Workshop Page

Last Updated 2010-12-07 at 21:30 UTC

1. Basic underlying theory and coordinate system(s)

Question PEP JPL (see note below) Hannover Paris
A. What coordinate origins are available in your code? Geocentric, heliocentric, and solar-system barycentric. Chosen according to accuracy requirements. Solar system barycentric coordinates are used in the integrator and for basic computation in fits. Integrator can also do Sun centered coordinates. Station positions are geocentric and reflector positions are Moon centered. Geocenter, solar system barycenter, selenocenter Solar System barycentric coordinates
B. What coordinate system do you use for LLR analysis? General relativistic isotropic. Solar system barycentric coordinates or their Earth-Moon differences are used for fits after space and time transformations. Mean ecliptical system J2000, origin in geocenter. Solar system barycentric coordinates or their Earth-Moon differences
C. What formulation do you use to transform times (e.g., the LLR time measurements in UTC)? Proper time (UTC) is converted to coordinate time (CT, a constant offset from TDB) via either an analytic approximation or an integrated time ephemeris, according to the accuracy requirements. Station UTC → TDB. Moyer dot product formulation used for time transformation. UTC > TAI > TT > TDB (Hirayama),
geocentric proper time TCG		 Transformations from UTC to UT1 or TT according to IERS Conventions 2003. Between TT and TDB, use of the transformation integrated together with equations of motion.
D. How do you handle mass contributions from kinetic energy of bodies? Implicit in the general relativistic n-body equations of motion. EIH N-body equations of motion. EIH N-body equations of motion. EIH N-body equations of motion.
E. What relativistic corrections do you apply in tandem with coordinate transformations (e.g., Lorentz contraction of Earth)? Lorentz contraction of Earth and Moon. Vector Lorentz transformation, constant (L_C for Earth) and potential on scale (including annual variation) for both Earth and Moon. Lorentz and Einstein contraction for Earth and Moon. None in the dynamical part, Lorentz and Einstein contraction for Earth and Moon in the LLR reduction of data.
F. Do you account for relativistic contributions from Earth rotation affecting position of stations in solar system? Can be thought of as skew in Lorentz transformation. Other than the obvious direct effect of diurnal rotation, no. Yes, for the big effect. Was something more subtle intended? No. No.
G. Is eccentricity of earth's orbit taken into account in the rescaling of proper distances of earth positions due to (1 + γ Usun/c²) factor? No. Yes, the Sun's distance is used in the potential so annual and smaller effects are present. Yes. Yes.
H. What is used for orientation of Earth for gravity field perturbation? Does orbit integration include figure-figure effects or other small interesting effects? Is there an anomalous eccentricity rate in the integration or solution? Earth orientation given by precession and IAU 2000 nutation series. Figure-figure interaction not included in orbit. No anomalous eccentricity rate. Orientation has constant angles, precession and obliquity expressions, and 18.6 yr nutation. No figure-figure effect on orbit. Anomalous eccentricity rate is in solution, but not in integration. Precession, Nutation, obliquity, GMST and the difference to the Earths principal axis system. Figure-figure effects only in the rotation of the Moon. No eccentricity rate taken into acount. Orientation is integrated together with other equation of motion. Figure-Figure effect only in the rotational motion of the Moon. No eccentricity rate taken into account.

Note: For JPL integrator documentation, see Standish and Williams, Chapter 8 for the Explanatory Supplement. The DE421 memos contain information on the solution used. Our 2001 paper discusses dissipation in the Moon.

2. Translational motions of solar system bodies

Question PEP JPL Hannover Paris
A. How do you handle post-Newtonian dynamics in your equations of motion? N-body equations of motion as shown in Will (1981) for isotropic coordinates. EIH equations of motion for Moon and planets. N-body Einstein-Infeld-Hoffmann equations of motion. EIH equations of motion for Moon, planets and 5 asteroids
B. How many asteroids do you treat as individuals, and how many diffuse distributions do you include to describe the asteroid population? 98 asteroids plus one ring. 8 of the 98 asteroid masses are estimated separately while the other 90 are modeled by five density classes with individual radii assigned from the best available information. About 300 asteroids, no ring. See Folkner et al DE421 memo. 7 asteroids as point masses. About 300 asteroids, one ring. Its orientation is integrated together with equations of motion.
C. Do you include any natural satellites of planets? Only the Moon. No planetary satellites other than Moon. Only the Moon. Only the Moon
D. Do you handle radiation pressure on Earth/Moon? How? Yes, assuming uniform albedo on each body, neglecting thermal inertia. No radiation pressure in integrator. Not in integrator. In the modelled range by adding 3.7mm*cos(synodic angle), as shown in Vokrouhlicky(1997) No.

3. Models of motions of bodies about their centers of mass

Question PEP JPL Hannover Paris
A. In the context of planet/system rotation, do you switch to a proper time description at the body that varies according to position? Yes for Mars only (the rotation being modeled as an essentially uniform angular velocity in proper time). The rotation of the Earth is determined empirically, rather than parametrically, and is thus tied to the time "flow" of the observations. Similarly, the integrated rotation of the Moon is tied to the coordinate time system of the integration. No proper time for Moon rotation. Earth stations use standard UTC→TDB. Moon retroreflectors and rotation use TDB. Other bodies' data analysis and rotations are in planetary software, not LLR software. Earth rotation empirically, neglected for the Moon. No.
B. From where do you get Earth orientation information? IERS UT1/PM from a JPL file. Precession, obliquity and nutations come from theories, but we solve for constant orientation angles, rates and 4 nutation periods (18.6 yr, 9.3 yr, 1 yr, 1/2 year) × 4 coefficients each. IERS C04 series Integrated together with equations of motion in the dynamical part. Fixed to IERS 2003 + C04 series in the reduction to observations.
C. What is your Earth gravity model (degree and order, which if any terms are fit parameters in your model)? Through fourth degree, zeroth order. No coefficients solved for. Values from SAO (1973) standard Earth model. Zonals degrees 2-4 for Earth. We are adding J2 and its secular rate as possible fit parameters, but that is not operational yet. Complete second degree, 3rd and 4th zonals. No coefficients solved for. Values from EGM 2008 Through fourth degree, zeroth order. J2 and J3 are solved for, J4 is fixed to EGM96 value.
D. What is your lunar gravity model (degree and order; which terms are fit vs. used from external sources)? What bodies apply torque to the Moon through the gravity field? Through third degree, third order. Solving for J2, J3, C22, C31, C32, C33, S31, S32, S33. Others taken to be zero. For Moon, J2, and degree,order 3,0-3 and 4,0-4. J2 and libration beta and gamma are used to compute C22. J2, beta, gamma, and all third-degree coefficients can be fit. LP150Q is source for some gravity coefficients, others are fit; fixed vs fit can vary. See DE421 memo for that set. Direct torques are from Earth, Sun, Venus, Mars, and Jupiter. Complete up to fourth order from LP165P. Solving for C20, C22, C30, C31, C32, C33, S32. Torques from Earth and Sun. Through fourth degree, fourth order. Solving for J2, C22, J3, C31, C33, S31, S32. Other fixed to LP150Q values.
E. How do you handle Earth tidal dissipation? Earth orientation is handled purely empirically. The effect of Earth tidal drag on the lunar orbit is modeled by a solved-for lag angle on the tidal response. The Standish and Williams paper gives our integrator Earth tide expression from Moon and Sun with 3 degree-2 Love numbers and 3 time delays (semidiurnal, diurnal, and zonal). We lately changed the diurnal and semidiurnal time delays to allow for linear frequency dependence adding 2 more parameters. The 5 time-delay parameters can be fit, but only 2 are typically fit. Solved for an lag angle. Variations of its inertia matrix and degree-2 potential coefficients are computed from delayed positions of tide generating bodies. 3 Love numbers and 3 time delays are taken into account.
F. What dissipation model do you have for describing lunar orientation? many fit parameters? Three different models. One uses constant time-delay response (two parameters); one has constant-Q response (two parameters, exclusive of previous model); one has a fluid core loosely coupled to the mantle (five parameters). In the integrator, we have a degree-2 Love number and time delay for tides on Moon, plus a fluid-core/solid-mantle boundary (CMB) dissipation term. During fits we can solve for or constrain up to 5 out-of-phase periodic libration terms to account for tidal dissipation vs frequency; currently we fit 3, constrain 1, and do not solve for 1. See our 2001 paper. Two models. constant time delay (two parameters) and a fluid core model (two parameters), not solved for core parameters at the moment Similar to Earth's one, but with only one Love number (k2) and one time delay (τ) for second degree harmonics, both fitted.
G. Do you handle geocenter motion, as seen by SLR? If so, what model/input? No. No geocenter motion, but we solve for 3-D vector rates for stations, which should absorb any secular geocenter rates. No. No.
H. What is tidal model for physical librations? Which bodies cause tides? What fluid core effects are present? Does integrator have figure-figure terms? Are there relativistic rotation or other effects? See 3F for tidal model and fluid core. Only Earth is considered for lunar tides. Figure-figure interaction is included in libration integration. No relativistic rotation effects. In the libration integration, solid-body tides on the Moon are from Earth, but not from Sun, which is <1% of Earth. We also have spin distortion, though that has very small variation. Libration model has fluid core moment, CMB dissipation and CMB oblateness. We have Earth-Moon degree-2 figure-figure effects, but no relativistic rotation effects. see 3F, time delayed variation of the tensor of inertia. Tides from Earth and Sun. Fluid core moment, CMB dissipation and Earth-moon degree2 figure-figure effects. Tidal model: see 3F. Tides are generated by Earth and Sun. No fluid core. Geodesical torque is taken into account.

4. Models of motions of Earth's surface

Question PEP JPL Hannover Paris
A. What is your Earth tidal model? What degree/order? How many Love numbers? Which ones are fit vs. used from external sources? Degree-independent response to perturbing potential characterized by two Love numbers and a time lag. The three parameters are fit. Degree-2 tidal displacements on Earth are from Moon and Sun with 2 displacement Love numbers. There is also a K1 amplitude correction and a pole tide contribution. Displacement Love numbers are not fit. IERS 2003 tidal corrections, 4 love numbers (2nd and 3rd degree) which are not fitted.
IERS 1992 tidal corrections due to polar motion		 IERS Conventions 2003 (V. Dehant's subroutine)
B. Do you handle continental drift, post-glacial rebound, and other local motions? How many parameters per station? Are these fit or used from external sources? Yes, Three solved-for velocity components per station. We can solve for 3 coordinate rates per station. We currently fit rates for McDonald, OCA and APO, while Haleakala and Matera motions are fixed to plate motion model. Yes, three solved-for velocity components per station. APOLLO modeled by NUVEL1A model Tectonic plate motion are taken into account. Values are fixed to ITRF's ones, the possibility to fit them is not used.
C. Do you model the fiducial point of the telescope? If necessary. The McD 2.7 m data has a constant bias applied to bring ranges to the intersection of axes. I believe the MLRS applies a range correction to refer to a marker. Other stations we assume intersection of axes when not told otherwise. We d o not apply any eccentricities. If necessary. No.
D. Do you handle thermal expansion of the telescope? If necessary. No telescope thermal expansion is modeled, but we would be interested if the stations have estimates or suggestions. Thermal expansion of telescope length should already be accounted for if there is internal calibration, but mount motion is open for improvement. No. No.
E. Do you handle atmospheric loading? No. No. Yes. Yes.
F. Do you handle ocean loading? No. No. Yes. Yes.
G. Do you handle hydrologic loading? No. No. No. No.

5. Models of motions of Moon's surface

Question PEP JPL Hannover Paris
A. What is your lunar tidal model? What degree/order? How many Love numbers? Degree-independent response to perturbing potential characterized by two Love numbers and a time lag. The three parameters are fit. Retroreflector tidal displacement has 2 degree-2 Love numbers. We normally solve for h2 and fix l2. Tides are from both Earth and Sun, but not from spin va riation. Degree-independent response to perturbing potential characterized by one Love number and a time lag. The two parameters are fit. Tides from Earth and Sun. Simplified version of IERS 2003's model. Two Love numbers h2 and l2, no time lag.
B. Do you handle thermal expansion of the reflectors on the Moon? No. No retroreflector thermal expansion. That should be significant for Lunokhod 1. No. No.

6. Partial derivatives

Question PEP JPL Hannover Paris
A. Do you obtain partial derivatives via integration of the variational equations, or use numerical difference methods. Integration. We have three programs for dynamical partials: 1 integrates variational equations and 2 use finite differences, where one differences separate integrations and the other integrates two cases simultaneously. Both methods. Numerical differences only.
B. If both are used, when is each method used, generally? Numerical differences are used only for checking the accuracy of the integrated partials. Long-time existing dynamical partials tend to be from variational equations, while newer partials are going into a finite difference program. We have had noise build up for some partials during the variational equations integration and have switched some partials. Numerical derivatives are used for relativistic parameters. N/A
C. For dynamical and geometrical partials, are the up and down legs calculated separately or is one leg calculated and doubled? Calculated Separately I think we do partials for both up and down legs in most cases, but we would have to check each partial in the code. Some partials, e.g. biases, do not depend on the up or down leg. One leg calculated and doubled. Calculated separately

7. Algorithms for integration

Question PEP JPL Hannover Paris
A. What is your integration method or methods? 15th-order Adams-Moulton. The starting procedure uses the Nordsieck method. We use the Krough variable step size variable order integrator running quadruple precision. Adams-Bashfort multistep with variable step size 12th-order Adams PECE. The starting procedure uses ODEX.
B. Do you simultaneously integrate the solar system and Moon? No. Do simultaneous integration of Moon, planets, Pluto and physical librations. Yes. Yes.
C. If not, how are these products merged? Three iterations of the cycle, first taking the Moon as given while integrating the rest of the solar system and then taking the solar system as given while integrating the Moon. The initial Moon is the Brown mean lunar theory. N/A N/A N/A
D. If not, what are the epochs for the two integrations corresponding to initial conditions? The same: May 1968. N/A N/A N/A
E. What is your integration step size (or sizes)? 1/8 day for the Moon, 1/2 day for the solar-system n-body integration, various sizes for other single-body integrations. Variable step size. Variable, output every 0.3 days. Fixed, about 0.055...
F. Do you simultaneously integrate the Moon orbit and rotation? Yes. Do simultaneous integration of Moon, planets, Pluto and physical librations. Yes. Yes.
G. If not, how are these products merged? N/A N/A N/A N/A
H. Do you determine the location of the Sun by integrating the equations of motion,or impose center of mass to dictate the Sun's position? Impose fixed center of mass. We use a relativistic center of mass expression to get the Sun. By integrating the equations of motion. Integrating its equations of motion.

8. Observables

Question PEP JPL Hannover Paris
A. What relativistic corrections do you apply to light propagation? Shapiro delay due to the Sun and, in the case of cislunar or lunar targets, due to the Earth, but not due to the Moon. Shapiro time delay is calculated for gravity of Sun and Earth. I thought we did Moon too, but do not see it in our code. Shapiro delay due to Sun and Earth. Shapiro delay due to the Sun and to the Earth, but not due to the Moon.
B. Do you apply the Shapiro light-time correction iteratively or in a single step? Are up and down legs computed separately, or is one leg calculated and doubled? Single step. Shapiro delay and atmospheric delay are iterated for up leg, but not for do wn leg (which is known much better at first iteration). Single Step. Iteratively. Up and downleg computed separately.
C. What is your atmospheric propagation model, and what input parameters do you require from the observation? Mendes and Pavlis (2004). Pressure, temperature, relative humidity -- obtained from site measurements when available and assumed to be nominal otherwise. Marini and Murray model is used for atmospheric delay. We use temperature, pressure, humidity and laser wavelength. Elevation angle is calculated internal ly. Mendes and Pavlis (2004). Pressure, temperature, relative humidity. Marini and Muray (1973), pressure, temperature, relative humidity, laser wavelength
D. How do you handle range bias? Do you have multiple bias parameters for each station (how many)? Solve-for constant bias parameters. As many parameters per station as needed (as determined by history of changes in measurement system). We have overall (full time span) range biases for stations MCD (2.7 m), OCA, HAL and APO plus shorter time biases where we suspect them from residuals or station information. Haleakala had several sets of biases depending on changes in different rings of lenses. We know the dates for those changes. Currently, our solutions do not solve for an overall bias for the MLRS using it as the zero standard. Our present solutions use 40 bias parameters, 39 biases and one slope. With zero overall MLRS bias, the other four overall biases are within ± 0.2 nsec. Solve-for constant (full time span) and time dependent bias parameters. All together 43 bias parameters and one slope. Possibility to solve for 40 biases (constant and drift for each). But only 2 constants are used up to now.
E. In order to extract LLR science, what additional observational inputs does your model process (this is separate from the use of importing analysis results, as in other questions)? Delay or delay-rate observations of other solar-system bodies, such as radar to Mercury and Venus, orbiter and lander tracking data on Mars, and flyby normal points for Jupiter and Saturn. Station specific inputs, in addition to range, range uncertainty and time, are pressure, temperature, humidity and laser wavelength. We can scale the normal point uncertainty with a factor and also add RSS noise. This adjusted uncertainty is used for weighting during the least-squares fit. The UT/PM table has uncertainties that limit adjustment of a stochastic UT/PM correction subject to a correlation time. Meteorology. Some parameters involved in the dynamical part are first fitted to planetary observations. Then a fit to LLR data is done, checking that it does not deteriorate planetary residuals.
F. What weighting scheme do you typically employ for your data sets? Do you scale errors, convolve them with a constant, etc.? Weighted least-squares using reported errors, except that errors may be scaled to balance residuals. We use weighted least-squares fits. We can fit LLR data alone or LLR + planetary, combining files from the planetary programs. Usually it is the former since the Earth's orbit is well known now, but for a public ephemeris like DE421 it is the latter. Linear constraints between parameters can be imposed. There are no a priori uncertainties. Matrix inversion uses a square root computation. We get solution parameter corrections and uncertainties. We also get the correlation matrix, and normal point residuals from linear corrections. After the solution we can use the residuals to get the weighted rms residual overall or by year with a choice of station and retroreflector. Amplitude spectra of residuals can be co mputed with a periodogram. We use an iterative, weighted least squares adjustment of the LLR data. The weighting is calculated with respect to the Normal Point uncertainty for every single observation. We can scale this uncertainty, e.g., depending on station or observation time. We introduce no a priori uncertainties. We get normal point residuals, correlation matrix, corrections to the solution parameters and their uncertainties. For LLR, weighting of a set of data is computed according the standard deviation of the residuals. Things are different for planetary observation (more information on request).

11. From Bob Reasenberg

Question PEP JPL Hannover Paris
A. Do you estimate the parameters of the lunar model simultaneously with the parameters of the solar-system model? (This requires incorporating the solar-system data in the LLR analysis, of course.) Yes. N/A No. First fit is done to planetary observations, then to LLR's ones.
B. What options do you have for working with and solving the normal equations (e.g: a priori parameter values, with an uncertainty for single parameters or a covariance for a related group; rank deficient solution; elimination of "uninteresting parameters" without distorting the estimate or covariance)? Parameters can be constrained with individual uncertainties or with full covariance of related parameters. Rank-deficient matrices can be inverted by eigenvalue elimination. Uninteresting parameters can be projected away in a preliminary stage, leaving a reduced set of normal equations that yields correct estimates and covariances for the interesting parameters. See 8F LLR fit: no options, just solving normal equations. For planetary fit, more options are available (more information on request).
C. What tools do you have for examining post-fit residuals (e.g., Fourier transform or other spectral analysis)? FFT, weighted Fourier transform, histograms. Nothing special, MatLab standard tools. None is used up to now.
D. What parameters can your software estimate? This is likely to be a very long list, with batches of parameters requiring explanatory notes. Briefly: masses, asteroid class densities, initial conditions, station positions and velocities, coordinates of fixed targets on Moon or planets, Earth precession and nutation coefficients, piecewise linear adjustments to polar motion and UT1, gravitational harmonic coefficients for all bodies (limited to J2 for the Sun), Moon moment-of-inertia ratios, Moon core and tidal dissipation parameters mentioned above, observation biases, radii and topography grids (and spins) for other planets, interplanetary plasma density, metric parameters beta and gamma, EP violation, de Sitter precession rate, G-dot, ad hoc coefficients of post-Newtonian terms in the integrated motions and in light propagation delay and in transformation between proper time and coordinate time. Info provided; will display separately. All together: about 220 can be solved, but not all are estimated in every analysis. I will prepare a list. For LLR: 188 can be solved! But only 59 of them are fitted to LLR: initial conditions, stations and reflector's positions, potential coefficients, tides parameters, biaises, ... Other parameters can be fitted to planetary observations: asteroid masses, Sun's J2, initial conditions ... more information on request.

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