My newest project page on the collapse of BE sphere's can be found ​here.

Bonnor Ebert Spheres

Bonnor Ebert spheres are ISOTHERMAL HYDROSTATIC equilibrium structures. They are of interest in the context of star formation, as they provide a basic model of the dense clumps from which stars form.

Theory

Statics

Within a given radius, one can describe the density profile of an isothermal sphere in hydrostatic equilibrium. This density contrast is derived from the Lane-Emden equation (or just the Emden equation, depending on the variables used). To arrive at the Lane-Emden equation, one begins with the equation of hydrostatic equilibrium, Poisson's equation for gravity, and the isothermal equation of state:

frac{-1}{\rho} \triangledown P - \triangledown\phi_g ~= ~0 \qquad (1)

triangledown^2\phi_g~=~4\pi G \rho \qquad (2)

~=~ \rho ~C_s^2 \qquad (3)

Plugging (3) into (1) for P shows:

n ~\rho + \frac{\phi_g}{C_s^2} ~= ~const. \Rightarrow

\rho(r) ~ =~ \rho_c~e^{(-\phi_g/C_s^2)}\qquad--for~spherical~geometry ~(4),

which, when inserted into (2), yields:

\frac {1}{r^2} \frac {d}{dr}~(r^2 \frac{d\phi_g}{dr}) ~= ~ 4\pi G \rho_c e^{{(-\phi_g/C_s^2)}}~ $\qquad--spherical coordinates

By making the following variable substitutions,

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

one arrives at the Lane-Emden equation:

\frac{1}{\xi^2}\frac{d}{d\xi}(\xi^2\frac{d\psi}{d\xi}) = e^{-\psi} $ \qquad(Lane-Emden Equation)(5)

Dynamics

There are a few competing ideas for the dynamics of gravitational collapse for the non-rotating, non-MHD Bonnor Ebert sphere. Of these the self-similar solutions developed first by Larson-Penston (1969) and then by Shu (1977) are the foremost theories. While controversy may continue to exist as to which more adequately describes the situation, a notable numerical study to distinguish the two was produced by Foster and Chevalier (1993), ​Gravitational Collapse of an Isothermal Sphere. These authors have found that the collapse most closely resembles the Larson-Penston solution. I give a brief summary of the paper and its findings ​here.

Numerics

Key Questions

1. What is the structure of a Bonnor Ebert Sphere in equilibrium? How does rotation and MHD affect this structure?

2. What are the conditions that lead to a stable Bonnor Ebert sphere (i.e. one that only ever oscillates about its equilibrium point, rather than collapses or expands without bound?). We will study how varying the initial conditions induces instability, triggering collapse.

3. What are the dynamics of collapse under various conditions?

Methods

• Non-Rotating, Non-MHD — Stability Tests

The Bonner Ebert problem module is located at: ​http://www.pas.rochester.edu/~erica/problem_f90.html

For large r, the Bonnor Ebert sphere is gravitationally dominated (p. 246, Stahler). That is, perturbations can more easily trigger gravitational collapse as the supporting thermal pressure is diminished for large r. (In the limit of small r, the configuration is by comparison known as pressure dominated). Perturbations that can lead to collapse are as follows: a) total mass > critical Bonnor-Ebert mass, b) increased external pressure, and c) increased central density (or identically— increased nondimensional radius).

In numerical simulations, it is meaningful to constrain Pin=Pext at the outer boundary of the sphere, as the Bonnor Ebert sphere is in pressure balance and no external pressures are considered in its derivation. This gives the following useful relationship:

P_i = P_o \Rightarrow $ $P_i = C_s^2 \times\rho_i = C_s^2\times\rho_o = P_o $

That is, by requiring that the pressures remain equal at the sphere's edge implies that reducing the density outside of the sphere concomitantly increases the external temperature (Fig. 1). This can lead to more realistic conditions, as well as prove helpful in managing the mass outside of the sphere during simulations (refer to Results: Non-Rotating, Non-MHD). Figure 2 shows the set-up without adjusting any external parameters.

Fig. 1. Simulation diagram. Set-up for a hotter, lighter ambient medium.

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Fig. 2. Simulation diagram. Set-up with constant rho, p, and T across boundary.

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Using an approximate ​analytical solution to the Lane Emden equation, Astrobear inputs nondimensional radius, dimensional radius, and central density to return outer density. This BE density function has been written into the clumps object in astrobear, and can be turned on utilizing this piece of code.

The clump was allowed to sit on the grid for 5 crossing times to test whether it remained static.

• Non-Rotating, Non-MHD — Collapse

Collapse was initiated by reducing the internal energy everywhere in the grid between frame 0 and frame 1. That is, after the grid was initialized, the thermal energy was decreased by 10% below equilibrium values. This was sufficient to induce collapse.

Results

Structure

• Non-Rotating, Non-MHD

The structure of a Bonnor Ebert sphere can be described in terms of its density alone; since the system is isothermal, the pressure is proportional to the density. The density has a characteristic flat region near the innermost region of the sphere. After some distance, this density falls off like \frac{1}{\xi^2}.

This curve is generated from the numerical solution of the Lane-Emden equation (derivation above, image from Stahler):

These are plots of the density for a simulated Bonnor Ebert sphere. The first is a 2D slice of a 3D Bonnor Ebert sphere, while the second shows the density profile as taken from the center of the sphere to the outer medium.

Stability

• Non-Rotating, Non-MHD

When the ambient density (hence pressure, hence temperature) as described by the Lane Emden equation remains constant across the outer edge boundary, we found the sphere to be gravitationally unstable. This was due to the in-fall of material from the ambient medium which exceeded the weight the sphere could hold up. We circumvented this unintentional instability by decreasing the ambient density. Since the Bonnor Ebert sphere in in pressure balance, we did so while requiring the pressure to remain constant across the outer boundary. This led to a stable configuration:

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The left panel above is a movie of the mass density of a bonnor ebert sphere with self-gravity turned on. This movie shows the sphere to remain upright for 5 crossing times. The right panel is a movie of the pressure.

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These movies (again mass density on the left, pressure on the right) contrast the situation when self-gravity is switched off. Under these conditions one expects the sphere to expand. The runs shown here depict an unstable sphere, one in which thermal pressure does indeed disperse the sphere into the ambient medium. However, the sphere remains within the box because it is contained by the boundary conditions. The boundary conditions on all sides of the box were set to step on the velocity, setting any velocities across this boundary to 0. Any outflow of material from the box was thus prevented; as the sphere expanded, pressure waves that moved outward were deflected off of the boundaries back toward the sphere. The collision of these waves with the sphere explains the spurious features that appear on the outer layers of the sphere as the simulation progressed.

Collapse

• Non-Rotating, Non-MHD

As can be seen from these line-outs of radial velocity and density the collapse proceeded in an outside-in fashion, as reported by Foster and Chevalier (1993) and Banerjee and Pudritz (2003). The density of the inner-flat region (of initial extent xi=3) moved inward with time, while the outer envelope developed a strong r^-2 distribution characteristic of hydrostatic equilibrium. The infall began at the edge of the flat core region of the sphere (xi~3) and moved inward with time, approaching ~ 1.5 Cs in agreement with B&P (2003). (Define t0).

Discussion

Non-Rotating, Non-MHD

The inflow features we found, qualitatively matched those of Foster and Chevalier (1992). The following bullet points are taken from their results:

• Inflow is initially marked by velocity proportional to r for small r, peaking at xi = 2, and turning back to 0 velocity out toward edge of sphere.

• Simulation approaches LP solution in center region.

• Velocity smoothly transitions from homologous inflow to an inflow rate of ~ -3Cs after core formation.

• Inner region evolves fast, outer evolves slow.

• Core formation is not a subsonic adjustment of density with radius as Shu said. The inner regions collapse to a core supersonically.