Observing the effects of different background media on the critical Bonnor Ebert sphere

This page describes the effects of placing a marginally stable Bonnor Ebert sphere in different ambient media. Two ambient mediums and their effects on BE spheres were measured - a) a uniform light ambient medium (rho=rho(Rbe)/100), and b) an ambient medium with density that matches the density at the sphere's outer edge (rho=rho(Rbe)). The aim was to see if the effect of the "matched" ambient would be induced collapse of the sphere, triggered by the ram pressure of the infalling ambient material gravitationally accelerated by the BE sphere. The ram pressure was calculated at the sphere's outer edge and indeed exceeded the critical threshold of external pressure on the BE sphere. However, Pram>Pcrit long after Pthermal>Pcrit, thus it seems the increase in thermal pressure was the primary trigger for collapse. In contrast, the sphere in the light ambient medium remained dynamically stable, oscillating slowly about its equilibrium values for ~ 5 crossing times.

Parameters

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Computational Scales

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Approximations

I have made a pdf of my calculations on the expected density and ram pressure at Rbe. Please see the attached pdf ​here. The results I have found were namely a predicted density at r=Rbe, rho(Rbe) = 2.315*10^-21 g/cm³, and ram pressure Pram(Rbe)= 5.52 (CU- computational units). The free fall time associated with the approximated uniform sphere was found to be tff=3.4 myrs.

Results:

Note the line marking 'dx' denotes the size of the smallest cells in the grid, those within the BE sphere's center most regions (dx~0.1 pc). The cells outside a radius of 150% Rbe (radius of BE sphere), however, made up the coarsest level grid (see below movies of the mesh) and had dx~1.62 pc for smaller grid and dx~3 pc for larger grid. The last time frame of data included in these plots and the movies below is the time before the collapse was no longer resolved adequately with the refinement criteria used. Since the aim of this study was to look at the initialization of the collapse, the refinement used here was sufficient. The time in which the sphere's Pram(Rbe) > Pcrit was between 2.3<tc<3 myr, ~ 67%<tff<88% (see calculation in approximations section).

Radial Velocity-

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The same wave of inward moving material is seen in each of these plots, except that the sphere in the lighter medium is more resilient to allowing the wave to pass through the boundary, whereas in the matched case there seems to be no boundary and the wave passes freely through Rbe. Thus it seems that in the light ambient case, while there is inward moving material, there is little momentum being imparted to the sphere's boundary and so, the velocity of the material inside doesn't acquire the general shape of the velocity curve outside of the sphere. The inward moving material in the matched case must have a greater momentum, allowing the flow of energy to enter into the sphere resulting in a continuous wave across the boundary.

Density-

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In the light ambient case, there is no shell of infalling matter building up on the boundary of the BE sphere. However, the density profile inside of the sphere is decreasing uniformly with time as it is "breathing" around its equilibrium values. Over time, these curves would increase back up to initial values.

In contrast, the matched ambient case shows a definite build up of material at the boundary, and by t=3.07 myrs, the sphere is no longer a BE sphere, with a density profile that markedly differs from the earlier density profiles. Additionally, while density is increasing at the boundary, it is not simply piling up at the sphere's initial surface (Rbe), rather material has caused the sphere to contract, building up further inside Rbe as t progresses.

By t=tc~2.3 myr, my ​calculations predicted a density to be of order rho~2.3x10^-21g/cm³. While this is off by ~ 2, it also should be noted that my calculation was considering a shell of mass piling up at the initial Rbe. Since by 2.3 myrs, material has passed within the initial Rbe, my calculation would presumably be larger than observed here.

Pressure Plots

The horizontal line in the following plots is the value of Pcrit, where Pcrit~1.14 Cs⁸/G³*M² and characterizes the maximum external pressure a critical BE sphere can withstand before collapsing. Note that both the thermal and ram pressure at the Bonnor Ebert surface, Rbe, is BELOW the threshold under light ambient medium conditions, whereas in the "matched" ambient medium case both of these pressures EXCEED the threshold.

Thermal Pressure-

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As the EOS used was effectively isothermal, the curves here are of the same shape as the density curves above. The x-axis is scaled as in the above plots, however, the y-axis is in computational units (CU's). To put into scaled units, one can multiply y-values by Pscale in the table above in the Scales section.

Of interest here is that in the light ambient case, the pressure at the sphere's edge (RBE), never exceeds Pcrit, which lies along the external pressure of the ambient material. However, as soon as the density exceeds equilibrium values as shown in above plot and proportional to these curves here, Pthermal>Pcrit. Thus the BE sphere is very early on UNSTABLE to collapse.

Ram Pressure-

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The ram pressure in the light ambient case does not build up outside of the sphere at all, which is in accordance with the above statement that the material in the ambient, while moving inward, is carrying no momentum. Inside the ram pressure is fluctuating in a manner that coincides with the radial velocity and density plots in this region. At all times, the ram pressure is well below the critical value.

In the matched ambient case, however, Pram>Pcrit only after t~2.3 myrs. By t =3 myrs, the Pram profile has taken on a very unique "cowboy-hat" appearance. Another look at the radial velocity plots and the density plots show that this is due to the high increase in density inside of the sphere by t=3myrs with small concomitant radial velocities, while outside of the sphere there are small density buildups coinciding with a large negative vrad, which when squared for Pram gives a large Pram outside as well.

Discussion

-It may be interesting to compare these plots to those of a similar collapse study in which the critical BE sphere in a light medium (same density and homogeneity as in the light ambient of this study) was triggered to collapse with a density enhancement above equilibrium values ​(click here). The sphere parameters are the same, except that in the B&P setup, collapse was induced by a 10% over density of the grid.

-To better understand the last time in the plot above of density in the Matched case, where the density left of the steepening shock drops below earlier values, see these movies of the radial velocity and density:

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-The ram pressure does not seem to be as dominant a factor in inducing collapse, rather the thermal pressure almost immediately exceeds Pcrit.. Thus it may be better to approximate tc as much earlier on.

-Next step may to better characterize Rbe, as it is moving in with time..

-size differences, computational difficulties, etc.

Movies

Left panel is sphere in the light ambient, right is the matched ambient.

Mesh-

The movies shown below are just of the time frames up until the collapse in the matched case became poorly resolved. The BE sphere in the light ambient was stable for 5 crossing times, to see an extended clip follow the link ​here.

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Density-

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The BE sphere in the light ambient was stable for 5 crossing times, to see a longer clip see ​here.