ACTA ASTRONOMICA Vol. 45 (1995) pp. 673±677

Determination of Masses of Ceres and Vesta from Their Perturbations on Four

by G. Sitarski and B. Todororovic-Juchniewicz Space Research Center, Polish Academy of Sciences, ul. Bartycka 18a, 00-716 Warsaw, Poland e-mail: [email protected]

Received September 1, 1995

ABSTRACT

Masses of 1 Ceres and were determined simultaneously from observations of four asteroids: 197 Arete, 203 Pompeja, 348 May, and 486 Cremona. We collected 785 observations of these four asteroids covering the observation interval 1879±1995; in this period the asteroids approached several times Ceres and Vesta closer than to 0.1 a.u. We applied the recurrent power series method to integrate numerically equations of motion of the four asteroids, including the perturbations caused by the nine major planets and by the four biggest asteroids: Ceres, Pallas, Vesta, and Hygiea. Twenty six normal equations, corresponding to 1536 observational equations, have been solved by means of the least squares method to determine the masses of Ceres and Vesta along with corrections

to the orbital elements of Arete, Pompeja, May, and Cremona. Thus we found the masses of Ceres

10 10

 )  (  )  and Vesta in the solar unit mass equal to (4 622 0 071 10 and 1 396 0 043 10 , respectively. Key words: Minor planets, asteroids

1. Introduction

Hertz (1966, 1968) was the ®rst who succeeded in determining a value of

10  the mass of Vesta (120 10 solar mass) from its on the motion of Arete. Schubart (1979) has repeated Hertz's determination for Vesta using his extended collection of the observations of Arete and found its mass equal to

10  138 10 . Adding all the observations of Arete obtained since its discovery to 1990 and using eight major planets and Ceres, Pallas, and Vesta as perturbing

10  planets, Gof®n (1991) calculated mass of Vesta as 133 10 solar mass. In the period 1970±1974 Schubart (1970, 1971, 1974) computed values of the mass of Ceres from the gravitational interaction between Ceres and Pallas caused by near 1:1 resonance between their mean motions; he included Vesta (with the 674 A. A.

Hertz's mass) in computing its action on the motion of Ceres. He found the mass

10  of Ceres equal to 5.9 and 5 1 10 solar mass. Landgraf (1988), using the combined observational material: normal positions and more extended set of ob-

10  servations of Pallas, found the mass of Ceres equal to 5 21 10 . The mass

10  of Ceres equal to 4 7 10 was derived by Gof®n (1991) from perturbation on Pompeja whereas Williams (1991) estimated the mass of Ceres from perturbations

10  on May as 443 10 . Sitarski and Todorovic-Juchniewicz (1992) calculated

10  the mass of Ceres (equal to 4 80 10 ) from its perturbations on Pompeja and May simultaneously. A different approachhas been taken by Standish and Hellings (1989) who estimated the masses of Ceres and Vesta by measuring their perturba-

10  tions upon the orbit of Mars; they found the mass of Ceres equalto 50 10

10  and of Vesta to 15 10 solar mass. In this paper we present the calculation of the masses of Ceres and Vesta from their simultaneous perturbations on four asteroids: Arete, Pompeja, May, and Cremona. Using the observations of the four asteroids together, we made the least squares correction of twenty six parameters: twenty four orbital elements of the four asteroids and two planetary masses of Ceres and Vesta. We used the same method that was successfully applied when correcting the mass of Ceres from observations of two asteroids, Pompeja and May, (Sitarski and Todorovic-Juchniewicz 1992).

2. Observational Material

We have chosen four asteroids which during the relatively long observation intervals approached closely Ceres and Vesta. The asteroids 203 Pompeja and 348 May approached Ceres several times while 197 Arete and 486 Cremona ap- proached Vesta closer than to 0.1 a.u. Orbital elements of the considered asteroids are given in Table 1, and minimum distances during the encounters with Ceres and

Table1

Orbital elements of the four asteroids for the osculation 1995 Jan. 3.0 ET = JD 2449720.5;

i the M and angular elements (referred to J2000.0) are in degrees, the

semimajor axis a is in a.u.

Element 197 Arete 203 Pompeja 348 May 486 Cremona

M 107.89415 133.21433 308.39349 258.93222

a 2.74003411 2.73646753 2.96888132 2.35166981

e 0.16276844 0.06032503 0.07328849 0.16393174

246.49494 56.46223 9.52347 124.15479

81.87946 348.27444 90.40736 94.41786

i 8.78955 3.18416 9.75635 11.08151 Vol. 45 675

Vesta in the observation intervals are presented in Tables 2 and 3. One should mention that the four considered minor planets encountered many times Ceres and Vesta though not as deeply as it is given in Tables 2 and 3; e.g.,on Oct. 8 1961 Cremona approached Ceres to within 0.191 a.u.

Table2 Dates of approaches of Pompeja and May to Ceres; the minimum distances are in a.u., the dates of the ®rst and last observations show the observational intervals of the asteroids (the number of observations is given below the name of ).

D a t e Pompeja May 232 obs. 132 obs. 1994 Jan. 9 Last obs. 1992 July 31 Last obs. 1984 Sep. 1 0.042 1969 Feb. 15 0.078 1948 Aug. 22 0.016 1938 Apr. 3 0.082 1892 Nov. 29 First obs. 1879 Sep. 26 First obs.

Table3 Dates of approaches of Arete and Cremona to Vesta; further explanations are the same as in the caption of Table 2.

D a t e Arete Cremona 322 obs. 99 obs. 1995 Jan. 1 Last obs. 1992 Jan. 1 Last obs. 1987 Aug. 16 0.054 1983 Dec. 29 0.038 1976 Jan. 27 0.035 1957 Dec. 9 0.035 1945 Dec. 23 0.072 1939 Oct. 17 0.032 1921 Aug. 25 0.029 1903 July 7 0.027 1902 May 28 First obs. 1885 May 14 0.018 1879 May 22 First obs. 676 A. A.

We collected all the astrometric observations of the asteroids since the dates of their discoveries. A table of observations of Pompeja made during 1879±1938 was published by Gof®n (1991); we used the set of observations given by Gof®n and completed it by the positions of the asteroid made up to 1994. The observations for each asteroid separately were selected according to the mathematically objective Bielicki's criterion derived on the assumption of a normal distribution of residuals and depending on the number of residuals used (Bielicki and Sitarski 1991).

3. TheResults

Equations of motion as well as variation equations for differential coef®cients have been integrated numerically by the recurrent power series of the 12th order. We took into account the planetary perturbations caused by thirteen disturbing bodies: the nine major planets of the Solar System and the four biggest asteroids: Ceres, Pallas, Vesta, and Hygiea. A set of coordinates of the biggest asteroids have been prepared earlier by numerical integration of equations of motion of asteroids in the time-span 1800±2019, starting from the initial data given by Bowell (1994). As output data of the integration the equatorial rectangular coordinates and velocity components of the biggest asteroids every forty days were stored. We can correct the mass of Ceres along with orbital elements of the disturbed asteroid using the observational equations either for Pompeja or for May, and similarly we can correct the mass of Vesta using the observations of Arete and Cremona. We can also join the normal equations for Pompeja with those for May to solve the set of thirteen normal equations corresponding to observational equations for both asteroids; similarly we can join the observational equations for Arete and Cremona to correct the mass of Vesta. Thus we improved the masses of Ceres and Vesta in various ways, getting the following solutions:

the mass of Ceres in 10 10 solar mass:



4560 0 081 from Pompeja



4926 0 163 from May

 4627 0 071 from Pompeja + May

the mass of Vesta in 1010 solar mass:



1399 0 043 from Arete



1417 0 227 from Cremona

 1399 0 042 from Arete + Cremona Finally we joined all the observational equations for the four asteroids solving by the least squares technique 1536 observational equations with 26 unknowns.

The mean RMS residual of 785 observations amounted to 1.00 67. We obtained the following solution for the masses of Ceres and Vesta expressed in the solar mass:

10

= (  )  mC 4 622 0 071 10

10

m (  )  V = 1 396 0 043 10 Vol. 45 677

The above values of masses were calculated along with corrections to 24 orbital elements of the four asteroids (the corrected orbital elements are presented

in Table 1). m

The ®nal mass of Ceres C , comparable with that as found from observations m

of Pompeja+May only, differs from C obtained earlier by Sitarski and Todorovic- Juchniewicz (1992). However, in the present work we applied the more complex method for planetary mass determination, used more extended observational ma- terial and included the four biggest asteroids as disturbing bodies when integrating the equations of motion of the minor planets.

Acknowledgements. We greatly appreciate Dr. Edward Bowell, Lowell Ob- servatory, help, who kindly sent us the exact values of orbital elements of the four biggest asteroids: Ceres, Pallas, Vesta, and Hygiea.

REFERENCES

Bielicki, M., and Sitarski, G. 1991, Acta Astron., 41, 309. Bowell, E. 1994, private communication. Gof®n, E. 1991, Astron. Astrophys., 249, 563. Hertz, H.G. 1966, I.A.U. Circ., 1983. Hertz, H.G. 1968, Science, 160, 299. Landgraf, W. 1988, Astron. Astrophys., 191, 161. Schubart, J. 1970, I.A.U. Circ., 2268. Schubart, J. 1971, Celestial Mechanics, 4, 246. Schubart, J. 1974, Astron. Astrophys., 30, 289. Sitarski, G. 1971, Acta Astron., 21, 87. Sitarski, G. 1979, Acta Astron., 29, 413. Sitarski, G., Todorovic-Juchniewicz, B. 1992, Acta Astron., 42, 139. Standish, E., Hellings, R. 1989, Icarus, 80, 326. Williams, G.V. 1991, in: Asteroids, , Meteors 1991, eds. Harris A.W., Bowell E., Northern Arizona University, Flagstaff, p. 641.