Power laws, scale invariance, and generalized Frobenius series: Applications to Newtonian and TOV stars near criticality

We present a self-contained formalism for analyzing 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. (Lane, Emden, and Chandrasekhar in the Newtonian case; 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.