determination of ppn and solar j 2 from asteroids orbits (update) d. hestroffer, j. berthier, s....
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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)
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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
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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
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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))
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One Trojan asteroidOne Trojan asteroid
Time
Space
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One Main-belt asteroidOne Main-belt asteroid
Time
Space
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Two NEOsTwo NEOs
Rank defficient!Rank defficient! NormalNormal
5 years 1/4 year
Space
Time
96 obs 11 obs
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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
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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
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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
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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°
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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!
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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?..
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A short comment A short comment On asteroids mass, taxonomy, On asteroids mass, taxonomy,
and densityand density
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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?
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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%
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