Encyclopedia of the Solar System 2nd ed

806 Encyclopedia of the Solar System

solar system seems to be chaotic yet stable. When the plan-
etary orbits are integrated forward for timescales for several
billion years using the averaged equations of motion, it is
found that there is a very small but finite probability that
the orbit of Mercury can become unstable and intersect the
orbit of Venus. Many challenges remain in understanding
how structural stability of planetary systems in the presence
of transient and intermittent chaos can be maintained, and
this subject remains a rich field for dynamical exploration.

7. Dissipative Forces and the Orbits of

Small Bodies

The foregoing sections describe the gravitational interac-
tions between the Sun, planets, and moons. Solar radia-
tion has been ignored, but this is an important force for
small particles in the solar system. Three effects can be dis-
tinguished: (1) the radiation pressure, which pushes parti-
cles primarily outward from the Sun (micron-sized dust);
(2) the Poynting–Robertson drag, which causes
centimeter-sized particles to spiral inward toward the Sun;
and (3) the Yarkovski effect, which changes the orbits of
meter- to kilometer-sized objects owing to uneven tempera-
ture distributions at their surfaces. The latter two effects are
relativistic and thus quite weak at solar system velocities, but
they can nonetheless be significant as they can lead to secu-
lar changes in orbital angular momentum and energy. Each
of these effects is discussed in the next three subsections
and then the effect of gas drag is examined. In the final sub-
section the influence of tidal interactions is discussed; this
effect (in contrast to the other dissipative effects described
in this section) is most important for larger bodies such as
moons and planets. [SeeSolarSystem Dust.]

7.1 Radiation Force (Micron-Sized Particles)

The Sun’s radiation exerts a force,Fr, on all other bodies of
the solar system. The magnitude of this force is

Fr=

LA

4 πcr^2

Qpr, (47)

whereAis the particle’s geometric cross section,Lis the
solar luminosity,cis the speed of light,ris the heliocentric
distance, andQpris the radiation pressure coefficient, which
is equal to unity for a perfectly absorbing particle and is of
order unity unless the particle is small compared to the
wavelength of the radiation. The parameterβis defined as
the ratio between the forces due to the radiation pressure

and the Sun’s gravity:

β≡

Fr

Fg

= 5. 7 × 10 −^5

Qpr

ρR

, (48)

where the radius,R, and the density,ρ, of the particle are in

c.g.s. units. Note thatβis independent of heliocentric dis-

tance and that the solar radiation force is important only for

micron- and submicron-sized particles. Using the parame-

terβ, a more general expression for the effective gravita-

tional attraction can be written:

Fgeff=

−(1−β)GmM

r^2

, (49)

that is, the small particles “see” a Sun of mass (1−β)M.It

is clear that small particles withβ>1 are in sum repelled

by the Sun, and thus quickly escape the solar system, unless

they are gravitationally bound to one of the planets. Dust

which is released from bodies traveling on circular orbits

at the Keplerian velocity is ejected from the solar system if

β>0.5.

The importance of solar radiation pressure can be seen,

for example, in comets. Cometary dust is pushed in the

antisolar direction by the Sun’s radiation pressure. The dust

tails are curved because the particles’ velocity decreases as

they move farther from the Sun, due to conservation of

angular momentum. [SeeCometaryDynamics;Physics

andChemistry ofComets.]

7.2 Poynting–Robertson Drag (Centimeter-Sized

Grains)

A small particle in orbit around the Sun absorbs solar radia-

tion and reradiates the energy isotropically in its own frame.

The particle thereby preferentially radiates (and loses mo-

mentum) in the forward direction in the inertial frame of

the Sun. This leads to a decrease in the particle’s energy and

angular momentum and causes dust in bound orbits to spi-

ral sunward. This effect is called the Poynting–Robertson

drag.

The net force on a rapidly rotating dust grain is given by

Frad≈

LQprA

4 πcr^2

[(

1 −

2 vr

c

)

rˆ−

vθ

c

θˆ

]

. (50)

The first term in Eq. (50) is that due to radiation pressure

and the second and third terms (those involving the velocity

of the particle) represent the Poynting–Robertson drag.

From this discussion, it is clear that small-sized dust

grains in the interplanetary medium are removed: (sub)-

micron sized grains are blown out of the solar system,

whereas larger particles spiral inward toward the Sun.