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| 1 | = Computing the Jeans Length = |

| 2 | |

| 3 | The first step in setting up this problem was to consider the Jeans Length for a given ambient density and temperature. Recall, |

| 4 | |

| 5 | [[latex($\lambda_J = C_s(\frac{\pi} { G \rho_0})^{1/2}$)]] |

| 6 | |

| 7 | I used the ambient density and temperature of the BP case for the Bonnor Ebert runs. This gave |

| 8 | |

| 9 | [[latex($\rho0 = 3.34 \times 10^{-23} g/cm^3$)]] |

| 10 | |

| 11 | [[latex($\ T = 100K $)]] |

| 12 | |

| 13 | Using the isothermal sound speed of, |

| 14 | |

| 15 | [[latex($C_s = (\frac{K_B*T}{m_H})^{1/2}$)]] |

| 16 | |

| 17 | where Kb is the boltzmann constant and mH is the mass of hydrogen, I calculated |

| 18 | |

| 19 | [[latex($\lambda_J = 1.08 \times 10^{20} cm \approx 35 pc $)]] |

| 20 | |

| 21 | = Setting up the problem domain = |

| 22 | |

| 23 | I decided on a 1D grid (a string of cells in x) that was much greater than the Jeans length: |

| 24 | |

| 25 | [[latex($L = 1550 ~ pc $)]] |

| 26 | |

| 27 | The Jeans Length needed to be resolved to prevent artificial fragmentation, so |

| 28 | |

| 29 | [[latex($\triangle x < \frac{1}{4} \lambda_J \Rightarrow $)]] |

| 30 | |

| 31 | [[latex($\triangle x < 9 ~pc $)]] |

| 32 | |

| 33 | Choosing 1550 cells in the x direction (small computational cost given 1D sim) satisfies this by having |

| 34 | |

| 35 | [[latex($\triangle x = 1~pc $)]] |

| 36 | |

| 37 | so there are about 35 cells/ Jeans length. |

| 38 | |

| 39 | = Seeding the perturbation, analytic solution = |

| 40 | |

| 41 | = The code = |