Michel Rapaport Observatoire de l’Université Bordeaux 1 33270 Floirac

1. Accuracy in the mass determinationof asteroids with Gaia Michel RapaportObservatoire de l’Université Bordeaux 133270 Floirac SSWG Gaia / Besançon 6-7 Novembre 2003

2. Target MaximalPerturbations Number ofobservations Mass (933)Moultona 278 (24.6±14.7)x10-12 2".1 (1023)Thomana 4".2 150 (17.9±15.5)x10-12 3".5 465 (85.1±16)x10-12 (84)Klio Ground based observations : (52)EUROPA 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. SSWG Gaia Besançon 6-7 Novembre 2003

3. Hipparcos observations (20)MASSALIA Mean standard errors of Hipparcos data : 15 mas Maximal expected perturbations : 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 : 15 mas Maximal expected perturbations : 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. SSWG Gaia Besançon 6-7 Novembre 2003

4. Several recent works on the determination of masses of asteroids (Table(3)) use an algorithm developed by Herget (1972). 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 Table(3) I now present a comment on Herget’s method. SSWG Gaia Besançon 6-7 Novembre 2003

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

6. (a) A Δx + B Δm = (O - C) (b) We start withΔm = Δm0 (c) The orbit is obtained with (d) B Δm = (O - C) - A Δx1 A Δx = (O - C) - BΔm0 Δ x1 gives the final massΔm1 Table(4) : I think that this method corresponds to the first iteration of the Gauss-Seidel method and must be iterated until convergence. SSWG Gaia Besançon 6-7 Novembre 2003

7. …………… m m 01 01 01 01 01 01 0101 01 01 01 01 A= S = 01 01 01 01 0101 n …………… 01 01 01 01 01 01 σ σ σ 1 2 6 n 01 01 01 01 01 01 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 byU t, the transformed equations are obtained (Table(5)) SSWG Gaia Besançon 6-7 Novembre 2003

8. σ x* + b*Δm = (O – C)* 1 1 1 1 σ x* + b*Δm = (O – C)* 2 2 2 2 (2) σ x* + b*Δm = (O – C)* 6 6 6 6 + b*Δm = (O – C)* 7 7 (1) + b*Δm = (O – C)* n n 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. SSWG Gaia Besançon 6-7 Novembre 2003

9. 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. SSWG Gaia Besançon 6-7 Novembre 2003

10. Maximal perturbations : 400 mas Interval of time : 5 years Observations : 71 Regularly distributed Accuracy of measurements : 1 mas σm : 2.6 x 10-13 Vesta - Tethys 1st example : Table(6) SSWG Gaia Besançon 6-7 Novembre 2003

11. Maximal perturbations : 50 mas Interval of time : 5 years Observations : 78 Accuracy of measurements : 1 mas σm : 1.05 x 10-12 (1)Ceres (45)Eugenia 2nd example : Table(7) SSWG Gaia Besançon 6-7 Novembre 2003

12. Maximal perturbations : 57 mas Interval of time : 5 years Observations : 103 Accuracy of measurements : 1 mas σm : 7.3 x 10-12 (1)Ceres (829)Academia 3rd example : Table(8) SSWG Gaia Besançon 6-7 Novembre 2003

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

14. Perspectives : a) Generalisation of the presented method to the global determination of the asteroids masses.This point should be theoritically not a very difficult task. 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) SSWG Gaia Besançon 6-7 Novembre 2003