Celestial Mechanics Analysis Workshop Dec. 9-10, 2010

1. LLR Analysis Workshop John Chandler CfA 2010 Dec 9-10

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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 2nd ring)

17. 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

18. 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

19. 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)

20. 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)

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

22. Proper time/coordinate time • Diurnal term from <site>·<velocity> • 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”

23. 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

24. 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

25. 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

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

27. 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

28. 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

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

30. 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