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 EMr

),,( ψ reflectorr

p

Further derivatives

Parameter estimation

p

f

p

f EM

EM

ψ

ψ

r

r,

Derivatives of orbit/rotation with respect to p

General - LUNAR

observations EO Ps

part. derivativesparam etersephem erides

EPH EM ER PAR ABL

PAR M O N D

…

Ephemeris integration

param etersephem erides

EPH EM ER

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

00Y

00Y

00Y

00Y

042Y

202

Y

4042Y02Y

02Y02Y

00Y

00Y

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

00Y00Y

202

Y

4042Y4042Y20

2Y

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)

20 2

1tGtGGG

)/( IG MM

Partial derivatives integration

part. derivativesparam eters

PAR ABL

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

ppEM

ψr

,pp

EM

ψr

,

00Y

3032Y

Parameter estimation

observations EO Ps

part. derivativesparam etersephem erides

PAR M O N D

…

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

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