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A Semi-Analytical Orbit Propagator Program for Highly Elliptical Orbits∗

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A Semi-Analytical Orbit Propagator Program for Highly Elliptical Orbits∗ A SEMI-ANALYTICAL ORBIT PROPAGATOR PROGRAM FOR HIGHLY ELLIPTICAL ORBITS∗ Martin Laray & Juan F. San-Juanz Denis Hautesserres University of La Rioja Centre National d’Etudes´ Spatiales GRUCACI – Scientific Computation Group Centres de Competence Technique 26004 Logrono,˜ Spain 31401 Toulouse CEDEX 4, France ABSTRACT and the coupling between J2 and moon’s disturbing effects are neglected. The paper outlines the semi-analytical theory. The algorithms used in the construction of a semi-analytical propagator for the long-term propagation of Highly Ellipti- cal Orbits (HEO) are described. The software propagates 2. DYNAMICS OF A SPACECRAFT IN HEO mean elements and include the main gravitational and non- gravitational effects that may affect common HEO orbits, as, Satellites in earth’s orbits are affected by a variety of perturba- for instance, geostationary transfer orbits or Molniya orbits. tions of a diverse nature. A full account of this can be found in textbooks on orbital mechanics like [1]. All known per- Index Terms— HEO, Geopotential, third-body perturba- turbations must be taken into account in orbit determination tion, tesseral resonances, SRP, atmospheric drag, mean ele- problems. But for orbit prediction the accuracy requirements ments, semi-analytic propagation are notably relaxed, and hence some of the disturbing effects may be considered of higher order in the perturbation model. 1. INTRODUCTION Furthermore, for the purpose of long-term predictions it is customary to ignore short-period effects, which occur on A semi-analytical orbit propagator to study the long-term evo- time-scales comparable to the orbital period. Thus, in the case lution of spacecraft in Highly Elliptical Orbits is presented. of the gravitational potential the focus is on the effect of even- The perturbation model taken into account includes the grav- degree zonal harmonics, which are known to cause secular itational effects produced by the first nine zonal harmonics effects. Odd-degree zonal harmonics may also be important and the main tesseral harmonics affecting to the 2:1 reso- because they originate long-period effects, whereas the effects nance, which has an impact on Molniya orbit-types, of Earth’s of tesseral harmonics in general average out to zero. The lat- gravitational potential, the mass-point approximation for third ter, however, can have an important effect in resonant orbits, body perturbations, which only include the Legendre polyno- as in the case of geostationary satellites (1 to 1 resonance) or mial of second order for the sun and the polynomials from GPS and Molniya orbits (2 to 1 resonance). second order to sixth order for the moon, solar radiation pres- The importance of each perturbation acting on an earth’s sure and atmospheric drag. Hamiltonian formalism is used to satellite depends on the orbit’s characteristics, and fundamen- model the forces of gravitational nature so as to avoid time- tally on the altitude of the satellite, but also on its mean mo- dependence issues the problem is formulated in the extended tion. Thus, for instance the atmospheric drag, which can have phase space. The solar radiation pressure and the atmospheric an important impact in the lower orbits, may be taken as a drag are added as generalized forces. The semi-analytical the- higher order effect for altitudes above, say, 800 km over the ory is developed using perturbation techniques based on Lie earth’s surface, and is almost negligible above 2000 km. transforms. Deprit’s perturbation algorithm is applied up to A sketch of the order of different perturbations when com- the second order of the second zonal harmonics, J2, includ- pared to the Keplerian attraction is presented in Fig.1 based ing Kozay-type terms in the mean elements Hamiltonian to on approximate formulas borrowed from [1, p. 114]. As il- get “centered” elements. The transformation is developed in lustrated in the figure, the non-centralities of the Geopotential closed-form of the eccentricity except for tesseral resonances have the most important effect in those parts of the orbit that are below the geosynchronous distance, where the J2 contri- ∗CNES CONTRACT REF. DAJ-AR-EO-2015-8181 bution is a first order effect and other harmonics cause second y Funded by the Ministry of Economic Affairs and Competitiveness of order effects. To the contrary, in those parts of the orbit that Spain: Projects ESP2013-41634-P and ESP2014-57071-R. zFunded by the Ministry of Economic Affairs and Competitiveness of are farther than the geosynchronous distance the gravitational Spain: Project ESP2014-57071-R. pull of the moon is the most important perturbation, whereas that of the sun is of second order when compared to the dis- tion, including at least the effects of the first few terms turbing effect of the moon, and perturbations due to J2 and of the Legendre polynomials expansion of the third- solar radiation pressure (SRP) are of third order. body perturbation for the moon, and at least the effect of the polynomial of the second degree for the sun; 0.01 Moon • solar radiation pressure effects; J2 10-4 Sun • and the circularizing effects of atmospheric drag affect- ing the orbit semi-major axis and eccentricity. magnitude -6 In what respects to the ephemeris of the sun and moon 10 SRP HAM = 0.01 m2kgL of needed for evaluating the corresponding disturbing poten- tials, the precision provided by simplified analytical formulas Order -8 J J 10 3 2,2 should be enough under the accuracy of a perturbation theory. For this reason truncated series from [5] of Chapront’s solar 2:1 1:1 1:2 1:3 and lunar ephemeris are enough at the precision of the mean 20 40 60 80 100 120 140 elements propagation and notably speed the computations. Distance from the Earth's center Hthousands of kmL 3. HAMILTONIAN ARRANGEMENT Fig. 1. Perturbation order relative to the Keplerian attraction. The averaged Hamiltonian is constructed by the Lie trans- Finally, we recall that integration of osculating elements forms technique [6], including the time dependence in the so- is properly done only in the True of Date system [2]. For this lution of the homological equation. reason Chapront’s solar and lunar ephemeris are used [3,4], Thus, the Hamiltonian is arranged as a power series in a which are directly referred to the mean of date thus including small parameter , viz. H = P (m=m!)H , with the effect of equinoctial precession. m≥0 m;0 In the case of a highly elliptic orbit (HEO) the distance H0;0 = −µ/(2a) of the satellite to the earth’s center of mass varies notably 2 H1;0 = −(µ/r)(R⊕=r) C2;0P2(sin ') along the orbit, a fact that makes particularly difficult to es- 1 X tablish the main perturbation over which to set up the correct H = −(µ/r) (R =r)mC P (sin ') + T + V + V 2 2;0 ⊕ m;0 m (j perturbation arrangement. This issue is aggravated by the im- m>2 portance of the gravitational pull of the moon on high altitude where µ is the earth’s gravitational parameter, (r; '; λ) are orbits, which requires taking higher degrees of the Legendre spherical coordinates and a is the semi-major axis of the satel- polynomial expansion of the third-body disturbing function lite’s orbit, R is the earth’s equatorial radius, P are Leg- into account. Since this expansion converges slowly, the size ⊕ m endre polynomials, and C are zonal harmonic coefficients. of the multivariate Fourier series representing the moon per- m;0 The tesseral disturbing function is turbation will soon grow enormously. Besides, for operational reasons, different HEOs may be synchronized with the earth m m µ X R⊕ X rotation, and hence, being notably affected by resonance ef- T =− Pm;n(sin ')[Cm;n cos nλ+Sm;n sin nλ] r rm fects. Also, a HEO satellite will spend most of the time in the m≥2 n=1 apogee region, where the solar radiation pressure may have in which P are associated Legendre polynomials while an observable effect in the long-term, yet depending on the m;n C and S , m 6= 0, are non-zonal harmonic coefficients. physical characteristics of the satellite. Finally, the perigee n;m n;m Under the assumption of point masses, of usual geostationary transfer orbits (GTO) will enter repeat- 2 edly the atmosphere, although only for short periods. V? = −(µ?=r?) r?=jjr − r?jj − r · r?=r? (1) Therefore, a long-term orbit propagator for HEO aiming at describing at least qualitatively the orbit evolution must where V? ≡ V(j for the moon, and V? ≡ V in the case of the consider the following perturbations: sun. The disturbing body is far away from the perturbed body when dealing with perturbed Keplerian motion. Then Eq. (1) • the effects of the main zonal harmonics of the Geopo- can be expanded in power series of the ratio r=r? tential as well as second order effects of J2; 2 3 X m V? = −β (n a =r?) (r=r?) Pm(cos ?); (2) • tesseral effects for the most common resonances. In ? ? m≥2 particular, the 2:1 affecting Molnya orbits, or super syn- chronous resonances affecting space telescope orbits; where β = m?=(m? + m) and • lunisolar perturbations in the mass-point approxima- cos ? = (xx? + yy? + zz?)=(rr?) (3) The semi-analytic theory only considers P2 in Eq. (2) for In the case of the moon, because the lunar inclination to the sun potential, whereas P2–P6 are taken for the moon. the equator is not constant, the orbit is rather referred to the The Lie transforms averaging is only developed up to the mean equator and equinox of the date by means of second order in the small parameter, so there is no coupling r = R (−") R (−Ω ) R (−J) R (−θ )(r ; 0; 0)τ between the different terms of the disturbing function.
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