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## PowerPoint Slideshow about ' LLR Analysis Workshop' - xerxes-rogers

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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

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