Speaker
Description
Many analytic solutions in astrophysics have been found by assuming certain symmetries. A classical example in accretion disk theory is the natural assumption of axial symmetry of the flow, which reduces the dimensionality of the problem from three to two. Additional assumptions -- often physically motivated rather than mathematically correct -- are still required to eliminate one more dimension and reduce the system to a one-dimensional problem that is relatively easy to solve. A typical approach is to assume strict vertical hydrostatic equilibrium, which allows integration of the fluid equations in the vertical direction. In this way, the originally two-dimensional problem is replaced by a one-dimensional one that may (or may not) retain many of the important properties of the former.
In this talk, we will consider a different approach based on assuming another type of symmetry: scaling invariance, which leads to self-similar solutions. After reviewing several existing examples of such solutions, I will focus on a classical case -- a cold, geometrically thin, and optically thick disk. I will show in which respects the structure of such a self-similar flow agrees with and differs from the classical Shakura–Sunyaev solution. The nature of the self-similar solution does not allow its application near compact objects, where general relativity -- naturally breaking scaling invariance -- plays an essential role. Nevertheless, I will briefly outline a perturbative method that may help overcome these difficulties.