LLR Analysis Workshop

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LLR Analysis Workshop. John Chandler CfA 2010 Dec 9-10. Underlying theory and coordinate system. Metric gravity with PPN formalism Isotropic coordinate system Solar-system barycenter origin Sun computed to balance planets Optional heliocentric approximation Explicitly an approximation - PowerPoint PPT Presentation

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  • LLR Analysis WorkshopJohn ChandlerCfA2010 Dec 9-10

  • Underlying theory and coordinate systemMetric gravity with PPN formalismIsotropic coordinate systemSolar-system barycenter originSun computed to balance planetsOptional heliocentric approximationExplicitly an approximationOptional geocentric approximationNot in integrations, only in observables

  • Free ParametersMetric parameter Metric parameter (two flavors)RELFCT coefficient of post-Newtonian terms in equations of motionRELDEL coefficient of post-Newtonian terms in light propagation delay

  • More Free ParametersATCTSC coefficient of conversion between coordinate and proper timeCoefficient of additional de Sitter-like precessionNordtvedt , where for Earth-Moon system is the difference of Earth and Moon

  • Units for IntegrationsGaussian gravitational constantDistance - Astronomical UnitAU in light seconds a free parameterMass Solar MassNo variation of mass assumedSolar Mass in SI units a derived parameter from Astronomical UnitTime Ephemeris Day

  • Historical Footnote to UnitsMoon integrations are allowed in Moon units in deference to traditional expression of lunar ephemerides in Earth radii not used anymore

  • Numerical Integration15th-order Adams-Moulton, fixed step sizeStarting procedure uses NordsieckOutput at fixed tabular intervalNot necessarily the same as step sizePartial derivatives obtained by simultaneous integration of variational equationsPartial derivatives (if included) are interleaved with coordinates

  • Hierarchy of Integrations, IN-body integration includes 9 planetsOne is a dwarf planetOne is a 2-body subsystem (Earth-Moon)Earth-Moon offset is supplied externally and copied to output ephemerisPartial derivatives not includedIndividual planetPartial derivatives includedEarth-Moon done as 2-body system as above

  • Hierarchy of Integrations, IIMoon orbit and rotation are integrated simultaneouslyPartial derivatives includedRest of solar system supplied externallyOther artificial or natural satellites are integrated separatelyPartial derivatives includedMoon and planets supplied externally

  • Hierarchy of Integrations, IIIIterate to reconcile n-body with MoonInitial n-body uses analytic (Brown) MoonMoon integration uses latest n-bodyMoon output then replaces previous Moon for subsequent n-body integrationThree iterations suffice

  • Step size and tabular intervalMoon 1/8 day, 1/2 dayMercury (n-body) 1/2 day, 2 daysMercury (single) 1/4 day, 1 dayOther planets (n-body) 1/2 day, 4 daysEarth-Moon (single) 1/2 day, 1 dayVenus, Mars (single) 1 day, 4 days

  • Evaluation of Ephemerides10-point Everett interpolationCoefficients computed as neededSame procedure for both coordinates and partial derivativesSame procedure for input both to integration and to observable calculation

  • Accelerations lunar orbitIntegrated quantity is Moon-Earth difference all accelerations are dittoPoint-mass Sun, planets relativistic (PPN)Earth tidal drag on MoonEarth harmonics on Moon and SunJ2-J4 (only J2 effect on Sun)Moon harmonics on EarthJ2, J3, C22, C31, C32, C33, S31, S32, S33

  • Accelerations lunar orbit (cont)Equivalence Principle violation, if anySolar radiation pressureuniform albedo on each body, neglecting thermal inertiaAdditional de Sitter-like precession is nominally zero, implemented only as a partial derivative

  • Accelerations librationEarth point-mass on Moon harmonicsSun point-mass on Moon harmonicsEarth J2 on Moon harmonicsEffect of solid Moon elasticity/dissipationk2 and lag (either constant T or constant Q)Effect of independently-rotating, spherical fluid coreAveraged coupling coefficient

  • Accelerations planet orbitsIntegrated quantity is planet-Sun difference all accelerations are dittoPoint-mass Sun, planets relativistic (PPN)Sun J2 on planetAsteroids (orbits: Minor Planet Center)8 with adjustable masses90 with adjustable densities in 5 classesAdditional uniform ring (optional 2nd ring)

  • Accelerations planets (cont)Equivalence Principle violation, if anySolar radiation pressure not includedEarth-Moon barycenter integrated as two mass points with externally prescribed coordinate differences

  • Earth orientationIAU 2000 precession/nutation seriesEstimated corrections to precession and nutation at fortnightly, semiannual, annual, 18.6-year, and 433-day (free core)IERS polar motion and UT1Not considered in Earth gravity field calc.Estimated corrections through 2003

  • Station coordinatesEarth orientation + body-fixed coordinates + body-fixed secular drift + Lorentz contraction + tide correctionTide is degree-independent response to perturbing potential characterized by two Love numbers and a time lag (all fit parameters)

  • Reflector coordinatesIntegrated Moon orientation + body-fixed coordinates + Lorentz contraction + tide correctionTide is degree-independent response to perturbing potential characterized by two Love numbers and a time lag (all fit parameters)

  • Planetary lander coordinatesModeled planet orientation in proper time + body-fixed coordinatesMars orientation includes precession and seasonal variations

  • Proper time/coordinate timeDiurnal term from Long-period term from integrated time ephemeris or from monthly and yearly analytic approximationsOne version of uses a secular drift in the relative rates of atomic (proper) time and gravitational (coordinate) timeCombination of above is labeled CTAT

  • Chain of times/epochsRecv UTC: leap seconds etc Recv TAIPEP uses A.1 internally (constant offset from TAI, for historical reasons)Recv TAI: Recv CTAT CTCT same as TDB, except for constant offsetRecv CT: light-time iteration Rflt CTRflt CT: light-time iteration Xmit CTXmit CT: Xmit CTAT Xmit TAIXmit TAI: leap seconds etc Xmit UTC

  • Corrections after light-time iterationShapiro delay (up-leg + down-leg)Effect of Sun for all observationsEffect of Earth for lunar/cislunar obsPhysical propagation delay (up + down)Mendes & Pavlis (2004) for neutral atmosphere, using meteorological dataVarious calibrations for radio-frequency obsMeasurement biasAntenna fiducial point offset, if any

  • Integrated lunar partialsMass(Earth,Moon), RELFCT, , metric ,Moon harmonic coefficientsEarth, Moon orbital elementsLunar core, mantle rotation I.C.sLunar core&mantle moments, couplingTidal drag, lunar k2, and dissipationEP violation, de Sitter-like precession

  • Integrated E-M-bary partialsMass(planets, asteroids, belt)Asteroid densitiesRELFCT, , Sun J2, metric ,Planet orbital elementsEP violation

  • Indirect integrated partialsPEP integrates partials only for one body at a timeDependence of each body on coordinates of other bodies and thence by chain-rule on parameters affecting other bodiesSuch partials are evaluated by reading the other single-body integrationsIterate as needed

  • Non-integrated partialsStation positions and velocitiesCoordinates of targets on Moon, planetsEarth precession and nutation coefficientsAdjustments to polar motion and UT1Planetary radii, spins, topography gridsInterplanetary plasma densityCT-rate version of Ad hoc coefficients of Shapiro delay, CTATAU in light-seconds

  • Partial derivatives of observationsIntegrated partials computed by chain ruleNon-integrated partials computed according to modelMetric , are both

  • SolutionsCalculate residuals and partials for all dataForm normal equationsInclude information from other investigations as a priori constraintsOptionally pre-reduce equations to project away uninteresting parametersSolve normal equations to adjust parameters, optionally suppressing ill-defined directions in parameter spaceForm postfit residuals by linear correction