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 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

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

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?

view in y-z plane view in x-y plane

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

Moon at Cassini-state transition

a = 0.7432b = 0.07605

constant gravity case

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

path of sub-Earth point at large obliquity

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

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