General solution of the Jeans equations for triaxial galaxies with separable potentials
General solution of the Jeans equations for triaxial galaxies with separable potentials
复制标题
可分离势三轴星系 Jeans 方程的通解
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发表时间：
2003
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影响因子：
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通讯作者：
ðð Ú¾º¾µ
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作者：
Èööòøøø;Öùùöý åae;Ä Ì øýðð;ðð Ú¾º¾µ
Jeans方程将二阶速度矩与恒星系统的密度和势能联系起来。对于一般的三维恒星系统，存在三个方程和六个独立的矩。通过假定势是三轴的、可分离的Stackel形式，混合力矩在共焦椭球坐标下消失。因此，这三个Jeans方程和三个剩余的非零矩组成了一个闭合的三个高度对称的三变量耦合一阶偏微分方程组。这些方程最早是由40多年前的林登-贝尔推导出来的，但一直拒绝用标准方法求解。我们在这里介绍一般的解决方案。
我们首先考虑二维极限情况。我们用一种叠加奇异解的新方法来求解他们的Jeans方程。奇异解是新的，是标准的Riemann-Green函数。由此得到的Jeans方程的解给出了整个系统的二阶矩，其形式是某些二阶矩的规定边界值。二维解适用于非轴对称圆盘、扁球体和长椭球，也适用于无标度三轴极限。对边界条件有一些限制，我们将详细讨论。然后，我们将奇异解的方法推广到三轴情形，并再次根据给定的二阶矩边值得到完整的解。这些边界值也有限制，但边界条件都可以在单个平面中指定。一般解可以用完全(超)椭圆积分来表示，这可以用一种简单的方式来计算，并且提供了在可分离的三轴势中支持三轴密度分布的完整的二阶矩集。
The Jeans equations relate the second-order velocity moments to the density and potential of a stellar system. For general three-dimensional stellar systems, there are three equations and six independent moments. By assuming that the potential is triaxial and of separable Stackel form, the mixed moments vanish in confocal ellipsoidal coordinates. Consequently, the three Jeans equations and three remaining non-vanishing moments form a closed system of three highly symmetric coupled first-order partial differential equations in three variables. These equations were first derived by Lynden-Bell, over 40 years ago, but have resisted solution by standard methods. We present the general solution here.
We consider the two-dimensional limiting cases first. We solve their Jeans equations by a new method which superposes singular solutions. The singular solutions, which are new, are standard Riemann–Green functions. The resulting solutions of the Jeans equations give the second moments throughout the system in terms of prescribed boundary values of certain second moments. The two-dimensional solutions are applied to non-axisymmetric discs, oblate and prolate spheroids, and also to the scale-free triaxial limit. There are restrictions on the boundary conditions that we discuss in detail. We then extend the method of singular solutions to the triaxial case, and obtain a full solution, again in terms of prescribed boundary values of second moments. There are restrictions on these boundary values as well, but the boundary conditions can all be specified in a single plane. The general solution can be expressed in terms of complete (hyper)elliptic integrals, which can be evaluated in a straightforward way, and provides the full set of second moments that can support a triaxial density distribution in a separable triaxial potential.