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LLR Analysis Workshop. John Chandler CfA 2010 Dec 9-10. Underlying theory and coordinate system. Metric gravity with PPN formalism Isotropic coordinate system Solar-system barycenter origin Sun computed to balance planets Optional heliocentric approximation Explicitly an approximation

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llr analysis workshop

LLR Analysis Workshop

John Chandler


2010 Dec 9-10

underlying theory and coordinate system
Underlying theory and coordinate system
  • Metric gravity with PPN formalism
  • Isotropic coordinate system
  • Solar-system barycenter origin
    • Sun computed to balance planets
    • Optional heliocentric approximation
      • Explicitly an approximation
    • Optional geocentric approximation
      • Not in integrations, only in observables
free parameters
Free Parameters
  • Metric parameter β
  • Metric parameter γ
  • Ġ (two flavors)
  • “RELFCT” coefficient of post-Newtonian terms in equations of motion
  • “RELDEL” coefficient of post-Newtonian terms in light propagation delay
more free parameters
More Free Parameters
  • “ATCTSC” coefficient of conversion between coordinate and proper time
  • Coefficient of additional de Sitter-like precession
  • Nordtvedt ηΔ, where Δ for Earth-Moon system is the difference of Earth and Moon
units for integrations
Units for Integrations
  • Gaussian gravitational constant
  • Distance - Astronomical Unit
    • AU in light seconds a free parameter
  • Mass – Solar Mass
    • No variation of mass assumed
    • Solar Mass in SI units a derived parameter from Astronomical Unit
  • Time – Ephemeris Day
historical footnote to units
Historical Footnote to Units
  • Moon integrations are allowed in “Moon units” in deference to traditional expression of lunar ephemerides in Earth radii – not used anymore
numerical integration
Numerical Integration
  • 15th-order Adams-Moulton, fixed step size
  • Starting procedure uses Nordsieck
  • Output at fixed tabular interval
    • Not necessarily the same as step size
  • Partial derivatives obtained by simultaneous integration of variational equations
  • Partial derivatives (if included) are interleaved with coordinates
hierarchy of integrations i
Hierarchy of Integrations, I
  • N-body integration includes 9 planets
    • One is a dwarf planet
    • One is a 2-body subsystem (Earth-Moon)
    • Earth-Moon offset is supplied externally and copied to output ephemeris
    • Partial derivatives not included
  • Individual planet
    • Partial derivatives included
    • Earth-Moon done as 2-body system as above
hierarchy of integrations ii
Hierarchy of Integrations, II
  • Moon orbit and rotation are integrated simultaneously
    • Partial derivatives included
    • Rest of solar system supplied externally
  • Other artificial or natural satellites are integrated separately
    • Partial derivatives included
    • Moon and planets supplied externally
hierarchy of integrations iii
Hierarchy of Integrations, III
  • Iterate to reconcile n-body with Moon
  • Initial n-body uses analytic (Brown) Moon
  • Moon integration uses latest n-body
  • Moon output then replaces previous Moon for subsequent n-body integration
  • Three iterations suffice
step size and tabular interval
Step size and tabular interval
  • Moon – 1/8 day, 1/2 day
  • Mercury (n-body) – 1/2 day, 2 days
  • Mercury (single) – 1/4 day, 1 day
  • Other planets (n-body) – 1/2 day, 4 days
  • Earth-Moon (single) – 1/2 day, 1 day
  • Venus, Mars (single) – 1 day, 4 days
evaluation of ephemerides
Evaluation of Ephemerides
  • 10-point Everett interpolation
  • Coefficients computed as needed
  • Same procedure for both coordinates and partial derivatives
  • Same procedure for input both to integration and to observable calculation
accelerations lunar orbit
Accelerations – lunar orbit
  • Integrated quantity is Moon-Earth difference – all accelerations are ditto
  • Point-mass Sun, planets relativistic (PPN)
  • Earth tidal drag on Moon
  • Earth harmonics on Moon and Sun
    • J2-J4 (only J2 effect on Sun)
  • Moon harmonics on Earth
    • J2, J3, C22, C31, C32, C33, S31, S32, S33
accelerations lunar orbit cont
Accelerations – lunar orbit (cont)
  • Equivalence Principle violation, if any
  • Solar radiation pressure
    • uniform albedo on each body, neglecting thermal inertia
  • Additional de Sitter-like precession is nominally zero, implemented only as a partial derivative
accelerations libration
Accelerations – libration
  • Earth point-mass on Moon harmonics
  • Sun point-mass on Moon harmonics
  • Earth J2 on Moon harmonics
  • Effect of solid Moon elasticity/dissipation
    • k2 and lag (either constant T or constant Q)
  • Effect of independently-rotating, spherical fluid core
    • Averaged coupling coefficient
accelerations planet orbits
Accelerations – planet orbits
  • Integrated quantity is planet-Sun difference – all accelerations are ditto
  • Point-mass Sun, planets relativistic (PPN)
  • Sun J2 on planet
  • Asteroids (orbits: Minor Planet Center)
    • 8 with adjustable masses
    • 90 with adjustable densities in 5 classes
    • Additional uniform ring (optional 2nd ring)
accelerations planets cont
Accelerations – planets (cont)
  • Equivalence Principle violation, if any
  • Solar radiation pressure not included
  • Earth-Moon barycenter integrated as two mass points with externally prescribed coordinate differences
earth orientation
Earth orientation
  • IAU 2000 precession/nutation series
    • Estimated corrections to precession and nutation at fortnightly, semiannual, annual, 18.6-year, and 433-day (free core)
  • IERS polar motion and UT1
    • Not considered in Earth gravity field calc.
    • Estimated corrections through 2003
station coordinates
Station coordinates
  • Earth orientation + body-fixed coordinates + body-fixed secular drift + Lorentz contraction + tide correction
  • Tide is degree-independent response to perturbing potential characterized by two Love numbers and a time lag (all fit parameters)
reflector coordinates
Reflector coordinates
  • Integrated Moon orientation + body-fixed coordinates + Lorentz contraction + tide correction
  • Tide is degree-independent response to perturbing potential characterized by two Love numbers and a time lag (all fit parameters)
planetary lander coordinates
Planetary lander coordinates
  • Modeled planet orientation in proper time + body-fixed coordinates
  • Mars orientation includes precession and seasonal variations
proper time coordinate time
Proper time/coordinate time
  • Diurnal term from <site>·<velocity>
  • Long-period term from integrated time ephemeris or from monthly and yearly analytic approximations
  • One version of Ġ uses a secular drift in the relative rates of atomic (proper) time and gravitational (coordinate) time
  • Combination of above is labeled “CTAT”
chain of times epochs
Chain of times/epochs
  • Recv UTC: leap seconds etc→ Recv TAI
    • PEP uses A.1 internally (constant offset from TAI, for historical reasons)
  • Recv TAI: “Recv CTAT”→ CT
    • CT same as TDB, except for constant offset
  • Recv CT: light-time iteration→ Rflt CT
  • Rflt CT: light-time iteration→ Xmit CT
  • Xmit CT: “Xmit CTAT”→ Xmit TAI
  • Xmit TAI: leap seconds etc→ Xmit UTC
corrections after light time iteration
Corrections after light-time iteration
  • Shapiro delay (up-leg + down-leg)
    • Effect of Sun for all observations
    • Effect of Earth for lunar/cislunar obs
  • Physical propagation delay (up + down)
    • Mendes & Pavlis (2004) for neutral atmosphere, using meteorological data
    • Various calibrations for radio-frequency obs
  • Measurement bias
  • Antenna fiducial point offset, if any
integrated lunar partials
Integrated lunar partials
  • Mass(Earth,Moon), RELFCT, Ġ, metric β,γ
  • Moon harmonic coefficients
  • Earth, Moon orbital elements
  • Lunar core, mantle rotation I.C.’s
  • Lunar core&mantle moments, coupling
  • Tidal drag, lunar k2, and dissipation
  • EP violation, de Sitter-like precession
integrated e m bary partials
Integrated E-M-bary partials
  • Mass(planets, asteroids, belt)
  • Asteroid densities
  • RELFCT, Ġ, Sun J2, metric β,γ
  • Planet orbital elements
  • EP violation
indirect integrated partials
Indirect integrated partials
  • PEP integrates partials only for one body at a time
  • Dependence of each body on coordinates of other bodies and thence by chain-rule on parameters affecting other bodies
  • Such partials are evaluated by reading the other single-body integrations
  • Iterate as needed
non integrated partials
Non-integrated partials
  • Station positions and velocities
  • Coordinates of targets on Moon, planets
  • Earth precession and nutation coefficients
  • Adjustments to polar motion and UT1
  • Planetary radii, spins, topography grids
  • Interplanetary plasma density
  • CT-rate version of Ġ
  • Ad hoc coefficients of Shapiro delay, CTAT
  • AU in light-seconds
partial derivatives of observations
Partial derivatives of observations
  • Integrated partials computed by chain rule
  • Non-integrated partials computed according to model
  • Metric β,γ are both
  • Calculate residuals and partials for all data
  • Form normal equations
  • Include information from other investigations as a priori constraints
  • Optionally pre-reduce equations to project away uninteresting parameters
  • Solve normal equations to adjust parameters, optionally suppressing ill-defined directions in parameter space
  • Form postfit residuals by linear correction