1. Michel Rapaport Observatoire de l’Université Bordeaux 1 33270 Floirac Accuracy in the mass determination of asteroids with Gaia SSWG Gaia / Besançon 6-7 Novembre 2003

2. Besançon 6-7 Novembre 2003 SSWG Gaia Ground based observations : (52)EUROPA Target Maximal Perturbations Number of observations Mass (933)Moultona 2".1 278(24.6±14.7)x10 -12 (1023)Thomana4".2150(17.9±15.5)x10 -12 (84)Klio 3".5465(85.1±16)x10 -12 Table(1) : The limits of ground based observations are illustrated with three recent independant determinations of the mass of Europa. These results well illustrate the accuracy which can be obtained with these observations, i.e about 10 -11 solar mass.

3. Besançon 6-7 Novembre 2003 SSWG Gaia Hipparcos observations (20)MASSALIA Mean standard errors of Hipparcos data : Maximal expected perturbations : 15 mas 59 mas 53 observations of the target Estimation of the mass (2.42 ± 0.41) x 10 -12 (1)CERES Mean standard errors of Hipparcos data : Maximal expected perturbations : 15 mas 59 mas 27 observations of the target Estimation of the mass (4.4 ± 0.6) x 10 -10 Table(2) : The main results using the Hipparcos data are shown in Table(2). It can be noticed that the value obtained for the mass of Ceres is smaller than all those obtained from ground based observations, and that the accuracy of this determination is rather low, probably due to the small number of observations.

4. Besançon 6-7 Novembre 2003 SSWG Gaia J.F. Bange : (A & A, 340, L1, 1998) An estimation of the mass of asteroïd (20)Massalia derived from the Hipparcos minor planets data M.J.S. Belton et al : (Nature, vol 374, 1995) Bulk density of asteroïd (243)Ida from the orbit of its satellite Dactyl P. Herget : I.A.U Symposium The motion, evolution of orbits, and origin of comets 1972 on the determination of planetary masses Several recent works on the determination of masses of asteroids (Table(3)) use an algorithm developed by Herget (1972). Table(3) I now present a comment on Herget’s method.

5. The equations associated to the observations are : A(Δx) + B(Δm) = (O - C) Besançon 6-7 Novembre 2003 SSWG Gaia where A is the matrix of partial derivatives with respect to the orbital elements, where B is the matrix of partial derivatives with respect to the mass. The unknown Δx and Δm are respectively the corrections to the orbital elements of the target and to the mass of the perturbing body. Herget’s method can be describe by the four steps presented in Table(4)

6. Besançon 6-7 Novembre 2003 SSWG Gaia The orbit is obtained with (c) gives the final mass Δm 1 (a) A Δx + B Δm = (O - C) (b) We start with Δm = Δm 0 A Δx = (O - C) - BΔm 0 Δ x 1 (d) B Δm = (O - C) - A Δx 1 Table(4) : I think that this method corresponds to the first iteration of the Gauss-Seidel method and must be iterated until convergence.

7. A = n …………… m S = n m 0 1 0 1 0 1 σ 1 σ 2 σ 6 Besançon 6-7 Novembre 2003 SSWG Gaia I present an alternative algorithm to determine the mass of the perturbing body. Starting with the same equation (Table(4),(a)), we determine the singular value decomposition of A = U S V t The equations become : U S V t (x) + BΔm = (0 - C) After the change of variables : V t x = x* and a left multiplication by U t, the transformed equations are obtained (Table(5))

8. + b * Δm = (O – C) * 77 (1) + b * Δm = (O – C) * nn Besançon 6-7 Novembre 2003 SSWG Gaia σ x * + b * Δm = (O – C) * 1111 σ x * + b * Δm = (O – C) * 6666 (2) σ x * + b * Δm = (O – C) * 2222 Table(5) : the transformed equations It can be easily proved that the least squares solution of the transformed equations for the mass is given by the solution of the part (1) of the system. If we considered « favourable close encounters », Δm will be well determined while the corrections to the orbital elements are obtained putting the solution for Δm in the part (2) of the system. So the determination of the orbital parameters of the target, and the mass of the perturbing body are really deconnected.

9. Besançon 6-7 Novembre 2003 SSWG Gaia The four following tables give the formal accuracy of the estimation of a mass for various close encounters. In the first case (Table(6)). I consider a set of observations regularly distributed in the 5 years interval of time considered. In this very « favourable case » we see that it would be possible to determine masses of ~10 -12 solar mass with a good accuracy. The next examples show that it is not always the same ; it will be necessary to search the « most efficient close encounters », if we will obtain a good relative accuracy on the determination of masses of about 10 -12 sm.

10. Besançon 6-7 Novembre 2003 SSWG Gaia Vesta - Tethys Maximal perturbations400 mas: Interval of time 5 years : Observations 71 Regularly distributed : Accuracy of measurements1 mas: σmσm :2.6 x 10 -13 1st example : Table(6)

11. Besançon 6-7 Novembre 2003 SSWG Gaia (1)Ceres (45)Eugenia Maximal perturbations50 mas: Interval of time 5 years : Observations 78 : Accuracy of measurements1 mas: σmσm :1.05 x 10 -12 2nd example : Table(7)

12. Besançon 6-7 Novembre 2003 SSWG Gaia (1)Ceres (829)Academia Maximal perturbations57 mas: Interval of time 5 years : Observations 103 : Accuracy of measurements1 mas: σmσm :7.3 x 10 -12 3rd example : Table(8)

13. Besançon 6-7 Novembre 2003 SSWG Gaia (1)Ceres (1765)Wrubel Maximal perturbations4 mas: Interval of time 5 years : Observations 53 : Accuracy of measurements1 mas: σmσm :1.3 x 10 -10 4th example : Table(9)

14. Besançon 6-7 Novembre 2003 SSWG Gaia Generalisation of the presented method to the global determination of the asteroids masses. This point should be theoritically not a very difficult task. Perspectives : a) Selection of the most favourable close encounters (from the works already developed, Fienga 2002), and of the sequences of dates of observations obtained by the simulations (Mignard), in order to determine the global statistical properties of the estimators of the masses. b)