Though the result obtained in the previous section is simple and in good agreement with the experimental result, the mechanism which determines the value of is not at all clear. In this section, we try to determine with more approximate approaches and to obtain qualitative understanding. First, we obtain the growth rate and growth mode of a typical planetesimal, using the collision probability obtained in section 2, and then derive the stationary distribution from them.
The growth rate of a planetesimal is obtained by integrating equation (5) over the range as Safronov1969,BargePellat1991 e.g.
By substituting equations (5) through (8) into equation (20) and neglecting the dependence on of and r and that on m of , we have
For the power-law mass distribution of equation (1), equation (21) is reduced to
where is the cut-off mass of the distribution function at the low-mass end. These two cases correspond to two different modes of accretion. If , the main mode of the growth is to eat small planetesimals, since the growth rate is determined by the low end of the integration over mass. If , on the other hand, the growth is driven by the collision with particles with similar masses. In other words, the growth is hierarchical.
Both the numerical results and the result obtained in the previous section suggest that . So we consider the case . The stationary state is realized if the mass flux does not depend on either the time or the mass, which is expressed as
Note that here the condition of the stationary distribution is the constant flux of the mass nm, and not the constant flux of the number of planetesimals n. This is because the dominant mode of the growth of the planetesimals is the hierarchical growth. The number of planetesimals would become smaller as they grow, since growth is driven by collisions of planetesimals of similar sizes.
If collisions are dominantly hierarchical, strictly speaking it does not make sense to use the growth equation expressed as the differential equation (20), since the change of the mass of planetesimals is not continuous. However, in this case we can use the mass doubling timescale
and consider the discrete mass bins with the ranges , , , ... , ... In this picture, collisions take place only between particles in the same mass bins. The product of the collision moves to the next mass bin.
In this case, the stationary state is realized when the incoming and outgoing flux for each mass bin are equal. For mass bin k, the total mass of planetesimals lost per unit time is
where the subscript k denotes the mass bin k and is the total mass of planetesimals in the mass bin k. Note that , since the width of the mass bin is proportional to m itself.
The incoming mass to bin k is what is lost from bin k-1. Thus, the condition for the stationary state is that the outgoing mass is the same for all mass bins,
Equations (24) through (26) leads to equation (23) as the condition for the stationary state. Thus, whether we assume continuous growth of the mass or discontinuous jumps in the mass, the condition for the stationary state is the same.
From equation (1), (22), and (23), we obtain
This value is the same as we obtained in section 2.
If we assume a constant flux of the number of planetesimals, we obtain , which is significantly smaller.
The reason why has a large negative value is now clear. As can be seen in equation (22), the growth rate of planetesimals has a strong positive dependence on their mass. This is, of course, why the runaway growth occurs . This strong dependence requires that the number of planetesimal goes down quickly as its mass increases, if a stationary distribution is to be realized. The strong dependence of the growth rate on the mass comes from the size, scale height and the effect of gravitational focusing, all of which enhance the growth rate for larger mass.
From equations (22) and (28), we obtain
which implies the mass of a planetesimal can reach infinity in a finite time. Of course, this cannot happen in reality, since the number of planetesimals with this infinite mass would go below unity. In addition, at a certain point the most massive particle grows so massive that its effect dominates the velocity dispersion of less massive particles, resulting in the slowing down of the growth . This slowing down causes the massive planetesimals of similar mass to be formed in roughly equal orbital separation. Thus, these massive particles cannot collide with each other . Nevertheless, equation (29) implies that the stationary distribution of the mass for a wide range of masses can be realized in a short time.
It should be noted that the growth rate of equation (29) is larger than what has been used for the timescale estimate for the runaway stage (see . e.g., Ida and Makino 1993). In previous estimates, uniform background distribution of planetesimals was assumed. Under this assumption, the power of m in equation (29) becomes .
In the case of the two-dimensional space, equation (5) becomes
which gives . Therefore. from equation (16) we obtain
However, with this formula, the total mass diverges if the high-mass end is taken infinite. Therefore, for the total mass to conserve, there must be a cutoff at the high-mass end, and no stationary solution is possible. In other words, the growth would be an orderly one.
The value of for two-dimensional case coincides with that for collisional cascade . This is simply because the value of is the same.
Note that the existence of the stationary solution is directly related to the occurrence of the runaway growth. The runaway implies that the heavy particles can become heavier by themselves, without being affected by lighter particles. Thus, the power-law behavior is observed because there is no characteristic scale. In the case of the two-dimensional simulation, the evolution of the heaviest particle is slower than that of lighter particles, and therefore all particles tend to have a mass similar to that of the heaviest particle. The power-law behavior is not observed in this case.