On the validity of the coagulation equation and the nature of runaway growth
On the validity of the coagulation equation and the nature of runaway growth
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DOI：
10.1006/icar.1999.6239
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发表时间：
2000-01-01
期刊：
影响因子：
3.2
通讯作者：
Lee, MH
中科院分区：
文献类型：
--
作者：
Lee, MH
凝结方程广泛用于模拟行星形成和其他天体物理问题中的生长，它是描述由于连续合并而导致的一组粒子的质谱演化的平均速率方程。开发了一种数值代码，可以通过合理数量的质量箱产生混凝方程的精确解，并用于研究混凝方程解的性质。我们考虑引力相互作用的合并率系数 A(ij) 的限制情况，质量半径关系的幂律指数 beta = 1/3（对于星子）和 2/3（对于恒星）。我们使用指数 lambda（用于两个质量相当的粒子之间的合并）以及指数 mu 和 nu（用于重粒子和轻粒子之间的合并）对 A(ij) 的质量依赖性进行分类。对于 nu 小于或等于 1 和 lambda 小于或等于 1 的两种情况，质谱以有序方式演化。对于 nu > 1 的其余情况，我们发现了强有力的数值和分析证据，表明凝血方程在任何时候都没有自洽的解。 nu > 1 情况的结果与众所周知的 A(ij) 与 ij 成比例的示例有质的不同。对于后一种情况，即 nu 小于或等于且 lambda > 1 的范围内，凝血方程有一个解析解，该解在有限时间 t(0) 内有效。我们讨论一个简化的合并问题，该问题说明了三类 A(ij) 的凝固方程解的质量差异。我们的结果强烈表明存在两种类型的失控增长。对于 nu 小于或等于 1 且 lambda > 1 的 A(ij)，失控增长从接近 t(0) 的 t(crit) 开始，此时凝固方程解变得无效。对于 nu > 1 的 A(ij)（包括预期显示失控增长的所有引力相互作用情况），失控增长开始的时间 t(crit) 对问题参数的依赖性尚未得到很好的理解，但有迹象表明 t(crit)（以 1/(n(0)A(11)) 为单位）可能会随着初始粒子总数 n(0) 的增加而缓慢减小到零。 (C) 2000 年学术出版社。
The coagulation equation, which is widely used for modeling growth in planet formation and other astrophysical problems, is the mean-rate equation that describes the evolution of the mass spectrum of a collection of particles due to successive mergers. A numerical code that can yield accurate solutions to the coagulation equation with a reasonable number of mass bins is developed, and it is used to study the properties of solutions to the coagulation equation. We consider limiting cases of the merger rate coefficient A(ij) for gravitational interaction, with the power-law index of the mass-radius relation beta = 1/3 (for planetesimals) and 2/3 (for stars). We classify the mass dependence of A(ij) using the exponent lambda for the merger between two particles of comparable mass, and the exponents mu and nu for the merger between a heavy particle and a light particle. For the two cases with nu less than or equal to 1 and lambda less than or equal to 1, the mass spectrum evolves in an orderly fashion. For the remaining cases, which have nu > 1, we find strong numerical and analytical evidence that there are no self-consistent solutions to the coagulation equation at any time. The results for the nu > 1 cases are qualitatively different from the well-known example with A(ij) proportional to ij. For the latter case, which is in the range nu less than or equal to and lambda > 1, there is an analytic solution to the coagulation equation that is valid for a finite amount of time t(0). We discuss a simplified merger problem that illustrates the qualitative differences in the solutions to the coagulation equation for the three classes of A(ij). Our results strongly suggest that there are two types of runaway growth. For A(ij) with nu less than or equal to 1 and lambda > 1, runaway growth starts at t(crit) approximate to t(0), the time at which the coagulation equation solution becomes invalid. For A(ij) with nu > 1, which include all gravitational interaction cases expected to show runaway growth, the dependence of the time t(crit) for the onset of runaway growth on the parameters of the problem is not yet well understood, but there are indications that t(crit) (in units of 1/(n(0)A(11))) may decrease slowly toward zero with increasing initial total number of particles n(0). (C) 2000 Academic Press.