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MASSES OF SOME LARGE MINOR PLANETS
Chernetenko Yu.A.(1), Kochetova O.M.
(1) (2) (2)

Institute of Applied Astronomy, nab. Kutuzova,10, 191187 S.Petersburg, Russia, E-mail: cya@quasar.ipa.nw.ru Institute of Applied Astronomy, nab. Kutuzova,10, 191187 S.Petersburg, Russia, E-mail:kom@quasar.ipa.nw.ru

ABSTRACT For estimating masses of minor planets 3, 7, 10, 15, 29, 52, 65, 511, 704 there were selected two groups of perturbed bodies. In the first one those minor planets were included which have close app-roaches with perturbing body. In the second group those minor planets were selected which are close to commensurability 1:1 with perturbing body. Preliminary determination of mass of perturbing planet was made from separate solution for each pair of planets. Perturbed planets giving solution with great errors were eliminated from subsequent consideration. Then observations of selected perturbed bodies were used in general solution for their orbital elements and mass of perturbing planet. The obtained results are compared with those by other authors. 1. INTRODUCTION Estimation of minor planet masses holds much favor in present time. Partly it can be explained by the fast growth of number of known minor planets and volume of their observations. Among newly numbered minor planets there are bodies having close approaches with large minor planets. On the other hand, astrophysicists would like to know the correct values of densities of minor planets instead of some mean values based on their taxonomic classes. Celestial mechanics need these data for construction accurate theory of motion of Solar system bodies, to calculate the trajectories of spacecraft with accounting for all gravitational perturbations. Densities of minor planets are of interest from the point of view of evolution of small bodies. Very low density can be result of `rubble pile' structure of some body suffered fragmentation and subsequent accretion. The other explanation of low density can be ascribed to cometary origin of some minor planets. The evaluation of percent of such objects in minor planet population is an important task for understanding evolution of small bodies. There are two methods of minor planet masses estimations: dynamical and astrophysical. Dynamical method is based on taking into account gravitational perturbations produced by some minor planet on the other bodies (major or minor planets, spacecraft). For a binary asteroids one can find mass using the third Kepler's law. Astrophysical method uses values of diameters of minor planets and their densities. The density estimations are based on studies of surface reflectance properties of minor planet and their comparison with laboratory investigations of meteorite

sumples. Astrophysical method starts from suppositions of spherical form of minor planets, its uniform structure, which can not be true in one or other case. In particular, for a minor planet which will prove to be binary, this method can give enlarged mass value by factor 1.4 (if two components have equal diameters). One of disadvantages of dynamical method is great relative errors of mass determination which are common for this method. They arise from insufficient accuracy of observations of perturbed minor planets and deficient number of close approaches when seek for mass produced perceptible perturbations in the motion of other bodies. As was shown in [1], the number of perturbed minor planets can be substantially enlarged as a result of consideration of bodies moving in the vicinity of commensurability of mean motions with perturbing minor planets. In this case one can take into consideration not only close (< 0.05 a.u.) encounters, but moderate as well. Usage of less stringent requirement than usually used 0.05 a.u. is here warrantable as the approaches of planets may occur over and over due to commensurability of their mean motions and mutual velocities of the bodies are rather small. Hoffmann [2] pointed out the fruitfull of such approach. So, both dynamical method and astrophysical one have their advantages and disadvantages and mass estimations obtained by these two methods should add and check each other. The aim of this paper is estimation of masses of some large asteroids using gravitational perturbations produced by them on the other minor planets. To obtain accuracy on the order of several units of 10-12 MSun which can be compared with the results of astrophysical approach [3], [4], we use positional observations of large number of perturbed minor planets. 2. ACCEPTED MODEL OF MOTION To estimate a mass of perturbing minor planet the deviations of observed positions from calculated ones of perturbed minor planets are considered. Observed positions were taken from catalogue supported by MPC. To calculate positions of perturbed minor planets their equations of motion were integrated together with these of perturbing minor planet. The perturbations from nine major planets in accordance with coordinates and masses from DE403 ephemeris were taken into account. Besides, the perturbations from 300 minor planets (including the minor planet with required mass) were considered. Masses of these minor planets were


taken in conformity with DE403, but their coordinates were obtained by numerical integration starting from the osculating elements published in [5]. The relativistic terms due to the Sun were included in equations of motion. The observations were corrected for gravitational deflection of light and for phase effect by Lommel-Zeeliger law of scattering. To calculate coefficients of equations of conditions the special equations for partial derivatives - coefficients of equations - were integrated along with the equations of motion. The LMS was used to fit conditional equations. All observations were supposed to have equal weights. 3. SELECTION OF PERTURBED MINOR PLANETS SUITABLE FOR MASS ESTIMATION OF PERTURBING MINOR PLANETS The most obvious criterion for selection of perturbed minor planet is the value of minimum distance, , between it and perturbing body: the less is the better. Hilton et al. [6] supposed and based that should be no greater than ~ 0.05 a.u. Besides, this perturbed minor planet should be observed before and after the encounter. Hilton et al. [6] proposed criterion of selection of pairs of minor planets based on estimation of deflection angle that is change of trajectory of perturbed body due to gravitation influence. The value of depends on the mass of perturbing body, minimum distance between approaching bodies and their relative velocity. By accounting for the value of this angle quality factor, ranging from 1 to 5 was obtained by Hilton et al. As they noted "the encounters with a high quality factor are ones most likely to yield reasonable determinations of the larger asteroid's mass". Using this criterion encounters for 4583 minor planets were calculated in [6] and the list of more promising for mass estimation encounters of 34 more large asteroids with 130 less massive during 1950 ­ 2017 was compiled. This list was used for a number of minor planet mass determinations. In particular, Kuznetsov [7] calculated values of deflection angle for time interval 1900 ­ 2000 for 1530 approaches of 30 massive asteroids with more than ten thousands minor planets. In the present paper for determining masses of some large minor planets, besides of single close encounters of perturbed and perturbing bodies (minimal distance no greater than 0.05 a.u.), more moderate encounters of minor planets close to commensurability 1:1 with each other were used. These last pairs were selected from the group of minor planets being in commen-surability 2:1 with Jupiter and from other groups. In last case sampling of the pairs was governed by criterion: minimum distance between planets is less or equal to 0.1 a.u. It is necessary to note that among the minor planets close to commensurability there are few encounters with minimal distances < 0.05 a.u. Such encounters were considered together whis others of the

second group. The number of perturbed minor planets selected in accordance with guideline of minimal distances is given in Table 1 for 9 perturbing minor planets under consideration in this paper. Table 1. The number of perturbed minor planets, selected in accordance with minimum distances and used in final general solutions (in brackets)
Perturbing minor planets 3 7 10 15 29 52 65 511 704 Having close approaches Close to commensurability 1:1


41 82 18 30 13 38 22 68 18

0.05 a.u (20) (26) (13) (8) (6) (11) (4) (16) (8)


160 80 219 87 63 239 279 96 116

0.10 a.u. (13) (4) (11) (6) (9) (15) (21) (10) (19)

Table 2 gives values of mean motion of perturbing minor planets (the second column) and limiting values of mean motion of selected perturbed bodies considered as being in commensurability with perturbing ones (the third column). Table 2. Mean motions of perturbing and perturbed minor planets
Perturbing minor planets 3 7 10 15 29 52 65 511 704 Mean motion (/day) 814 962 638 825 869 650 557 628 662 Limiting values of mean motion of perturbed planets 8 9 6 8 8 6 6 6 6 (/day) 07 ­ 823 54 ­ 971 28 ­ 646 19 ­ 859 25 - 876 13 - 640 21 - 646, 563 for (790) 23 - 648 51 - 670

At the second stage the observations of each perturbed planet were used in solution for corrections to their orbital elements and that of mass of perturbing planets. For example, in the Table 3 some results for perturbing minor planet (10) Hygeia are given. In this table the columns 1 - 6 contain the number of perturbed minor planet, , date of encounter (with precision of 1 day), the minimum distance in a.u., obtained value of mass error in units of 10-11 Msun, deflection angle in arcsec [7] for these encounters, quality factor (F) calculated by Hilton et al. in [6] Obtained range of values of mass reflects not only potential possibility to estimate this mass, but real, depending on number and distribution of observations,


correlation between some parameters under determination, unknown possible errors of positional observations as well. On the other hand, deflection angle and quality factor reflects potential possibility only. One can compare, for example, mass and values of deflection angle for minor planets (829) and (3946): values of deflection angle differ more than 5 times for them, whereas mass for these planets has practical-ly equal values. Conclusions follow that i) mass error gives more reliable information about `convenience' of encounter for mass estimation; ii) common solution can be done only after preliminary analysis of error of mass Table 3. Estimation of mass of minor planet (10) Hygeia using close encounters with perturbed minor planets
Minor planet 4 4 5 8 52 65 79 29 59 87 50 44 80 25 65 86 39 79 Date of encounter 19 19 19 19 19 19 19 19 19 19 19 19 20 19 19 19 19 19 19 19 19 19 50 95 56 27 59 84 88 89 33 78 84 83 00 56 83 72 83 95 89 51 98 95 09 12 03 05 04 02 06 05 08 02 05 03 10 06 07 10 12 02 12 02 03 02 11 26 12 19 26 11 28 27 11 18 14 15 15 28 13 03 05 24 11 11 30 07 . . . . . . . . . . . . . . . . . . . . . .

The results are given in the Table 4. The number of perturbed minor planets used in each solution is given in the Table 1in brackets. In the Table 5 are presented Table 4. Estimation of masses from different sets of perturbed minor planets
Minor planet 3 7 10 15 29 52 65 Commensurability 1:1 (10-11 Msun ) 1.64 1.53 4.50 1.91 1.60 0.81 0.57 2.78 0.86 Close approaches (10-11 MSun ) 2.26 1.40 4.93 0.18 0.93 1.58 0.60 2.48 Combined solution (10-11 MSun) 2.09 1.41 5.01 1.22 0.77 1.28 0.58 2.40 0.81


(a.u.) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 502 380 449 059 378 345 263 443 280 614 431 296 214 493 314 991 454 578 224 445 144 092



mass
-11

(10 Msun ) 2.21 1.46 1.56 1.05 3. 2. 14. 2. 1. 1. 13. 20. 10. 1. 78 14 58 59 50 67 65 29 91 83

()

F

511 704

±. ± ± ± ± ± ± ± ±

0.78 0.63 0.76 0.18 0.52 0.39 0.16 1.54 0.54

.

1.05

± ± ± ± ± ± ± ± ±

0.38 0.16 0.48 0.50 0.14 0.33 0.43 0.24 0.65

±. ± ± ± ± ± ± ± ±

0.35 0.14 0.41 0.16 0.12 0.25 0.15 0.24 0.42

Table 5. Results of other authors.
0.116 0.011 3 Minor planet 3 7 3 10 15 29 0 0 0 0 . . . . 006 018 017 109 52 65 3 4 511 704 0.594 0.108 2.5 3.9 2.8 5.1 1.3 Dynamical method [8] (10-11 MSun ) Dynamical method [9] (10-11 MSun ) 2.33 1.62 Astrophys. method [3] (10-11 MSun ) 1.01 0.58 2.79 1.40 0.74 1.08 0.52 1.36 1.30

12 12 15 16 17 18 19 19 23 24

0.039

± ± ± ± ±

0 .5 0.3

1.82 1.78 2.31

0.7

3.91 3.58

26 26 39 60

19 90 46 06

2. 358. 0. 3.

95 8 96 32

0.3 0.8

1.74

± ± ± ± ± ± ± ±

0.97 0.38 0.69 0.18 0.20 0.48 2.08 0.45

± ± ± ± ± ± ± ± ±

0.1 0.1 0.1 0.2 0.1 0.06 0.03 0.07 0.07

obtained from observations of each perturbed bodies. The results were separated by the value of the error of mass determination. Those minor bodies with unit close approach that gave error greater than 510-11 MSun as well as those with several approaches that gave error greater than 10 10-11 MSun were eliminated from subsequent consideration. 4. ESTIMATION OF MASSES After that three different solutions were found for each perturbing planet using i) only perturbed minor planets close to commensurability; ii) perturbed planets having close approaches; iii) combined solution. All solutions were done for orbital elements of perturbed minor planets and mass of corresponding perturbing planet.

some results of other authors who used dynamical or astrophysical approaches. The comparison of the results in the Table 5 obtained by dynamical method shows that errors of our determination are smaller than obtained in [8] and [9]. The authors of [8] and [9] used different from this paper approaches for choosing perturbed planets. The main ideas of astrophysical approach is given in [4], and values of mass for individual minor planets are given in [3]. In these papers analysis of radar measurements of the martian landers Viking-1,2 and Pathfinder was used to estimate the mean density of asteroids belonging to the three averaged taxonomic classes: C, S, and M. The corresponding values of density were obtained with great accuracy ­ 0.04, 0.02, and 0.13. IRAS diameters were used to calculate the masses of 357 minor planets. The mass errors of [3] given in the Table 5 correspond to the densities and diameters errors. Our mass values


for minor planets (3), (15), (29), (52), (65), (704) are in satisfactory agreement with results in [3]. As to mass values of (7), (10), (511), the disagreements are more than triple errors. Possible explanation can consist in the fact that taxonomic types of these minor planets (and their real densities) do not correspond to the density of one of three averaged types C, S, and M evaluated in [3]. Therefore improvement of the masses of these minor planets is still desirable. 5. CONCLUSIONS · Masses of minor planets 3, 7, 10, 15, 29, 52, 65, 511, 704 were obtained by dynamical method which uses the gravitational perturbations produced by minor planet on the other minor planets. · It was shown that more accurate values of asteroid masses can be obtained in studies of several perturbed bodies. However the preliminary selection is needed. The error of mass determination from the solution for one perturbed minor planet can be criterion for selection. · Expansion of number of perturbed bodies due to minor planets close to commensurability with perturbing one gives possibility to obtain more reliable mass estimations despite the usage of less close encounters. · Masses of minor planets 3, 15, 29, 52, 65, 704 are in a satisfactory agreement with their astrophysical values. It can mean that these minor planets have uniform structure and that their real densities are in agreement with those of averaged taxonomic classes prescribed to them. · The reasons of disagreement of dynamical and astrophysical values of masses for 7, 10, 511 can be ascribed to some systematic errors of positional observations of perturbed bodies or not uniform structure of perturbing bodies or variation of their real densities from those adopted in [3]. 6. ACKNOWLEDGEMENTS We thank V.A. Shor (IAA) for assistance in work and valuable advices. 7. REFERENCES 1. Kochetova O., Chernetenko Yu., In: Dynamics of Natural and Artificial Celestial Bodies. Eds. H.PretkaZiomek, E.Wnuk, P.K.Seidelmann, D.Richardson, 333334, 2001. 2. Hoffmann M., Asteroid mass determination: present situation and perspectives. In: Asteroids II. Eds:Richard P. Binzel, Tom Gehrels, Mildred Shapley Matthews, 228-239, 1989.

3. Krasinsky G.A,, Pitjeva E.V., Vasilyev M.V., Yagudina E.I., Estimating Masses of Asteroids, Communications of IAA, N 139, 2001.

4 Krasinsky G.A,, Pitjeva E.V., Vasilyev M.V., Yagudina E.I. Hidden mass in the asteroid belt. Icarus, 158, 98-105, 2002 5 Shor V.A. (Ed.). Ephemerides of Minor planets for 2003, St.Petersburg, IAA RAS, 2002.
6. Hilton J.L., Seidelman P.K., Middour J. Prospects for determining asteroid masses. Astron. Journal, Vol. 112, 2319-2329, 1996.. 7. Kuznetsov V.B., Catalogue of asteroid approaches. Communication of IAA, N 136, 2000 (in Russian). 8. Michalak G.,. Determination of masses of six asteroids from close asteroid-asteroid encounters. In: Dynamics of Natural and Artificial Celestial Bodies. Eds: Halina Pretka-Ziomek, Edwin Wnuk, P. Kenneth Seidelmann, David Richardson, 281-282, 2001. 9. Kuznetsov V.B., Determination of masses of 108 asteroids. Communication of IAA, N 138, 2001 (in Russian).