LLR Analysis Workshop
-
Upload
xerxes-rogers -
Category
Documents
-
view
31 -
download
2
description
Transcript of LLR Analysis Workshop
![Page 1: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/1.jpg)
LLR Analysis Workshop
John Chandler
CfA
2010 Dec 9-10
![Page 2: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/2.jpg)
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
![Page 3: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/3.jpg)
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
![Page 4: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/4.jpg)
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
![Page 5: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/5.jpg)
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
![Page 6: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/6.jpg)
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
![Page 7: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/7.jpg)
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
![Page 8: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/8.jpg)
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
![Page 9: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/9.jpg)
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
![Page 10: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/10.jpg)
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
![Page 11: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/11.jpg)
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
![Page 12: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/12.jpg)
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
![Page 13: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/13.jpg)
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
![Page 14: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/14.jpg)
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
![Page 15: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/15.jpg)
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
![Page 16: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/16.jpg)
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)
![Page 17: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/17.jpg)
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
![Page 18: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/18.jpg)
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
![Page 19: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/19.jpg)
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)
![Page 20: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/20.jpg)
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)
![Page 21: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/21.jpg)
Planetary lander coordinates
• Modeled planet orientation in proper time + body-fixed coordinates
• Mars orientation includes precession and seasonal variations
![Page 22: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/22.jpg)
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”
![Page 23: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/23.jpg)
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
![Page 24: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/24.jpg)
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
![Page 25: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/25.jpg)
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
![Page 26: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/26.jpg)
Integrated E-M-bary partials
• Mass(planets, asteroids, belt)
• Asteroid densities
• RELFCT, Ġ, Sun J2, metric β,γ
• Planet orbital elements
• EP violation
![Page 27: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/27.jpg)
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
![Page 28: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/28.jpg)
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
![Page 29: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/29.jpg)
Partial derivatives of observations
• Integrated partials computed by chain rule
• Non-integrated partials computed according to model
• Metric β,γ are both
![Page 30: LLR Analysis Workshop](https://reader036.fdocuments.net/reader036/viewer/2022062516/56812bc5550346895d900a1e/html5/thumbnails/30.jpg)
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