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
view of south polar region: permanent shadow regions
view of south polar region: permanent shadow regions
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
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 sˆ maintains a constant
inclination to ecliptic pole kˆ
3. spin pole sˆ, orbit pole nˆ , and
ecliptic pole kˆ , all remain coplanar
outline
• Bill Ward’s lunar obliquity history
• what is a Cassini state?
• basics of orbit and spin precession
• influence of tides on obliquity history
William R. Ward,
Past orientation of the lunar spin axis,
Science, 189, 377-379, 1975.
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 general, spin and orbit poles have complicated relative motion
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
orbit pole precession
two torques act on the lunar orbit plane:
• torque from Sun
• orbit pole precesses about ecliptic pole
• rate increases with distance from Earth
d
3 n 
  ne  e  cos[ e ]   W1 am3 / 2
dt
4  nm 
• torque from Earth’s oblate figure
• orbit pole precesses about Earth’s spin pole
• rate decreases with distance from Earth
2
R 
d
3
  nm J 2e  e    W2 am5 / 2
dt
2
 am 
spin pole precession
• orbit pole
nˆ
precesses about ecliptic pole
kˆ
 
d nˆ
  kˆ  nˆ
dt
• spin pole sˆ precesses about the orbit pole
nˆ
d sˆ
  nˆ  sˆ    nˆ  sˆ 
dt
with rate parameters
3  J 2  C2, 2  3  4(c  a)  (b  a) 
  n 
  n 

2 
c
2
4
c



3  C2, 2 

  n 
2  2c 
3 ba
 n

2  8c 
(connection to gravity)
2/n is orbital period
{a, b, c} are dimensionless
principal moments
spin pole precession
• in orbit-fixed reference frame, spin pole motion is
 
d sˆ
  nˆ  sˆ    nˆ  sˆ    kˆ  sˆ
dt
• along spin trajectory, Hamiltonian is constant
H

2
nˆ  sˆ 2   nˆ  sˆ    kˆ  sˆ 
what is a Cassini state?
constraints on spin pole unit vector sˆ
• unit vector:
sx2  s y2  sz2  1
• Hamiltonian (energy)
1
2
H  a s x  s z  b 
2
(parabola)
with
  sin[i] 
a 




  cos[i]   
b




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
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
what is a Cassini state?
view in y-z plane
view in x-y plane
what is a Cassini state?
transition from 4 to 2 steady states
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:
a 2 / 3  b2 / 3  1
near to transition
what is a Cassini state?
transition from 4 to 2 steady states
when state 1 disappears,
dissipation will drive
spin pole to state 2
view in xy-plane
Moon at Cassini-state transition
constant gravity case
a = 0.7432
b = 0.07605
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
tides
spin rate
spin direction
gravity field
obliquity
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
path of sub-Earth point at large obliquity
hydrostatic model for lunar gravity
hydrostatic contribution: at distance a and obliquity 
J 2 [a,  ]  F1[a,  ]
C2,2[a,  ]  F2[a,  ]
 F1[a,  ]  1  R 

 
 
F
[
a
,

]
16
q
a
 2

3
11  8q  9 cos[ 2 ] 


4
 6 cos[  / 2]

where q = Mm/Me = 1/81.3 is mass ratio
Moon is far from hydrostatic
• gravity coefficients
C  ( A  B) / 2
J2 
M R2
C2 , 2 
B A
4M R 2
• current values:
J 2  203.67  0.07106
C2,2  22.19  0.01106
• hydrostatic values:
6
J 2  9.3810
C2,2  2.83106
simple model for past variation
hydrostatic plus constant offset
J 2 [a,  ]  F1[a,  ]  J 2
C2,2[a,  ]  F2[a,  ]  C2,2
offsets: difference between observed and current hydrostatic
J 2  J 2obs  J 2[a0 ,  0 ]  194.39106
6
C2,2  C2obs

C
[
a
,

]

19
.
41

10
,2
2, 2
0
0
less simple models
hydrostatic plus linear offset
a
J 2 [a,  ]  F1[a,  ]  J 2  
 a0 
a
C2, 2 [a,  ]  F2 [a,  ]  C2, 2  
 a0 
hydrostatic plus quadratic offset
a
J 2 [a,  ]  F1[a,  ]  J 2  
 a0 
2
a
C2, 2 [a,  ]  F2 [a,  ]  C2, 2  
 a0 
2
lunar gravity: constant bias case
lunar gravity: linear bias case
lunar gravity: quadratic bias case
conclusions
• obliquity during Cassini state transition
– very different that at present
– strongly dependent upon lunar gravity
• prospects for improving knowledge
– grim