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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