Origin of the Martian MoonsJoe BurnsCornell UniversityOctober 26, 2015

Surfaces are cratered, dark albedoes, no strong spectral absorptions.Porous densities (formed in low g envir).P’s Orbit first suggested in ’45 to evolve.Might P&D be captured, and be evolving via tides? But low e and i argue against.

Orbital Characteristics

r ~ 1.87, 1.48 g/cm3

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Tides

• Tides are (differential) gravitational forces caused by an external body. They distort the object’s shape.

• The Earth feels tides from the Sun & Moon.

But the Earth is not a point; some places closer to the Moon than others.

Differential => ~(r/R)(1/R2) ~ r/R3(

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Difference in local gravity relative to that needed for orbit determines distortion.

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Gravity difference leads to semidiurnal tide

half-spin period due to Earth’s rotation

Dobrovolskis & Burns, 1980

Tidal Strain ~ k2

(m/M)(r/R)3. where k2 = Love no. ~Gr2 r2/m

PHOBOS’ TIDAL DISTORTION

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Tidal Effects

• Planet does not respond instantaneousls; max distortion occurs after max force.

• Tidal bulge leads (trails for n>W) by sin 2e ~ 1/Q.

• Non-aligned bulge causes torque b/w Moon & Earth

• By conservation of angular momentum, planet’s rotation slows, and moon is pulled forward, so it recedes from planet.

• But little transfer of angular momentum to Mars; spin constant

W> n

Q~100

(Deimos)

Tidal Effects: Phobos vs. Deimos

Burns, In Mars, 1992.

Tidal distortion of Phobos.~k2 (m/M)(r/a))3 r ~ 10 m

Rotational slowing = (9/8)( k2 G m2 /2wQ)(R5/a6)

Time for tidal de-spin = 104k2/ Q yrs PHOBOS

= 107 k2/ Q yrs DEIMOS Peale, 1977

Synchronous rotation and quick damping to align with minimum energy state(long axis to Mars)

Because of Phobos’ small mass, little effect on Mars’ spin, unlike Earth-Moon case.

Libration due to forcing by uneven speed along satellite’s eccentric orbit

=>( B-A)/C (natural libration freq),which suggests homogeneous interior

Phobos’ Rotational State

From M. Tiscareno, 2011 EGS

Man

y oth

ers f

rom

Mar

s Exp

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,

P&SS

, 201

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Orbital Evolution

Orbital Energy = KE + PE = - GM/2a = E ORBIT SIZE

da/dt =(2a2/GM)dE/dt= (2a2/GM) v.dF , work done

Energy Loss => Smaller orbits, even if loss causes orbital speed-up.

Forces in Orbit plane (in direction of velocity; i.e., power) only affect a

Orbital Collapse with energy loss (atmospheric drag, tidal drag, PR drag)----------

Orbital Angular Momentum = H = [Gma (1- e2)]1/2 ORBIT SHAPEor …de/dt = (e2-1)(2H’/H + E’/E)/(2e)or…Orbits circularize when H is constant and E is lost (satellite tides)

Only Forces in orbit plane can change shape. Burns, Am. Jnl. Phys. 1976

Tidal Evolution of Circular Orbits

da/dt = 3 (G/M)1/2 m k2 a-11/2R5/Q

(a/ao)13/2 =1 – (13/3) (n’o/no) (t – to) ~ tens of millions of years for Phobos,much longer for Deimos

Phobos a, e

Planetary tides

Satellite tides, too

Tidal Evolution (Higher-Order Terms)

e runs away

Phobos a, e

Planetary tides

Satellite tides, too

Deimos a, e

…but high eccentricity orbits cannot have happened.

Once orbits become interlaced, mutual collisions occur quickly.

Yoder, 1982

Szeto, 1983

Resonance Passage Effects on Evolution

Yoder, Icarus, 1982

3:1 spin:orbit2:1 spin: orbit

2:1 solar evection

Moon tides;

e-0

Orbit Orientation

Orbit tilt depends on Angular Momentum’s direction. Moments change orientation; Just forces normal to orbit plane cause such reorientations. Not important in tides. Gas drag forces the body’s orbit to adopt that of medium.

If orbit evolves slowly, it keeps a constant angle relative to mean (Laplace) plane.Distant orbits precess around planet’s heliocentric orbit plane; close-in orbits precess around equatorial plane. Thus moon orbits follow Mars’ axial precession and itschaotic obliquity oscillations.

Approach (Relative) Velocity

In slow tidal evolution, orbit inclination relative to Laplace plane isroughly constant (Goldreich 1965) because orbit precession (oblatenessor solar tides) is relatively fast .

Orbital evolution under tides suggests low inclinations (relative toMars’ orbit plane) when captured in the past.

Low inclinations on capture are very rare (poles of orbit planes fill littlesolid angle).

Relative velocity:

asteroidThink meteor streams (Draconids: Oct 8; Orionids: Oct 21-22; Taurids:

early Nov; Leonids: Nov 17-18); Do they all arrive on equator??

DVMars

Phobos’ Inclination History

A. Cazenave et al. Icarus, 1980F. Mignard, MNRAS, 1981

Phobos a, e, i

N.B. small inclinations

Deimos a, e, i

Atmospheric Drag

Pollack, Burns, Tauber Icarus, 1979Hunten, Icarus, 1979Sasaki, LPSC, 1990

Can capture quickly w/ dense disk,but orbit evolves quickly too.

Periodic impacts into disk cause energy loss (lower a), drop angular momentum (lower e) and regularize i.

Cuk & Burns 2004

Exponential decay

V = Vi e –b Cd

Bottom Line: in situ origin

• Capture is difficult, at best, whether tidally or thru atmospheric drag. Hard to dissipate energy and to circularize/flatten orbit to equator.

• Tidal scenarios suggest both Martian moons formed closer to synchronous orbit and with e & i ~ 0

• Formation by impact or within disk (cf. Canup)

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Tidal Effects

• This model of tides on the Earth is oversimplified:– Earth not completely covered by oceans.– The solid Earth distorts and bulges from tides as well.– The Earth distorts the Moon into an ellipsoid.

• Tidal forces change the Earth-Moon system:– Tidal Friction is slowing the Earth's spin (~ msec/century).– Conservation of angular momentum, which is transferred

from Earth’s rotation to lunar orbit: Moon is receding from Earth (3.8 cm/year).

• Tides can affect the spin and internal heating of some solar system bodies.