The role of a massive ambient medium in triggering collapse of a Bonnor Ebert sphere: deviation from the classic outside-in collapse

Abstract

This paper describes a series of 3D hydrodynamic simulations that explored the effects of placing a marginally stable Bonnor Ebert sphere in different ambient media. Various uniform density media were initialized that were in pressure equilibrium with the Bonnor Ebert sphere. These densities ranged from a traditional light ambient (rho=0.01rho(Rbe)) to an ambient with density that matched that at the sphere’s outermost edge (rho=rho(Rbe)). The main aims were to a) discern whether a massive ambient medium would be sufficient to induce collapse of the sphere, b) whether this collapse would be triggered by the ram pressure of infalling ambient material gravitationally accelerated by the BE sphere, and c) how this collapse would compare with the classic cases of outside-in and inside-out collapse of previous models. [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. These results are qualitatively different than those found previously by authors such as who found a very nice outside-in collapse.

-[Results should go here, but not sure if this is the line of reasoning we want to follow] -Maybe add something about Hannebelle et al. here, describe the collapse more clearly as triggered by a compression wave? -Add to the list of runs the 10% overdensity case — B&P run? -Should we run these same set-ups, but with a sub-critical sphere?

Introduction Outline

1. Classic outside-in collapse features vs. SIS
2. Relation to star formation
3. BE sphere definitions
4. BE spheres are out there in space, SIS discounted.
5. Modeling BE spheres, the old
   1. Why was the ambient neglected before?
6. Non-realistic to have a sharp discontinuity in density at spheres-ambient interface
7. More realistic to have ambient=rho(Rbe)
8. Modeling BE spheres, the new
9. How does the inward pull of gravity of the ambient medium interact with the sphere? A critical sphere is close to collapse, how is it perturbed out of eq., ram pressure?, mass?
10. The code/the model
11. Thesis Paragraph

The collapse of the isothermal, hydrostatic sphere has been studied extensively, albeit with focus mainly on 2 special cases: the self-similarity solutions of the collapsing singular isothermal sphere (Shu), and the more modern numerical studies of collapsing marginally stable spheres (F&C, B&P). However, the latter, and today generally more popular, simple model for protostar formation has been largely limited to the placement of a critical BE sphere in an ambient medium that was a) in pressure equilibrium with the sphere, b) of uniform density many times lighter than that of the sphere’s outermost edge, and c) perturbed out of equilibrium by some artificial trigger such as an overdensity of the sphere and surrounding medium. This led to the present objective of exploring the effect of whether collapse of a marginally stable sphere could be induced by its environment with no ad hoc perturbations, and how that collapse may differ from previous qualitative features.

Methods Outline

1. Astrobear/Numerical Setup
   1. 3D-Eulerian grid, with self-gravity, hydro equations? ii.. AMR, refinement criteria used
2. larger grid, ~30 RBe across corresponding to ~45 pc across
3. Resolution
4. Lane Emden !parameters/BE specific quantities for initialization
5. xi, rho_central, r to give rho_outer.
6. EOS, ideal, gamma=1.0001
8. The runs we did
   1. B&P - 10% overdensity
   2. Stabillity Check - B&P without overdensity, 5 crossing times.
   3. Cloud Crushing runs - the 3 or so that I have sent Phil?
   4. Maybe a table with the runs and params for each?

Results/Analysis

1. Movies

B&P, classic outside-in collapse

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Stability

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Matched

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1. Approximations/Quantitative Estimates?
2. Quantitative calculations on sim. specific quantities?
3. Figures
   1. Mesh
   2. Lineouts of fluid vars iii.

Discussion: -Further directions: non-uniform collapse? Rotation? MHD? -Hannebelle?

I ran a series of simulations to study the deviation from the 'classic' collapse studies of the BE sphere - a slight increase in density values above equilibrium of a critical sphere (xi=6.5) in pressure equilibrium with its environment (Foster and Chevalier, 1993, Banerjee and Pudritz, 2003). My simulations were all variations of the B&P setup, but with a sphere in a domain twice the size (~30 Rbe), and varying ambient densities. These runs are as follows:

1) Stability test of the critical BE sphere placed in an ambient rho=0.01rho(Rbe), with sphere and ambient in pressure equilibrium at r=Rbe. This is just the B&P sphere, with no density enhancement, in a box twice as large. My apologies, but the movie is broken into 2, with the first half here: ​https://clover.pas.rochester.edu/trac/astrobear/attachment/wiki/u/BonnorEbertMatched/rhoCroppedLight.gif and second half here: ​https://clover.pas.rochester.edu/trac/astrobear/attachment/wiki/u/BEmoviesLonger/rhoSecondHalf.gif. This set-up was stable to collapse for at least 5 crossing times, where 1 tc= 0.2 in computational units, as can be seen from this 2d slice through the center of the sphere. Here the sphere is breathing about its equilibrium initial condition.

2) 'Classic B&P collapse' of a BE sphere. That is, 10% density enhancement of sphere above equilibrium values, in pressure equilibrium with a light (rho=0.01rho(Rbe) ) ambient. This, as well as the following plots, show lineouts through the center of the sphere of the quantities: rho, absolute value(radial velocity), absolute value(mach), and pressure, all normalized to their initial values in the grid. The x-axis is logarithmic and begins at the length of the smallest zone dx. The y-axis is linear: ​http://www.pas.rochester.edu/~erica/BPJuly22.gif . The collapse here resembles the B&P results, namely collapse proceeds in an outside-in fashion and remains subsonic most of the way through.

3) BE sphere in an ambient rho= rho(Rbe). This is the same setup as above, except now there is no 10% rho enhancement and the ambient has same density as BE sphere at r=Rbe. ​http://www.pas.rochester.edu/~erica/MatchedJuly26.gif Here collapse seems to be proceeding in a very different manner. At small times, material begins to pile up at the sphere's outer edge. After enough material has accumulated, the pressure at the sphere's outer edge exceeds critical values and moves inward with time in a compression wave. The collapse is supersonic most of the way through. Note, radial velocity is now the actual quantity, to show clearly the sign (<0 is moving inward, >0 is moving outward).

4) 3 simulations at equally spaced ambient densities between the light and matched cases: ​http://www.pas.rochester.edu/~erica/Light105July22.gif ​http://www.pas.rochester.edu/~erica/Light70July22.gif ​http://www.pas.rochester.edu/~erica/Light35July22.gif

In summary, I would say that the runs in which collapse was NOT induced to occur through a density enhancement, seem to proceed much differently. They resembled qualitatively the runs by Hannebelle et al., in which a pressure wave moved through the sphere, causing collapse into a sink particle. The resolution of these sims became poor by the time the apparent wave has moved into the sphere, so the simulation was killed ostensibly before a sink formed. I can run these simulations out longer and at higher resolution as a next step if you suggest. This however, will be postponed until after I take the preliminary PhD written exam, due to take place at the end of August.