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

– Optional geocentric approximation• Not in integrations, only in observables

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

• “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

• 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

• Moon integrations are allowed in “Moon units” in deference to traditional expression of lunar ephemerides in Earth radii – not used anymore

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

• 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

• 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

• 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

• 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

• 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

• 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)

• 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

• 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

• 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)

• 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

• 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

• 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

• 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

• Modeled planet orientation in proper time + body-fixed coordinates

• Mars orientation includes precession and seasonal variations

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

• 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

• 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

• 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

• Mass(planets, asteroids, belt)

• Asteroid densities

• RELFCT, Ġ, Sun J2, metric β,γ

• Planet orbital elements

• EP violation

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

• 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

• Integrated partials computed by chain rule

• Non-integrated partials computed according to model

• Metric β,γ are both

Solutions

• 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