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2 Theory

  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.

next up previous
Next: 3 Interpretation Up: No Title Previous: 1 Introduction

Jun Makino
Thu Jul 2 18:05:36 JST 1998