Encyclopedia of the Solar System 2nd ed

Solar System Dynamics: Regular and Chaotic Motion 807

Typical decay times (in years) for circular orbits are given
by

τP−R≈ 400

r^2

β

, (51)

with the distancerin AU.
Particles that produce the bulk of the zodiacal light (at in-
frared and visible wavelengths) are between 20 and 200μm,
so their lifetimes at Earth orbit are on the order of 10^5 yr,
which is much less than the age of the solar system. Sources
for the dust grains are comets as well as the asteroid belt,
where numerous collisions occur between countless small
asteroids.

7.3 Yarkovski Effect (Meter-Sized Objects)

Consider a rotating body heated by the Sun. Because
of thermal inertia, the afternoon hemisphere is typically
warmer than the morning hemisphere, by an amountT
T. Let us assume that the temperature of the morning hemi-
sphere isT−T/2, and that of the evening hemisphere
T+T/2. The radiation reaction upon a surface element
dA,normal to its surface, isdF= 2 σT^4 dA/ 3 c.For a spher-
ical particle of radiusR, the Yarkovski force in the orbit
plane due to the excess emission on the evening side is

FY=

8

3

πR^2

σT^4

c

T

T

cosψ, (52)

whereσis the Stefan–Boltzmann constant andψis the par-
ticle’sobliquity, that is, the angle between its rotation axis
and orbit pole. The reaction force is positive for an object
that rotates in the prograde direction, 0<ψ< 90 ◦, and
negative for an object with retrograde rotation, 90◦<ψ<
180 ◦. In the latter case, the force enhances the Poynting–
Robertson drag.
The Yarkovski force is important for bodies ranging in
size from meters to several kilometers. Asymmetric out-
gassing from comets produces a nongravitational force
similar in form to the Yarkovski force. [SeeCometary
Dynamics.]

7.4 Gas Drag

Although interplanetary space generally can be considered
an excellent vacuum, there are certain situations in plane-
tary dynamics where interactions with gas can significantly
alter the motion of solid particles. Two prominent exam-
ples of this process are planetesimal interactions with the
gaseous component of the protoplanetary disk during the
formation of the solar system and orbital decay of ring
particles as a result of drag caused by extended planetary
atmospheres.

In the laboratory, gas drag slows solid objects down un-

til their positions remain fixed relative to the gas. In the

planetary dynamics case, the situation is more complicated.

For example, a body on a circular orbit about a planet loses

mechanical energy as a result of drag with a static atmo-

sphere, but this energy loss leads to a decrease in semima-

jor axis of the orbit, which implies that the body actually

speeds up! Other, more intuitive effects of gas drag are the

damping of eccentricities and, in the case where there is a

preferred plane in which the gas density is the greatest, the

damping of inclinations relative to this plane.

Objects whose dimensions are larger than the mean free

path of the gas molecules experience Stokes’ drag,

FD=−

CDAρv^2

2

, (53)

wherevis the relative velocity of the gas and the body,ρis

the gas density,Ais the projected surface area of the body,

andCDis a dimensionless drag coefficient, which is of order

unity unless theReynolds numberis very small. Smaller

bodies are subject to Epstein drag,

FD=−Aρvv′ (54)

wherev′is the mean thermal velocity of the gas. Note that

as the drag force is proportional to surface area and the

gravitational force is proportional to volume (for constant

particle density), gas drag is usually most important for the

dynamics of small bodies.

The gaseous component of the protoplanetary disk in

the early solar system is believed to have been partially sup-

ported against the gravity of the Sun by a negative pressure

gradient in the radial direction. Thus, less centripetal force

was required to complete the balance, and consequently

the gas orbited less rapidly than the Keplerian velocity. The

“effective gravity” felt by the gas is

geff=−

GMS

r^2

−(1/ρ)

dP

dr

. (55)

To maintain a circular orbit, the effective gravity must be

balanced by centripetal acceleration,rn^2. For estimated

protoplanetary disk parameters, the gas rotated∼0.5%

slower than the Keplerian speed.

Large particles moving at (nearly) the Keplerian speed

thus encountered a headwind, which removed part of their

angular momentum and caused them to spiral inward

toward the Sun. Inward drift was greatest for mid-sized

particles, which have large ratios of surface area to mass yet

still orbit with nearly Keplerian velocities. The effect dimin-

ishes for very small particles, which are so strongly coupled

to the gas that the headwind they encounter is very slow.

Peak rates of inward drift occur for particles that collide