Comparison of Numerical Integration and 'Gas Dynamic' Modelling of Runaway Planetesimal Growth
Comparison of Numerical Integration and 'Gas Dynamic' Modelling of Runaway Planetesimal Growth
复制标题
失控星子生长的数值积分和“气体动力学”模型的比较
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
J. Chambers
中科院分区：
文献类型：
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作者：
G. Wetherill;E. Kokubo;S. Ida;J. Chambers
[摘要。介绍了3000颗小行星的轨道和碰撞演化，它们最初的相对速度很低(1)。这些都与使用“气体动力学”技术计算同一问题进行了比较，在“气体动力学”技术中，小行星就像气体运动论中的分子一样对待。结果表明，两者具有很好的一致性。这一比较支持了使用气体动力学方法来解决许多不适用于详细数值积分的问题的有效性。]在行星形成理论的发展中，一个基本问题是在日心轨道上运动的一组初始行星的动力学演化和碰撞积累问题(2)。总而言之，这个问题太复杂了，不能用简单的方法计算。出于这个原因，调查人员必须利用模型，试图捕捉问题的一些基本特征，代价是未能治疗其他可能具有重要意义的问题。在其他方法中，有两种方法是：(1)通过对单个天体的运动方程进行直接的数值积分来跟踪小行星组合的三维演化(1)。这有一个优点，就是以一种非常准确的方式包含了许多基础物理知识。一个缺点是，当天体数量足够少(例如，5100个)，使计算工作在足够长的时间尺度(107年至108年)上是可管理的时，行星增长过程的后期阶段受到实际限制。(2)利用与气体运动论的类比，在这个类比中，即使当物体的数量太大而不能单独考虑它们时，也可以确定一组物体(分子或行星体)的平均方面(3，4，5)。这种方法在计算上不受限制。此外，将碎裂的重要物理影响计入计算也很简单，尽管与这一现象有关的天体数量非常多。然而，它确实需要一些有问题的假设，比如使用平均量，以及包括三体效应和长程力的方式的有效性。出于这些原因，这两种方法或任何其他现有的替代方法都不足以单独解决这个问题。然而，通过对两种技术都可以处理的特殊问题的比较计算，可以洞察第二种方法的有效性程度。最近报道了一个此类问题的数值积分(1)。初始条件为3000个10~~体，介于0.98~1.02 AU之间，初始偏心率为两个Hill单位(e=5.1x，倾角为一个Hill单位)。为了利用可用的(主要)计算机资源获得有用的较大程度的进化，有必要人为地增加天体的半径，相当于它们具有0.016克/厘米~3的材料密度。这个系统的轨道和碰撞演化被计算了2万年，在更现实的密度下相当于数十万年。使用“气体动力学”程序(6)计算了完全相同的问题，包括使用低材料密度。对该程序进行了修改，仅从计算中删除了数值积分中未包括的碰撞破碎和气体阻力等现象。这两次计算的结果非常相似。20000年后的累积质量分布如图1所示。此时，两种情况下的天体总数都已从3000个减少到约1000个。在这些星体中，大多数(600个)仍有其原始质量，占总质量的20%。大多数质量处于一个非常陡峭的(Dnldr)mr-5.5分布的“堆积级联”中，其中大部分质量集中在低质量。在分布的高质量端，有几个逃逸体从陡峭的堆积瀑布中分离出来。对于数值积分计算，跑道占总质量的21%；对于气体动力学计算，跑道占总质量的33%。这些失控天体的大小和数量受到随机波动的影响。很有可能温和但
[Summary. The orbital and collisional evolution of 3000 planetesimals, initially at very low relative velocities has been presented (1). These are compared with calculations of the identical problem using the "gas dynamic" technique, in which the planetesimals are treated like molecules in the kinetic theory of gases. Comparison of the results shows excellent agreement. This comparison supports the validity of using the gas dynamic approach to address the many problems that are not amenable to detailed numerical integration.] In the development of theories of planetary formation, a fundamental problem is that of the dynamical evolution and collisional accumulation of an initial assemblage of planetesimals moving in heliocentric orbits (2). In its full generality, this problem is too complex to calculate in a straightforward way. For this reason, investigators must make use of models that try to capture some essential features of the problem, at the cost of failing to treat others that are likely to be of importance. Among others, two such approaches are: (1) Following the three-dimensional evolution of the planetesimal assemblage by straightforward numerical integration of the equations of motion of the individual bodies (1). This has the advantage of including much of the fundamental physics in a very accurate way. A disadvantage is the practical limitation to a late stage in the planetary growth process when the number of bodies is small enough (e.g., 5 100) to permit the computational effort to be manageable on a sufficiently long timescale (lo7 to lo8 years). (2) Making use of an analogy with the kinetic theory of gases, in which averaged aspects of an assemblage of bodies (molecules or planetesimals) can be determined even when the number of bodies is too large to permit them to be considered individually (3,4,5). This approach is not limited computationally. In addition, it is simple to include the physically important effects of fragmentation into the calculations, despite the extremely large number of bodies associated with this phenomenon. It does require, however, questionable assumptions regarding the validity of such things as the use of averaged quantities, and of the way in which 3-body effects and long range forces are included. For these reasons, neither of these approaches, nor any other existing alternatives, are alone adequate to treat the problem. Insight into the degree of validity of the second aproach can be obtained, however, by comparative calculations of special problems that are amenable to treatment by both techniques. A numerical integration of one such problem has been reported recently (1). The initial conditions are 3000 1 0 ~ ~ ~ bodies, beetween 0.98 and 1.02 AU, and with initial eccentricities of two Hill units (e = 5.1 x and inclinations of one Hill unit. In order to obtain a usefully large degree of evolution with the available (major) computer resources, it was necessary to artificially increase the radius of the bodies, equivalent to their having a material density of 0.016 g/cm3. The orbital and collisional evolution of this system was calculated for 20,000 years, equivalent to several hundred thousand years at more realistic densities. Exactly the same problem was calculated using a "gas dynamic" program (6), including use of the low material density. This program was modified only by removal from the calculation of phenomena such as collisional fragmentation and gas drag that were not included in the numerical integration. The results of the two calculations were remarkably similar. The cumulative mass distributions after 20,000 years are shown in Fig. 1. At this time, in both cases the total number of bodies had decreased from 3000 to about 1000. Of these, the majority (600) still have their original mass, comprising 20% of the total mass. Most of the mass is in a very steeply (dnldr) m r-5.5 distributed "accumulation cascade" in which most of the mass is concentrated at low masses. At the high mass end of the distribution there are a few > 1 0 ~ ~ ~ runaway bodies detached from the steep accumulation cascade. The runaways contain 21% of the total mass for the numerical integration calculation and 33% of the total mass for the gas dynamic calculation. The size and number of these runaway bodies is subject to stochastic fluctuations. It is quite possible that moderate but