Big Picture

A Bonnor Ebert sphere is just a clump "object" with a Bonnor Ebert density profile. The Bonnor Ebert density profile is coded into astrobear using an approximate analytic solution to the Lane-Emden equation. This function solves for the density contrast of the sphere from center to outer edge, as a function of the non-dimensional radius,xi, where xi is given by:

xi ~=~ (\frac{4 \pi G \rho_c}{C_s^2})^\frac{1}{2}~r

, r is the dimensional radius of the sphere, Cs is the isothermal sound speed of the sphere (function of the temperature), and rho_c is the central density of the sphere. Once the user specifies r, rho_c, and xi for the sphere (see next section), astrobear sets both a) the temperature of the sphere using the above equation, and b) the outer density of the sphere. One can then use the ideal EOS in physics.data with a gamma = 1.0001 to approximate the grid as isothermal. Given P = nKT, the pressure of the sphere then drops away from the center of the sphere with the same gradient as rho.

Now, that all explains the clump object itself, but there is an additional "object" in the BE problem module that controls the ambient medium in which the clump resides. This ambient object is by default set to have a uniform density, equal to the density at the sphere's outer most edge, rho(Rbe). Additionally, it is also in pressure equilibrium with the sphere by default. That is, Pamb=P(Rbe). Now, since the simulation is isothermal, this condition means the temperature is discontinuous at the sphere-ambient interface, given by P/n = KT, where n is the number-density of the gas.

To summarize, the BE sphere is a clump object that is intialized with the BE density profile. It is placed in an ambient medium object. How to control the different parameters of these objects is described next.

Problem.data

The BE module requires the following 3 user-specified parameters to set up a BE sphere on the grid:

1. central density of the sphere (central_rho),
2. non-dimensional radius (xi),
3. dimensional radius (clump_rad).

These 3 parameters are to be set in problem.data, and are the 3 ingredients that uniquely and completely specify the BE sphere (reference here).

Now, to recap, the BE sphere is actually a clump object. It is controlled externally from the problem.f90 file by specifying the parameters (1-3) above in physics.data. However, the controls for the ambient medium object are internal to the problem.f90 file. It's treatment is provided next.

Ambient Object

As stated above, the clump object is embedded inside another object, the ambient medium. This ambient medium begins outside of the clump, and the lines of code that initialize this object are described as follows. Both the ambient object's density and pressure are specified inside of problem.f90. The module defines the ambient pressure, pOut, and the ambient density, rhoOut as follows:

That is, the module takes the outer density and pressure of the BE sphere (rho(Rbe), P(Rbe)) and copies these values to the ambient object. There is a weight factor attached to rhoOut, however, that controls how light the ambient is compared to the sphere's outer most edge. This weight is easily modified in the problem.data file. Next, the B-field and V-field are set; their values are read in from problem.data, where they =0 by default.

Units

The 3 parameters the user sets in problem.data should be in units of: [central_rho]:= particles/cm³, [xi]:= unit-less, but should be chosen based on desired stability u/BonnorEbert see, [clump_rad]:= cm. The scales should be chosen accordingly.

Standard Out

The BE module comes with a few lines of helpful information printed out at the beginning of astrobear's standard out. This includes the central density of the BE sphere in cgs, the crossing time of the sphere in computational units, the isothermal sound speed of the sphere (note, this will be different in the ambient if there is a temperature discontinuity across the sphere-ambient interface) in cgs, and whether or not the BE sphere should be stable if no perturbations are present.