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obliquity-oblateness feedback at the Moon Bruce G. Bills 1 with help from William B. Moore 2 Matthew A. Siegler 3 William I. Newman 3 1 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 2 Department of Atmospheric and Planetary Sciences, Hampton University, Hampton, VA 3 Department of Earth and Space Sciences, UCLA, Los Angeles, CA

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view of south polar region: permanent shadow regions

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summary obliquity is the angular separation of spin and orbit poles, and controls polar radiation balance during the Moon’s orbital evolution away from Earth, lunar obliquity has changed significantly details of that change depend upon the degree two lunar gravity field tides and spin rate variations perturbed the early gravity field past lunar response is unknown we examine several possible histories

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history of lunar obliquity studies 1693 Giovanni Domenico Cassini –announced 3 observed “laws” of lunar rotation 1966 Giuseppe (Bepi) Colombo –explained Cassini’s laws 2 and 3 1969 Stan Peale –generalized to triaxial case 1975 Bill Ward –applied theory to past lunar history

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Cassini’s laws 1.spin rate equals mean orbit rate 2.spin pole maintains a constant inclination to ecliptic pole 3.spin pole, orbit pole, and ecliptic pole, all remain coplanar

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outline Bill Ward’s lunar obliquity history what is a Cassini state? basics of orbit and spin precession influence of tides on obliquity history

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William R. Ward, Past orientation of the lunar spin axis, Science, 189, 377-379, 1975.

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why did the Moon do that? what is a Cassini state? why is the Moon in such a state? what causes obliquity to change? what did Ward leave out?

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what is a Cassini state? in general, spin and orbit poles have complicated relative motion

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what is a Cassini state? in a tidally damped “Cassini state”, the spin pole adjusts distance from orbit pole, so as to remain coplanar with the other two poles

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two torques act on the lunar orbit plane: torque from Sun orbit pole precesses about ecliptic pole rate increases with distance from Earth torque from Earth’s oblate figure orbit pole precesses about Earth’s spin pole rate decreases with distance from Earth orbit pole precession

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spin pole precession orbit pole precesses about ecliptic pole spin pole precesses about the orbit pole with rate parameters 2 /n is orbital period {a, b, c} are dimensionless principal moments (connection to gravity)

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spin pole precession in orbit-fixed reference frame, spin pole motion is along spin trajectory, Hamiltonian is constant

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constraints on spin pole unit vector unit vector: Hamiltonian (energy) what is a Cassini state? with (parabola)

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what is a Cassini state? “a” is radius of curvature at vertex “b” is position of axis for given “a” and “b” there is a family of parabolas, each with a different H, or energy

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what is a Cassini state? view in x-z plane when parabola intersects sphere at tangent point, spin pole trajectory collapses to a fixed point

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what is a Cassini state? view in y-z planeview in x-y plane

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what is a Cassini state? transition from 4 to 2 steady states near to transition when the radius of curvature at the state 4 intersection point equals 1, states 1 and 4 merge for larger radii, only 2 states exist transition criterion:

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what is a Cassini state? transition from 4 to 2 steady states view in xy-plane when state 1 disappears, dissipation will drive spin pole to state 2

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Moon at Cassini-state transition a = 0.7432 b = 0.07605 constant gravity case

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what did Ward leave out? lunar gravity field (J 2 and C 2,2 ) –influences spin precession rate obliquity –depends upon distance from Earth (included) obliquity (not included)

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primary connections orbit spin direction obliquity gravity field spin ratetides spin direction spin rate orbit

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tidal and rotational gravity as the Moon moved away from Earth, –the tidal and rotational potentials changed, –which changed the lunar mass distribution, –which changed the spin precession rate, –which changed the obliquity, –which changed the tidal potential….. obliquity-oblateness feedback

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tidal and rotational gravity rotation flattens Moon –symmetric about spin axis –faster rotation yield more flattening tides stretch Moon –symmetric about Earth-Moon line –stronger when close –obliquity “smears” the pattern

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path of sub-Earth point at large obliquity

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hydrostatic contribution: at distance a and obliquity hydrostatic model for lunar gravity where q = M m /M e = 1/81.3 is mass ratio

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Moon is far from hydrostatic gravity coefficients current values: hydrostatic values:

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simple model for past variation hydrostatic plus constant offset offsets: difference between observed and current hydrostatic

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less simple models hydrostatic plus linear offset hydrostatic plus quadratic offset

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lunar gravity: constant bias case

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lunar gravity: linear bias case

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lunar gravity: quadratic bias case

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conclusions obliquity during Cassini state transition – very different that at present –strongly dependent upon lunar gravity prospects for improving knowledge –grim

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