越盘轨道的包络和垂直振幅
Envelopes and vertical amplitudes of disk-crossing orbits
摘要：
我们最近发现，在轴对称星系盘的规则轨道有其包络$Z(R)$精确描述的关系$Z(R)\propto[Σ_I(R)]^{-1/3}$，如果它们的振幅是可比的磁盘厚度，其中$Σ_I$是表面密度的磁盘（整合在其整个垂直范围）。此外，通常的绝热近似给出了一个很好的描述低垂直振幅的轨道的包络。然而，这两种方法显然是脱节的，因为它们的表达方式在性质上有所不同。我们在本文中的目的是填补这一空白，这些以前的公式扩展到定期轨道内的磁盘任意垂直振幅。我们比较现有的$Z(R)$估计：薄磁盘的情况下，绝热近似（低振幅轨道在三维磁盘），和综合表面密度估计（高振幅轨道在三维磁盘），以建立它们之间的联系。这里提出的公式以一种优雅和连续的方式将上述结果联系起来，对盘的整个垂直延伸的垂直振幅都有效，并且具有对所有区域都具有相同形式的表达式。本形式的优点是$Z(R)$只依赖于可观测的量，即磁盘的垂直积分表面密度，而不需要获得系统的引力势。
英文摘要：
We recently found that regular orbits in axially symmetric galactic disks have their envelopes $Z(R)$ accurately described by the relation $Z(R)\propto[Σ_I(R)]^{-1/3}$, if their amplitudes are comparable to the disk thickness, where $Σ_I$ is the surface density of the disk (integrated over its whole vertical range). Moreover, the usual adiabatic approximation gives a good description of the orbits' envelopes for low vertical amplitudes. However, these two approaches are apparently disconnected, since their expressions differ qualitatively. Our purpose in this paper is to fill this gap by extending these previous formulae to regular orbits with arbitrary vertical amplitudes inside the disk. We compare existing $Z(R)$ estimates: the razor-thin disk case, the adiabatic approximation (low-amplitude orbits in three-dimensional disks), and the integrated surface-density estimate (high-amplitude orbits in three-dimensional disks) in order to establish a connection between them. The formula presented here links the aforementioned results in an elegant and continuous way, being valid for vertical amplitudes throughout the whole vertical extension of the disk and with an expression which has the same form for all regimes. The advantage of the present formalism is the dependence of $Z(R)$ only on observable quantities, namely the disk's vertically integrated surface density, without the need to obtain the gravitational potential for the system.
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