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

Determination of Determination of PPN PPN and Solar J and Solar J22

from asteroids orbits (update)from asteroids orbits (update)

D. Hestroffer, J. Berthier, S. Mouret D. Hestroffer, J. Berthier, S. Mouret (IMCCE, Paris Observatory)(IMCCE, Paris Observatory)

&&

F. MignardF. Mignard (OCA, Nice)(OCA, Nice)

Orbit perturbationsOrbit perturbations

PPN PPN : only : only oror dd/dt/dt a a-5/2-5/2 . (1-e . (1-e22))-1-1 . .

Solar JSolar J2 2 : perihelion : perihelion , node , node , and mean , and mean anomaly Manomaly M (consider Sun equator = ecliptic)(consider Sun equator = ecliptic)

dd/dt, d/dt, d /dt /dt a a-7/2-7/2 . (1-e . (1-e22))-2-2 . . JJ22

dMdM a a-7/2-7/2 . (1-e . (1-e22))-3/2-3/2 . . JJ22

1500 asteroids (NEAs + MBAs + Trojans)1500 asteroids (NEAs + MBAs + Trojans) distribution in (e,a) planedistribution in (e,a) plane

Relativistic perihelion driftRelativistic perihelion drift

~1500 asteroids

Trojans, MBAs,

and NEOs

astorb.dat + simulated astorb.dat + simulated

population NEOspopulation NEOs

sensitivity sensitivity

e.de.d Mercury Mercury

~e.d~e.d Icarus Icarus

Orbit improvementOrbit improvement Assume LLS, or iterativeAssume LLS, or iterative dd=[∂=[∂/∂c].dc=(O-C)/∂c].dc=(O-C)

Partial derivatives Partial derivatives [∂[∂/∂c] = P/∂c] = Pzz.[∂x/∂c].[∂x/∂c]

11stst. closed form from 2-body osculating elts. . closed form from 2-body osculating elts. (Brouwer & (Brouwer &

Clemence 1961) Clemence 1961) ((da/a, de, dlo+dr, dp, dq, e.dr)

22ndnd. variational equations + numerical integration of motion. variational equations + numerical integration of motion

(also for J(also for J22, , , mass, etc.), mass, etc.)

33rdrd. test wrt direct calculation of PD . test wrt direct calculation of PD [x(c[x(coo+h)-x(c+h)-x(coo-h)]/-h)]/22hh

in progress: in progress: [∂x/∂c] [∂x/∂c] for orbit ok, but not for global parametersfor orbit ok, but not for global parameters

(integrator, step,?.. (integrator, step,?.. TBCTBC))

One Trojan asteroidOne Trojan asteroid

Time

Space

One Main-belt asteroidOne Main-belt asteroid

Time

Space

Two NEOsTwo NEOs

Rank defficient!Rank defficient! NormalNormal

5 years 1/4 year

Space

Time

96 obs 11 obs

Rank defficiency for some NEAsRank defficiency for some NEAs Better for MBAs, Hildas and even TrojansBetter for MBAs, Hildas and even Trojans

Trojans & Hildas

MBAs

NEOs

rank defficiency

Global parametersGlobal parameters

PPN PPN (with (with known) and known) and JJ22

Also global rotation vector Also global rotation vector WW00 and rateand rate WW11==

ddWW00/dt/dt (ecliptic and equinox): (ecliptic and equinox):

dxdx|rot|rot==WW x xx ; dx ; dx|rot|rot==WW11 x xx . (t-t . (t-too))

total of total of N=8N=8 global parameters with ~ 1500 global parameters with ~ 1500 target asteroids target asteroids (1500*6 unknowns)(1500*6 unknowns)

Sparse matrix, solve a Sparse matrix, solve a NxNNxN matrix problem matrix problem

ResultsResults

No frame rotation (No frame rotation (, J, J22))

With frame rotation With frame rotation WW in in as and as and as/yearas/year

JJ22 corrcorr..

6.E-56.E-5 1.E-81.E-8 0.7750.775

JJ22 corrcorr.. WW00 WW11 corrcorr(max)(max)

6.E-56.E-5 1.E-81.E-8 0.7750.775 3, 3, 113, 3, 11 3, 3, 53, 3, 5 -0.08-0.08

(i.e. 1.5E-11 -- 2.5E-11 rd/yr >> 1E-13

Sun aspect angle Sun aspect angle decrease from decrease from

50° down to 50° down to 45° in new design45° in new design

go to less higher go to less higher elongations (IEO)elongations (IEO)

=45°

Sun aspect angle Sun aspect angle decrease from decrease from

50° down to 50° down to 45° in new design45° in new design

go to less higher go to less higher elongations (IEO)elongations (IEO)

loose interesting loose interesting targets?targets? =45°

Sergei’s champion!

Global parametersGlobal parameters

= 45° not to constraining? may loose most = 45° not to constraining? may loose most

powerful (a,e) targets. Need to know mag, nobs.powerful (a,e) targets. Need to know mag, nobs.

Only 1500 targets in this run (instead of ~500,000) Only 1500 targets in this run (instead of ~500,000) butbut not much more NEOs with large eccentricity not much more NEOs with large eccentricity

howeverhowever better for rotation rate better for rotation rate (<10(<10-11-11rad/yr)rad/yr)

Can we observe secular changes of JCan we observe secular changes of J22?... ?...

i.e. di.e. d22/dt/dt22 (2011 Solar activity) (2011 Solar activity)

Get Nordtvedt effect Get Nordtvedt effect , strong test of PE , strong test of PE with with

Trojans Trojans (e.g. Orellana 1993)(e.g. Orellana 1993)? Time scales?..? Time scales?..

A short comment A short comment On asteroids mass, taxonomy, On asteroids mass, taxonomy,

and densityand density

On asteroids mass and densityOn asteroids mass and density

Important for modern planetary ephemeris Important for modern planetary ephemeris (Standish & Fienga, Pitjeva, ...)(Standish & Fienga, Pitjeva, ...)

The ‘astrophysical method’ The ‘astrophysical method’ volume * bulk-density = massvolume * bulk-density = mass absolute mag. H => diamabsolute mag. H => diam…........…........ (H,G) system? (H,G) system? size/diam => volumesize/diam => volume.………..…...………..….. generally ≈ ok generally ≈ ok Taxonomy => densityTaxonomy => density………………………… bulk-porosity? bulk-porosity?

.………... .………... surface/interior?surface/interior?

Asteroid porosityAsteroid porosityfrom Consolmagno & Britt L&PSc XXXIII (2002)

Error (systematic) large for rubble-pile asteroids : Error (systematic) large for rubble-pile asteroids : bulk porosity ~ 20-50%bulk porosity ~ 20-50%