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

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

2002-10-31

arxiv.org/abs/gr-qc/0211001v1

Int.J.Mod.Phys. A18 (2003) 3433-3468

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

35 pages; IJMPA style file

Scientific paper

10.1142/S0217751X03013892

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.

Affiliated with

Also associated with

No associations

Rating

Power laws, scale invariance, and generalized Frobenius series: Applications to Newtonian and TOV stars near criticality does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Power laws, scale invariance, and generalized Frobenius series: Applications to Newtonian and TOV stars near criticality, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Power laws, scale invariance, and generalized Frobenius series: Applications to Newtonian and TOV stars near criticality will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-432257