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

Contents  General  Ephemeris integration  Integration of partial derivatives  Parameter estimation

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)

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

General - LUNAR …

Ephemeris integration

 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 Ephemeris integration – translational motion

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 (k 2 ) -Dissipation: time delay – only effect from Earth -Fluid core moment, CMB dissipation  Earth orientation -Empirically -Precession, nutation according to IAU resolutions GMST with offset to the principal axis system

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/r 2 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)

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 Partial derivatives integration

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 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)

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

Parameter estimation  Estimation process -Weighted least squares adjustment -We use ca NP up to now  how many NP exist?  CDDIS approx 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

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 k 2  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

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 k 2 and time lag -GM EM -C20 sun (fixed to -2x10 -7 ) -Relativistic parameters

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