1. Origin of the Martian Moons Joe Burns Cornell University
October 26, 2015
I thank the organizers (Dan Britt, Jim Head, Carle Pieters, Ken Ramsley). for their invitation. I had hoped to visit Providence and Brown (always enjoyed) but not in new world.
Good to see Phobos and Deimos finally getting their due. My story: First real extraterrestrial bodies (how bizarre? What a preview for how surprising satellites have been seen to be via planetary exploration. Wrote dynamical review before Mariner 9, in orbit on Nov 1971
Had written paper for Mars in early 90s. Reviewed by Stan Peale (who died six months ago) and Dan Britt/Scott Murchie. Different opinions from the dynamicist and the planetary geologists.

2. 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.
Nico S, Andy Rivkin and Sascha Basilevsky all stated these are ancient surfaces but still 40 years on distinguieh. Dessicated phyllosilicates or space weathered surface? Still two possibilities

3. Orbital Characteristics
Note: Orbits lie in orbit plane and are nearly circular: Like mid-sized satellites of giant planets, normally ascribed to growth in the nebulae. Not captured, like irregular satellites of giant planets (high e, I, large a). Or like our Moon, formed from impact-derived ejecta disk.
Orbits lie within and exterior to synchronous, at 6.02 R.
From recent measurements, mostly Mars Express: Q ~100; k2 ~0.16
r ~ 1.87, 1.48 g/cm3

4. 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.
Let’s discuss the plausibility of capture at an elementary level. Tides provide a natural simple explanation for the evolution that is noted already.
Of course, same true for Martian moons, and these forces are reciprocal: Moon distorts just like Earth does.
Moreover, we’re used to thinking in terms of fluid tides but in solid body too, altho not distorted as much.
Differential implies closer is (much) better (i.e., Moon vs Sun). But also for Phobos. And Phobos much more affected than Deimos,
Differential => ~(r/R)(1/R2) ~ r/R3(

5. Difference in local gravity relative to that
needed for orbit determines distortion.
Top: Not quite right, b/c total orce should be equal, but also every place feels force
Far side too, but less
Much of this gravitational attraction is needed to cause synchronous orbit, as shown at bottom right
It’s the difference that matters, to our current story.

6. Gravity difference leads to semidiurnal tide
Error: As body distorts, it must keep total volume, so should be shown as polar flattening.
Complications: 1)The actual distortion depends on interior strength
2) As body distorts (greatly exagerated here), it no longer is a point mass. Example of Rhode Island coast.
half-spin period due to Earth’s rotation

7. PHOBOS’ TIDAL DISTORTION
Here we have added the potentials of point mass Mars, ellipsoidal Phobos, and centrifugal potential due to spin of coordinate system.
Dobrovolskis & Burns, 1980
Tidal Strain ~ k2(m/M)(r/R)3. where k2 = Love no. ~Gr2 r2/m

8. 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
(Deimos)
Q ~100
> n
Q~100

9. Tidal Effects: Phobos vs. Deimos
Recall Phobos is at 2.7 R, Deimos at 6.9 R; synchronous at 6.02R
Burns, In Mars, 1992.

10. Phobos’ Rotational State
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

11. Many others from Mars Express,
More recently. After Mars Express
: Willner, Shi & Oberst (P&SS, 2013) get 1.14o
Witasse et al. (P&SS, 2013) also get 1.2 (+/- 0.15)o.
Oberst et al. (P&SS, 2014) get 1.09o
Many others from Mars Express,
P&SS, 2014
From M. Tiscareno, 2011 EGS

12. 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 SHAPE
or …
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
Note planetary tides usually cause e’>0

13. 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

14. Phobos a, e
Planetary tides
Satellite tides, too

15. Tidal Evolution (Higher-Order Terms)
e runs away

16. Phobos a, e
Planetary tides
Satellite tides, too

17. Deimos a, e

18. …but high eccentricity orbits cannot have happened.
Once orbits become interlaced, mutual collisions occur quickly.
Yoder, 1982
Szeto, 1983

19. Resonance Passage Effects on Evolution
2:1 solar evection
Moon tides;
e-0
3:1 spin:orbit
2:1 spin: orbit
The eccentricities that we see today are not primordial. Rather they are remnants of past resonance passages.
Yoder, Icarus, 1982

20. 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 its
chaotic obliquity oscillations.

21. Approach (Relative) Velocity
In slow tidal evolution, orbit inclination relative to Laplace plane is
roughly constant (Goldreich 1965) because orbit precession (oblateness
or solar tides) is relatively fast .
Orbital evolution under tides suggests low inclinations (relative to
Mars’ orbit plane) when captured in the past.
Low inclinations on capture are very rare (poles of orbit planes fill little
solid angle).
Relative velocity:
asteroid
Think meteor streams (Draconids: Oct 8; Orionids: Oct 21-22; Taurids:
early Nov; Leonids: Nov 17-18); Do they all arrive on equator??
Mars DV

22. Phobos’ Inclination History
A. Cazenave et al. Icarus, 1980
F. Mignard, MNRAS, 1981

23. Phobos a, e, i
N.B. small inclinations

24. Deimos a, e, i

25. Atmospheric Drag Exponential decay V = Vi e –b Cd
Pollack, Burns, Tauber Icarus, 1979
Hunten, Icarus, 1979
Sasaki, 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

26. 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)

28. 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.