Orbital Characteristics
r ~ 1.87, 1.48 g/cm3
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
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PHOBOS’ TIDAL DISTORTION
Dobrovolskis & Burns, 1980
Tidal Strain ~ k (m/M)(r/R)3. where k
2
2
= Love no. ~Gr2 r2/m
Tidal Effects
W >n
Q~100
• 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
(Deimos)
• 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
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Tidal Effects: Phobos vs. Deimos
Burns, In Mars, 1992.
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
Libration of Phobos
• Phobos was the
first extraterrestrial moon
with detected
librations
• Predictions and
observations
seem to follow
each other
• No need for them
to match!
– A difference
would indicate
internal structure
From M. Tiscareno, 2011 EGS
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
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
2:1 solar evection
3:1 spin:orbit
2:1 spin: orbit
Yoder, Icarus, 1982
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.
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:
Mars
asteroid
Think meteor streams (Draconids: Oct 8; Orionids: Oct 21-22; Taurids:
early Nov; Leonids: Nov 17-18); Do they all arrive on equator??
Phobos’ Inclination History
A. Cazenave et al. Icarus, 1980
F. Mignard, MNRAS, 1981
Phobos a, e, i
N.B. small inclinations
Deimos a, e, i
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.
Cuk & Burns 2004
Periodic impacts into disk cause
energy loss (lower a), drop angular
momentum (lower e) and regularize
i.
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)
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.
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