Collision risk assessment for perturbed orbits via rigorous global optimization

A method intended to assess the occurrence of impacts between satellites and space debris is presented. The method is based on the computation of the minimum orbit intersection Distance (MOID) between two perturbed orbits. The MOID is obtained by means of a global optimization, using a global optimizer based on Taylor models. The position of the orbiting objects is described through analytical solutions that take into account zonal harmonics and atmospheric drag. The global optimizer searches the global minimum of the square distance between the two orbits, that is a function of the two true anomalies and time. The optimization is capable of providing tight enclosures of the global minimum. The method is applied to the case of a Sun-synchronous orbit and an equatorial elliptic orbit. Due to the effect of the Earth zonal harmonics, the orbital plane of the Sun-synchronous orbit performs a complete rotation around Earth's North Pole direction. The relative geometry of the two orbits is such that four intersections occur within one year window. A second test-case demonstrates the ability to predict the effect of drag perturbations. Two LEO orbits are considered, one of which is strongly influenced by atmospheric drag. The decrease of semi-major axis leads to an intersection between the two orbits in the considered time span. The method is capable to correctly identify the intersections between two orbits and, thus, the conditions in which close approaches between satellites and debris occur. The method is intended to perform a fast screening of all possible combinations of orbiting object, since the use of analytical theory cuts down simulation time.

[1]  E. Wnuk.  Prediction of artificial satellite positions on the basis of the second order theory of motion , 2002 .

[2]  Konstantin V. Kholshevnikov,et al.  On the Distance Function Between Two Keplerian Elliptic Orbits , 1999 .

[3]  B. Fornberg.  Generation of finite difference formulas on arbitrarily spaced grids , 1988 .

[4]  Martin Berz,et al.  COSY INFINITY Version 9 , 2006 .

[5]  Tianhe Xu,et al.  Analytical solution of a satellite orbit disturbed by lunar and solar gravitation , 2011 .

[6]  Martin Berz,et al.  Long-term stability of the Tevatron by verified global optimization , 2006 .

[7]  Martin Berz,et al.  The method of power series tracking for the mathematical description of beam dynamics , 1987 .

[8]  Norman Davids,et al.  Table of the zeros of the Legendre polynomials of order 1-16 and the weight coefficients for Gauss’ mechanical quadrature formula , 1942 .

[9]  M. Berz.  High‐order computation and normal form analysis of repetitive systems , 1992 .

[10]  J. J. F. Liu,et al.  Advances in orbit theory for an artificial satellite with drag. , 1982 .

[11]  Felix R. Hoots,et al.  Reformulation of the Brouwer geopotential theory for improved computational efficiency , 1981 .

[12]  F. Hoots.  An analytical satellite theory using gravity and a dynamic atmosphere , 1982 .

[13]  Yoshihide Kozai,et al.  The motion of a close earth satellite , 1959 .

[14]  Martin Berz,et al.  5. Remainder Differential Algebras and Their Applications , 1996 .

[15]  P. Dybczyński,et al.  On the minimum distance between two Keplerian orbits with a common focus , 1986 .

[16]  T. S. No,et al.  Analytical solution to perturbed geosynchronous orbit , 2005 .

[17]  S. Coffey,et al.  An analytical theory for tesseral gravitational harmonics , 2000 .

[18]  Kyoko Makino,et al.  Rigorous analysis of nonlinear motion in particle accelerators , 1998 .

[19]  R. Baker Kearfott,et al.  Introduction to Interval Analysis , 2009 .

[20]  Dirk Brouwer,et al.  Theoretical evaluation of atmospheric drag effects in the motion of an artificial satellite , 1961 .

[21]  A. Griewank,et al.  Automatic differentiation of algorithms : theory, implementation, and application , 1994 .

[22]  H. Klinkrad,et al.  Modeling of the Current Space Debris Environment , 2006 .

[23]  Victor A. Brumberg,et al.  Analytical Techniques of Celestial Mechanics , 1995 .

[24]  T. S. Kelso,et al.  Revisiting Spacetrack Report #3 , 2006 .

[25]  Martin Berz,et al.  Efficient Control of the Dependency Problem Based on Taylor Model Methods , 1999, Reliab. Comput..

[26]  The fifth-order analytical solution of the equations of motion of a satellite in orbit around a non-spherical planet , 1995 .

[27]  Kaare Aksnes,et al.  On the use of the Hill variables in artificial satellite theory - Brouwer's theory. , 1972 .

[28]  Dirk Brouwer,et al.  SOLUTION OF THE PROBLEM OF ARTIFICIAL SATELLITE THEORY WITHOUT DRAG , 1959 .

[30]  Martin Berz,et al.  Computing the critical points of the distance function between two Keplerian orbits via rigorous global optimization , 2010 .

[31]  P. W. Hawkes,et al.  Modern map methods in particle beam physics , 1999 .

[32]  Richard S. Hujsak.  A restricted four-body solution for resonating satellites with an oblate Earth. , 1979 .

[33]  F. A. Abd El-Salam,et al.  An efficient algorithm for orbital evolution of artificial satellite , 2007, Appl. Math. Comput..

[34]  Wu Chen,et al.  Analytical solution of a satellite orbit disturbed by atmospheric drag , 2011 .

[35]  K. Aksnes,et al.  Short-period and long-period perturbations of a spherical satellite due to direct solar radiation , 1976 .

[36]  H. Kinoshita.  Third-order solution of an artificial-satellite theory , 1977 .

[37]  M. Berz,et al.  TAYLOR MODELS AND OTHER VALIDATED FUNCTIONAL INCLUSION METHODS , 2003 .

[38]  K. Cranford,et al.  An improved analytical drag theory for the artificial satellite problem , 1969 .

[39]  R. R. Allan.  On the motion of nearly synchronous satellites , 1965, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[40]  W. M. Kaula,et al.  Theory of Satellite Geodesy: Applications of Satellites to Geodesy , 2000 .

[41]  G. E. Cook.  Perturbations of satellite orbits by tesseral harmonics in the earth's gravitational potential , 1963 .

[42]  Martin Berz,et al.  Verified global optimization with Taylor model based range bounders , 2005 .

[43]  M. Tigner,et al.  Handbook of Accelerator Physics and Engineering , 2013 .

[44]  Paul W. Schumacher,et al.  History of Analytical Orbit Modeling in the U.S. Space Surveillance System , 2004 .

[45]  Linda L. Crawford,et al.  An analytic method to determine future close approaches between satellites , 1984 .

[46]  N. M. Harwood,et al.  Long-periodic and secular perturbations to the orbit of a spherical satellite due to direct solar radiation pressure , 1995 .

[47]  K. Aksnes.  A second-order artificial satellite theory based on an intermediate orbit , 1970 .

[48]  D. Kessler,et al.  Collision frequency of artificial satellites: The creation of a debris belt , 1978 .

[49]  M.K.M. Ahmed,et al.  The Effect of Direct Solar Radiation Pressure on a Spacecraft of Complex Shape , 1998 .

[50]  Oliver Montenbruck,et al.  Satellite Orbits: Models, Methods and Applications , 2000 .

[51]  Nicholas L. Johnson,et al.  History of on-orbit satellite fragmentations , 1984 .