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Institute of Applied Astronomy Russian Academy of Sciences

Method of Determining the Orbits of the Small Bodies in the Solar System Based on an Exhaustive Search of Orbital Planes
Dmitrii Vavilov, Yurii Medvedev
Introduction to the problem: Let we have n 3 positional observations of a body: points in time ( = 1, ) , right ascensions , and declinations . Then, unit vectors pointing to the body in the topocentric equatorial coordinate system have the following form: = cos cos , sin cos , sin , = 1, . The relationship between the heliocentric and topocentric vectors of the celestial body positions is determined by the equations: = , + , = 1, , where are the heliocentric vectors of the celestial body positions, are the topocentric distances, and are the heliocentric vectors of the observer 's position, and are inclination and longitude of the ascending node of the asteroid's orbit, correspondingly. Note that is a function of and . Therefore, it is necessary to know in order to determinate topocentric distances . Normally, these vectors are determined by solving a system of nonlinear equations. These equations are generally solved using iterations where a type of orbit is assigned beforehand since different formulas are used to calculate different types of orbits, which is not always convenient. Moreover, iterations can diverge in some unfavorable cases, such as the nonuniform distribution or the presence of erroneous observations, making it impossible to determine orbital elements of a small body. Description of the method: Exhaustive search for orbital planes (inclinations and longitudes of the ascending node of the heliocentric orbit) is fulfiled. For each plane we make the following operations: · Calculation according to: (, ) = , (, ) where = sin sin, -sin cos, cos is the normal vector to the orbital plane. · Aberration correlations are taken into account: 1 = - , where is the velocity of light. · Two reference observations are chosen (generally the first and the last observations). · Determination of the orbit using the Gauss method of determining orbital elements based on two heliocentric positions and points in time. · The differences between the observed and calculated positions of the body (O­C) are calculated and rms of the observation fit is determined: = 1 2


-
=0

2

cos + -

2

2

,

where , are the calculated equatorial coordinates of the celestial body. We choose that plane, which has the least , and improve the orbit, corresponding to the plane, using the differential method.

Designation 2010 2010 2011 2010 2011 2011 2011 2011 2010 2010 2011 2010 2010 2011 2011 2010 2010 2011 2010 2011 2010 2011 2011 2011 2010 2010 2011 2011 2011 2010 2010 2010 2011 2010 SJ SX11 KO17 SS3 KR12 KJ15 KD11 KH4 SY3 SA12 KC15 SG13 SL13 JA8 KF9 SP3 OC127 KW15 PT66 KK15 RG137 KQ12 KP17 JP29 BK118 SV3 KW19 KN17 KQ19 SE12 SR3 SD13 KP16 SH13

- 226 118 114 59 57 34 13 9 9 6 4 3 2 2 1 1 1 1 0 0 0 0 0 0 -1 -1 -1 -1 -2 -12 -15 -16 -27 -41

673 101 52 15 157 366 59 29 28 6 21 27 5 5 21 99 1 117 1 16 19 16 10 1 3 6 27 8 15 111 73 34 43 97

651 300 354 31 54 231 50 122 74 213 48 4042 3140 102 13 28 2 1187 1 4 246 210 61 43 132 23486 106 4 25 83 1064 748 44 105

895 715 1124 78 75 251 61 199 96 410 55 3242 5362 231 14 28 2 534 2 8 269 209 57 22 126 12344 96 3 23 74 757 385 8 63

1 0.97 1.24 1.32 1.96 1.93 1.74 1.09 1.68 2.77 2.98 1.99 2.12 10.15 19.74 1.83 0.58 59.31 1.87 33.13 1.63 14.66 4.71 19.58 23.78 235.22 1.66 3.08 1.72 3.00 2.03 1.96 1.40 1.78 2.11

20.1 21.4 188.8 22.9 17.8 12.6 27.8 59.9 43.9 144.8 85.7 340.0 2360.1 227.8 34.0 13.9 104.3 361.5 57.0 36.9 147.9 60.8 244.9 230.9 877.3 543.9 53.8 23.0 30.8 19.1 167.7 111.7 16.2 33.1

Gauss -

-

-

-

References: Yu.S. Bondarenko, D.E. Vavilov, Yu.D. Medvedev Method of Determining the Orbits of the Small Bodies in the Solar System Based on an Exhaustive Search of Orbital Planes, Solar System Research, 2014, Vol. 48, No. 3, pp. 212­216.

JournИes 2014 , 22 ­ 24 September