obliquity-oblateness feedback at the Moon
Bruce G. Bills1
with help from
William B. Moore2
Matthew A. Siegler3
William I. Newman3
1Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA
2Department of Atmospheric and Planetary Sciences, Hampton University, Hampton, VA
3Department of Earth and Space Sciences, UCLA, Los Angeles, CA
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 theearly gravity field
• past lunar response is unknown
• we examine several possible histories
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
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
s
ks
nk
outline
• Bill Ward’s lunar obliquity history
• what is a Cassini state?
• basics of orbit and spin precession
• influence of tides on obliquity history
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?
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
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
2/52
2
22
3
m
m
eem aW
a
RJn
td
d
2/31]cos[
4
3me
m
ee aWn
nn
td
d
spin pole precession
• orbit pole precesses about ecliptic pole
• spin pole precesses about the orbit pole
with rate parameters
nkdt
ndˆˆˆ
snsndt
sdˆˆˆˆ
ˆ
n k
s n
c
abacn
c
CJn
4
)()(4
2
3
2
3 2,22
c
abn
c
Cn
82
3
22
3 2,2
2/n is orbital period
{a, b, c} are dimensionless principal moments
(connection to gravity)
spin pole precession
• in orbit-fixed reference frame, spin pole motion is
• along spin trajectory, Hamiltonian is constant
sksnsndt
sdˆˆˆˆˆˆ
ˆ
sksnsnH ˆˆˆˆˆˆ2
2
constraints on spin pole unit vector
• unit vector:
• Hamiltonian (energy)
what is a Cassini state?
1222 zyx sss
2
2
1bssaH zx
with
]cos[]sin[ i
bi
a
s
(parabola)
what is a Cassini state?
“a” is radius of curvature at vertex
“b” is position of axis
for given “a” and “b” thereis a family of parabolas, each with a different H, or energy
what is a Cassini state?view in x-z plane
when parabola intersectssphere at tangent point,
spin pole trajectory collapsesto a fixed point
what is a Cassini state?transition from 4 to 2 steady states
near to transition
when the radius of curvatureat the state 4 intersection pointequals 1, states 1 and 4 merge
for larger radii, only 2 states exist
transition criterion:
13/23/2 ba
what is a Cassini state?transition from 4 to 2 steady states
view in xy-plane
when state 1 disappears,dissipation will drivespin pole to state 2
what did Ward leave out?
lunar gravity field (J2 and C2,2)– influences
• spin precession rate• obliquity
– depends upon• distance from Earth (included)
• obliquity (not included)
primary connections
orbit
spin direction
obliquity
gravity field
spin ratetides
spin direction
spin rate
orbit
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
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
hydrostatic contribution: at distance a and obliquity
hydrostatic model for lunar gravity
],[],[ 12 aFaJ
],[],[ 22,2 aFaC
4
3
2
1
]2/cos[6
]2cos[9811
16
1
],[
],[
q
a
R
qaF
aF
where q = Mm/Me = 1/81.3 is mass ratio
Moon is far from hydrostatic
• gravity coefficients
• current values:
• hydrostatic values:
22
2/)(
RM
BACJ
62 1007.067.203 J
22,2 4 RM
ABC
62,2 1001.019.22 C
62 1038.9 J
62,2 1083.2 C
simple model for past variation
212 ],[],[ JaFaJ
2,222,2 ],[],[ CaFaC
hydrostatic plus constant offset
offsets: difference between observed and current hydrostatic
600222 1039.194],[ aJJJ obs
6002,22,22,2 1041.19],[ aCCC obs
less simple models
02,222,2
0212
],[],[
],[],[
a
aCaFaC
a
aJaFaJ
hydrostatic plus linear offset
hydrostatic plus quadratic offset
2
02,222,2
2
0212
],[],[
],[],[
a
aCaFaC
a
aJaFaJ