Computing the Jeans Length

The first step in setting up this problem was to consider the Jeans Length for a given ambient density and temperature. Recall,

\lambda_J = C_s(\frac{\pi} { G \rho_0})^{1/2}

I used the ambient density and temperature of the BP case for the Bonnor Ebert runs. This gave

\rho0 = 3.34 \times 10^{-23} g/cm^3

\ T = 100K

Using the isothermal sound speed of,

C_s = (\frac{K_B*T}{m_H})^{1/2}

where Kb is the boltzmann constant and mH is the mass of hydrogen, I calculated

\lambda_J = 1.08 \times 10^{20} cm \approx 35 pc

Setting up the problem domain

I decided on a 1D grid (a string of cells in x) that was much greater than the Jeans length:

L = 1550 ~ pc

The Jeans Length needed to be resolved to prevent artificial fragmentation, so

\triangle x < \frac{1}{4} \lambda_J \Rightarrow

\triangle x < 9 ~pc

Choosing 1550 cells in the x direction (small computational cost given 1D sim) satisfies this by having

\triangle x = 1~pc

so there are about 35 cells/ Jeans length.

Seeding the perturbation, analytic solution

The code