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Matlab & Mathematica

Consider the equilibrium structure of a rotating, self-gravitating, isothermal gas cloud. Such an object obeys the partial differential equations.

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Consider the equilibrium structure of a rotating, self-gravitating, isothermal gas cloud. Such an object obeys the partial differential equations:

where Φ = 0 at the origin. where:

  • Φ is the gravitational potential;
  • ρ is the density;
  • P is pressure;
  • G is the gravitational constant;
  • cs is the sounds speed;
  • and Ω is the angular frequency of

Work in cylindrical coordinates and assume that the cloud is axially symmetric about the z axis: so that r=r(r,z), P=P(r,z), and Φ = Φ (r,z). Work in units where 4πG = cs = ρc = Pc = 1, where ρc and Pc are the density and pressure at the centre of the cloud respectively.

Assume that the cloud is truncated by an external pressure 0 < Ps < 1, such that ρ = 0 when P< Ps. The medium external cloud is “hot” gas, with pressure Ps everywhere, effectively zero density, and no rotation.

Solid-body rotation:

 The simplest model has Ω = const, but this will lead to a net outward force – and severe numerical instability – when r is large. Highly pressure-truncated cores will be OK, but there will be numerical instabilities when the core is not very truncated.

Your task is to devise an iterative numerical scheme to solve our system of equations self consistently and provide a range of solutions for varying values of Ω and Ps. I suggest the following algorithm:

 

  1. Assume an initial pressure-truncated density
  2. Solve eq. 1 using any method discussed in class. (Successive Over Relaxation would also be a good choice. Please do not use MATLAB’s PDE )
  3. Use eq. 4 to obtain a new estimate of ρ.
  4. Exit if converged, else go to

Your writeup should include a grid of figures (I would suggest contour plots) showing how the density structure changes as Ω and Ps are varied. I suggest at least a 5x5 grid over a sufficient range of values to show how the structure changes. Some combinations of Ω and Ps will not permit a solution, due to instability effects.

Also, calculate the XX and ZZ components of the moment of inertia tensor, where Z is the symmetry axis. The ratio Izz/Ixx is a good way to describe the deviation of the object from spherical symmetry. Make a well-labelled contour plot of Izz/Ixx as a function of Ω and Ps.

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