Spin-Orbit Resonance Planetary Physics EP 420

Spin-Orbit Resonancein Planetary Bodies

Harsh Menon

Spin-Orbit Resonance Planetary Physics EP 420

Overview

• An Introduction to Tidal Forces

• Spin-Orbit Resonance

• Resonance Criteria

• Capture into Resonance

• Surface of Section

• Conclusion

Spin-Orbit Resonance Planetary Physics EP 420

Introduction to Tidal Forces

• Force produced due to “gravitational gradient or variation of gravitational force across the body”

• Proportional to radius of larger body

• Inversely proportional to distance between the centers of the bodies

Spin-Orbit Resonance Planetary Physics EP 420

Spin-Orbit Resonance

• Interaction between a planet’s orbital angular momentum and a satellite’s gravitational permanent quadrupole moment

• Resonance results when there is a simple integer or near integer relationship between the spin period and orbital period of the satellite

Spin-Orbit Resonance Planetary Physics EP 420

Spin-Orbit Resonance

pM

Spin-Orbit Resonance Planetary Physics EP 420

Spin-Orbit Resonance

• Result in oscillations called librations• http://aa.usno.navy.mil/graphics/Moon_mo

vie.gifThe equation of motion of the satellite is

given by:

• Analogous to equation of motion for a pendulum

2sin2

1))],(([ 2

0epHsign

Spin-Orbit Resonance Planetary Physics EP 420

Spin-Orbit Resonance

• Libration Frequency

5.0

0 ),(3

epHC

ABn

43

3423

44

16

533),3(

48

845),

2

5(

16

123

2

7),

2

3(

16

1

2

51),1(

16

1

2

1),

2

1(

0),0(48

1),

2

1(

24

1),1(

eeHeeH

eeeHeeeHeeeH

eHeeHeeH

Spin-Orbit Resonance Planetary Physics EP 420

Spin-Orbit Resonance

• Physical Meaning of γ

• “Orientation of the long axis of the satellite on the passage of the satellite through the pericenter or that it is a stroboscopic angle that is evaluated when M=0”

Spin-Orbit Resonance Planetary Physics EP 420

Spin-Orbit Resonance

http://cseligman.com/text/planets/DayOnMercury.swf

Spin-Orbit Resonance Planetary Physics EP 420

Resonance Criteria

• Loop is closed

• p is integer multiple of ½

• Torque acting on satellite >Torque due to tidal drag

Spin-Orbit Resonance Planetary Physics EP 420

Capture into Resonance

• Pendulum Analogy• Capture into Resonance • Variation of average tidal torque with dγ/dt

Spin-Orbit Resonance Planetary Physics EP 420

Surface of Section

Spin-Orbit Resonance Planetary Physics EP 420

Surface of Section

Spin-Orbit Resonance Planetary Physics EP 420

Surface of Section

• Chirikov’s Criterion

• Chaotic behavior depends on eccentricity and asphericity parameter

• Linearly on e, exponentially on α

C

AB )(3

Spin-Orbit Resonance Planetary Physics EP 420

Conclusions

• Spin-Orbit Resonance

• Resonance Criteria

• Capture into Resonance

• Surface of Section

Spin-Orbit Resonance Planetary Physics EP 420

Recommendations

• Current theory “naïve”

• More accurate theory - Hamiltonian Theory on Tidal Forces

Spin-Orbit Resonance Planetary Physics EP 420

References

• Murray, C.D. & Dermott, S.F. (1999). Solar System Dynamics. New York: Cambridge University Press.