Determination of asteroid masses

Astron. Astrophys. 360, 363–374 (2000)

ASTRONOMY AND ASTROPHYSICS

Determination of asteroid masses I. (1) Ceres, (2) Pallas and (4) Vesta G. Michalak Wrocláw University Observatory, Kopernika 11, 51-622 Wrocláw, Poland ([email protected]) Received 2 November 1999 / Accepted 26 April 2000

Abstract. The masses of the three largest asteroids: (1) Ceres, (2) Pallas and (4) Vesta were determined from gravitational perturbations exerted on respectively 25, 2, and 26 selected asteroids. These masses were calculated by means of the leastsquares method as weighted means of the values obtained separately from the perturbations on single asteroids. Special attention was paid to the selection of the observations of the asteroids. For this purpose, a criterion based on the requirement that the post-selection distribution of the (O − C) residuals should be Gaussian was implemented. The derived masses are: (4.70 ± 0.04) × 10−10 M , (1.21 ± 0.26) × 10−10 M , and (1.36 ± 0.05) × 10−10 M for (1) Ceres, (2) Pallas and (4) Vesta, respectively. We also show how the fact that a statistically substantial number of perturbed asteroids is used in the determination of the mass of (1) Ceres and (4) Vesta increases the reliability of their mass determination because effects like the flaws of the dynamical model and/or the observational biases cancel out. In case of Ceres and Vesta, these effects have a very small influence on the final result. The number of acceptable mass determinations of Pallas is much smaller, but can be increased in the future when the dynamical model is improved. We indicate some promising encounters with Pallas. Key words: minor planets, asteroids – astrometry – planets and satellites: individual: 1 Ceres, 2 Pallas, 4 Vesta

quently that the differences between independent determinations of the masses are large in comparison with their formal errors. A good example of such a case is shown in Table 1: the difference between two recent values of the mass of (1) Ceres (Viateau & Rapaport 1998; Hilton 1999) is about 16 and 9 times greater than the errors given in these papers. Possible causes of such discrepancies are discussed in detail in Sect. 2.3. Most mass determinations shown in Table 1 are based on the perturbations exerted on a single asteroid, even if many more asteroids underwent close approaches with the massive minor planet. As we shall show later, the use of as many perturbed asteroids as possible is crucial for reliable estimation of the asteroid mass. We therefore performed a search for suitable asteroids and used about 25 of them for the determination of the mass of (1) Ceres and (4) Vesta. The situation is much less favourable when there is only one or a few such asteroids, like for (2) Pallas. All that can be done in this case is to improve the dynamical model using the available data and postpone a more reliable mass determination for the future. For the reasons given above, new mass determinations are highly desirable. This is the main goal of this series of papers. In the present paper the method of calculation of asteroid masses is presented and new masses for (1) Ceres, (2) Pallas and (4) Vesta are derived and discussed. Next papers of the series will be devoted to the determination of masses of other, relatively massive asteroids and to improvement of the dynamical model used in calculations.

1. Introduction

2. Method

As is well known, many large asteroids from the main belt produce detectable gravitational perturbations on the orbits of both minor and major planets. In the construction of the DE403 ephemerides (Standish et al. 1995), for example, perturbations from 300 asteroids were taken into account. Up to now, the masses of only 17 asteroids were determined directly from the gravitational perturbations; many of them are still of poor accuracy. The current status of the asteroid mass determinations is presented in Table 1. It is clearly seen from this table that only five asteroids, viz. (1) Ceres, (2) Pallas, (4) Vesta, (11) Parthenope and (253) Mathilde, have masses determined with formal errors smaller than 5%. Moreover, it happens fre-

The best method used for the determination of an asteroid mass consists in taking its gravitational perturbations on the orbits of many other (perturbed) asteroids. This method was used by Sitarski & Todorovic-Juchniewicz (1992) for (1) Ceres, Sitarski (1995) for (1) Ceres and (4) Vesta, and Hilton (1999) for (1) Ceres, (2) Pallas and (4) Vesta. In an easier and more frequently applied method, the mass of the perturbing body is derived as a weighted mean of the masses found separately from perturbations on a few single asteroids. It could be shown that if all individual determinations are independent, both methods are equivalent. In the present paper, we use the second method. In this approach, the value of

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G. Michalak: Determination of asteroid masses. I

Table 1. Current status of asteroid mass determination. Mass σ [in 10−10 M ] (1) Ceres: 6.7 0.4 5.1 5.9 0.3 4.99 0.09 5.0 5.21 0.07 5.0 0.2 4.9 0.15 4.74 0.04 4.43 4.796 0.085 4.8 0.22 4.85 0.06 4.92 0.07 5.04 0.10 4.64 4.78 0.06 4.622 0.071 4.67 0.09 4.26 4.71 0.05 4.4 0.6 4.7 4.759 0.023 4.39 0.04

Perturbed bodies (2) Pallas (4) Vesta (2) Pallas (2) Pallas (2) Pallas (2) Pallas Mars (2) Pallas (203) Pompeja (348) May (203) & (348) (348) May (32)(91)(203)(324)(348)(534) (91)(203)(348)(534) (2) Pallas DE403 solution (2) & (203) (203) & (348) (91)(203)(324)(348)(534) (203) (32)(91)(203)(324)(348)(534)(2572) (63) Ausonia DE405 solution (2)(4)(16)(32)(91)(203)(324)(348)(534) (2) & (4)

Reference (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25)

(2) Pallas: 1.3 0.4 1.14 0.22 1.4 0.2 1.05 1.0 1.59 0.05

(1) Ceres (1) Ceres Mars DE403 solution DE405 solution (1) & (4)

(3) (26) (7) (16) (23) (25)

(4) Vesta: 1.2 1.38 0.12 1.5 0.3 1.33 1.34 1.396 0.043 1.3 1.69 0.11

(197) Arete (197) Arete Mars (197) Arete DE403 solution (197) & (486) DE405 solution (1) & (2)

(27) (28) (7) (9) (16) (18) (23) (25)

Mass σ [in 10−10 M ]

Perturbed bodies

Reference

(10) Hygiea: 0.47 0.23 0.21 0.13 0.55 1.02

(829) Academia (829) Academia ´ (1259) Ogyalla

(29) (9) (30)

(17) Thetis

(31)

(15) Eunomia: 0.042 0.011

(1313) Berna

(32)

(16) Psyche: 0.087 0.026

(94) Aurora

(33)

(20) Massalia: 0.0242 0.0041

(44) Nysa

(34)

(24) Themis: 0.289 0.126

(2296) Kugultinov

(30)

(45) Eugenia: 0.03

satellite S/1998(45)1

(35)

(46) Hestia: 0.109 0.068

(19) Fortuna

(22)

(121) Hermione: 0.047 0.008

(278) Paulina

(33)

(11) Parthenope: 0.0258 0.0010

(243) Ida: 2.1 0.3

[mass in 10−14 M ] S/1993(243)1 Dactyl

(36)

(253) Mathilde: 5.293 0.221

[mass in 10−14 M ] NEAR tracking data

(37)

(433) Eros: 0.36 0.09

[mass in 10−14 M ] NEAR tracking data

(38)

(704) Interamnia: 0.37 0.17

(993) Moultona

(39)

(804) Hispania: 0.05 0.04

(1002) Olbersia

(39)

References to Table 1: (1) Schubart (1970), (2) Schubart (1971), (3) Schubart (1974), (4) Landgraf (1984), (5) Goffin (1985), (6) Landgraf (1988), (7) Standish & Hellings (1989), (8) Schubart (1991), (9) Goffin (1991), (10) Williams (1991), (11) Sitarski & Todorovic-Juchniewicz (1992), (12) Williams (1992), (13) Bowell et al. (1994), (14) Muinonen et al. (1994), (15) Viateau & Rapaport (1995), (16) Standish et al. (1995), (17) Viateau (1995), (18) Sitarski (1995), (19) Carpino & Kneˇzević (1996a), (20) Kuzmanoski (1996), (21) Carpino & Kneˇzević (1996b), (22) Bange & Bec-Borsenberger (1997), (23) Standish (1998), (24) Viateau & Rapaport (1998), (25) Hilton (1999), (26) Schubart (1975), (27) Hertz (1966), (28) Schubart & Matson (1979), (29) Scholl et al. (1987), (30) Garcia et al. (1996), (31) Viateau & Rapaport (1997), (32) Hilton (1997), (33) Viateau (2000), (34) Bange (1998), (35) Merline et al. (1999), (36) Belton et al. (1995), (37) Yeomans et al. (1997), (38) Yeomans et al. (1999), (39) Landgraf (1992).

the mass of a given massive asteroid is determined along with the six orbital elements of the usually smaller and much less

massive perturbed minor planet by means of the least-squares method. This calculation is carried out for many different, care-

G. Michalak: Determination of asteroid masses. I Table 2. The asteroids used in our dynamical model. If the adopted mass was not the same as that used by Viateau & Rapaport (1998), a reference is given. Asteroid

Mass [in 10−10 M ]

(1) Ceres (2) Pallas (4) Vesta (10) Hygiea (11) Parthenope (52) Europa (511) Davida (704) Interamnia

4.8 (preliminary solution from this work) 1.2 1.396 (Sitarski 1995) 0.47 0.026 0.14 0.18 0.35

fully selected (see Sect. 2.2) perturbed asteroids hereafter called test asteroids. The more an orbit of a test asteroid is perturbed by the massive one, the better it is for the mass calculation. The dynamical model used in our calculations includes all nine planets and eight asteroids as the perturbing bodies. The asteroids we used were the same as those used by Viateau & Rapaport (1998). Their names and adopted masses are listed in Table 2. Observations of minor planets were provided by the Minor Planet Center through the Extended Computer Service. In this paper we did not use Hipparcos observations of the minor planets. In the calculations of positions of the test asteroids for each date of observation we used the Bulirsh and Stoer variable step numerical integrator (Bulirsh & Stoer 1966). Initial conditions and masses for all planets were taken from the DE405 planetary ephemerides (Standish 1998). The initial conditions for the minor planets were derived from the osculating elements given in the ‘Ephemerides of Minor Planets for 1998’ (Batrakov 1997). The partial derivatives of the observed quantities (right ascension and declination) with respect to the unknowns (corrections to the initial conditions of the orbit of a test asteroid and the mass of the perturbing body) were calculated by means of the Romberg method. 2.1. Data selection and weighting Many observations of minor planets are of rather poor quality and a method for rejecting outliers is needed. An effort was made to find a reliable selection procedure. Since observational errors should have a Gaussian (normal) distribution, the selection criterion had to retain observations with (O − C) residuals having normal distribution and reject those with large values of (O − C). During the test stage of our calculations we tried iterative Chauvenet-type criteria in which the largest accepted residual depends on the number of observations, as well as kσtype criteria (k = 1, 2, 2.5...), frequently used by other authors. In some cases (e.g. when the scatter of the residuals is large), residuals finally accepted with those two kinds of criteria were not found to have a normal distribution. The agreement between the actual distribution and the normal distribution was checked by means of the χ2 test. Therefore, we implemented χ2 test as the selection procedure. To begin with, for each asteroid, the observations were divided into sev-

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eral groups with different dispersions. Next, for each group the mean value m and the standard deviation σ of the residuals were calculated. Two histograms: theoretical from normal distribution, N (m, σ), and observational from the (O − C) residuals, were then calculated and compared. From the differences between these two histograms the value of χ2 and significance level α were determined. Small significance level meant large discrepancy between the two distributions. In the next step, the largest residuals were successively rejected (m and σ were recalculated after each rejection) until α > αcrit , where αcrit determines the selection criterion used in this procedure. Greater αcrit means stronger selection. For all cases αcrit = 0.01 was adopted as the upper limit. This method of selection of usable observations will be called hereafter the normal selection. Some observations in our sample were obtained photographically. For these observations we rejected both residuals (in right ascension and declination) if one of them was to be rejected. For the other observations the selection in right ascension and declination was made independently. Having rejected the outliers by the normal selection, each group of observations was given a weight equal to the reciprocal of the variance in this group. The normal selection was performed each time when the correction of the six orbital elements of the test asteroid and the mass of the perturbing body was made. Residuals rejected in the i-th iteration were not used in the (i + 1)-st iteration. This guaranteed convergence of the iterative process, avoiding the situation in which the solution oscillates around a given value because some observation(s) are alternately included and rejected in successive iterations. It can also happen that if the initial orbit of the test asteroid is grossly incorrect, a too strong selection at the start of the iterative orbit improvement could reject some quite good points. Therefore, we started iterations with αcrit = 0.00001 and then gradually increased this value to 0.01 during the next iterations. 2.2. Searching for asteroids suitable for mass determination A good selection of the suitable test asteroids is an important step in the mass determination process. Many different methods were used for this purpose in the past. Usually, the scattering angle between the path of an asteroid before and after the close encounter with a more massive body was used as a selection parameter (see e.g. Hilton et al. 1996). In this work, the suitable asteroids were found in a different way. The orbit of each numbered asteroid up to number 4500 was integrated backwards with and without the massive asteroid for the whole time interval covered by the observations. The outcome of this procedure was the list of dates of the closest encounters of the massive asteroid with 4500 test asteroids, as well as the maximum difference in right ascension and declination between the perturbed and unperturbed orbit of the test asteroid. If the difference was large (typically, larger than 100 in right ascension) and the available observations covered long enough time before and after the encounter, the test asteroid was selected as a good candidate for mass determination. As will be shown below, all test asteroids used by previous investigators (Table 1) were found

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by our searching procedure. This confirms the reliability of the selection process. 2.3. Reliability of the mass determination The mass of an asteroid, calculated from the perturbations exerted on a given test asteroid can be biased because of: (i) systematic errors in the observations of the test asteroid, (ii) small number, inhomogeneous distribution or short time span covered by the observations, (iii) incompleteness of the dynamical model, (iv) the method of selection and weighting of observations. If there are several determinations of the mass of a given massive asteroid, the final value of its mass can be calculated as a weighted mean of the independent determinations. When the number of determinations is large, the effects caused by factors (i)–(iv) can be assumed to cancel out and not to affect seriously the final mass. The problem which arises here is the following: how large is the actual uncertainty of the final mass? Let us assume that we have determined n independent masses mi ± σi of a given asteroid. If the formal errors, σi , represent the actual uncertainties, the standard deviation of the weighted mean is given by the straightforward formula: σm = s

1 n P

i=1

,

(1)

wi

where m is the weighted mean and wi = 1/σi2 are the weights. However, the errors σi , and in consequence m, are very often underestimeted (see Schubart 1974, Goffin 1991, Hilton 1997, Viateau 2000). This is also illustrated below in this paper (see Tables 3–5), where the masses of the three massive asteroids under investigation were calculated twice: with and without seven perturbing asteroids from Table 2 included in the dynamical model. Comparison of the results of this exercise shows that the formal errors are in both cases similar, whereas the values of the derived masses are often quite different. In order to get a more realistic estimate of the mass uncertainty, we can calculate σm from the scatter of mi , using the formal errors σi for weighting only, according to the equation: v uP u n w (m − m)2 u i i u . (2) σm = u i=1 n P t (n − 1) wi i=1

Unlike in Eq. (1), in this equation the weights wi can be scaled with no effect on σm . However, Eq. (2) can be used only if the number of individual determinations, n, is large enough, as in case of Ceres and Vesta in this paper. Otherwise, like for Pallas, Eq. (1) has to be applied.

2.4. Selection of derived masses When we have many individual mass determinations, the weighted mean of the values is assumed to be close to the true value of mass. Some of the individual determinations can give a result for the mass that differs from the mean by more than 3σi (where 3σi is their own formal error). If we assume that: (1) the distribution of the (O − C) residuals is normal and there are no systematic errors in the observations and (2) the dynamical model is appropriate, then the formal error of the individual value of a mass is its actual uncertainty. Consequently, in our work all the results which differed from the mean by more than 3σi were rejected, because assumptions (1) and/or (2) were not satisfied, which means that the mass was significantly influenced by systematic factors. For this reason an iterative 3σ selection was made on the individual mass determinations obtained for (1) Ceres and (4) Vesta, i.e. masses for which d=

|mi − m| >3 σi

(3)

were rejected. When using the complete dynamical model, only one iteration was required. 3. The asteroid masses and densities 3.1. (1) Ceres The largest asteroid in the main asteroid belt, (1) Ceres, causes very strong perturbations on almost all minor planets and thus the knowledge of its mass is of great importance. Fortunately, owing to its large mass and the orbit lying well within the main asteroid belt, its close encounters with other asteroids are relatively frequent. With the help of our searching procedure, described in Sect. 2.2, we have initially selected 86 test asteroids for the determination of the mass of Ceres. Because we do not know a priori which encounters lead to the best (in the sense of the smallest formal error) mass estimation, the masses of Ceres found from individual test asteroids were calculated for all of them and then sorted according to the increasing mass error. Finally, we accepted only those mass estimates, for which the formal error was smaller than 0.5 × 10−10 M (about 10% of the asteroid mass). There were 31 such asteroids, that are listed in Table 3. A comparison of this table with Table 1 shows that many asteroids we found were not used previously for the determination of the mass of Ceres. One of the best new test asteroids, with formal error comparable to that of (203) Pompeja, is (454) Mathesis. This asteroid, as well as (1646) Rosseland and (1847) Stobbe, were indicated to be useful for this task by Hilton et al. (1996). For the best eight new test asteroids from Table 3, Fig. 1 shows the perturbation effect in right ascension caused by Ceres and the mutual distance between Ceres and a given test asteroid. The perturbation effect in right ascension was derived from the difference between perturbed and not perturbed orbit of the test asteroid when integrated backward. After a selection described in Sect. 2.4, 25 individual determinations of the mass of (1) Ceres remained. The weighted mean of these values representing the final mass of Ceres is

6 5 4 3 2 1 0 1960

1980

1920

1940

1960

10

(76) Freja

10 0

6 4 2 0

1980

1910

1930

1950

1970

5

6 5 4 3 2 1 0

(14) Irene

0

1990

1860

1880

1900

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["]

1890

["]

1870

30

30

|∆α cos δ|

(347) Pariana

20 10 0

Mutual distance [AU]

|∆α cos δ|

0

|∆α cos δ|


|∆α cos δ|

20


6 5 4 3 2 1 0

(1642) Hill

["]

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1920

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0

20


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1900

6 5 4 3 2 1 0 1940

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(792) Metcalfia

20 10 0 6 5 4 3 2 1 0

1980

1910

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["]

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["]

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|∆α cos δ|

(850) Altona

20 10 0

(1847) Stobbe

15 10 5 0


|∆α cos δ|

40

(454) Mathesis

40



|∆α cos δ|

60


367

["]

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6 5 4 3 2 1 0 1920

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6 5 4 3 2 1 0 1900

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Fig. 1. Perturbations in right ascension caused by (1) Ceres on the eight selected test asteroids from Table 3 (upper panel of each plot), and the mutual distance between a given test asteroid and (1) Ceres (lower panel of each plot). Solid circles correspond to the dates of observations.

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Table 3. Results of the mass determinations for (1) Ceres from perturbations on individual test asteroids. Mp is the largest perturbation effect in right ascension multiplied by cos δ. In the column denoted by d we give the difference (in standard deviations of each solution) between the weighted mean of all solutions and each solution. Solutions rejected after iterative 3σ selection are marked with ‘!’. Masses given in the last column were obtained when neglecting asteroidal perturbations other than from Ceres. An asterisk after the name of a test asteroid means that the details of perturbations are shown in Fig. 1.

Test asteroid

Date of the closest approach

Min. dist. [AU]

Mp [00 ]

Time interval covered

(91) Aegina (348) May (203) Pompeja (4) Vesta (454) Mathesis * (2) Pallas (1642) Hill * (324) Bamberga (76) Freia * (534) Nassovia (14) Irene * (347) Pariana * (792) Metcalfia * (850) Altona * (16) Psyche (1847) Stobbe * (1646) Rosseland (32) Pomona (908) Buda (621) Werdandi (2572) Annschnell (741) Botolphia (27) Euterpe (63) Ausonia (811) Nauheima (18) Melpomene (548) Kressida (10) Hygiea (488) Kreusa (3344) Modena (563) Suleika

1973.09.13 1984.09.02 1948.08.22 1893.03.08 1971.11.23 1825.05.16 1925.11.25 1944.04.01 1957.08.05 1975.12.24 1961.03.29 1943.05.29 1950.07.25 1970.02.22 1975.11.12 1958.09.07 1954.07.12 1975.11.25 1959.11.13 1962.05.01 1971.03.26 1940.11.07 1863.01.17 1990.02.01 1968.10.09 1903.11.16 1982.07.13 1950.07.26 1963.07.17 1980.09.27 1964.11.14

0.033 0.042 0.016 0.186 0.022 0.188 0.012 0.020 0.062 0.023 0.092 0.078 0.014 0.026 0.198 0.079 0.049 0.025 0.072 0.050 0.012 0.101 0.203 0.194 0.031 0.142 0.049 0.184 0.252 0.021 0.116

62 104 83 6 54 26 39 11 25 59 10 30 27 32 10 17 10 29 18 23 22 15 9 2 16 9 17 4 17 20 10

1866–1997 1892–1998 1879–1998 1827–1998 1900–1998 1825–1997 1908–1997 1892–1998 1864–1998 1904–1998 1851–1997 1892–1997 1915–1998 1917–1996 1852–1997 1902–1998 1937–1998 1864–1997 1918–1998 1906–1998 1950–1998 1909–1998 1853–1997 1861–1998 1906–1998 1856–1997 1904–1998 1849–1997 1901–1998 1955–1999 1905–1998

Number of obs.: available, accepted, % of accepted

[10−10 M ]

d [σi ]

Mass (no perturb.) [10−10 M ]

67% 79% 79% 90% 76% 91% 90% 87% 75% 74% 88% 81% 62% 77% 85% 84% 90% 75% 83% 81% 99% 73% 78% 81% 84% 91% 74% 91% 76% 99% 70%

4.74 ± 0.06 4.98 ± 0.06 4.69 ± 0.08 4.25 ± 0.09 4.75 ± 0.10 3.96 ± 0.10 4.79 ± 0.14 4.68 ± 0.17 4.29 ± 0.17 4.90 ± 0.19 4.42 ± 0.19 5.21 ± 0.22 5.72 ± 0.24 4.91 ± 0.28 5.18 ± 0.29 4.44 ± 0.30 4.35 ± 0.31 4.78 ± 0.34 4.56 ± 0.34 4.12 ± 0.34 4.52 ± 0.35 5.81 ± 0.35 2.77 ± 0.36 5.26 ± 0.36 4.72 ± 0.37 4.71 ± 0.37 4.52 ± 0.39 4.66 ± 0.41 3.65 ± 0.47 3.61 ± 0.48 4.53 ± 0.50

1.2 5.0! 0.3 4.7! 0.9 7.1! 0.9 0.1 2.2 1.3 1.3 2.4 4.3! 0.9 1.8 0.7 1.0 0.3 0.3 1.6 0.4 3.3! 5.3! 1.7 0.1 0.1 0.4 0.0 2.1 2.2 0.3

4.76 ± 0.06 4.94 ± 0.06! 4.84 ± 0.08 4.32 ± 0.08! 4.68 ± 0.10 4.17 ± 0.07! 4.89 ± 0.13 4.63 ± 0.17 4.50 ± 0.17 5.07 ± 0.18 0.15 ± 0.19! 5.23 ± 0.22 4.91 ± 0.14 4.53 ± 0.27 4.78 ± 0.29 4.17 ± 0.29 3.98 ± 0.35 5.14 ± 0.38 4.67 ± 0.33 4.26 ± 0.28 4.56 ± 0.35 5.57 ± 0.35 4.79 ± 0.35 5.15 ± 0.36 4.68 ± 0.36 3.34 ± 0.40! 4.41 ± 0.55 5.03 ± 0.41 3.77 ± 0.46 3.49 ± 0.48 4.43 ± 0.57

Weighted mean of all solutions: Weighted mean without the values marked with ‘!’:

4.66 ± 0.07 4.70 ± 0.04

880 468 570 13184 660 13708 266 1714 1112 452 2798 458 306 320 3110 246 178 818 278 352 164 316 2772 1782 438 6924 350 3912 518 174 402

(4.70 ± 0.04) × 10−10 M . For comparison, the formal error of the mean defined by Eq. (1) equals in this case 0.03 × 10−10 M . From the last column of Table 3 we also see that even if no perturbations from minor planets other than Ceres are taken into account in the dynamical model, the average mass of Ceres is not significantly affected. Some individual masses are, however, quite different, such as that when (14) Irene is the test asteroid. Note that in this case the formal error is the same with and without perturbations from the massive asteroids included. The mass we obtained for Ceres agrees with most of the recent determinations (see Table 1). Since the mass of this asteroid was determined most frequently, some of the individual determinations need to be commented upon. First, the individual

592 371 448 11861 500 12500 239 1489 829 334 2470 372 189 247 2631 206 161 616 230 285 163 232 2176 1440 366 6301 258 3556 396 172 280

Mass

4.59 ± 0.12 4.76 ± 0.05

masses of Ceres found by Viateau & Rapaport (1998) agree with ours, except for those from Pallas and Vesta, for which we obtained values significantly smaller than the mean. The smaller masses of Ceres from Pallas and Vesta were also derived by Hilton (1999). The maximum perturbation on Vesta caused by Ceres amounts to only 600 (see Table 3) and it is possible that some systematic errors in Vesta’s observations and/or unmodeled perturbations can influence the derived mass of Ceres. We found that asteroids: (29) Amphitrite, (15) Eunomia and (532) Herculina can be responsible for some of the unmodeled perturbations on Vesta. On the other hand, the underestimated mass of Ceres from perturbations on Pallas can probably be explained by obser-


vational errors. In our solution we used observations of Pallas covering the period 1825–1997, while some other authors used also earlier observations, starting from 1802. Because a series of close approaches between Ceres and Pallas occurred in the years 1800–1830, the oldest observations of Pallas were the most sensitive to the perturbations by Ceres. Because we found no significant perturbers of Pallas other than those used in our dynamical model, the most probable explanation for the low mass of Ceres obtained from Pallas are the systematic errors in the observations of the latter asteroid for the earliest epochs in the data we used. Surprisingly, the mass of Ceres obtained from (348) May, an asteroid very often used for this purpose, was not used in the calculation of the mean mass of Ceres because it was rejected as too high during the final selection. The search for possible perturbers of May among the 2256 asteroids (all having absolute magnitudes smaller than 10 mag supplemented by those with available diameters) gave no result. We therefore checked whether systematic errors in the observations of May are responsible for the discrepancy. First, the mass of Ceres was calculated without the oldest observations of May made between 1892 and 1942, yielding a value of (5.02 ± 0.08) × 10−10 M , not very different from the value in Table 3. Next, the observations made in 1942–1983 were rejected, while those made in 1892–1942 were retained. The resulting mass of Ceres dropped to (4.65 ± 0.05) × 10−10 M in very good agreement with our mean value. This means that some observations made in 1942–1983 can be responsible for the discrepant mass of Ceres obtained from May. In most cases, all solutions from Table 3 which differ distinctly from the weighted mean (in comparison with their formal errors) can be explained either by perturbations by asteroids not taken into account or by systematic errors in the observations. Assuming the shape of (1) Ceres to be that of an oblate spheroid with equatorial radius of 479.6 ± 2.4 km and polar radius of 453.4 ± 4.5 km (Millis et al. 1987), we found its mean density to be 2.14 ± 0.04 g cm−3 . 3.2. (4) Vesta As for (1) Ceres, we found many useful test asteroids for the determination of the mass of (4) Vesta. The result of our search gave over 70 asteroids which could be potentially valuable for this task. After examination, we left 31 of them with formal errors of the derived mass not greater than 0.5 × 10−10 M . The results are listed in Table 4. Similarly to (1) Ceres, the results which have not passed the iterative 3σ selection, are labeled with ‘!’. The final value of the mass of Vesta, determined as a weighted mean of the results for the remaining 26 test asteroids, is equal to (1.36 ± 0.05) × 10−10 M . For this mass, the formal error of the mean obtained with Eq. (1) is 0.03 × 10−10 M . The mass of Vesta derived agrees with all determinations made so far except that of Hilton (1999). It can also be noticed that, like for (1) Ceres, the mass derived from the perturbations on (27) Euterpe, appears to be systematically smaller than the

369

mean. The plots with the eight best previously not used test asteroids from Table 4 are given in Fig. 2. Assuming a triaxial ellipsoid shape of Vesta with radii 280 ± 12 km, 272 ± 12 km and 227 ± 12 km determined from HST images (Thomas et al. 1997), we obtain the mean density of the asteroid to be equal to 3.7 ± 0.3 g cm−3 . 3.3. (2) Pallas Because of the highly inclined and eccentric orbit of this asteroid, close encounters of (2) Pallas with other asteroids are rare. Consequently, the mass of this third most massive asteroid is so far much less reliably determined than those of Ceres and Vesta. Our search, (see Sect. 2.2), however, yielded a few additional suitable candidates, never used before for the determination of the mass of Pallas. In Table 5 we present the solutions for the mass of Pallas with the formal errors not greater than approximately half the mass of this minor planet. Unfortunately, because of the small number of test asteroids found, it is not possible to make the same kind of selection of the results as for Ceres and Vesta. Relying on the results of other authors who found the mass of Pallas to be in the range (1.1–1.6) × 10−10 M , we accept only two solutions: (582) Olympia and (9) Metis. The weighted mean of the two results amounts to (1.21 ± 0.26) × 10−10 M . In this case, the error was calculated from Eq. (1). There are six other test asteroids for which we obtained relatively small formal errors, but the mass of Pallas appeared to be either too large or too small. The second part of Table 5 contains information on these test asteroids which could be useful in the future for the derivation of the mass of Pallas, if supplemented by new observations and/or additional perturbing asteroids in the dynamical model. The effects of the perturbations from Pallas on the test asteroids are presented in Fig. 3. We point out here the unusual case of (9) Metis for which the closest approaches with Pallas took place at a distance of about 1 AU. The large sensitivity of this asteroid to the perturbations from Pallas is due to the resonance, because the quite distant approaches between these two asteroids happen every five revolutions of (9) Metis around the Sun, approximately in the same place in space. In spite of the small formal error, the solution obtained from Ceres as the test asteroid cannot be accepted because the resulting mass is too large. The best encounters between Ceres and Pallas took place in the 19th century, the perturbation effect was small, and therefore the large value found for the mass can probably be explained by the errors of the old observations. Moreover, some additional perturbations can be important in this case. We have found that at least five additional asteroids can influence the orbit of Ceres. These are: (16) Psyche, (121) Hermione, (423) Diotima, (76) Freia and (532) Herculina. The mass of Pallas obtained from perturbations on (27) Euterpe appears to be unusually low. As we mentioned above, this asteroid also gives underestimated values of the mass of Ceres (Table 3) and Vesta (Table 4). This could mean that some large systematic errors in the observations of this asteroid exist or it is additionally perturbed by another body(ies). A very interest-

["]


["]

370

10

4 2

|∆α cos δ|

(14) Irene

6

5


|∆α cos δ|

8

6 5 4 3 2 1 0


0 6 5 4 3 2 1 0 1900

1920

1940

1960

1980

1850

10 5



0 6 5 4 3 2 1 0 1910

1930

1950

1970

1910

1930

1950

1970

1990

(77) Frigga

10 5 0 6 5 4 3 2 1 0

1990

1880

5 4 3 2 1 0

1900

1920

1940

20

|∆α cos δ|

(27) Euterpe

1960

1980

(206) Hersilia

15 10 5 0


|∆α cos δ| Mutual distance [AU]

1890

["]

1890

["]

1870

6 5 4 3 2 1 0 1890

1910

1930

1950

1970

6 5 4 3 2 1 0

1990

1880

1900

1920

1940

1960

1980

["]

1870

["]

1850 8

40

(346) Hermentaria

6

|∆α cos δ|

|∆α cos δ|

1870

15

(109) Felicitas

15

|∆α cos δ|

|∆α cos δ|

20

4 2

(67) Asia

30 20 10 0


0


0

["]

1880

["]

1860

(8) Flora

6 5 4 3 2 1 0 1890

1910

1930 1950 Year

1970

1990

6 5 4 3 2 1 0 1870

Fig. 2. Same as in Fig. 1 but for (4) Vesta, and the selected test asteroids from Table 4.

1890

1910

1930 Year

1950

1970

1990

10

1.0

|∆α cos δ|

0

|∆α cos δ|

0.5


6 5 4 3 2 1 0


(582) Olympia

5

6 5 4 3 2 1 0

1950

1970

1990

1860

|∆α cos δ|

2.5 2.0 1.5 1.0 0.5 0.0 6 5 4 3 2 1 0

(1) Ceres

3 2 1 0


0.0


|∆α cos δ|

4

6 5 4 3 2 1 0

1900

1920

1940

1960

1980

1870

1890

1910

1930

1950

1970

1990

["] |∆α cos δ|

|∆α cos δ|

4

1.0 0.8 0.6 0.4 0.2 0.0 6 5 4 3 2 1 0

(1829) Dawson

3 2 1 0


1880

(27) Euterpe

1850


["]

1830 1850 1870 1890 1910 1930 1950 1970 1990

6 5 4 3 2 1 0 1970

1990

(354) Eleonora

1890

1910

1930

1950

1970

1990

["]

1950

["]

1930 1.0 0.8 0.6 0.4 0.2 0.0

2.0

|∆α cos δ|

(14) Irene

(40) Harmonia

1.5 1.0 0.5 0.0


|∆α cos δ|

(9) Metis

["]

1930

["]

1910


371

["]

["]


6 5 4 3 2 1 0 1850

1870

1890

1910 1930 Year

1950

1970

1990

Fig. 3. Same as in Fig. 1 but for (2) Pallas, and the test asteroids from Table 5.

6 5 4 3 2 1 0 1860

1880

1900

1920 1940 Year

1960

1980

372


Table 4. Same as in Table 3 but for (4) Vesta. Plots in Fig. 2 refer to the test asteroids with asterisk after their names.

Test asteroid


Min. dist. [AU]

Mp [00 ]


(197) Arete (14) Irene * (8) Flora * (109) Felicitas * (486) Cremona (77) Frigga * (27) Euterpe * (206) Hersilia * (346) Hermentaria * (67) Asia * (56) Melete (163) Erigone (1667) Pels (564) Dudu (1793) Zoya (66) Maja (460) Scania (935) Clivia (3231) Mila (1707) Chantal (854) Frostia (1393) Sofala (782) Montefiore (1063) Aquilegia (739) Mandeville (2778) Tangshan (498) Tokio (905) Universitas (855) Newcombia (1055) Tynka (428) Monachia

1885.05.14 1963.06.29 1963.03.10 1959.04.15 1983.12.29 1955.06.07 1968.05.10 1889.08.03 1925.10.09 1991.01.20 1923.11.14 1934.04.24 1981.01.02 1960.02.12 1982.03.18 1933.09.02 1973.01.04 1989.10.04 1966.04.19 1975.04.06 1958.03.21 1981.03.17 1956.09.17 1935.10.15 1962.12.26 1978.10.22 1954.11.05 1995.08.14 1963.02.27 1963.09.02 1926.08.19

0.018 0.081 0.226 0.019 0.038 0.025 0.203 0.102 0.059 0.031 0.112 0.196 0.128 0.019 0.028 0.026 0.064 0.024 0.009 0.078 0.114 0.081 0.115 0.038 0.037 0.044 0.060 0.044 0.229 0.063 0.041

74 7 12 20 6 13 5 18 8 40 13 10 10 18 10 12 6 16 7 7 16 12 5 5 2 5 4 19 12 8 4

1879–1997 1851–1997 1847–1996 1869–1997 1902–1998 1879–1997 1853–1997 1879–1997 1892–1997 1864–1997 1865–1998 1892–1997 1930–1998 1905–1998 1932–1998 1861–1997 1900–1998 1920–1998 1949–1998 1906–1998 1931–1997 1928–1996 1911–1998 1902–1998 1915–1998 1948–1998 1900–1998 1918–1997 1916–1997 1902–1998 1898–1998

Number of obs.: available, accepted, % of accepted

[10−10 M ]

d [σi ]


78% 88% 87% 77% 90% 83% 78% 68% 76% 76% 83% 77% 85% 81% 97% 82% 74% 82% 97% 87% 87% 96% 68% 71% 82% 97% 77% 80% 90% 72% 88%

1.26 ± 0.04 1.52 ± 0.07 1.69 ± 0.08 1.66 ± 0.11 1.50 ± 0.13 1.28 ± 0.15 0.60 ± 0.15 1.44 ± 0.16 0.88 ± 0.17 1.39 ± 0.19 1.18 ± 0.20 0.78 ± 0.27 1.51 ± 0.30 0.90 ± 0.30 0.40 ± 0.32 1.82 ± 0.33 0.98 ± 0.33 2.17 ± 0.34 1.05 ± 0.36 2.32 ± 0.36 0.40 ± 0.37 0.37 ± 0.42 2.00 ± 0.43 2.94 ± 0.44 2.12 ± 0.45 1.40 ± 0.45 1.43 ± 0.46 1.99 ± 0.47 1.31 ± 0.49 2.02 ± 0.50 3.51 ± 0.50

2.7 2.3 3.8! 2.5 1.0 0.7 5.1! 0.4 2.8 0.1 1.0 2.2 0.4 1.6 3.1! 1.3 1.2 2.3 0.9 2.7 2.6 2.4 1.4 3.5! 1.7 0.1 0.1 1.3 0.1 1.3 4.3!

1.34 ± 0.04! 0.03 ± 0.06! 0.73 ± 0.09! 1.56 ± 0.11! 2.04 ± 0.10! 1.59 ± 0.16 2.64 ± 0.14! 1.36 ± 0.23 0.99 ± 0.19 1.55 ± 0.20 1.07 ± 0.18 −0.79 ± 0.34! 1.99 ± 0.30 0.98 ± 0.26 0.37 ± 0.32 2.47 ± 0.33! 0.82 ± 0.33 1.93 ± 0.34 1.14 ± 0.36 2.73 ± 0.37 0.24 ± 0.35! 0.00 ± 0.42! 1.74 ± 0.49 3.07 ± 0.43! 1.50 ± 0.44 1.59 ± 0.45 0.99 ± 0.52 1.82 ± 0.43 0.41 ± 0.57 1.95 ± 0.50 4.17 ± 0.57!

Weighted mean of all solutions: Weighted mean without solutions marked with ‘!’:

1.38 ± 0.06 1.36 ± 0.05

738 2798 3050 468 262 1190 2772 630 536 646 728 530 248 124 400 860 370 290 114 254 198 192 232 232 324 140 488 240 132 236 252

ing case is (1829) Dawson. This asteroid had a close encounter with Pallas in August 1949. We had only 5 pairs of observations (α,δ) with earlier dates in our database. After selection, only half of them were accepted. In spite of this small number of observations, the formal error is quite small. If additional old observations of this asteroid are found, then this could result in a much more reliable estimate of the mass of Pallas. Asteroid (14) Irene has over 200 observations before encounter with Pallas in 1911, but probably is perturbed by other bodies. The comparison of the masses of Pallas, calculated with and without the massive asteroids included in the dynamical model shows that all masses, except for (582) Olympia, are affected. In other words, Pallas is the most important asteroidal perturber for Olympia. In order to make sure that this is indeed the case, we searched the 2256 asteroids as possible perturbers for Olympia. We have found that only (1) Ceres has a relatively large perturbing effect on this asteroid (1.400 in right ascension).

579 2474 2663 361 235 990 2163 426 408 492 602 410 212 100 389 705 274 237 111 220 173 184 158 165 266 136 374 193 119 171 223

Mass

1.15 ± 0.13 1.31 ± 0.10

Because Ceres is included in the dynamical model, the mass of Pallas obtained from Olympia seems to be reliable provided that there are no systematic effects in the observations. Assuming the IRAS diameter of (2) Pallas (Tedesco et al. 1989) to be 523 ± 20 km, the mean density of the asteroid amounts to 3.2 ± 0.8 g cm−3 . 4. Summary and conclusions Using the searching procedure for finding asteroids useful for the determination of asteroid masses from mutual perturbations (described in Sect. 2.2), we found a large number of asteroids whose orbits had been strongly perturbed by (1) Ceres and (4) Vesta, and several perturbed by (2) Pallas. We then calculated the masses of Ceres, Pallas, and Vesta independently for all test asteroids using the least squares method and a specially implemented normal selection of the observations (Sect. 2.1).


373

Table 5. Same as in Table 3 but for (2) Pallas. The column with the values of d, as irrelevant, was omitted.

Test asteroid


Min. dist. [AU]

Mp [00 ]


Number of observations: available, accepted, % of accepted

(582) Olympia (9) Metis

1936.07.14 1991.06.04

0.033 1.037

10 1

1906–1996 1849–1997

(1) Ceres (27) Euterpe (1829) Dawson (354) Eleonora (14) Irene (40) Harmonia

Additional potentially useful test asteroids: 1830.01.02 0.202 4 1830–1997 12216 1926.09.21 0.061 2 1853–1997 2772 1949.09.07 0.011 4 1929–1998 200 1950.01.13 0.109 1 1893–1998 2270 1911.04.22 0.142 1 1851–1997 2798 1917.07.15 0.056 2 1860–1998 6164

Mass [10−10 M ]


85% 88%

1.22 ± 0.27 1.10 ± 0.76

1.19 ± 0.22 −3.18 ± 0.81

Weighted mean of the above 2 solutions:

1.21 ± 0.26

0.89 ± 0.21

1.75 ± 0.09 0.05 ± 0.18 2.81 ± 0.26 2.22 ± 0.33 0.22 ± 0.49 −0.40 ± 0.75

1.05 ± 0.09 0.93 ± 0.27 2.05 ± 0.41 −0.46 ± 0.34 −0.40 ± 0.47 −2.43 ± 0.74

556 2708

Many of these perturbed test asteroids were never used before for this purpose; they gave quite good estimates of the mass of the massive minor planet. After examination, we accepted only those solutions which gave the smallest formal errors. For Ceres and Vesta we made an additional selection of results rejecting those that were found significantly influenced by systematic factors. The resulting mass of (1) Ceres, determined from the perturbations on 25 asteroids, is equal to (4.70 ± 0.04) ×10−10 M , that of (4) Vesta found with the use of 26 perturbed asteroids, is equal to (1.36 ± 0.05) ×10−10 M . The mass of (2) Pallas is calculated from the perturbations on only two asteroids. In practice, the final mass of (2) Pallas, (1.21 ± 0.26) ×10−10 M , is determined by the value obtained from the close encounter with (582) Olympia in 1936. The mean densities of the asteroid we derived using their published radii indicate that Ceres (density of 2.14 ± 0.04 g cm−3 ) is less dense than Vesta and Pallas (3.7 ± 0.3, and 3.2 ± 0.8 g cm−3 , respectively). Generally, the masses we found agree with the best recent results of other authors and indicate that the mass of (1) Ceres appears to be smaller by about 5% than the value of 5.0 × 10−10 M , recommended by the IAU. We also point out that the masses of (1) Ceres and (2) Pallas, determined from their mutual perturbations, differ significantly from the mean values. We indicate the possible explanation for these discrepancies. It is obvious that refining the dynamical model will improve the accuracy of the mass determination of massive asteroids and explain at least some of the discrepancies between the mean and individual masses indicated above. In the next paper of the series we are going to determine masses of other massive asteroids, mainly those which are included in our dynamical model (Table 2). Acknowledgements. I would like to thank Dr. A. Pigulski for discussions and help in the creating the paper and Prof. M. Jerzykiewicz for critical reading the manuscript. I also thank an anonymous referee for her/his helpful remarks and suggestions. This work was supported by the Wrocláw University grant No. 2262/W/IA/99.

474 2374

10906 2187 174 1960 2473 5678

89% 79% 87% 86% 88% 92%

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