ASTRONOMY AND ASTROPHYSICS Determination of asteroid masses

Transcription

1 Astron. Astrophys., 7 () ASTRONOMY AND ASTROPHYSICS Determination of asteroid masses I. () Ceres, () Pallas and () Vesta G. Michalak Wrocĺaw University Observatory, Kopernika, - Wrocĺaw, Poland (michalak@astro.uni.wroc.pl) Received November 999 / Accepted April Abstract. The masses of the three largest asteroids: () Ceres, () Pallas and () Vesta were determined from gravitational perturbations exerted on respectively,, and 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: (.7 ±.) M, (. ±.) M, and (. ±.) M for () Ceres, () Pallas and () 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 () Ceres and () 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: Ceres, Pallas, Vesta 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 DE ephemerides (Standish et al. 99), for example, perturbations from asteroids were taken into account. Up to now, the masses of only 7 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. It is clearly seen from this table that only five asteroids, viz. () Ceres, () Pallas, () Vesta, () Parthenope and () Mathilde, have masses determined with formal errors smaller than %. Moreover, it happens frequently 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 : the difference between two recent values of the mass of () Ceres (Viateau & Rapaport 998; Hilton 999) is about and 9 times greater than the errors given in these papers. Possible causes of such discrepancies are discussed in detail in Sect... Most mass determinations shown in Table 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 of them for the determination of the mass of () Ceres and () Vesta. The situation is much less favourable when there is only one or a few such asteroids, like for () 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 () Ceres, () Pallas and () 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.. Introduction. Method 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 (99) for () Ceres, Sitarski (99) for () Ceres and () Vesta, and Hilton (999) for () Ceres, () Pallas and () 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

2 G. Michalak: Determination of asteroid masses. I Table. Current status of asteroid mass determination. Mass σ Perturbed Mass σ Perturbed [in M ] bodies Reference [in M ] bodies Reference () Ceres: () Hygiea:.7. () Pallas ().7. (89) Academia (9). () Vesta ().. (89) Academia (9).9. () Pallas ().. (9) Ógyalla ().99.9 () Pallas (). () Pallas () () Parthenope:..7 () Pallas ().8. (7) Thetis ().. Mars (7).9. () Pallas (8) () Eunomia:.7. () Pompeja (9).. () Berna (). (8) May ().79.8 () & (8) () () Psyche:.8. (8) May ().87. (9) Aurora ().8. ()(9)()()(8)() ().9.7 (9)()(8)() () () Massalia:.. () Pallas ().. () Nysa (). DE solution ().78. () & () (7) () Themis:..7 () & (8) (8).89. (9) Kugultinov ().7.9 (9)()()(8)() (9). () () () Eugenia:.7. ()(9)()()(8)()(7) (). satellite S/998() ().. () Ausonia ().7 DE solution () () Hestia:.79. ()()()()(9)()()(8)() ().9.8 (9) Fortuna ().9. () & () () () Hermione: () Pallas:.7.8 (78) Paulina ().. () Ceres ().. () Ceres () () Ida: [mass in M ].. Mars (7).. S/99() Dactyl (). DE solution (). DE solution () () Mathilde: [mass in M ].9. () & () ().9. NEAR tracking data (7) () Vesta: () Eros: [mass in M ]. (97) Arete (7)..9 NEAR tracking data (8).8. (97) Arete (8).. Mars (7) (7) Interamnia:. (97) Arete (9).7.7 (99) Moultona (9). DE solution ().9. (97) & (8) (8) (8) Hispania:. DE solution ().. () Olbersia (9).9. () & () () References to Table : () Schubart (97), () Schubart (97), () Schubart (97), () Landgraf (98), () Goffin (98), () Landgraf (988), (7) Standish & Hellings (989), (8) Schubart (99), (9) Goffin (99), () Williams (99), () Sitarski & Todorovic-Juchniewicz (99), () Williams (99), () Bowell et al. (99), () Muinonen et al. (99), () Viateau & Rapaport (99), () Standish et al. (99), (7) Viateau (99), (8) Sitarski (99), (9) Carpino & Knežević (99a), () Kuzmanoski (99), () Carpino & Knežević (99b), () Bange & Bec-Borsenberger (997), () Standish (998), () Viateau & Rapaport (998), () Hilton (999), () Schubart (97), (7) Hertz (9), (8) Schubart & Matson (979), (9) Scholl et al. (987), () Garcia et al. (99), () Viateau & Rapaport (997), () Hilton (997), () Viateau (), () Bange (998), () Merline et al. (999), () Belton et al. (99), (7) Yeomans et al. (997), (8) Yeomans et al. (999), (9) Landgraf (99). 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-

3 G. Michalak: Determination of asteroid masses. I Table. The asteroids used in our dynamical model. If the adopted mass was not the same as that used by Viateau & Rapaport (998), a reference is given. Asteroid Mass [in M ] () Ceres.8 (preliminary solution from this work) () Pallas. () Vesta.9 (Sitarski 99) () Hygiea.7 () Parthenope. () Europa. () Davida.8 (7) Interamnia. fully selected (see Sect..) 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 (998). Their names and adopted masses are listed in Table. 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 9). Initial conditions and masses for all planets were taken from the DE planetary ephemerides (Standish 998). The initial conditions for the minor planets were derived from the osculating elements given in the Ephemerides of Minor Planets for 998 (Batrakov 997). 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... 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 =,,...), 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 χ test. Therefore, we implemented χ test as the selection procedure. To begin with, for each asteroid, the observations were divided into several 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 χ 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>α crit, where α crit determines the selection criterion used in this procedure. Greater α crit means stronger selection. For all cases α crit =. 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 +)-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 α crit =. and then gradually increased this value to. during the next iterations... 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. 99). In this work, the suitable asteroids were found in a different way. The orbit of each numbered asteroid up to number 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 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 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 ) were found

4 G. Michalak: Determination of asteroid masses. I by our searching procedure. This confirms the reliability of the selection process... 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) (ii) systematic errors in the observations of the test asteroid, 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 m i ± σ i ofagiven asteroid. If the formal errors, σ i, represent the actual uncertainties, the standard deviation of the weighted mean is given by the straightforward formula: σ m =, () n w i i= where m is the weighted mean and w i =/σi are the weights. However, the errors σ i, and in consequence m, are very often underestimeted (see Schubart 97, Goffin 99, Hilton 997, Viateau ). This is also illustrated below in this paper (see Tables ), where the masses of the three massive asteroids under investigation were calculated twice: with and without seven perturbing asteroids from Table 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 σ m from the scatter of m i, using the formal errors σ i for weighting only, according to the equation: n w i (m i m) i= σ m = (n ) n. () w i i= Unlike in Eq. (), in this equation the weights w i can be scaled with no effect on σ m. However, Eq. () 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. () has to be applied... 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 σ i (where σ i is their own formal error). If we assume that: () the distribution of the (O C) residuals is normal and there are no systematic errors in the observations and () 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 σ i were rejected, because assumptions () and/or () were not satisfied, which means that the mass was significantly influenced by systematic factors. For this reason an iterative σ selection was made on the individual mass determinations obtained for () Ceres and () Vesta, i.e. masses for which d = m i m > () σ i were rejected. When using the complete dynamical model, only one iteration was required.. The asteroid masses and densities.. () Ceres The largest asteroid in the main asteroid belt, () 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.., we have initially selected 8 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. M (about % of the asteroid mass). There were such asteroids, that are listed in Table. A comparison of this table with Table 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 () Pompeja, is () Mathesis. This asteroid, as well as () Rosseland and (87) Stobbe, were indicated to be useful for this task by Hilton et al. (99). For the best eight new test asteroids from Table, Fig. 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.., individual determinations of the mass of () Ceres remained. The weighted mean of these values representing the final mass of Ceres is

5 G. Michalak: Determination of asteroid masses. I 7 () Mathesis () Hill (7) Freja () Irene (7) Pariana (79) Metcalfia (8) Altona (87) Stobbe Year Year Fig.. Perturbations in right ascension caused by () Ceres on the eight selected test asteroids from Table (upper panel of each plot), and the mutual distance between a given test asteroid and () Ceres (lower panel of each plot). Solid circles correspond to the dates of observations.

6 8 G. Michalak: Determination of asteroid masses. I Table. Results of the mass determinations for () Ceres from perturbations on individual test asteroids. M p 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 σ 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.. Date of Min. Time Number of obs.: Mass Mass Test the closest dist. M p interval available, accepted, d (no perturb.) asteroid approach [AU] [ ] covered % of accepted [ M ] [σ i] [ M ] (9) Aegina %.7 ±...7 ±. (8) May %.98 ±..!.9 ±.! () Pompeja %.9 ±.8..8 ±.8 () Vesta %. ±.9.7!. ±.8! () Mathesis * %.7 ±..9.8 ±. () Pallas %.9 ±. 7.!.7 ±.7! () Hill * %.79 ± ±. () Bamberga %.8 ±.7.. ±.7 (7) Freia * %.9 ±.7.. ±.7 () Nassovia %.9 ±.9..7 ±.8 () Irene * %. ±.9.. ±.9! (7) Pariana * %. ±... ±. (79) Metcalfia * %.7 ±..!.9 ±. (8) Altona * %.9 ±.8.9. ±.7 () Psyche %.8 ± ±.9 (87) Stobbe * %. ±..7.7 ±.9 () Rosseland %. ±...98 ±. () Pomona %.78 ±... ±.8 (98) Buda %. ±...7 ±. () Werdandi %. ±... ±.8 (7) Annschnell %. ±... ±. (7) Botolphia %.8 ±..!.7 ±. (7) Euterpe %.77 ±..!.79 ±. () Ausonia %. ±..7. ±. (8) Nauheima %.7 ±.7..8 ±. (8) Melpomene %.7 ±.7.. ±.! (8) Kressida %. ±.9.. ±. () Hygiea %. ±... ±. (88) Kreusa %. ± ±. () Modena %. ±.8..9 ±.8 () Suleika %. ±... ±.7 Weighted mean of all solutions:. ±.7.9 ±. Weighted mean without the values marked with! :.7 ±..7 ±. (.7 ±.) M. For comparison, the formal error of the mean defined by Eq. () equals in this case. M. From the last column of Table 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 () 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 ). Since the mass of this asteroid was determined most frequently, some of the individual determinations need to be commented upon. First, the individual masses of Ceres found by Viateau & Rapaport (998) 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 (999). The maximum perturbation on Vesta caused by Ceres amounts to only (see Table ) 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: (9) Amphitrite, () Eunomia and () 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-

7 G. Michalak: Determination of asteroid masses. I 9 vational errors. In our solution we used observations of Pallas covering the period 8 997, while some other authors used also earlier observations, starting from 8. Because a series of close approaches between Ceres and Pallas occurred in the years 8 8, 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 (8) 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 asteroids (all having absolute magnitudes smaller than 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 89 and 9, yielding a value of (. ±.8) M, not very different from the value in Table. Next, the observations made in 9 98 were rejected, while those made in 89 9 were retained. The resulting mass of Ceres dropped to (. ±.) M in very good agreement with our mean value. This means that some observations made in 9 98 can be responsible for the discrepant mass of Ceres obtained from May. In most cases, all solutions from Table 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 () Ceres to be that of an oblate spheroid with equatorial radius of 79. ±. km and polar radius of. ±. km (Millis et al. 987), we found its mean density to be. ±. g cm... () Vesta As for () Ceres, we found many useful test asteroids for the determination of the mass of () Vesta. The result of our search gave over 7 asteroids which could be potentially valuable for this task. After examination, we left of them with formal errors of the derived mass not greater than. M. The results are listed in Table. Similarly to () Ceres, the results which have not passed the iterative σ selection, are labeled with!. The final value of the mass of Vesta, determined as a weighted mean of the results for the remaining test asteroids, is equal to (. ±.) M. For this mass, the formal error of the mean obtained with Eq. () is. M. The mass of Vesta derived agrees with all determinations made so far except that of Hilton (999). It can also be noticed that, like for () Ceres, the mass derived from the perturbations on (7) Euterpe, appears to be systematically smaller than the mean. The plots with the eight best previously not used test asteroids from Table are given in Fig.. Assuming a triaxial ellipsoid shape of Vesta with radii 8 ± km, 7 ± km and 7 ± km determined from HST images (Thomas et al. 997), we obtain the mean density of the asteroid to be equal to.7 ±.gcm... () Pallas Because of the highly inclined and eccentric orbit of this asteroid, close encounters of () 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..), however, yielded a few additional suitable candidates, never used before for the determination of the mass of Pallas. In Table 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 (..) M, we accept only two solutions: (8) Olympia and (9) Metis. The weighted mean of the two results amounts to (. ±.) M. In this case, the error was calculated from Eq. (). 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 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.. We point out here the unusual case of (9) Metis for which the closest approaches with Pallas took place at a distance of about 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 9th 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: () Psyche, () Hermione, () Diotima, (7) Freia and () Herculina. The mass of Pallas obtained from perturbations on (7) Euterpe appears to be unusually low. As we mentioned above, this asteroid also gives underestimated values of the mass of Ceres (Table ) and Vesta (Table ). 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-

8 7 G. Michalak: Determination of asteroid masses. I 8 () Irene (8) Flora (9) Felicitas (77) Frigga (7) Euterpe () Hersilia () Hermentaria (7) Asia Year Year Fig.. Same as in Fig. but for () Vesta, and the selected test asteroids from Table.

9 G. Michalak: Determination of asteroid masses. I 7 (8) Olympia... (9) Metis () Ceres (7) Euterpe (89) Dawson () Eleonora () Irene..... () Harmonia Year Year Fig.. Same as in Fig. but for () Pallas, and the test asteroids from Table.

10 7 G. Michalak: Determination of asteroid masses. I Table. Same as in Table but for () Vesta. Plots in Fig. refer to the test asteroids with asterisk after their names. Date of Min. Time Number of obs.: Mass Mass Test the closest dist. M p interval available, accepted, d (no perturb.) asteroid approach [AU] [ ] covered % of accepted [ M ] [σ i] [ M ] (97) Arete %. ±..7. ±.! () Irene * %. ±.7.. ±.! (8) Flora * %.9 ±.8.8!.7 ±.9! (9) Felicitas * %. ±... ±.! (8) Cremona %. ±... ±.! (77) Frigga * %.8 ±..7.9 ±. (7) Euterpe * %. ±..!. ±.! () Hersilia * %. ±... ±. () Hermentaria * %.88 ± ±.9 (7) Asia * %.9 ±.9.. ±. () Melete %.8 ±...7 ±.8 () Erigone %.78 ± ±.! (7) Pels %. ±...99 ±. () Dudu %.9 ±...98 ±. (79) Zoya %. ±..!.7 ±. () Maja %.8 ±...7 ±.! () Scania %.98 ±...8 ±. (9) Clivia %.7 ±...9 ±. () Mila %. ±..9. ±. (77) Chantal %. ±..7.7 ±.7 (8) Frostia %. ±.7.. ±.! (9) Sofala %.7 ±... ±.! (78) Montefiore %. ±...7 ±.9 () Aquilegia %.9 ±..!.7 ±.! (79) Mandeville %. ±..7. ±. (778) Tangshan %. ±...9 ±. (98) Tokio %. ±...99 ±. (9) Universitas %.99 ±.7..8 ±. (8) Newcombia %. ±.9.. ±.7 () Tynka %. ±...9 ±. (8) Monachia %. ±..!.7 ±.7! Weighted mean of all solutions:.8 ±.. ±. Weighted mean without solutions marked with! :. ±.. ±. ing case is (89) Dawson. This asteroid had a close encounter with Pallas in August 99. We had only 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 () Irene has over observations before encounter with Pallas in 9, 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 (8) 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 asteroids as possible perturbers for Olympia. We have found that only () Ceres has a relatively large perturbing effect on this asteroid (. in right ascension). 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 () Pallas (Tedesco et al. 989) to be ± km, the mean density of the asteroid amounts to. ±.8 g cm.. Summary and conclusions Using the searching procedure for finding asteroids useful for the determination of asteroid masses from mutual perturbations (described in Sect..), we found a large number of asteroids whose orbits had been strongly perturbed by () Ceres and () Vesta, and several perturbed by () 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..).

11 G. Michalak: Determination of asteroid masses. I 7 Table. Same as in Table but for () Pallas. The column with the values of d, as irrelevant, was omitted. Date of Min. Time Number of observations: Mass Mass Test the closest dist. M p interval available, accepted, (no perturb.) asteroid approach [AU] [ ] covered % of accepted [ M ] [ M ] (8) Olympia %. ±.7.9 ±. (9) Metis %. ±.7.8 ±.8 Weighted mean of the above solutions:. ±..89 ±. Additional potentially useful test asteroids: () Ceres %.7 ±.9. ±.9 (7) Euterpe %. ±.8.9 ±.7 (89) Dawson %.8 ±.. ±. () Eleonora %. ±.. ±. () Irene %. ±.9. ±.7 () Harmonia %. ±.7. ±.7 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 () Ceres, determined from the perturbations on asteroids, is equal to (.7 ±.) M, that of () Vesta found with the use of perturbed asteroids, is equal to (. ±.) M. The mass of () Pallas is calculated from the perturbations on only two asteroids. In practice, the final mass of () Pallas, (. ±.) M, is determined by the value obtained from the close encounter with (8) Olympia in 9. The mean densities of the asteroid we derived using their published radii indicate that Ceres (density of. ±. g cm ) is less dense than Vesta and Pallas (.7 ±., and. ±.8gcm, respectively). Generally, the masses we found agree with the best recent results of other authors and indicate that the mass of () Ceres appears to be smaller by about % than the value of. M, recommended by the IAU. We also point out that the masses of () Ceres and () 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 ). 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 Wrocĺaw University grant No. /W/IA/99. References Bange J., 998, A&A, L Bange J.F, Bec-Borsenberger A., 997, Determination of the masses of minor planets. In: Hipparcos Symposium 997, Venice, Italy, p. 9 Batrakov Y.V., 997, Ephemerides of Minor Planets for 998, Institute of Theoretical Astronomy, St. Petersburg Belton M.J.S., Chapman C.R., Thomas P.C., et al., 99, Nat 7, 78 Bowell E., Muinonen K., Wasserman L.H., 99, Asteroid mass determination from multiple asteroid-asteroid encounters. In: Abstracts for Small Bodies in the Solar System and their Interactions with the Planets (Mariehamn, Finland), 9 Bulirsh R., Stoer J., 9, Numerische Mathematik 8, Carpino M., Knežević Z., 99a, Asteroid mass determination: () Ceres. In: Ferraz-Mello S., Morando B., Arlot J.-E. (eds.), IAU Symp. 7, Dynamics, Ephemerides and Astrometry of the Solar System, Paris, France, p. Carpino M., Knežević Z., 99b, Determination of asteroid masses from mutual close approaches. In: Proc. of the First Italian Meeting of Planetary Science, Bormio, Italy, Garcia A.L., Medvedev Y.D., Fernandez J.A.M, 99, Using close encounters of minor planets for their improvement of their masses. In: Wytrzyszczak I.M., Lieske J.H., Feldman R.A. (eds.), Proc. IAU Coll., Dynamics and Astrometry of Natural and Artificial Celestial Bodies, Poznań, Poland, p. 99 Goffin E., 98, private communication to Landgraf (99) Goffin E., 99, A&A 9, Hertz H.G., 9, IAU Circ. 98 Hilton J.L., 997, AJ, Hilton J.L., 999, AJ 7, 77 Hilton L., Seidelman P. K., Moddour J., 99, AJ, 9 Kuzmanoski M., 99, A method for asteroid mass determination. In: Ferraz-Mello S., Morando B., Arlot J.-E. (eds.), IAU Symp. 7, Dynamics, Ephemerides and Astrometry of the Solar System, Paris, France, p. 7 Landgraf W., 98, MPC 8 Landgraf W., 988, A&A 9, Landgraf W., 99, A determination of the mass of (7) Interamnia from observations of (99) Moultona. In: Ferraz-Mello S. (ed.), Proc. IAU Symp., Chaos, resonance and collective dynamical phenomena in the Solar System, Dordrecht: Kluwer, p. 79 Merline W.J., Close L.M., Dumas C., et al., 999, Nat,

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