Power Laws, Scale Invariance, and Generalized Frobenius Series: Applications to Newtonian and Tov Stars near Criticality

We present a self-contained formalism for calculating the background solution, the linearized solutions, and a class of generalized Frobenius solutions to a system of scale invariant differential equations. We first cast the scale invariant model into its equidimensional and autonomous forms, find its fixed points, and then obtain power-law background solutions. After linearizing about these fixed points, we find a second linearized solution, which provides a distinct collection of power laws characterizing the deviations from the fixed point. We prove that generically there will be a region surrounding the fixed point in which the complete general solution can be represented as a generalized Frobenius-like power series with exponents that are integer multiples of the exponents arising in the linearized problem. This Frobenius-like series can be viewed as a variant of Liapunov’s expansion theorem. As specific examples we apply these ideas to Newtonian and relativistic isothermal stars and demonstrate (both numerically and analytically) that the solution exhibits oscillatory power-law behaviour as the star approaches the point of collapse. These series solutions extend classical results; as exemplified for instance by the work of Lane, Emden, and Chandrasekhar in the Newtonian case, and that of Harrison, Thorne, Wakano, and Wheeler in the relativistic case. We also indicate how to extend these ideas to situations where fixed points may not exist — either due to “monotone” flow or due to the presence of limit cycles. Monotone flow generically leads to logarithmic deviations from scaling, while limit cycles generally lead to discrete self-similar solutions.