A&A 370, 602–609 (2001) Astronomy DOI: 10.1051/0004-6361:20010222 & c ESO 2001 Astrophysics Mass and density of asteroids (4) Vesta and (11) Parthenope B. Viateau and M. Rapaport Observatoire de Bordeaux, UMR 5804, CNRS, BP 89, 33270 Floirac, France Received 29 August 2000 / Accepted 7 February 2001 Abstract. As mentioned in a previous paper (Viateau & Rapaport 1997), the orbit of asteroid (17) Thetis is strongly perturbed by two large minor planets, (4) Vesta and (11) Parthenope. These strong gravitational perturbations enabled us to determine both the mass of Vesta and the mass of Parthenope. We also independently derived the mass of Vesta from observations of (197) Arete. The weighted mean of the two results gives the value −10 3 (1.306 0.016) 10 M (solar mass) for the mass of Vesta, and its derived mean density is (3.3 0.5) g/cm . −12 3 For (11) Parthenope, the values (2.56 0.07) 10 M and (2.3 0.2) g/cm were obtained, respectively for its mass and its mean density. Key words. minor planets, asteroids – astrometry – ephemerides 1. Introduction perturbation of Vesta on the orbit of Thetis, only a few years of additional ground-based observations of the latter Large asteroids induce significant gravitational perturba- asteroid would be sufficient to obtain a very precise deter- tions on the orbits of other solar system bodies. The un- mination of the mass of Vesta, which is a rare phenomenon certainties on their mass are a problem for the calcula- in asteroid mass determination. tion of accurate planetary ephemerides (Standish et al. This determination of the mass of Vesta from obser- 1995; Hilton et al. 1996). As an example, these uncertain- vations of Thetis is now possible. In order to improve the ties cause problems in improving the determination of the accuracy of the determination of this mass, we also used relativistic precession of the perihelion of Mars’ orbit, be- the observations of a second target asteroid, (197) Arete. cause of long-period resonances between the planet and Arete was the first asteroid used for the determination of some large asteroids (Eubanks 1996). (4) Vesta is one of the mass of Vesta, by Hertz (1966). A few other determi- the strongest perturbers of Mars’ orbit and one of those nations of this mass have been made since, most of them causing the largest uncertainties on its ephemeris. Vesta from perturbations on Arete (Table 1), which now appears also induces strong gravitational perturbations on the or- to be the second-best target asteroid after Thetis for the bits of many other asteroids. For example, Bange (1998) determination of the mass of Vesta. The final result pre- showed that the mass of asteroid (20) Massalia obtained sented in this paper is obtained from the resolution of the from perturbations on (44) Nysa was strongly correlated equations associated with the observations of Thetis and with the mass of Vesta. Thus, a very accurate value of the Arete. These equations are treated as a single system, and mass of Vesta is needed. solved by the Gauss-Seidel iterative method, so that the The mass of a large asteroid can directly be deter- masses of Vesta and Parthenope are determined together, mined from its gravitational perturbations on other bod- with their correlation coefficient. ies, named “target” bodies. Most of the time, the target bodies are other asteroids. In a previous paper (Viateau & Rapaport 1997), hereafter referred to as Paper I, we 2. Description of the close encounters pointed out that recent very close encounters occurred between asteroid (17) Thetis and two larger minor plan- 2.1. (4) Vesta–(17) Thetis ets, (4) Vesta and (11) Parthenope, respectively in 1996 and 1997. In that paper, a first value of the mass of The evolution with time of the distance between Thetis Parthenope was obtained from the analysis of the observa- and Vesta is shown in Fig. 1a. It can be seen that both tions of Thetis, thanks to a previous very close encounter asteroids come close to each other every 55 years. The in 1968. We also pointed out that, due to the very strong last encounter in 1996 was unusual, because both asteroids had almost the same osculating elements and consequently Send offprint requests to: M. Rapaport, the distance between the asteroids remained smaller e-mail: [email protected] than 0.4 AU from March 1995 to June 1998 (Fig. 1b). Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20010222 B. Viateau and M. Rapaport: Mass and density of asteroids (4) Vesta and (11) Parthenope 603 25 Table 1. Status of the mass determination of Vesta, before the 20 work presented here 15 10 Author perturbed mass σ cos (") 5 −10 ∆α δ bodies (10 M ) 0 Hertz (1966) (197) Arete 1.17 0.10 1995 2000 2005 2010 2015 2020 date Hertz (1968) (197) Arete 1.2 0.08 Schubart & Matson (1979) (197) Arete 1.38 0.12 Fig. 2. Effect of the gravitational perturbation of (4) Vesta on Standish & Hellings (1989) Mars 1.5 0.3 the orbit of (17) Thetis in right ascension, starting forward nu- Goffin (1991) (197) Arete 1.33 merical integration at epoch JD 2450000.5 = 1995 Oct. 10.0 TT −10 Sitarski (1995) (197) & (486) 1.396 0.043 and assuming a mass of 1.35 10 M for Vesta Standish et al. (1995) DE403 solution 1.34 Standish et al. (1998) DE405 solution 1.3 Hilton (1999) (1) & (2) 1.69 0.11 Michalak (2000) 25 asteroids 1.36 0.05 (a) 6 5 4 3 Fig. 3. Distance between (11) Parthenope and (17) Thetis 2 distance (AU) from 1800 to 2050 1 0 1850 1900 1950 2000 date a relative velocity of 2.3 km s−1. This is this encounter (b) that allows the mass of Parthenope to be determined, 00 0,4 with a gravitational perturbation up to 15 in right ascen- 0,3 sion (Fig. 4). The second encounter took place in January 0,2 1997 with a minimal distance of 0.0054 AU and a rela- 0,1 −1 distance (AU) tive velocity of 2.4 km s . This recent encounter was not 0 1995 1996 1997 1998 1999 yet effective when the first determination of the mass of Fig. 1. Distance between (17) Thetis and (4) Vesta. a) From Parthenope was made (Paper I), and allows now suppress 1850 to 2023. b) From 1995 to 1999 a more precise determination of this mass. 0.0016 AU is the smallest minimal distance between two asteroids we had to deal with in asteroid mass deter- The minimum was 0.0194 AU in June 1996, with a relative mination. Moreover, both asteroids are not very different velocity of 1.2 km s−1. A much smaller relative velocity of in size: the IRAS diameter of Parthenope is 162 km, and 0.2 km s−1 was reached in March 1997. Since Vesta is the the diameter of Thetis is 93 km. The closer two aster- second most massive asteroid of the main belt, with a mass oids come to each other, the more accurately their mutual equal to about 30% of the mass of Ceres (Table 2), such distance must be calculated. The necessity of taking into an encounter induced a very strong gravitational pertur- account the reciprocal perturbation of Thetis when the bation on the orbit of Thetis, up to 100 per year on average orbit of Parthenope was computed by numerical integra- (Fig. 2). Since the best ground-based asteroid observations tion was considered. However, the maximal deviation of currently have an accuracy better than 0.100,afewyears the orbit of Parthenope caused by Thetis was about 200, of such data are sufficient to obtain the value of the mass which caused a too small relative change on the mutual of Vesta with a very small standard deviation. distance for the determination of the mass of Parthenope Reciprocally, Thetis, with an assumed mass of 5 × to be affected. We therefore neglected this effect. The de- −13 00 10 M (its diameter measured by the IRAS satellite is viation of 2 of the orbit of Parthenope was also too small 93 km and its mean density was assumed to be 2.4 g/cm3), for the mass of Thetis to be determined. is almost 300 times less massive than Vesta, and its per- turbation on the orbit of Vesta is always less than 0.200 in 2.3. (4) Vesta–(197) Arete right ascension, too small for its mass to be determined. Vesta and Arete come close to each other every 18 years 2.2. (11) Parthenope–(17) Thetis (Fig. 5). Among the 7 close encounters experienced by the two asteroids in the period 1879–1999, the first one Since its discovery in 1852, Thetis has experienced two in May 1885 was the closest, with a minimal distance of close encounters with (11) Parthenope (Fig. 3). The first 0.018 AU and a relative velocity of 2.2 km s−1.Sincethat one, in February 1968, was extremely close, with a min- date, the minimal distance tends to increase at each en- imal distance of 0.0016 AU (i.e. about 240 000 km) and counter. The last encounter between the two asteroids 604 B. Viateau and M. Rapaport: Mass and density of asteroids (4) Vesta and (11) Parthenope 15 and Parthenope. Calculation of singular values of M shows 10 that the problem is well conditioned, the condition num- ber being equal to 440.