808 Encyclopedia of the Solar System
with roughly their own mass of gas in one orbital period.
Meter-sized bodies in the inner solar nebula drift inward at
a rate of up to 10^6 km/yr! Thus, the material that survives
to form the planets must complete the transition from cen-
timeter to kilometer size rather quickly, unless it is confined
to a thin dust-dominated subdisk in which the gas is dragged
along at essentially the Keplerian velocity.
Drag induced by a planetary atmosphere is even more
effective for a given density, as atmospheres are almost
entirely pressure supported, so the relative velocity between
the gas and particles is high. As atmospheric densities drop
rapidly with height, particles decay slowly at first, but as they
reach lower altitudes, their decay can become very rapid.
Gas drag is the principal cause of orbital decay of artificial
satellites in low Earth orbit.
7.5 Tidal Interactions and Planetary Satellites
Tidal forces are important to many aspects of the structure
and evolution of planetary bodies:
1.On short timescales, temporal variations in tides (as
seen in the frame rotating with the body under consider-
ation) cause stresses that can move fluids with respect to
more rigid parts of the planet (e.g., the familiar ocean tides)
and even cause seismic disturbances (though the evidence
that the Moon causes some earthquakes is weak and dis-
putable, it is clear that the tides raised by Earth are a major
cause of moonquakes).
2.On long timescales, tides cause changes in the orbital
and spin properties of planets and moons. Tides also de-
termine the equilibrium shape of a body located near any
massive body; note that many materials that behave as solids
on human timescales are effectively fluids on very long ge-
ological timescales (e.g., Earth’s mantle).
The gravitational attraction of the Moon and Earth on
each other causes tidal bulges that rise in a direction close
to the line joining the centers of the two bodies. Particles
on the nearside of the body experience gravitational forces
from the other body that exceed the centrifugal force of the
mutual orbit, whereas particles on the far side experience
gravitational forces that are less than the centripetal forces
needed for motion in a circle. It is the gradient of the grav-
itational force across the body that gives rise to the double
tidal bulge.
The Moon spins once per orbit, so that the same face
of the Moon always points toward Earth and the Moon is
always elongated in that direction. Earth, however, rotates
much faster than the Earth–Moon orbital period. Thus, dif-
ferent parts of Earth point toward the Moon and are tidally
stretched. If the Earth was perfectly fluid, the tidal bulges
would respond immediately to the varying force, but the
finite response time of Earth’s figure causes the tidal bulge
to lag behind, at the point on Earth where the Moon was
overhead slightly earlier. Since Earth rotates faster than
the Moon orbits, this “tidal lag” on Earth leads the posi-
tion of the Moon in inertial space. As a result, the tidal
bulge of Earth accelerates the Moon in its orbit. This causes
the Moon to slowly spiral outward. The Moon slows down
Earth’s rotation by pulling back on the tidal bulge, so the
angular momentum in the system is conserved. This same
phenomenon has caused most, if not all, major moons to
be in synchronous rotation: the rotation and orbital periods
of these bodies are equal. In the case of the Pluto–Charon
system, the entire system is locked in a synchronous ro-
tation and revolution of 6.4 days. Satellites in retrograde
orbits (e.g., Triton) or satellites whose orbital periods are
less than the planet’s rotation period (e.g., Phobos) spiral
inward toward the planet as a result of tidal forces.
Mercury orbits the Sun in 88 days and rotates around its
axis in 59 days, a 3:2 spin–orbit resonance. Hence, at every
perihelion one of two locations is pointed at the Sun: the
subsolar longitude is either 0◦or 180◦. This configuration is
stable because Mercury has both a large orbital eccentricity
and a significant permanent deformation that is aligned with
the solar direction at perihelion. Indeed, at 0◦longitude
there is a large impact crater, Caloris Planitia, which may
be the cause of the permanent deformation.
3.Under special circumstances, strong tides can have
significant effects on the physical structure of bodies. Gen-
erally, the strongest tidal forces felt by solar system bodies
(other than Sun-grazing or planet-grazing comets) are those
caused by planets on their closest satellites. Near a planet,
tides are so strong that they rip a fluid (or weakly aggregated
solid) body apart. In such a region, large moons are unsta-
ble, and even small moons, which could be held together
by material strength, are unable to accrete because of tides.
The boundary of this region is known asRoche’s limit.
Inside Roche’s limit, solid material remains in the form of
small bodies and rings are found instead of large moons.
The closer a moon is to a planet, the stronger is the tidal
force to which it is subjected. Let us consider Roche’s limit
for a spherical satellite in synchronous rotation at a distance
rfrom a planet. This is the distance at which a loose particle
on an equatorial subplanet point just remains gravitationally
bound to the satellite. At the center of the satellite of mass
mand radiusRs, a particle would be in equilibrium and soGM
r^2=n^2 r, (56)where M(m) is the mass of the planet. However, at
the equator, the particle will experience (i) an excess
gravitational or centrifugal force due to the planet, (ii) a
centrifugal force due to rotation, and (iii) a gravitational
force due to the satellite. If the equatorial particle isjustin
equilibrium, these forces will balance and−d
dr(
GM
r^2)
Rs+n^2 r=Gm
R^2 s. (57)