具有自转的等温Lane-Emden方程的精确解及其对行星和卫星形成的意义
Exact Solutions of the Isothermal Lane--Emden Equation with Rotation and Implications for the Formation of Planets and Satellites
摘要：
本文导出了圆柱几何中有无转动的等温Lane-Emden方程的精确解。相应的流体静力平衡是最相关的动力学的原太阳星云之前和期间的行星和卫星的形成阶段。质量密度的非旋转解是解析的、非奇异的、随半径单调递减的，并且它很容易满足中心处通常的物理边界条件。当微分旋转添加到莱恩-埃姆登方程，一类完整的精确解的质量密度出现。我们也通过分析确定了所有这些解。在这一类中，幂律或幂律组合的解不能够满足相关的边值问题，但它们仍然非常重要，因为它们构成了“基线”，当施加中心边界条件时，实际的解会接近基线。执行这种物理边界条件的数值积分表明，实际的径向平衡密度分布出现在接近非旋转解的中心，但是一旦它们在相应的基线以下交叉，它们就不再是单调的。实际解被迫永久地围绕基线解振荡，而不会沉降到基线解上，因为中心边界条件严格禁止这种沉降，即使在大半径的渐近状态下也是如此。根据我们的研究结果，我们预计准静态演化的原行星盘应发展振荡密度分布在其等温阶段的中平面。在这些配置文件中的峰值对应于当地潜在的最小值和它们的位置是理想的网站为形成的原行星。
英文摘要：
We have derived exact solutions of the isothermal Lane--Emden equation with and without rotation in a cylindrical geometry. The corresponding hydrostatic equilibria are most relevant to the dynamics of the protosolar nebula before and during the stages of planet and satellite formation. The nonrotating solution for the mass density is analytic, nonsingular, monotonically decreasing with radius, and it satisfies easily the usual physical boundary conditions at the center. When differential rotation is added to the Lane--Emden equation, an entire class of exact solutions for the mass density appears. We have determined all of these solutions analytically as well. Within this class, solutions that are power laws or combinations of power laws are not capable of satisfying the associated boundary--value problem, but they are nonetheless of profound importance because they constitute "baselines" to which the actual solutions approach when the central boundary conditions are imposed. Numerical integrations that enforce such physical boundary conditions show that the actual radial equilibrium density profiles emerge from the center close to the nonrotating solution, but once they cross below the corresponding baselines, they cease to be monotonic. The actual solutions are forced to oscillate permanently about the baseline solutions without ever settling onto them because the central boundary conditions strictly prohibit such settling, even in the asymptotic regime of large radii. Based on our results, we expect that quasistatically--evolving protoplanetary disks should develop oscillatory density profiles in their midplanes during their isothermal phase. The peaks in these profiles correspond to local potential minima and their locations are ideal sites for the formation of protoplanets ...
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