Download Hofmann Talk - UCSD Department of Physics

Institut für Erdmessung
Hannover LLR analysis
software „LUNAR“
Franz Hofmann, Jürgen Müller, Institut für Erdmessung, Leibniz Universität Hannover
Contents
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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 Y20 4  Moon Y00
Moon Y2044  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 Y2044
• Sun Y00  Moon Y2044
• 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 Y2033
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)
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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