The presence of the power-law distribution implies that the evolution of the mass distribution function follows an approximately self-similar solution. Instead of trying to obtain the self-similar solution, however, here we try to find a stationary solution, with the supply of the planetesimals at the low-mass end and the removal of the massive planetesimals at the high-mass end. This is a usual exercise in studying the distribution function analytically.

The change in the surface number density of planetesimals with
mass **m** is
expressed as

where and are the incoming and outgoing fluxes. They are defined as

Here, is the probability that two planetesimals of masses
**m** and collide in a unit time. This is the classical
Smoluchowski-Safronov coagulation equation [] integrated
over the velocity distribution. The integration over velocity is
justified on the ground that the relaxation timescale is shorter than
the coagulation timescale.

Note that we assume that when two planetesimals collide they always stick together (perfect accretion). This assumption is okay for planetesimals since the velocity dispersion is lower than the escape velocity.

The collision probability is expressed roughly as

where **H** is the
scale height, **r** is the radius of a planetesimal, is the
escape velocity given by

and is the average relative velocity given by

Here **G** is the gravitational constant.
Note that , , and denote corresponding quantities for
the planetesimal of mass .

In the following, we assume that the velocity distribution of the planetesimals satisfies the thermal equilibrium

and that the velocity is in the extreme gravitational focusing region,

Here we assumed that the two-body formalism can be used to obtain the collision cross section. This assumption is valid since the velocity dispersion is in the ``dispersion dominant'' regime while the runaway growth takes place.

Our goal here is to obtain the mass distribution function for
which for all values of **m**. Before trying
to obtain the solution, let us first investigate the characteristics
of the collision probability **P**. From equations (5)
to (8), we can find the behavior of **P** in two limiting cases

To put it in a slightly different way, we can express **P** as

with . Here is a function which has the asymptotic behavior of:

Note that equation (11) combined with equation
(12) is equivalent to equation
(10), but still is exact as long as we
retain the function **h**.

For the power-law mass distribution of equation (1), equation (10) implies that both and diverge in the low-mass end if , and diverges also at the high-mass end if . In other words, for any value of , either of or diverges. In particular, for the experimental result of , both diverge at the low-mass end.

The divergence at the low-mass end is, however, not a physical
reality, but a mathematical artifact caused by the inadequate form of
equations (3) and
(4). In the limit of ,
the apparent flux expressed in equations
(3) and
(4) diverges. However, the mass of
particles which originally had a mass of **m** or does not change
in the limit of . In other words, there is an
infinite flux, in between the two mass bins with infinitesimal
separation. The net effect of the product of the infinite and the
infinitesimal must be carefully examined.

A convenient way to avoid this difficulty of the apparent divergence
of **f** is to introduce the ``mass flux'' **F**, in such a way that its
partial derivative in mass space gives the net change of the
distribution function as

Tanakaetal1996 used this form to study the collision cascade
process. The formal derivation here is essentially the same as theirs.
Since equation (13) implies that **F** is
the integral
of , one might think it should behave in exactly the same
way. However, as we'll see shortly, **F** does not diverge at the
low-mass end, even though and diverge. This is because
the contribution of the low-mass end to the mass flux **F** is small. As
we stated earlier, the change of mass vanishes at the low-mass
end. Thus, even though and diverge, their contribution to
**F** vanishes at the low-mass end.

The mass flux **F** is calculated as

It is straightforward to prove that the combination of equations (13) and (14) is formally equivalent to equations (2) through (4).

As shown by Tanakaetal1996, for a
collision rate of the form of equation (11),
**F** is reduced to

where and .

The stationary solution corresponds to the case , which is realized when

As stressed by Tanakaetal1996 this result does not depend on the functional form of , as far as can be expressed in the form of equation (11).

The double integral in equation (15) should have a finite value. To determine if it is the case or not, it is more convenient to rewrite the double integral in a slightly different form as

where .

Without losing generality, we can assume that **h** has the following
asymptotic expression

Since is symmetric, the limiting behavior of **h** in either
limit determines the behavior in the other limit. Equation
(12) corresponds to the case of .

The condition that **F** is finite is given by

The first inequality comes from the condition that the integrand
should approach zero faster than in the limit of . The second comes from the condition that the integrand should
not diverge as or faster in the limit of . This
inequality also guarantees that the integrand does
not diverge as or faster in the limit of . Note that it can diverge faster than , since the range of the
integration over **x** is proportional to **z**.

The criterion (19) is different from what
is shown in the Appendix of Tanakaetal1996. Their
derivation did not incorporate the effect of the power of **h** on the convergence criterion correctly.

For , we have . As noted above, . This set of values satisfies the convergence criteria. Thus,
we found a stationary solution of the coagulation equation expressed
as . This is in quite good agreement with the
numerical results obtained by **N**-body or Fokker-Planck calculations.

Thu Jul 2 18:05:36 JST 1998