Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical Sciences1991-86152310-7081Samara State Technical University2077210.14498/vsgtu1344Original ArticleA method for the fast MOID computation for two confocal heliocentric orbitsDerevyankaAndrey EPostgraduate Student, Dept. of Applied Mathematics & Computer ScienceAndrDerev@gmail.comSamara State Technical University1512201418414415618022020Copyright © 2014, Samara State Technical University2014The paper is on the problem of classiﬁcation an asteroid as potentially hazardous (PHA), namely the estimation of the MOID parameter. Minimum Orbital Intersection Distance describes the minimal distance between two confocal heliocentric orbits. Analytical, numerical and hybrid methods used for the MOID estimation are reviewed. A brief description of the K. V. Kholshevnikov and G. F. Gronchi analytical methods, which are considered to be classical, is given. The task of calculating the MOID parameter for a large number of asteroids (more than 10,000) with a maximum calculating speed and the ability to parallelize the process is set. A numerical method based on geometrical considerations concerning the location of the bodies on their orbits is proposed. Let us consider two bodies A and E. Since only the minimum distance between two orbits is required, the information on the actual position of the bodies on their orbits is insigniﬁcant. The idea is to calculate one full revolution of the body A. For each position of body A the corresponding position of the body E is calculated under the following assumption. Consider a plane P , comprising the body A and the Sun. Therefore, plane P is perpendicular to the orbital plane of the body E. Of the two points at which the plane P intersects the orbit of the body E, E is considered to be at the point that is the nearest the body A. Thus, the position of the body E will depend on the position of the body A. As a result, from the geometric assumptions on the triangle formed by the Sun and two bodies, the distance between A and E is calculated. When one complete revolution of the body A with a certain step is calculated, we receive a set of the distances between two orbits, from which we can identify the areas of the local minima of the discrete representation of the distance function (the distance between the orbits of A and E). Then, the procedure of tuning is carried out to verify and precise the values of local minima of discrete representation of the distance function. As a result, the smallest value of the local minima is considered to be the estimation of the Minimum Orbital Intersection Distance (MOID) takes. Pros of the suggested method are as follows: high speed and adjustable calculation accuracy, the suitability to the use of parallel computing. Comparative tests of the described method were carried out. The results received are consistent with the classical G. F. Gronchi method.MOIDMOIDpotentially hazardous asteroidsorbital elementscelestial mechanicsнебесная механикаэлементы орбитпотенциально опасные астероиды[Tancredi G. A criterion to classify asteroids and comets based on the orbital parameters // Icarus, 2014. vol. 234. pp. 66-80. doi: 10.1016/j.icarus.2014.02.013.][Milani A., Chesley S. R., Valsecchi G. B. Asteroid close encounters with Earth: Risk assessment // Planetary and Space Science, 2000. vol. 48, no. 10. pp. 945-954. doi: 10.1016/s0032-0633(00)00061-1.][Milani A. The asteroid identiﬁcation problem I. Recovery of lost asteroids // Icarus, 1999. vol. 137, no. 2. pp. 269-292. doi: 10.1006/icar.1999.6045.][Sitarski G. Approaches of the parabolic comets to the outer planets // Acta Astronomica, 1968. vol. 18, no. 2. pp. 171-195.][Milani A., Chesley S. R., Valsecchi G. B. Asteroid Close Approaches: Analysis and Potential Impact Detection / Asteroids III ; eds. W. Bottke, A. Cellino, P. Paolicchi, and R. P. Binzel: University of Arizona Press, 2002. pp. 55-69.][Kholshevnikov K. V., Vassiliev N. N. On the distance function between two Keplerian elliptic orbits // Celestial Mechanics and Dynamical Astronomy, 1999. vol. 75, no. 2. pp. 75-83. doi: 10.1023/A:1008312521428.][Baluyev R. V., Kholshevnikov K. V. Distance between two arbitrary unperturbed orbits // Celestial Mechanics and Dynamical Astronomy, 2005. vol. 91, no. 3-4. pp. 287-300. doi: 10.1007/s10569-004-3207-1.][Gronchi G. F., Tommei G., Milani A. Mutual geometry of confocal Keplerian orbits: uncertainty of the MOID and search for virtual PHAs // Proceedings of the International Astronomical Union, 2006. vol. 2, no. S236. pp. 3-14. doi: 10.1017/s1743921307003018.][Gronchi G. F. An Algebraic Method to Compute the Critical Points of the Distance Function Between Two Keplerian Orbits // Celestial Mechanics and Dynamical Astronomy, 2005. vol. 93, no. 1-4. pp. 295-329. doi: 10.1007/s10569-005-1623-5.][Gronchi G. F. On the stationary points of the squared distance between two ellipses with a common focus // SIAM J. Sci. Comput., 2002. vol. 20, no. 1. pp. 61-80. doi: 10.1137/s1064827500374170.][Armellin R., Di Lizia P., Berz M., Makino K. Computing the critical points of the distance function between two Keplerian orbits via rigorous global optimization // Celestial Mechanics and Dynamical Astronomy, 2010. vol. 107, no. 3. pp. 377-395. doi: 10.1007/s10569-010-9281-7.][Wićniowski T., Rickman H. Fast Geometric Method for Calculating Accurate Minimum Orbit Intersection Distances (MOIDs) // Acta Astronomica, 2013. vol. 63, no. 2. pp. 293-307.][Vasile M., Colombo C. Optimal Impact Strategies for Asteroid Deﬂection // Journal of Guidance, Control and Dynamics, 2008. vol. 31, no. 4. pp. 858-872. doi: 10.2514/1.33432.][Besse I. M., Rhee N. H. A numerical method for calculating minimum distance to Near Earth Objects // Applied Mathematics and Computation, 2014. vol. 237. pp. 274-281. doi: 10.1016/j.amc.2014.03.115.][Maršeta D., Segan S. The distributions of positions of Minimal Orbit Intersection Distances among Near Earth Asteroids // Advances in Space Research, 2012. vol. 50, no. 2. pp. 256-259. doi: 10.1016/j.asr.2012.04.005.][Carusi A., Dotto E. Close Encounters of Minor Bodies with the Earth // Icarus, 1996. vol. 124, no. 2. pp. 392-398. doi: 10.1006/icar.1996.0216.][Milisavljevic S. The proximities of asteroids and critical points of the distance function // Serbian Astronomical Journal, 2010. vol. 180. pp. 91-102. doi: 10.2298/saj1080091m.][Segan S., Milisavljević S., Maršeta D. A combined method to compute the proximities of asteroids // Acta Astronomica, 2011. vol. 61, no. 3. pp. 275-283.][Hoots F. R., Crawford L. L., Roehrich R. L. An analytical method to determine future close approaches between satellites // Celestial Mechanics and Dynamical Astronomy, 1984. vol. 33, no. 2. pp. 143-158. doi: 10.1007/bf01234152.][Dybczyński P. A., Jopek T. J., Seraﬁn R. A. On the minimum distance between two Keplerian orbits with a common focus // Celestial Mechanics and Dynamical Astronomy, 1986. vol. 38, no. 4. pp. 345-356. doi: 10.1007/bf01238925]