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Institut für Erdmessung Hannover LLR analysis software „LUNAR“ Franz Hofmann, Jürgen Müller, Institut für Erdmessung, Leibniz Universität Hannover 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 ψ ( , , ) rreflector Earth-Moon-Vector rEM f rEM f ψ , rEM p ψ p Derivatives of orbit/rotation with respect to p Further derivatives Parameter estimation p General - LUNAR EPHEMER PARABL parameters ephemerides part. derivatives PARMOND observations EOPs… Ephemeris integration EPHEMER ephemerides parameters 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 Y20 4 Moon Y00 Moon Y2044 Earth Y00 0 Y00 Earth Y2 Sun Moon Y20 2 Sun Y00 Sun Y20 Earth, Moon Y00 Sun Y20 Mercure to Saturn Y00 Tidal acceleration Ephemeris integration - rotation Lunar orientation - Integrated together with translational motion - Basis: Euler equations - Torques from Earth and Sun • Earth Y00 Moon Y2044 • Sun Y00 Moon Y2044 • Earth Y20 2 Moon Y20 2 - 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 Ephemeris integration Further model extensions (implemented, e.g. for special tests) 1 2 - Time variable G: G G0 G t G t 2 - Geodetic precession of the lunar orbit in addition to EIH - Violation of equivalence principle ( M G / M I ) - 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) Partial derivatives integration PARABL parameters part. derivatives Partial derivatives integration Dynamical partials of orbit/rotation - rEM ψ r ψ , determined by integrating EM , , 414 derivatives p p p p - 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 Y00 Moon Y2033 Parameter estimation parameters ephemerides part. derivatives PARMOND observations EOPs… 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. 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 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 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 Thank you for your attention