Hannover LLR analysis software „LUNAR“

1. Franz Hofmann, Jürgen Müller, Institut für Erdmessung, Leibniz Universität Hannover Hannover LLR analysis software „LUNAR“ Institut für Erdmessung

2. Contents • General • Ephemeris integration • Integration of partial derivatives • Parameter estimation

3. General • Coded in FORTRAN90, quadruple precision • Integrator • Adams-Bashfort algorithm • Multi step integration method • Variable step size • Output every 0.3 days • Coordinate systems • Barycentric ecliptical for ephemeris and analysis • Stations geocentric (ITRF) • Reflectors selenocentric (principal axis system) • Time • UTC  TAI  TT  TDB (Hirayama + station dependent term)

4. Derivatives of orbit/rotation with respect to Further derivatives Parameter estimation  General - LUNAR Ephemerides of the Moon (solar system) Eulerian angles Earth-Moon-Vector

5. General - LUNAR …

6. Ephemeris integration

7. Ephemeris integration – translational motion • Integration of EIH equations of motion • Barycentric ecliptical system • Sun, Moon, all planets, Ceres, Vesta, Pallas, Juno, Iris, Hygiea, Eunomia • Inititial values planets: DE421 • Initial values Asteroids: JPL/Horizons (DE405) • No radiation pressure • Additional non-relativistic accelerations • Earth  Moon • Moon  Earth • Earth  Sun • Moon  Sun • Sun  Earth, Moon • Sun  Mercure to Saturn • Tidal acceleration

8. Ephemeris integration - rotation • Lunar orientation • Integrated together with translational motion • Basis: Euler equations • Torques from Earth and Sun • Earth  Moon • Sun  Moon • Earth  Moon • Relativistic torques (geodetic and Lense-Thirring) from Sun and Earth • Elasticity: variation in the tensor of inertia with one Love number (k2) • Dissipation: time delay – only effect from Earth • Fluid core moment, CMB dissipation • Earth orientation • Empirically • Precession, nutation according to IAU resolutions 2006 • GMST with offset to the principal axis system

9. Ephemeris integration • Further model extensions (implemented, e.g. for special tests) • Time variable G: • Geodetic precession of the lunar orbit in addition to EIH • Violation of equivalence principle • Acceleration due to dark matter in the galactic center (violation of equivalence principle) • Yukawa term for modifying Newtons 1/r2 law of gravity • Preferred frame effects 1, 2 and metric parameters ,  (Will, 1993) • Gravitomagnetic effects (Soffel et al., 2008) • Optional spin-orbit coupling (Brumberg/Kopeikin)

10. Partial derivatives integration

11. Partial derivatives integration • Dynamical partials of orbit/rotation • determined by integrating , 414 derivatives • Therefore: calculating a simplified ephemeris • Only Newtonian equations of motion, Sun  Neptun point masses • Translational motion: Earth‘s, Moon‘s grav. field up to degree 3 • Tidal accelerations • Rotation: Earth  Moon

12. Parameter estimation …

13. Parameter estimation • Partials • Computation of complete derivatives from single contributions • Dynamical • Geometrical direct from observation equation (reflector/station coordinates) • Numerical (relativistic parameters) • Partials calculated at reflection time (Lagrangian interpolation, degree 10) and doubled • Modelling of the observed pulse travel time • Time-trafo UTC (NP)  TAI  TT  TDB (Hirayama + station dependent term which is not included in Hirayama) • Coordinate-trafo ITRF, SRF, barycentric • Ephemeris interpolation for transmission-, reflection-, reception-time with Lagrangian interpolation, degree 10

14. Parameter estimation • Computation of station coordinates + corrections • Earth‘s orientation with high accuracy (IERS Conv. 2003, C04): Pole coordinates, pole offsets, dUT1 with longperiodic, diurnal and sub-diurnal variations Precession + nutation (IAU resolutions 2006) • Longperiodic latitude variation (before 1983, Dickey et al., 1985) • Lunisolar tides of elastic Earth (IERS Conv. 2003) • Tidal effects due to polar motion (IERS 1992) • Ocean loading (IERS Conv.1996) • Atmospheric loading • Continental drift rates (NUVEL1A or estimated) • Lorentz and Einstein-contraction of coordinates (also reflector coordinates)

15. Parameter estimation • Reflector coordinates transformed with integrated Eulerian angles • Light propagation • Atmospheric time delay from Mendes and Pavlis (2004) • Shapiro delay due to Sun and Earth • Biases • Radiation pressure from Vokrouhlicky (1997) • Weighting • From normal point uncertainty for every single observation • Scaling is possible (e.g., station, time span) • Variance component analysis in preparation

16. Parameter estimation • Estimation process • Weighted least squares adjustment • We use ca. 17000 NP up to now  how many NP exist?  CDDIS approx. 12000 NP?  reference data set with all original observations • Outlier test by ratio residuals/accuracy of residuals (not in every iteration) • Iterative process (ephemeris integration  parameter estimation) • Output • NP residuals • Correlation matrix • Corrections to the parameters + uncertainties

17. Parameter estimation • Possible solve-for parameters: • Earth related parameters • Station coordinates (McDonald as one station with local ties) • Station velocity components • Biases for every station (whole time span) • Biases for shorter time spans • 4 nutation periods with 4 coefficients each (18.6yr, 9.3yr, 1 yr, ½yr) • Precession rate • Earth k2 for tidal acceleration • Additional rotations for transformation terrestrial  inertial • Corrections to initial Earth position and velocity • Coefficients for longperiodic latitude variation before 1983 • Optional pole coordinates for nights with > 10 normal points

18. Parameter estimation • Lunar related parameters • Lunar initial position, velocity, rotation vector, Eulerian angles • Lunar gravity field coefficients up to degree 4 (degree 4, S31, S33 fixed on LP165P values) • Reflector coordinates • Dynamical flattening  and  • Lunar k2 and time lag • GMEM • C20sun (fixed to -2x10-7) • Relativistic parameters

19. Thank you for your attention