A perturbation solution to Vinti’s dynamical problem
Keywords:
artificial satellite theory, intermediaries, oblate spheroid, integrability, perturbations, Vinti’s dynamical problemAbstract
For the short times in which Vinti’s analytical solution is suitable for real Earth orbital problems, we propose to replace the exact orbital propagation algorithm by an analytical perturbation approximation. In practice, the approximation provides analogous accuracy to that of the exact solution, is also free from essential small divisors, and is of straightforward evaluation. As a side effect, our approach presents another example of how the truncation of an integrable problem can lead to the illusion of chaotic behavior.
References
VINTI, J. P., New method of solution for unretarded satellite orbits, Journal of Research of the National Bureau of Standards, 62B, 2, pp. 105–162, 1959.
VINTI, J. P., Theory of an accurate intermediary orbit for satellite astronomy, Journal of Research of the National Bureau of Standards, 65B, 3, pp. 169–201, 1961.
BROUWER, D., CLEMENCE, G. M., Methods of celestial mechanics, New York and London, Academic Press, 1961.
AKSENOV, Y. P., GREBENIKOV, Y. A., DEMIN, V. G., General solution of the problem of the motion of an artificial satellite in the normal field of the earth’s attraction, Planetary and Space Science, 9, 8, pp. 491–498, 1962.
VINTI, J. P., Invariant properties of the spheroidal potential of an oblate planet, Journal of Research of the National Bureau of Standards, 70B, 1, pp. 1–16, 1966.
VINTI, J. P., Inclusion of the third zonal harmonic in an accurate reference orbit of an artificial satellite, Journal of Research of the National Bureau of Standards, 70B, 1, pp. 17–46, 1966.
IZSAK, I. G., A second-order solution of Vinti’s dynamical problem, Smithsonian Contributions to Astrophysics, 6, pp. 81–107, 1963.
BONAVITO, N. L., Computational procedure for Vinti’s accurate reference orbit with inclusion of the third zonal harmonic, Technical Report NASA-TN-D-3562, Goddard Space Flight Center, Greenbelt, Maryland, Aug. 1966.
DER, G. J., An elegant state transition matrix, The Journal of the Astronautical Sciences, 45, 4, pp. 371–390, 1997.
BONAVITO, N. L., WATSON, S., WALDEN, H., An accuracy and speed comparison of the Vinti and Brouwer orbit prediction methods, Technical Report NASA TN D-5203, Goddard Space Flight Center, Greenbelt, Maryland, May 1969.
GORDON, R. A, MISTRETTA, G. D., WATSON, J. S., A comparison of classical analytic theories for the motion of artificial satellites, Journal of Guidance Control Dynamics, 2, 3, pp. 184–189, 1979.
BROUWER, D., Solution of the problem of artificial satellite theory without drag, The Astronomical Journal, 64, pp. 378–397, 1959.
LYDDANE, R. H., Small eccentricities or inclinations in the Brouwer theory of the artificial satellite, Astronomical Journal, 68, 8, pp. 555–558, 1963.
VINTI, J. P., Orbital and celestial mechanics, Vol. 177 of “Progress in Astronautics and Aeronautics” (edited by G. J. Der, N. L. Bonavito), Reston, Virginia: American Institute of Aeronautics and Astronautics, 1998.
WIESEL, W. E., Numerical solution to Vinti’s problem, Journal of Guidance Control Dynamics, 38, pp. 1757–1764, Sept. 2015.
WRIGHT, S., Orbit determination using Vinti’s solution, PhD thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, Sept. 2016.
BIRIA, A. D., RUSSELL, R. P., A satellite relative motion model including J2 and J3 via Vinti’s intermediary, Celestial Mechanics and Dynamical Astronomy, 130, p. 23, March 2018.
BIRIA, A. D., RUSSELL, R. P., Analytical solution to the Vinti problem in oblate spheroidal equinoctial orbital elements, The Journal of the Astronautical Sciences, 67, pp. 1–27, March 2020.
VINTI, J. P., Zonal harmonic perturbations of an accurate reference orbit of an artificial satellite, Journal of Research of the National Bureau of Standards, 67B, 4, pp. 191–222, 1963.
CUSHMAN, R., Reduction, Brouwer’s Hamiltonian, and the critical inclination, Celestial Mechanics, 31, 4, pp. 401–429, 1983.
COFFEY, S. L., DEPRIT, A., MILLER, B. R., The critical inclination in artificial satellite theory, Celestial Mechanics, 39, 4, pp. 365–406, 1986.
JUPP, A. H., The critical inclination problem – 30 years of progress, Celestial Mechanics, 43, 1-4, pp. 127–138, 1988.
LARA, M., On inclination resonances in Artificial Satellite Theory, Acta Astronautica, 110, pp. 239–246, 2015.
DEPRIT, A., MILLER, B. R., Normalization in the face of integrability, Annals of the New York Academy of Sciences, 536, 1, pp. 101–126, 1988.
FARRELLY, D., UZER, T., Normalization and the detection of integrability: the generalized van der Waals potential, Celestial Mechanics and Dynamical Astronomy, 61, 1, pp. 71–95, 1995.
COFFEY, S. L., DEPRIT, A., DEPRIT, E., Frozen orbits for satellites close to an earth-like planet, Celestial Mechanics and Dynamical Astronomy, 59, 1, pp. 37–72, 1994.
CONTOPOULOS, G., POLYMILIS, C., Approximations of the 3-particle Toda lattice, Physica D: Nonlinear Phenomena, 2, 1, pp. 328–342, 1987.
LARA, M., Solution to the main problem of the artificial satellite by reverse normalization, Nonlinear Dynamics, 101, 2, pp. 1501–1524, 2020.
STERNE, T. E., The gravitational orbit of a satellite of an oblate planet, The Astronomical Journal, 63, pp. 28–40, 1958.
GARFINKEL, B., An improved theory of motion of an artificial satellite, The Astronomical Journal, 69, 3, pp. 223–229, 1964.
AKSNES, K., On the dynamical theory of a near-earth satellite, I., Astrophysica Norvegica, 10, pp. 69–77, Aug. 1965.
CID, R., LAHULLA, J. F., Perturbaciones de corto periodo en el movimiento de un sat´elite artificial, en funci´on de las variables de Hill, Publicaciones de la Revista de la Academia de Ciencias de Zaragoza, 24, pp. 159–165, 1969.
DEPRIT, A., The elimination of the parallax in satellite theory, Celestial Mechanics, 24, 2, pp. 111–153, 1981.
OBERTI, P., A simple intermediary orbit for the J2 problem, Astronomy & Astrophysics, 437, pp. 333–338, 2005.
KOZAI, Y., Second-order solution of artificial satellite theory without air drag, The Astronomical Journal, 67, pp. 446–461, 1962.
DEPRIT, A., ROM, A., The main problem of artificial satellite theory for small and moderate eccentricities, Celestial Mechanics, 2, pp. 166–206, June 1970.
LARA, M., SAN-JUAN, J. F., LO´ PEZ-OCHOA, L.M., Proper averaging via aarallax elimination, in: “Astrodynamics 2013”, Vol. 150 of Advances in the Astronautical Sciences (San Diego, California), pp. 315–331, Univelt, Inc., 2014.
DEPRIT, A., The main problem in the theory of artificial satellites to order four, Journal of Guidance Control Dynamics, 4, 2, pp. 201–206, 1981.
COFFEY, S. L., DEPRIT, A., Third-order solution to the main problem in satellite theory, Journal of Guidance, Control and Dynamics, 5, 4, pp. 366–371, 1982.
ALFRIEND, K. T., COFFEY, S. L., Elimination of the perigee in the satellite problem, Celestial Mechanics, 32, pp. 163–172, Feb. 1984.
COFFEY, S., ALFRIEND, K. T., An analytical orbit prediction program generator, Journal of Guidance, Control and Dynamics, 7, 5, pp. 575–581, 1984.
LARA, M., SAN-JUAN, J. F., LO´ PEZ-OCHOA, L.M., Delaunay variables approach to the elimination of the perigee in artificial satellite theory, Celestial Mechanics and Dynamical Astronomy, 120, 1, pp. 39–56, 2014.
LARA, M., Brouwer’s satellite solution redux, arXiv, vol. eprint 2009.10665 [math.DS], pp. 1–20, Sept. 2020.
IZSAK, I. G., A note on perturbation theory, The Astronomical Journal, 68, pp. 559–561, Oct. 1963.
AKSNES, K., On the use of the Hill variables in artificial satellite theory, Astronomy and Astrophysics, 17, 1, pp. 70–75, 1972.
LARA, M., Efficient formulation of the periodic corrections in Brouwer’s gravity solution, Mathematical Problems in Engineering, article ID 980652, 2015.
LARA, M., LEO intermediary propagation as a feasible alternative to Brouwer’s gravity solution, Advances in Space Research, 56, 3, pp. 367–376, 2015.
LARA, M., VILHENA DE MORAES, R., SANCHEZ, D. M., PRADO, A. F. B. A., Efficient computation of short-period analytical corrections due to third-body effects, AAS/AIAA Spaceflight Mechanics Meeting 2015, Vol. 155 of Advances in the Astronautical Sciences (San Diego, California), pp. 437–455, Univelt, Inc., 2015.
LARA, M., Exploring sensitivity of orbital dynamics with respect to model truncation: the frozen orbits approach, Stardust Final Conference, Vol. 52 of Astrophysics and Space Science Proceedings (Cham), pp. 69–83, Springer International Publishing, 2018.
CHURCHILL, R. C., KUMMER, M., ROD, D. L., On averaging, reduction, and symmetry in hamiltonian systems, Journal of Differential Equations, 49, 3, pp. 359–414, 1983.
H´ENON, M., HEILES, C., The applicability of the third integral of motion: Some numerical experiments, The Astronomical Journal, 69, p. 73, Feb. 1964.
VERHULST, F., Nonlinear differential equations and dynamical systems, Universitext, Berlin Heidelberg, Springer-Verlag, 2 ed., 2000.
LYDDANE, R. H., COHEN, C. J., Numerical comparison between Brouwer’s theory and solution by Cowell’s method for the orbit of an artificial satellite, The Astronomical Journal, 67, pp. 176–177, 1962.
HAUTESSERRES, D., LARA, M., Intermediary LEO propagation including higher order zonal harmonics, Celestial Mechanics and Dynamical Astronomy, 127, pp. 505–526, 2017.
PERSSON, S., JACOBSSON, B., GILL, E., PRISMA – Demonstration Mission for Advanced Rendezvous and Formation Flying Technologies and Sensors, Proceedings of the 56th International Astronautical Congress (IAC), October 17-21 2005, Fukuoka, Japan, pp. 1–10, International Astronautical Federation, 2005.
LARA, M., SAN-JUAN, J. F., HAUTESSERRES, D., HEOSAT: a mean elements orbit propagator program for highly elliptical orbits, CEAS Space Journal, 10, pp. 3–23, 2018.
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