A&A 391, 1115-1122 (2002)
DOI: 10.1051/0004-6361:20020810

Asteroid encounters suitable for mass determinations

A. Galád 1 - B. Gray 2


1 - Astronomical Institute, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovak Republic
2 - Project Pluto, 168 Ridge Road, Bowdoinham, ME 04008, USA

Received 19 December 2001 / Accepted 28 May 2002

Abstract
As in a previous paper (Galád 2001), the search for effective perturbers among asteroids is done using the same method and during the same time period. The only difference is in the number of asteroids that were processed - 24 599 instead of 9511. Special attention is paid to comparison between perturbations due to (2) Pallas and (10) Hygiea. It is confirmed that the latter has a larger effect on the motion of main belt asteroids, perhaps by a factor of three. This is a reason to include its mass in asteroid orbit determinations.

In addition to the Big Four main belt asteroids - (1) Ceres, (2) Pallas, (4) Vesta, (10) Hygiea - the masses of many other large asteroids, such as (11) Parthenope, (13) Egeria, (15) Eunomia, (16) Psyche, (24) Themis, (29) Amphitrite, (39) Laetitia,(45) Eugenia, (52) Europa, (65) Cybele, (121) Hermione, (451) Patientia, and (511) Davida, could be achieved by the end of this decade using astrometric data. In general, over the next decade small asteroids (with much higher numbers than above) will be used more thoroughly for mass determination of large asteroids.

Key words: minor planets, asteroids - astrometry - ephemerides


1 Introduction

The number of papers devoted to mass determinations of large asteroids is growing in recent years. There are several opportunities that enable such a determination - the discovery of satellites of asteroids (e.g. Merline et al. 1999; Merline et al. 2000; Merline et al. 2001; Brown & Margot 2001; Storrs et al. 2001), measurements by space probes that visited asteroids (e.g. Thomas et al. 1999). Also important are the growing number of astrometrical measurements and their increased precisions (see e.g. Carpino & Knezevic 1996; Hilton et al. 1996; Viateau & Rapaport 2001; Micha\lak 2001 and references therein).

It was realized that after a small asteroid had passed close enough to a large one, the perturbing effects of the latter (on the motion of the former) could be significant. Large asteroid mass should therefore not be neglected - it can be determined through the study of the motion of the affected small asteroid. The three largest main belt asteroids - (1) Ceres, (2) Pallas, and (4) Vesta - contain about half the mass of the belt. Thus, it is believed their masses are sufficient enough to be included (along with planetary masses) for high precision orbit computations of the other asteroids in the inner part of the Solar System. Their masses have been accurately determined by this astrometric method.

However, it would be beneficial to determine masses of the other large main belt asteroids similarly. Not only is this useful for physically characterizing asteroids; it is helpful for precise long-term orbit determinations of several smaller objects that come close enough to large asteroids to be measurably affected. The discovery rate of smaller asteroids is rapidly growing, and these may greatly aid in future mass determinations, as their deviations from the ephemeris positions are evident several years (or even decades) before and after an encounter with larger objects.

The aim of this paper is to renew the previous list by Galád (2001) (hereafter referred as Paper I) of close encounters suitable for asteroid mass determination by astrometric methods. It reflects the role of the increased number of known asteroids and the larger amount of astrometric data. Included are only the most important encounters. In Paper I, a database of the first 9511 numbered asteroids were used. Here, the first 24 599 numbered asteroids are discussed, excluding 29 Centaurs and transneptunian objects. Asteroid masses are not computed here.


 

 
Table 1: Asteroids heavily perturbed by (1) Ceres ( $P \ge 60
~{\rm km~ s}$, $D = 913 ~{\rm km}$), which were well observed before and after an encounter. The dates of the first astrometric measurements of the perturbed asteroid available in MPC data are given in the last column.
number name date r v P obs
    year/m/d ${\rm [AU]}$ ${\rm [km~ s^{-1}]}$ ${\rm [km~ s]}$ year
(2572) Annschnell 1971/ 3/26 0.012011 4.786 88.5 1950
(454) Mathesis 1971/11/23 0.021613 2.928 80.4 1900
(3643) Tienchanglin 1972/ 9/11 0.008282 2.765 222.2 1937
(11 137) 1996 XE19 1975/11/ 4 0.007778 6.615 98.9 1949
(5564) 1991 VH2 1975/12/21 0.030697 1.446 114.6 1951
(534) Nassovia 1975/12/24 0.022585 2.752 81.8 1904
(3344) Modena 1980/ 9/27 0.021136 2.386 100.9 1955
(7298) 1992 WM5 1982/ 5/23 0.043757 1.394 83.4 1953
(786) Bredichina 1983/ 4/10 0.028102 2.598 69.7 1914
(6325) 1991 EA1 1983/ 9/18 0.015592 3.603 90.6 1955
(348) May 1984/ 9/ 2 0.042422 0.794 151.0 1892
(5303) Parijskij 1996/ 9/11 0.005537 2.736 335.8 1971



 

 
Table 2: Asteroids heavily perturbed by (4) Vesta ( $P \ge 20
~{\rm km~ s}$, $D = 501 ~{\rm km}$) which were well observed before and after an encounter.
number name date r v P obs
    year/m/d ${\rm [AU]}$ ${\rm [km~ s^{-1}]}$ ${\rm [km~ s]}$ year
(5205) 1988 CU7 1977/ 5/12 0.002858 3.731 78.8 1954
(21 225) 1995 GQ1 1981/ 1/12 0.014154 1.651 36.0 1972
(3057) Malaren 1983/ 1/19 0.024393 1.646 20.9 1952
(8331) Dawkins 1988/ 1/19 0.007964 1.475 71.6 1982
(17) Thetis 1996/ 6/16 0.019377 1.168 37.1 1852


2 Orbital elements

The initial osculating orbital elements for epoch April 1, 2001 (JD 2 452 100.5), were taken from Bowell et al. (1994). Perturbations due to all planets, Moon, and the three largest main belt asteroids - (1) Ceres, (2) Pallas, (4) Vesta - were taken into account.

We derived orbital elements for several other epochs to the past and the future from the initial elements. However, perturbations due to asteroids were not included. Adjacent epochs differ by 200 days. (Planetary and lunar positions were computed using analytic Poisson series. The details of how the orbital elements were derived can be found at http://www.projectpluto.com/jpl_eph.htm.) Then, the positions of asteroids were computed using orbital elements for the nearest epoch as in a two-body problem. This procedure is convenient as a first approximation. Since actual asteroids positions may differ from their computed ones far from the initial epoch due to several asteroid - asteroid encounters, we used only the same (short) period as in Paper I - from November 6, 1967 (JD 2 439 800.5) to September 13, 2023 (JD 2 460 200.5). In fact, some asteroids might have close encounters with any of the planets and thus their positions could be miscalculated using this simple approximation. However, this has a minor importance for our purposes due to the following three reasons:

1. Near terrestrial planets asteroids move very fast and approaches among asteroids occur at high velocities. Beyond the main belt, asteroids rarely encounter Jupiter. Just (5201) Ferraz-Mello and (6144) 1994 EQ3 were within a distance of $0.6 ~{\rm AU}$ from that planet.

2. Perturbations due to asteroids are negligible in comparison to those due to planets especially in the vicinity of the planet.

3. For the time just between the epochs - 100 days away - asteroid positions and their mutual distances were calculated twice - using both closest epochs. If the difference between mutual distances computed using different epochs exceeded $0.00005 ~{\rm AU}$ for a given asteroid pair and if this pair was about to reach its minimum separation distance ( $r < 0.005 ~{\rm AU}$ for asteroids with H > 9.5or $r < 0.05 ~{\rm AU}$ for asteroids with at least one member brighter than absolute magnitude H = 9.5) then orbital elements for a much closer epoch were used. There was practically no need to do that.


 

 
Table 3: Asteroids perturbed by (2) Pallas ( $P \ge 6 ~{\rm km~ s}$, $D = 523 ~{\rm km}$). Emphasized asteroids may probably be used for mass determination with current data.
number name date r v P obs
    year/m/d ${\rm [AU]}$ ${\rm [km~ s^{-1}]}$ ${\rm [km~ s]}$ year
(6995) 1996 BZ1 1968/ 4/ 4 0.009244 14.687 7.0 1978
(2204) Lyyli 1968/ 5/ 3 0.023200 6.780 6.1 1943
(16 972) 1998 WK11 1968/ 5/ 9 0.009665 14.352 6.9 1994
(21 522) 1998 MX11 1972/11/ 2 0.013167 11.338 6.4 1991
(7671) Albis 1977/ 6/ 5 0.003497 12.473 21.9 1969
(5470) 1988 BK5 1979/12/13 0.006511 12.837 11.4 1977
(20 929) 2050 T-1 1984/11/10 0.009646 7.405 13.4 1971
(3131) Mason-Dixon 1984/12/ 4 0.011895 10.845 7.4 1922
(20 555) 1999 RC115 1986/ 9/25 0.004325 9.671 22.9 1990
(11 161) 1998 BA8 1991/ 4/23 0.009354 12.804 8.0 1982
(23 830) 1998 QZ85 1993/10/27 0.008444 16.832 6.7 1992
(18 849) 1999 RK55 1994/ 3/ 3 0.007465 10.756 11.9 1987
(6752) Ashley 2000/ 6/25 0.015028 8.093 7.9 1971
(21 266) 1996 HL25 2009/10/ 2 0.009026 12.782 8.3 1996
(17 278) 2000 LK27 2014/ 4/14 0.006341 14.418 10.5 1991
(3219) Komaki 2021/ 8/16 0.011104 11.284 7.6 1934
(18 544) 1997 AA2 2021/11/ 1 0.008950 10.474 10.2 1984



 

 
Table 4: Asteroids perturbed by (10) Hygiea before January 1, 2000. $P \ge 6 ~{\rm km~ s}$, $D = 429 ~{\rm km}$. Emphasized asteroids may probably be used for mass determination with current data.
number name date r v P obs
    year/m/d ${\rm [AU]}$ ${\rm [km~ s^{-1}]}$ ${\rm [km~ s]}$ year
(14 375) 1989 SU 1971/ 5/ 2 0.027451 2.592 7.4 1976
(11 924) 1992 WS3 1972/10/27 0.022385 2.801 8.4 1990
(16 036) 1999 GV8 1972/12/ 8 0.008438 4.939 12.7 1951
(9268) 1978 VZ2 1972/12/15 0.025907 2.474 8.2 1978
(16 919) 1998 FF35 1973/ 5/18 0.008637 1.434 42.6 1990
(11 288) 1990 XU 1973/ 5/31 0.005207 5.764 17.6 1990
(4407) Taihaku 1978/ 1/13 0.052457 1.522 6.6 1970
(16 593) 1992 UB3 1978/ 2/ 4 0.012803 5.421 7.6 1955
(13 266) 1998 QY30 1978/ 5/31 0.014689 2.403 14.9 1992
(7362) Rogerbyrd 1978/ 6/11 0.022725 3.634 6.4 1960
(10 818) 1993 FK81 1979/ 5/29 0.022368 3.798 6.2 1979
(22 108) 2000 PD 1982/ 3/25 0.014267 3.757 9.9 1955
(3951) Zichichi 1983/ 8/16 0.027718 2.765 6.9 1938
(21 248) 1995 YP1 1983/ 9/23 0.030090 1.154 15.2 1995
(6143) Pythagoras 1983/12/19 0.018749 2.464 11.4 1951
(1259) Ógyalla 1984/ 2/11 0.034542 2.050 7.4 1928
(1780) Kippes 1984/ 5/14 0.043080 1.901 6.4 1906
(20 638) 1999 TV108 1988/ 7/26 0.016438 1.714 18.7 1993
(20 371) 1998 KE30 1988/11/ 8 0.047163 1.467 7.6 1981
(20 540) 1999 RV86 1989/ 3/ 3 0.018664 2.723 10.4 1990
(12 141) 4112 P-L 1989/ 5/20 0.041714 1.631 7.8 1960
(11 559) 1993 FS23 1989/ 6/ 4 0.020666 2.514 10.2 1990
(18 215) 4792 P-L 1989/ 6/12 0.033893 1.829 8.5 1960
(2619) Skalnaté Pleso 1989/12/11 0.022379 1.707 13.8 1975
(12 777) 1994 QA1 1990/ 2/11 0.013101 4.643 8.7 1990
(5457) Queen's 1993/ 6/10 0.041282 1.864 6.9 1948
(22 880) 1999 RL224 1994/ 9/ 3 0.007061 3.701 20.2 1994
(17 109) 1999 JF52 1994/10/31 0.022559 3.619 6.5 1977
(6006) Anaximandros 1995/ 2/ 7 0.009236 2.668 21.4 1972
(10 788) 1991 UC 1995/ 5/ 3 0.032804 2.419 6.7 1990
(11 215) 1999 HN10 1995/ 5/31 0.005374 3.441 28.5 1978
(465) Alekto 1995/12/25 0.038037 1.509 9.2 1901
(15 487) 1999 CC63 1996/ 5/25 0.040184 1.297 10.1 1994
(3946) Shor 1998/ 3/30 0.014400 0.917 40.0 1950
(2061) Anza 1999/ 4/ 4 0.006464 9.419 8.7 1960



 

 
Table 5: Asteroids perturbed by (10) Hygiea after January 1, 2000. $P \ge 6 ~{\rm km~ s}$, $D = 429 ~{\rm km}$. Most of them will probably be used for mass determination in the future.
number name date r v P obs
    year/m/d ${\rm [AU]}$ ${\rm [km~ s^{-1}]}$ ${\rm [km~ s]}$ year
(1965) van de Kamp 2000/10/15 0.021421 3.289 7.5 1927
(10 380) Berwald 2001/ 1/ 5 0.013185 3.364 11.9 1949
(5941) Valencia 2001/ 2/15 0.017958 2.964 9.9 1972
(15 187) 2112 T-2 2003/12/17 0.008236 4.096 15.6 1973
(17 207) 2000 AW126 2004/ 5/22 0.018852 3.549 7.9 1954
(24 433) 2000 CF83 2004/ 6/19 0.003165 4.216 39.5 1988
(3030) Vehrenberg 2005/ 7/24 0.021885 3.695 6.5 1962
(75) Eurydike 2005/ 9/16 0.013484 5.647 6.9 1864
(11 054) 1991 FA 2005/11/ 3 0.014154 6.250 6.0 1937
(7487) 1994 YM 2006/ 6/ 8 0.011351 6.883 6.8 1954
(15 567) Giacomelli 2006/ 6/14 0.026796 1.915 10.3 1994
(5957) Irina 2007/ 1/ 3 0.006555 7.622 10.6 1988
(6579) 1981 ES4 2010/ 5/29 0.014936 5.274 6.7 1953
(11 054) 1991 FA 2011/ 5/23 0.009320 6.301 9.0 1937
(14 094) 1997 OJ1 2011/12/19 0.005030 3.140 33.4 1996
(12 936) 2549 P-L 2012/ 3/21 0.032637 2.593 6.2 1960
(13 646) 1996 HC12 2012/ 3/23 0.010860 2.138 22.7 1971
(10 018) 1979 MG4 2015/ 5/24 0.023737 1.425 15.6 1979
(20 331) 1998 HH45 2016/ 6/ 3 0.041641 2.052 6.2 1997
(11 328) 1995 UL 2016/10/30 0.023988 2.188 10.1 1986
(4803) Birkle 2017/ 4/ 5 0.011922 2.069 21.4 1950
(22 769) 1999 BD4 2019/10/28 0.040339 1.315 9.9 1978
(16 323) 1107 T-2 2020/12/23 0.017165 2.594 11.9 1973
(1160) Illyria 2022/ 1/24 0.016898 4.939 6.3 1929
(13 634) 1995 WY41 2022/ 6/10 0.018540 2.944 9.7 1970
(12 095) 1998 HE102 2022/12/11 0.049546 1.672 6.4 1992



 

 
Table 6: Perturbers (first column) other than big four main belt asteroids and perturbed asteroids (third column) before January 1, 1980 ( $P \ge 5 ~{\rm km~ s}$). Masses of emphasized asteroids may probably be determined with current data. D is (assumed) perturber's effective diameter.
  N1 D   N2 date r v P obs
    ${\rm [km]}$     year/m/d ${\rm [AU]}$ ${\rm [km~ s^{-1}]}$ ${\rm [km~ s]}$ year
(29) Amphitrite 219 (13 892) 1266 T-2 1967/11/24 0.007938 1.578 5.6 1973
(704) Interamnia 333 (14 100) Weierstrass 1968/ 7/ 2 0.002303 6.456 16.6 1991
(39) Laetitia 159 (2416) Sharonov 1968/ 1/3 0.002459 1.976 5.5 1916
(11) Parthenope 162 (17) Thetis 1968/ 2/18 0.001796 2.297 6.9 1852
(511) Davida 337 (7191) 1993 MA1 1969/ 7/16 0.004306 5.979 9.9 1949
(45) Eugenia 214 (17 763) 1998 EG 1969/ 7/23 0.007365 0.577 15.4 1994
(24) Themis 205 (22 491) 1997 GX32 1969/ 9/ 9 0.005852 1.395 7.1 1997
(45) Eugenia 214 (15 167) 2000 GS135 1970/ 1/21 0.009186 0.895 8.0 1990
(3) Juno 244 (6817) Pest 1970/ 4/19 0.001563 5.441 11.4 1982
(409) Aspasia 168 (9347) 1991 RY21 1970/ 5/12 0.000623 4.274 11.9 1979
(15) Eunomia 272 (16 693) 1994 YC2 1970/ 5/20 0.017941 1.173 6.4 1955
(52) Europa 312 (1023) Thomana 1971/ 5/31 0.006537 3.763 8.3 1924
(3) Juno 244 (16 670) 1994 AS2 1971/12/ 7 0.003455 4.688 6.0 1994
(173) Ino 159 (23 094) 1999 XF143 1973/ 5/ 5 0.001138 2.658 8.9 1990
(16) Psyche 264 (13 206) 1997 GC22 1974/ 7/18 0.003876 0.743 42.7 1960
(324) Bamberga 242 (11 482) 1988 BW 1974/10/30 0.001517 6.893 9.1 1988
(511) Davida 337 (1960) Guisan 1975/ 2/19 0.007197 6.713 5.3 1955
(15) Eunomia 272 (23 389) 1181 T-3 1975/ 7/20 0.005183 3.090 8.4 1977
(24) Themis 205 (2296) Kugultinov 1975/12/23 0.015699 0.438 8.4 1941
(13) Egeria 215 (20 175) 1996 XJ27 1976/ 6/ 7 0.001442 7.023 6.6 1996
(15) Eunomia 272 (19 524) Acaciacoleman 1976/ 8/ 1 0.024084 0.715 7.8 1975
(52) Europa 312 (13 180) 1996 HV19 1977/ 3/ 6 0.008204 3.441 7.2 1976
(76) Freia 190 (10 383) 1996 SR7 1977/ 3/16 0.000653 5.181 13.6 1994
(52) Europa 312 (24 290) 1999 XS190 1977/ 8/31 0.006515 4.904 6.4 1980
(12) Victoria 117 (24 367) 2000 AC126 1978/ 9/ 4 0.000259 6.956 5.9 1986
(532) Herculina 231 (18 014) 1999 JC121 1979/ 2/25 0.002822 5.173 5.6 1993
(356) Liguria 135 (20 567) 1999 RS129 1979/ 4/27 0.000414 6.497 6.1 1977
(16) Psyche 264 (6852) 1985 CN2 1979/ 9/10 0.006830 1.956 9.2 1952
(324) Bamberga 242 (16 035) 1999 FX32 1979/10/10 0.001969 8.211 5.9 1986



 

 
Table 7: Perturbers (first column) other than big four main belt asteroids and perturbed asteroids (third column) from 1980 to 1997 ( $P \ge 5 ~{\rm km~ s}$). Masses of emphasized asteroids may probably be derived in this decade.
  N1 D   N2 date r v P obs
    ${\rm [km]}$     year/m/d ${\rm [AU]}$ ${\rm [km~ s^{-1}]}$ ${\rm [km~ s]}$ year
(15) Eunomia 272 (19 994) 1990 TJ15 1980/ 4/18 0.018923 1.352 5.3 1985
(511) Davida 337 (18 085) 2000 JZ14 1980/ 9/ 8 0.010761 4.594 5.2 1986
(128) Nemesis 194 (22 347) 1992 SE13 1980/10/18 0.004070 2.393 5.0 1981
(24) Themis 205 (11 474) 1982 SM2 1980/11/13 0.001083 3.393 15.7 1982
(24) Themis 205 (8700) 1993 JL1 1981/ 2/ 1 0.008747 0.614 10.7 1975
(65) Cybele 245 (22 920) 1999 TF94 1982/ 3/31 0.007481 2.462 5.3 1989
(121) Hermione 217 (5750) Kandatai 1982/ 6/ 9 0.001114 4.545 13.5 1970
(145) Adeona 155 (6173) Jimwestphal 1982/12/ 1 0.003096 1.120 7.2 1977
(52) Europa 312 (16 192) 2000 AU207 1983/ 4/ 6 0.007176 3.276 8.6 1978
(11) Parthenope 162 (18 801) 1999 JO76 1983/ 7/ 5 0.001105 1.509 17.0 1992
(29) Amphitrite 219 (17 396) 1981 EK45 1983/ 8/19 0.004021 2.198 7.9 1981
(423) Diotima 217 (12 146) 6035 P-L 1984/ 1/13 0.002637 3.845 6.7 1960
(16) Psyche 264 (9473) Ghent 1984/ 5/17 0.009724 2.151 5.9 1993
(65) Cybele 245 (526) Jena 1984/ 6/24 0.005867 3.330 5.0 1901
(16) Psyche 264 (10 908) Kallestroetzel 1985/ 4/21 0.011500 1.862 5.7 1958
(29) Amphitrite 219 (6904) McGill 1985/ 6/15 0.009335 0.558 13.5 1990
(29) Amphitrite 219 (14 809) 1981 ES28 1986/ 1/17 0.005391 2.467 5.3 1981
(451) Patientia 230 (18715) 1998 HE121 1986/ 2/20 0.005351 1.254 12.1 1969
(15) Eunomia 272 (20 784) 2000 RN56 1986/ 7/ 7 0.006294 3.081 6.9 1981
(704) Interamnia 333 (21 868) 1999 TK291 1987/ 1/11 0.008407 5.565 5.3 1997
(16) Psyche 264 (20 874) 2000 VL49 1987/ 3/20 0.010296 2.064 5.8 1992
(52) Europa 312 (4140) Branham 1987/10/ 8 0.010997 3.295 5.6 1929
(9) Metis 184 (7684) Marioferrero 1989/ 2/15 0.002479 2.752 6.1 1983
(15) Eunomia 272 (3591) Vladimirskij 1989/ 2/27 0.003816 5.553 6.3 1932
(9) Metis 184 (9362) 1992 FE1 1989/ 7/26 0.005891 1.174 6.0 1992
(45) Eugenia 214 (673) Edda 1990/ 8/13 0.003766 2.548 6.8 1908
(16) Psyche 264 (19 462) Ulissedini 1991/ 8/19 0.029378 0.762 5.5 1980
(511) Davida 337 (11 985) 1995 VG 1991/10/14 0.004396 6.155 9.5 1981
(7348) 1993 FJ22 18.2 (7562) Kagiroino-Oka 1993/ 3/26 0.000001 1.673 24.2a 1986
(15) Eunomia 272 (10 324) Vladimirov 1993/ 7/10 0.006067 3.264 6.8 1990
(52) Europa 312 (124) Alkeste 1993/10/17 0.012436 2.585 6.3 1872
(52) Europa 312 (13 151) Polino 1993/10/28 0.002112 4.250 22.6 1995
(19) Fortuna 166 (18 162) 2000 PX15 1993/12/ 1 0.001684 3.608 5.0 1993
(29) Amphitrite 219 (987) Wallia 1994/ 3/ 3 0.002452 3.198 9.0 1899
(56) Melete 117 (22 960) 1999 UE4 1994/ 9/23 0.000344 4.402 7.1 1994
(19) Fortuna 166 (3486) Fulchignoni 1996/ 5/14 0.002135 2.304 6.2 1952
(704) Interamnia 333 (7461) Kachmokiam 1997/ 5/31 0.007478 5.285 6.2 1984
(13) Egeria 215 (14 689) 2000 AM2 1997/ 7/21 0.000801 4.015 20.7 1973

$\textstyle \parbox{15.2cm}{
$^{{a}}$ ~Probably an upper limit. A very uncertain value
due to $r <<$\space and assumed albedo of $0.04$ .}$


3 Selection process

The astrometric method of mass determination is done by measuring the deflection in position of the perturbed body from its ephemeris. The magnitude of the change in its mean motion $\vert\Delta{n}\vert$depends on several quantities. Neglecting its orbital eccentricity and its mass we have

\begin{eqnarray*}\vert\Delta{n}\vert \simeq \frac{6 ~G ~m ~\epsilon}{a ~r ~v},
\end{eqnarray*}


where G is the universal constant of gravitation, m is the mass of the perturber (depends on its diameter Dand bulk density $\rho$), $\epsilon$ is the energy efficiency of the close approach, the fraction of the change in the heliocentric velocity, which occurs in the tangential direction, a is the semimajor axis of the perturbed asteroid, r is the minimum separation distance between asteroids, and v is their encounter velocity (see Carpino & Knezevic 1996). Three quantities, D3, r and v, may differ by several orders of magnitude from one asteroid pair to the other. Thus, they play the major role in the $\vert\Delta{n}\vert$ determination.

We used the same method as in Paper I to find possible asteroid pairs that could be used for mass determinations. We searched for the largest values of the auxiliary P parameter defined as

\begin{eqnarray*}P = \frac {D^{3}}{r ~v},
\end{eqnarray*}


where D is in ${\rm km}$, r is in ${\rm km}$, v is in ${\rm km~ s^{-1}}$, and P is in ${\rm km~ s}$. The method sensitivity to orbital eccentricity was not analysed.

Since D was used mostly from Tedesco (1989) with several large asteroid diameters missing, other sources or guesses were also used occasionally, as in Paper I. For example, (64) Angelina is of type E, so its albedo is probably high; diameters for (88) Thisbe and (776) Berbericia were used from Bowell et al. (1994) containing the new IRAS catalog of asteroid diameters. In general, the new values for diameters are a bit smaller than those in Tedesco (1989) (e.g. diameter of (1) Ceres is about $848 ~{\rm km}$ in comparison to $913 ~{\rm km}$). But the results from other techniques, such as occultation of (2) Pallas, or HST images of (4) Vesta, indicate the new ones may be underestimated values. Moreover, the results of P would not be lowered substantially even using the new diameters.

It must be noted that the P parameter is not valid outside the sphere of action of the perturber. Its computation is just an approximation allowing comparisons between the encounters. For example, Hill's radius $R_{\rm H}$ is considered as

\begin{eqnarray*}R_{\rm H} = \sqrt[3]{\frac {m}{3M}}.r_{\rm S},
\end{eqnarray*}


where m is the mass of the body, M is the mass of Sun, and $r_{\rm S}$ is the distance from Sun (semimajor axis a or perihelion distance q). $R_{\rm H}$ values are approximately $0.0100 ~{\rm AU}$ for Earth, $0.0072 ~\rm {AU}$ for Mars, $0.3552 ~{\rm AU}$ for Jupiter, but only $0.0015 ~\rm {AU}$for (1) Ceres, $0.0010 ~\rm {AU}$ for (2) Pallas, $0.0009 ~\rm {AU}$ for (4) Vesta.

4 The big four main belt asteroids

The mass of (1) Ceres has been determined by the astrometric method many times (e.g. Williams 1992; Carpino & Knezevic 1996; Micha\lak 2000). There is no need to list pairs with small P here. The total number of those with $P \ge 6 ~{\rm km~ s}$is about 1000 in the period under study. It is only required that heavily perturbed asteroids due to (1) Ceres be listed and used to lower the uncertainty in its mass. Only some asteroids with $P \ge 60
~{\rm km~ s}$ are listed in Table 1. All of these were observed quite well before and after the encounter and are therefore emphasized in Table 1. Similarly, observations of several other asteroids may improve our knowledge of the mass of (4) Vesta, which seems to be the second largest perturber in the main belt. In Table 2, we chose five heavily perturbed asteroids due to (4) Vesta ( $P \ge 20
~{\rm km~ s}$) that were observed before and after the encounter. In fact, more than 300 pairs were found to have $P \ge 6 ~{\rm km~ s}$.

As was mentioned in Paper I the fourth largest main belt asteroid (10) Hygiea may displace (2) Pallas in the total number of significant encounters. Indeed, this is confirmed if we compare the perturbations due to these two asteroids. While we found only 17 pairs with (2) Pallas (Table 3), there were 61 pairs with (10) Hygiea for $P \ge 6 ~{\rm km~ s}$ (Tables 4 and 5). The summation of P over all these encounters is $174.5 ~{\rm km~s}$ for the former, and $754.6 ~{\rm km~s}$ for the latter. Thus, asteroid (10) Hygiea may be more than three times more effective on the motion of main belt asteroids (though comparison of the bulk densities may lower the difference). This is because the orbit of (2) Pallas is highly inclined to the ecliptic. As a result, asteroid encounters with (2) Pallas tend to occur at high speed and rarely result in significant perturbations. Emphasized asteroids in the tables are likely to be used for mass determination at present. Some of them have still not been tested by other authors. We used the combination of P, and time periods of astrometric observations pre- and post- encounter for a given pair, respectively, in order to denote asteroids suitable for mass determination. Neglecting the perturber's mass (here (2) Pallas and (10) Hygiea) emphasized asteroids may be deflected from their ephemerides by about $1 \hbox{$^{\prime\prime}$ }$ or more. We did not emphasize any asteroid without pre-encounter observations, nor asteroids that will be perturbed significantly in the future. The tables also contain unusual asteroids - (2204) Lyyli encountered (2) Pallas, while (2061) Anza encountered (10) Hygiea. (11 054) 1991 FA will have two encounters with (10) Hygiea.

5 Other large perturbers

The observed positions of some small numbered main belt asteroids may be deflected from their ephemerides even after perturbations due to the four largest main belt asteroids are included. This is because there are several other large perturbers in the belt, the masses of which are usually neglected. In general, we need precise long-term (pre- and post-encounter) astrometric observations of both encountered asteroids to reveal deflection and determine perturber's mass. However, special care must be taken not to neglect perturbations due to other sources during such extended periods of time.

Our search for large perturbers (other than the four largest ones) within the main belt was limited to bodies with absolute magnitude $H \le 9.5$. We found a lot of pairs with $P \ge 5 ~{\rm km~ s}$. Those that occurred before 1980 are summarized in chronological order in Table 6, while those that occurred between 1980 and 1997 (including these years) are in Table 7. The rest (in 1998 and later) are listed at http://www.uniba.sk/~galad/massdet.htm. These could only be used for mass determination after several years. The policy of denoting asteroids in Table 6 is similar to the one used in the previous section except for the fact that we emphasized the perturbers instead of perturbed asteroids. In Table 7, we emphasized asteroids suitable for mass determination this decade (before 2010). All asteroids listed in the tables are from the main belt. Unfortunately, no pairs were found among the Trojans.

Altogether we emphasized 13 different asteroids (asteroid (16) Psyche is emphasized twice and (52) Europa three times) that could be tested for mass determination at present or by the end of this decade. This inference is based only on P values and the date of the first astrometrical data available in the Minor Planet Center for both asteroids (last column in tables). However, some measurements are of low quality, unreliable, or poorly distributed before the encounter. In such cases, masses could not be determined and the total number of candidates will probably be less than 13. In general, uncertainties in mass determination are high from single encounters, but they tend to be lower from multiple ones.

The limit for P is lower than in Paper I, but it is not guaranteed that all such pairs are found here. Some changes in P (in comparison to Paper I) are due to changes in D, which was previously computed from albedo; e.g. pair (65) Cybele - (526) Jena had $P < 6 ~{\rm km~ s}$ and would be missing here using the same limit. Some pairs were not used for mass determination by other authors. They are usually concentrated on low numbered asteroid encounters that occured several decades ago. If some positions are found in old archived photographic plates for high numbered asteroids, they could then be used for mass determination. The results from Paper I were compared to that of Hilton et al. (1996). The mass of (45) Eugenia computed by the astrometrical method could be of special interest for comparison to the mass derived by other ground-based technique, as a satellite of this object has been discovered.

Most of the asteroids are continually observed near their oppositions. The last observation dates are, therefore, variable and not listed in tables.

6 Small perturbers

Masses of asteroids with H > 9.5 are difficult to reveal by the astrometrical method, as they are smaller than $100 ~{\rm km}$in diameter (their albedos are probably higher than 0.03). Thus, only extremely close and slow encounters could cause large P values. In the previous section, we found only one asteroid candidate for mass determination with H < 9.5 that is smaller than $100 ~{\rm km}$ in diameter - in 2014 (445) Edna will be encountered by (1764) Cogshal at a minimum distance of just $0.000052 ~{\rm AU}$. In spite of the fact that we know little about effective diameters of small perturbers with H > 9.5, it seems that only one pair was found with $P \ge 5 ~{\rm km~ s}$. This is the pair (7348) 1993 FJ22 - (7562) Kagiroino-Oka, which had the closest encounter of all computed pairs. In 1993 they were only about $200 ~{\rm km}$ apart! (It should be noted that the uncertainties in the ephemerides for the objects are probably several times greater than this.) The pair is included in Table 7. However, the sparse astrometric data for (7562) Kagiroino-Oka prior to the encounter, combined with the extreme uncertainty in the minimum separation, render this encounter unsuitable for mass computation.

7 Conclusions

We tried to find asteroids for which masses could be determined by the astrometric method. We concentrated on those small asteroids that could manifest quick ephemeris deviations after a close encounter to a larger asteroid. The masses of the largest perturbers among main belt asteroids - (1) Ceres and (4) Vesta - could be determined more reliably from several heavily perturbed bodies. In the case of the other two large perturbers we conclude that perturbations due to (10) Hygiea are more effective on the motion of main belt asteroids than those due to (2) Pallas by a factor of three or more. At this time, however, the latter is used in asteroid orbit determinations along with the planets, Moon, and two large asteroids (1) Ceres and (4) Vesta while the former is normally not (except for special cases). We found also several small asteroids that could be used for reliable mass determination of (10) Hygiea.

In addition to the four largest main belt asteroids, there are others - (11) Parthenope, (13) Egeria, (15) Eunomia, (16) Psyche, (24) Themis, (29) Amphitrite, (39) Laetitia, (45) Eugenia, (52) Europa, (65) Cybele, (121) Hermione, (451) Patientia, (511) Davida - the masses of which could probably be achieved by the end of this decade using astrometric data of smaller asteroids. However, if the astrometric measurements of the small asteroids before the important encounter with the perturber are unreliable, of low quality, or poorly distributed, it would result in reducing the number of perturbers listed above. In general, high numbered asteroids will be used more thoroughly for mass determination of quite large asteroids over the next decade. However, in some cases, useful data for them may be found in old photographic plates.

Acknowledgements
This work was supported by Grant VEGA 7157 and Comenius University Grant (No. UK/100/2001).

References

 


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