双子宇宙天体物理学与宇宙学 鬼物质 天体物理学.6. 螺旋结构 (第2页)
图4：星系被束缚在洞穴中，周围是炽热的鬼物质，示意图。
图中洞穴的边界是突变的。更详细的图像可参见参考文献[22]。见图5。
图5：星系被束缚在洞穴中，周围是炽热的鬼物质，示意图。数值计算结果 [22]。
以目前的系统处理此类三维问题似乎相当困难，这需要大量相互作用的质量点。
我们来考察一个不现实的星系情况，即被一个环形结构所束缚。为了构建有用的初始条件，我们首先引入一些理论工具。
原文（英文）
twin universe astrophysics and cosmology Matter ghost matter astrophysics.6. Spiral structure.(p2)
Fig.4 : Schematic representation of a galaxy confined in a hole, in the surrounding hot ghost matter.
In the figure the border of the hole is abrupt. A more detailed image can be found in reference [22]. See figure 5.
Fig.5 : Schematic representation of a galaxy confined in a hole, in the surrounding hot ghost matter. Computational result [22].
It looks fairly difficult to deal with such a 3D problem, with today's systems. It would require a huge number of interacting mass-points.
Let us consider the case of a non-realistic galaxy, confined by a ring. In order to build useful initial conditions we will first bring some theoretical machinery.