Do We Understand Fission?

Issues:

• Stable bars should undergo further secular evolution

• Different initial conditions may lead to grossly different outcomes (e.g., star/disk systems and one-armed spirals).

• Different gas physics applies during different evolution phases (e.g, diatomic vs. monatomic gases).

• Thermodynamic effects

Solution:

Perform full-blown numerical solutions with all relevant physics following the collpase from Molecular Cloud densities (10^-18 g cm^-3) to stellar densities (100 g cm^-3). Not feasible. In the 1960s, 1-dimensional simulations were performed (Larson 1969). Thirty years later, in the late 1990s, one 3-dimensional simulation has been published (Bate 1998). Even then, the simulation did not use an energy equation. It used a barotropic equation-of-state (an equation-of-state where the pressure is a function only of the density). The calculation was performed using the numerical technique known as Smoothed Particle Hydrodynamics.

What's A Boy to Do?:

We investigate fission during the quasi-static portions of the star formation process using linear, quasi-linear, and nonlinear techniques. We consider the linear stability properites of star forming regions and the quasi-linear and nonlinear development of fission-type instabilities.

Mathematical Models:

At the moment, we are concerned primarily with the hydrodynamic instabilities which arise in the star formation process. As such, we use simplified treatments of the thermal physics. We consider barotropic equations-of-state. In particular, we investigate polytropic gases, i.e., gases where


and K and n are constants. A monatomic gas has n = 3/2 and a diatomic gas has n = 5/2. Bate (1998), using a parametrized EOS which mimiced an isothermal gas, a monatomic gas, and a diatomic gas in different density regimes, reproduced 1-dimensional simulations which included realistic thermodynamic models.

Nonlinear Simulations:

Given the assumption of a polytropic gas, we need only solve the mass and momentum conservation equations,


The fully nonlinear simulations use the 3-dimensional forms of the hydrodynamic equations. The nonlinear problem is solved as an initial value problem by following the evolution of axisymmetric equilibrium states which have been loaded into the code. The equations are solved on a cylindrical grid with a transport scheme which is 2nd order in space and time.

Linear Simulations:

The set of linear evolution equations is found using perturbations of the form,


The time is carried implicitly, different than in most normal mode approaches. The set of linearized evolution equations are



The linearized problem is solved on a cylindrical grid. The spatial derivatives are discretized while the time evolution is performed using a Runge-Kutta algorithm. The code follows the evolution of initially random perturbations away from equilibrium.

Quasi-Linear Evolution

We bridge our linear and nonlinear simulations using a quasi-linear theory. The quasi-linear evolution is studied by calculating the gravitational self-interaction torque of the linear eigenfunction (the spiral pattern leads to the torque). The torque has the form