description of the tests performed in the validation 10 SRP HAM = 0.01 m2kgL of of the HEOSAT software is given in [22]. Order -8 10 J3 J2,2 2 Dynamics of a spacecraft in HEO 2:1 1:1 1:2 1:3 20 40 60 80 100 120 140 Satellites in earth’s orbits are affected by a variety of per- Distance from the Earth's center Hthousands of kmL turbations of different nature. A full account of this can be Fig. 1 Perturbation order relative to the Keplerian attraction (after [1]). found in textbooks on orbital mechanics (see, for instance, An area to mass ratio of 0:01m2=kg was taken for estimating the SRP. −6 [32,35]). All known perturbations must be taken into ac- The horizontal, dashed line corresponds to 3:33 × 10 times the Kep- lerian attraction. count in orbit determination problems. However, for orbit prediction the accuracy requirements are notably relaxed, and hence some of the disturbing effects may be considered Finally, we recall that integration of osculating elements of higher order in the perturbation model. is properly done only in the True of Date system [9]. For this Furthermore, for the purpose of long-term predictions it reason Chapront’s solar and lunar ephemeris are used [4,3], is customary to ignore short-period effects, which occur on which are directly referred to the mean of date in this way time-scales comparable to the orbital period. Thus, in the including the effect of equinoctial precession. case of the gravitational potential the focus is on the ef- In the case of a highly elliptical orbit the distance from fect of even-degree zonal harmonics, which are known to the satellite to the earth’s center of mass varies notably along cause secular effects. Odd-degree zonal harmonics may also the orbit, a fact that makes particularly difficult to establish HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits 3 the main perturbation over which to hinge the correct pertur- which is written in terms of the x and X, into a new Hamil- bation arrangement. This issue is aggravated by the impor- tonian tance of the gravitational pull of the moon on high altitude em Hf= ∑ H0;m; (2) orbits, which requires taking higher degrees of the Legendre m≥0 m! polynomial expansion of the third-body disturbing function in the new, prime variables, which is free from short-period into account. Because this expansion converges slowly, the terms. size of the multivariate Fourier series representing the moon The transformation is derived from a generating function perturbation will soon grow enormously.