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Precise Mean Orbital Elements Determination for Leo Monitoring and Maintenance

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Precise Mean Orbital Elements Determination for Leo Monitoring and Maintenance PRECISE MEAN ORBITAL ELEMENTS DETERMINATION FOR LEO MONITORING AND MAINTENANCE Sofya Spiridonova(1), Michael Kirschner(2), and Urs Hugentobler(3) (1)DLR / GSOC, Munchener¨ Str. 20, 82234 Weßling, Germany, , Tel. +49(8153)28-3492, [email protected] (2)DLR / GSOC, Munchener¨ Str. 20, 82234 Weßling, Germany, , Tel. +49(8153)28-1385, [email protected] (3)Technische Universitat¨ Munchen¨ (TUM), Arcisstraße 21, 80333 Munich, Germany, , Tel. +49(89)289-231-95, [email protected] Abstract: Mean orbital elements can be used for monitoring the satellite’s long-term behavior and for maneuver planning. In this paper, an analytical algorithm for conversion of osculating orbital elements into mean orbital elements is introduced and evaluated on several satellite missions. The accuracy of the mean orbital elements is estimated and the applications of the algorithm for determination of the relative satellite’s motion are discussed. Keywords: Mean Orbital Elements, Orbital Perturbations, Analytical Orbital Theories. 1. Introduction New satellite applications and concepts in satellite mission design pose challenging requirements for the accuracy of orbit determination and maintenance. In particular, data collection of satellite formations like GRACE 1 & 2 or TerraSAR-X and TanDEM-X introduces strict formation keeping requirements. For formation monitoring and maintenance as well as for orbit monitoring of single satellites, so-called mean orbital elements are often used. These elements are free from short-period perturbations and reveal the long-term behavior of the satellite’s orbit. The mean orbital elements can be also used for maneuver planning and calculation. The straightforward computation of Dv from the required change in the semi-major axis Da is the most direct approach to altitude corrections for satellites in circular orbits. For instance, in the case of the GRACE satellites, the required change in the mean semi-major axis of the maneuvering satellite is normally about 15-30 meters. Considering an acceptable error margin of 3% this leads to the requirement that Da of the two satellites has to be resolved better than to 1 meter. The topic of mean orbital elements has already been discussed many times in literature. Among the older publications are the papers [10] and [11], where two iterative procedures are described for the determination of the mean elements for the famous Brouwer’s analytical theory, [4]. The newer publications on the topic include [5] and [13]. As the older analytical theories often do not deliver the required accuracy, and the implementation of the newer theories requires access to the internal documentation of other space agencies or journal papers of limited access, other available theories have been investigated to be implemented and operationally used at GSOC. The method proposed in this paper is a combination of theories developed by Eckstein, Ustinov and Kaula ([1], [2] and [3]), and will be further referred to as EUK. It allows to calculate analytically first order perturbations due to arbitrary degree and order, as well as second-order perturbations 1 due to the oblateness term J2. The remaining higher-order variations in the mean semi-major axis are restricted to ±1:5 meters. Thus, the required accuracy is achieved analytically, with no time-expenses of numerical averaging and no problems of inaccurate orbit sampling. Further numerical averaging over one orbit proves to be sufficient to achieve a sub-meter accuracy in the determination of the mean semi-major axis. In the case of satellites in a formation (e.g. GRACE), their Da, required for the planning of the station-keeping cycles, can be calculated to the accuracy of a few centimeters. Section 2. contains an overview of the applied perturbation theories and describes the suggested algorithm. Some results of the algorithm evaluated on TOPEX/Poseidon, GRACE and TerraSAR-X propagated orbital elements are presented in section 3. Section 4. demonstrates the performance of the algorithm in the determination of Da of the GRACE satellites at the end of several station- keeping cycles as well as gives an insight into the typical errors of the algorithm applied to maneuver post-processing. Finally, section 5. provides a summary of the accomplished work. 2. Theoretical background 2.1. Kaula’s theory Kaula’s linear perturbation theory (LPT), developed in [1], has been applied since 1966 in many geodetic areas, e.g. ground-tracking of the satellites, altimetry, SST, and space-borne gravity gradiometry. It is based on the well-known Lagrange planetary equations da 2 ¶R = dt na¶M p de 1 − e2 ¶R 1 − e2 ¶R = − dt na2e ¶M na2e ¶w di cosi ¶R 1 ¶R = p − p dt na2 1 − e2 sini¶w na2 1 − e2 sini¶W dW 1 ¶R = p dt na2 − e2 i ¶i p 1 sin dw 1 − e2 ¶R cosi ¶R = − p dt na2e ¶e na2 1 − e2 sini ¶i dM 1 − e2 ¶R 2 ¶R = n − − ; dt na2e ¶e na ¶a which describe the satellite’s motion in terms of its Keplerian elements a, e, i, W, w, M, and the so-called perturbing potential R. Without going into the details of derivation, the perturbing potential R can be represented as a function of Keplerian orbital elements L l l l q=+¥ X mae X X X R = F (i) G (e)S (w;M;W;q); al+1 lmp lpq lmpq l=2 m=0 p=0 q=−¥ 2 where ae is the Earth’s equatorial radius, q is Greenwich sidereal time, Flmp(i) and Glpq(e) are the inclination and eccentricity functions. Here ( Clm cosylmpq + Slm sinylmpq; if l − m is even; Slmpq = (1) −Slm cosylmpq +Clm sinylmpq; if l − m is odd; and ylmpq = (l − 2p)w + (l − 2p + q)M + m(W − q): (2) The exact formulas for the inclination functions Flmp and the eccentricity functions Glpq can be found in [1]. As we can see from (1), every elementary potential Rlmpq is a harmonic function of ylmpq. Several different cases can be considered depending on the period of ylmpq. Elementary potentials Rlmpq with l − 2p + q 6= 0 have frequencies that are multiples of the satellite revolution frequency. Corre- sponding orbit variations are referred to as short-period perturbations. Another type of perturbations is caused by Rlmpq with l − 2p + q = 0 and m 6= 0. They have frequencies that are multiples of the Earth revolution frequency, and a period equal to or shorter than the Earth’s rotation period, therefore, such perturbations are referred to as medium-period or m-daily perturbations. Elementary potentials Rlmpq with l − 2p + q = 0, m = 0, and l − 2p 6= 0 lead to long-period perturbations, whereas terms where all coefficients in (2) are zero, cause secular variations. The integration of the Lagrange equations under the assumption that the time dependency on the right side comes only from the secular rates of the angular elements, delivers the following expressions for perturbations: l ma Slmpq Da = e 2F G (l − 2p + q) lmpq l+2 lmp lpq ˙ n0a0 ylmpq l ma q Slmpq De = e F G 1 − e2(l − 2p + q) − (l − 2p) lmpq l+3 lmp lpq 0 ˙ n0a0 e0 ylmpq l mae Slmpq Dilmpq = q FlmpGlpq [(l − 2p)cosi0 − m] l+3 2 y˙lmpq n0a0 1 − e0 sini0 l mae ¶Flmp Selmpq DWlmpq = q Glpq l+3 2 ¶i y˙lmpq n0a0 1 − e0 sini0 2q 3 l − e2 ma 1 0 ¶Glpq coti0 ¶Flmp Selmpq = e F − G Dwlmpq l+3 4 lmp q lpq5 n0a e0 ¶e 2 ¶i y˙lmpq 0 1 − e0 2 q 3 l − e2 ma 1 0 ¶Glpq Selmpq DM = e − + 2(l + 1)G F ; lmpq l+3 4 e e lpq5 lmp ˙ n0a0 0 ¶ ylmpq where all the functions on the right-hand side are evaluated on the mean precessing ellipse. All the used notations and derivations of the formulas can be found in [1]. 3 2.2. Eckstein-Ustinov’s theory The Eckstein-Ustinov’s analytical satellite theory was first developed by Ustinov in [3] and later corrected by Eckstein in [2]. The advantage of the theory is that instead of the usual Keplerian elements it employs non-singular elements a, h, l, i, W, l, where h = esinw l = ecosw l = w + M: The Lagrange equations expressed in terms of these elements are da 2 ¶R = ; dt na¶l p " # dh 1 − h2 − l2 ¶R h ¶R = − p dt na2 ¶l 1 + 1 − h2 − l2 ¶l l coti ¶R − p na2 1 − h2 − l2 ¶i p " # dl 1 − h2 − l2 ¶R l ¶R = + p dt na2 ¶h 1 + 1 − h2 − l2 ¶l hcoti ¶R − p (3) na2 1 − h2 − l2 ¶i ! di coti ¶R ¶R ¶R csci ¶R = p l − h + − p dt na2 1 − h2 − l2 ¶h ¶l ¶l na2 1 − h2 − l2 ¶W dW csci ¶R = p dt na2 1 − h2 − l2 ¶i p ! dl 2 ¶R 1 − h2 − l2 1 ¶R ¶R = n − + p h + l dt na ¶a 1 + 1 − h2 − l2 na2 ¶h ¶l coti ¶R − p : na2 1 − h2 − l2 ¶i We can observe that the eccentricity has disappeared from the denominators, and thus the Eckstein- Ustinov’s theory, as opposed to Kaula’s, is valid for zero eccentricity. Another advantage of the Ekstein-Ustinov’s theory is that it is among the very few theories where not only long-periodic, but also short-periodic perturbations in the semi-major axis are developed to the second order. Removing second-order short-periodic perturbations from the osculating semi-major axis is essential for meeting our accuracy requirements, but is often neglected by propagator-type theories, where secular and long-period perturbations affecting long-term behavior of the elements are considered of greater interest.
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