## Project for ASTR7741

Fall Semester, 2007

Gravitational Wave Strain Curve for Collapsing, Rotating Stellar Cores

Before proceeding to study this problem in a relatively simplistic fashion, you should be aware that some quite sophisticated models of rotating core collapse are already in the literature. Most notable is the series of articles reporting on simulations by a group at the Max Planck Institute (MPI) in Garching, Germany. Here is the URL of their home page:

www.mpa-garching.mpg.de/Hydro/

From this page, you can click on their "Waveform Catalogs" pages ( Newtonian Models or GR Models) to see some of their published strain curves. You'll be trying to reproduce only the very earliest portion of these strain curves.

The accompanying PDF document called newstrain.pdf contains some notes that a former student of mine (Ravi Kumar Kopparapu) put together to illustrate how relatively simple formulae for the strain h(t) and associated gravitational-wave luminosity of a dynamically evolving system can be derived using the post-Newtonian quadrupole formulation.

Part I: Spherical, Free-fall Collapse of a nonrotating, n = 0 Polytrope

Work through the spherical "free-fall" problem described in the accompanying PDF file. Note that the first page of this file defines the problem; the second page contains the solution. You should try to derive the solution yourself before reading the solution.

Write a program (or use Mathematica) to also derive r(t), v(t), and r(t) for this spherical free-fall problem; compare your results with the analytic answer by plotting both results on top of one another.

Part II: Free-fall Collapse of a Rotating (and nonrotating), n = 0 Spheroid

First, you need to generalize the spherical free-fall collapse problem by (numerically) deriving the collapse solution for a freely falling (pressure-free) spheroid. You should set the problem up so that you can study both nonrotating and rotating spheroids. The solution to this problem has previously been published in three key papers:

• Lynden-Bell, D. 1962, Proc. Cambridge Phil. Soc., 50, 709
• Lin, C.C., Mestel, L. & Shu, F.H. 1965, ApJ, 142, 1431
• Hutchins, J.B., 1976, ApJ, 205, 103
You need to write a numerical algorithm that will follow this collapse, starting with the dynamical equations outlined in any one of these papers, or as given by eqs. in the "Maclaurin Spheroid" chapter of my on-line textbook. If handled in cylindrical coordinates, the problem is two-dimensional and (at least in the absence of rotation) is governed by the equation,
dv/dt = - ÑF = - ev (2p GrA1) v - ez (2p GrA3) z.

Part III: Gravitational-Wave Strain Curve for Nonspherical Collapses

Once you have a numerical algorithm that follows the free-fall collapse of a uniform-density spheroid, you need to add to it a routine that calculates the two principal moments of inertia of the spheroid at various points during its collapse. The second time-derivative of these moments of inertia can then be used to determine the strain curve.

The only published work that I know comes close to providing this desired solution is the following paper by Thuan & Ostriker: