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 –Optional geocentric approximation Not in integrations, only in observables

Free Parameters Metric parameter β Metric parameter γ Ġ (two flavors) “RELFCT” coefficient of post-Newtonian terms in equations of motion “RELDEL” coefficient of post-Newtonian terms in light propagation delay

More Free Parameters “ATCTSC” coefficient of conversion between coordinate and proper time Coefficient of additional de Sitter-like precession Nordtvedt ηΔ, where Δ for Earth-Moon system is the difference of Earth and Moon

Units for Integrations Gaussian gravitational constant Distance - Astronomical Unit –AU in light seconds a free parameter Mass – Solar Mass –No variation of mass assumed –Solar Mass in SI units a derived parameter from Astronomical Unit Time – Ephemeris Day

Historical Footnote to Units Moon integrations are allowed in “Moon units” in deference to traditional expression of lunar ephemerides in Earth radii – not used anymore

Numerical Integration 15th-order Adams-Moulton, fixed step size Starting procedure uses Nordsieck Output at fixed tabular interval –Not necessarily the same as step size Partial derivatives obtained by simultaneous integration of variational equations Partial derivatives (if included) are interleaved with coordinates

Hierarchy of Integrations, I N-body integration includes 9 planets –One is a dwarf planet –One is a 2-body subsystem (Earth-Moon) –Earth-Moon offset is supplied externally and copied to output ephemeris –Partial derivatives not included Individual planet –Partial derivatives included –Earth-Moon done as 2-body system as above

Hierarchy of Integrations, II Moon orbit and rotation are integrated simultaneously –Partial derivatives included –Rest of solar system supplied externally Other artificial or natural satellites are integrated separately –Partial derivatives included –Moon and planets supplied externally

Hierarchy of Integrations, III Iterate to reconcile n-body with Moon Initial n-body uses analytic (Brown) Moon Moon integration uses latest n-body Moon output then replaces previous Moon for subsequent n-body integration Three iterations suffice

Step size and tabular interval Moon – 1/8 day, 1/2 day Mercury (n-body) – 1/2 day, 2 days Mercury (single) – 1/4 day, 1 day Other planets (n-body) – 1/2 day, 4 days Earth-Moon (single) – 1/2 day, 1 day Venus, Mars (single) – 1 day, 4 days

Evaluation of Ephemerides 10-point Everett interpolation Coefficients computed as needed Same procedure for both coordinates and partial derivatives Same procedure for input both to integration and to observable calculation

Accelerations – lunar orbit Integrated quantity is Moon-Earth difference – all accelerations are ditto Point-mass Sun, planets relativistic (PPN) Earth tidal drag on Moon Earth harmonics on Moon and Sun –J2-J4 (only J2 effect on Sun) Moon harmonics on Earth –J2, J3, C22, C31, C32, C33, S31, S32, S33

Accelerations – lunar orbit (cont) Equivalence Principle violation, if any Solar radiation pressure –uniform albedo on each body, neglecting thermal inertia Additional de Sitter-like precession is nominally zero, implemented only as a partial derivative

Accelerations – libration Earth point-mass on Moon harmonics Sun point-mass on Moon harmonics Earth J2 on Moon harmonics Effect of solid Moon elasticity/dissipation –k2 and lag (either constant T or constant Q) Effect of independently-rotating, spherical fluid core –Averaged coupling coefficient

Accelerations – planet orbits Integrated quantity is planet-Sun difference – all accelerations are ditto Point-mass Sun, planets relativistic (PPN) Sun J2 on planet Asteroids (orbits: Minor Planet Center) –8 with adjustable masses –90 with adjustable densities in 5 classes –Additional uniform ring (optional 2 nd ring)

Accelerations – planets (cont) Equivalence Principle violation, if any Solar radiation pressure not included Earth-Moon barycenter integrated as two mass points with externally prescribed coordinate differences

Earth orientation IAU 2000 precession/nutation series –Estimated corrections to precession and nutation at fortnightly, semiannual, annual, 18.6-year, and 433-day (free core) IERS polar motion and UT1 –Not considered in Earth gravity field calc. –Estimated corrections through 2003

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

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

Planetary lander coordinates Modeled planet orientation in proper time + body-fixed coordinates Mars orientation includes precession and seasonal variations

Proper time/coordinate time Diurnal term from · Long-period term from integrated time ephemeris or from monthly and yearly analytic approximations One version of Ġ uses a secular drift in the relative rates of atomic (proper) time and gravitational (coordinate) time Combination of above is labeled “CTAT”

Chain of times/epochs Recv UTC: leap seconds etc→ Recv TAI –PEP uses A.1 internally (constant offset from TAI, for historical reasons) Recv TAI: “Recv CTAT”→ CT –CT same as TDB, except for constant offset Recv CT: light-time iteration→ Rflt CT Rflt CT: light-time iteration→ Xmit CT Xmit CT: “Xmit CTAT”→ Xmit TAI Xmit TAI: leap seconds etc→ Xmit UTC

Corrections after light-time iteration Shapiro delay (up-leg + down-leg) –Effect of Sun for all observations –Effect of Earth for lunar/cislunar obs Physical propagation delay (up + down) –Mendes & Pavlis (2004) for neutral atmosphere, using meteorological data –Various calibrations for radio-frequency obs Measurement bias Antenna fiducial point offset, if any

Integrated lunar partials Mass(Earth,Moon), RELFCT, Ġ, metric β,γ Moon harmonic coefficients Earth, Moon orbital elements Lunar core, mantle rotation I.C.’s Lunar core&mantle moments, coupling Tidal drag, lunar k2, and dissipation EP violation, de Sitter-like precession

Integrated E-M-bary partials Mass(planets, asteroids, belt) Asteroid densities RELFCT, Ġ, Sun J2, metric β,γ Planet orbital elements EP violation

Indirect integrated partials PEP integrates partials only for one body at a time Dependence of each body on coordinates of other bodies and thence by chain-rule on parameters affecting other bodies Such partials are evaluated by reading the other single-body integrations Iterate as needed

Non-integrated partials Station positions and velocities Coordinates of targets on Moon, planets Earth precession and nutation coefficients Adjustments to polar motion and UT1 Planetary radii, spins, topography grids Interplanetary plasma density CT-rate version of Ġ Ad hoc coefficients of Shapiro delay, CTAT AU in light-seconds

Partial derivatives of observations Integrated partials computed by chain rule Non-integrated partials computed according to model Metric β,γ are both

Solutions Calculate residuals and partials for all data Form normal equations Include information from other investigations as a priori constraints Optionally pre-reduce equations to project away uninteresting parameters Solve normal equations to adjust parameters, optionally suppressing ill-defined directions in parameter space Form postfit residuals by linear correction