1. Determination of PPN  and Solar J 2 from asteroids orbits (update) D. Hestroffer, J. Berthier, S. Mouret (IMCCE, Paris Observatory) & F. Mignard (OCA, Nice)

2. Orbit perturbations  PPN  : only  or    d   /dt  a -5/2. (1-e 2 ) -1.   Solar J 2 : perihelion , node , and mean anomaly M (consider Sun equator = ecliptic)  d   /dt, d  /dt  a -7/2. (1-e 2 ) -2. J 2  dM  a -7/2. (1-e 2 ) -3/2. J 2 ä 1500 asteroids (NEAs + MBAs + Trojans)  distribution in (e,a) plane

3. Relativistic perihelion drift ä ä ~1500 asteroids ä ä Trojans, MBAs, and NEOs astorb.dat + simulated population NEOs  sensitivity e.d  Mercury ~e.d  Icarus

4. Orbit improvement  Assume LLS, or iterative d =[∂ /∂c].dc=(O-C)  Partial derivatives [∂ /∂c] = P z.[∂x/∂c]  1 st. closed form from 2-body osculating elts. (Brouwer & Clemence 1961) (  1 st. closed form from 2-body osculating elts. (Brouwer & Clemence 1961) (da/a, de, dl o +dr, dp, dq, e.dr)  2 nd. variational equations + numerical integration of motion (also for J 2, , mass, etc.)  3 rd. test wrt direct calculation of PD [x(c o +h)-x(c o -h)]/ 2 h  in progress: [∂x/∂c] for orbit ok, but not for global parameters (integrator, step,?.. TBC )

5. One Trojan asteroid Time Space

6. One Main-belt asteroid Time Space

7. Two NEOs ä Rank defficient! ä Normal 5 years 1/4 year Space Time 96 obs11 obs

8. ä Rank defficiency for some NEAs ä Better for MBAs, Hildas and even Trojans Trojans & Hildas MBAs NEOs rank defficiency

9. Global parameters  PPN  (with  known) and J 2  Also global rotation vector W 0 and rate W 1 = d W 0 /dt (ecliptic and equinox):  dx |rot = W x ; dx |rot = W 1 x. (t-t o )  dx |rot = W x x ; dx |rot = W 1 x x. (t-t o ) ä total of N=8 global parameters with ~ 1500 target asteroids (1500*6 unknowns) ä Sparse matrix, solve a NxN matrix problem

10. Results  No frame rotation ( , J 2 )  With frame rotation W in  as and  as/year   J 2  corr. 6.E-51.E-80.775   J 2  corr.  W 0   W 1  corr (max) 6.E-51.E-80.775 3, 3, 11 3, 3, 5 -0.08 (i.e. 1.5E-11 -- 2.5E-11 rd/yr >> 1E-13

11.  Sun aspect angle decrease from  50° down to 45° in new design ä go to less higher elongations (IEO) ä  =45°

12.  Sun aspect angle decrease from  50° down to 45° in new design ä go to less higher elongations (IEO) ä loose interesting targets?  =45° Sergei’s champion!

13. Global parameters   = 45° not to constraining? may loose most powerful (a,e) targets. Need to know mag, nobs. ä Only 1500 targets in this run (instead of ~500,000)  but not much more NEOs with large eccentricity  however better for rotation rate (<10 -11 rad/yr) ä Can we observe secular changes of J 2 ?... i.e. d 2  /dt 2 (2011 Solar activity)  Get Nordtvedt effect , strong test of PE with Trojans (e.g. Orellana 1993) ? Time scales?..

14. A short comment On asteroids mass, taxonomy, and density

15. On asteroids mass and density ä Important for modern planetary ephemeris (Standish & Fienga, Pitjeva,...) ä The ‘astrophysical method’  volume * bulk-density = mass  absolute mag. H => diam…........ (H,G) system?  size/diam => volume.………..….. generally ≈ ok  Taxonomy => density…………… bulk-porosity?.………... surface/interior?.………... surface/interior?

16. Asteroid porosity from Consolmagno & Britt L&PSc XXXIII (2002) Error (systematic) large for rubble-pile asteroids : bulk porosity ~ 20-50%