Boundary Effects

Boundary noise can manifest when subsonic waves near the boundary cause errors that may propagate inward with speed less or equal to the sound speed. Thus to consider if such a disturbance has time enough to travel from the boundary to the BE sphere/ambient boundary, I put together this table that shows the sound speed of the ambient, the final time of the simulation, and the distance that a sonic wave would travel in this time. For all cases, except for the stability and 1/30 ambient cases, sonic disturbances from the boundary do not have time enough to propagate from the boundary to the sphere's surface. Note, I have taken the boundary to be the sphere of radius = 15, centered on the origin. The physical boundary conditions on the box are set to extrapolating with perpendicular velocities set to 0 using winds on all outward faces, and reflective on the inward faces. The Poisson boundary conditions are set to multipole on outward faces, and reflective on inward faces. Every spatial unit is in terms of Rbe, the Bonnor Ebert sphere's radius. So 1 computational unit on the domain = 1 Rbe. Further, the domain is 15 Rbe long in each dimension.

Case	Sound Speed Ambient	Tsim	Distance sound wave from boundary can travel	Does flow at boundary become supersonic?
Matched	4.491	0.216	0.97 RBe ~ 1/15 Domain	yes
BP	42.82	0.192	8.2 RBe ~ ½ Domain	no
Stability	44.91	1.1	44 RBe ~ 3*Domain	no
1/3 Ambient	8	0.28	2.25 RBe ~ 1/7 Domain	yes
1/10 Ambient	14.2	0.4	5.6 RBe ~ 1/3 Domain	no
1/30 Ambient	25	0.7	18 RBe ~ 1.2*domain	no

Below are 2D slices of pseudocolor plots showing rho and vrad/cs.

Matched

As you can see in the left plot of rho, the density in the ambient, within a fictitious sphere of r=15, is increasing as the simulation progresses, going from darkest blue to lighter shades. This is coincident with increasing speeds in this region, likely due to the gravitational acceleration of a homologous collapse, of sphere r=15, rho=rho_ambient. The free fall time for the ambient is ~ 4Myr, or 0.25 in computational time (compare to the tsim = 0.216 above). The boundary of the fictitious sphere is set up I believe from the Poisson boundary conditions or solver… I believe new material is not supplied at either this boundary or any physical boundaries after it is diminished, causing a rarefaction wave to be set up. Any winds coming in or going out from the physical boundary is just set to 0. Along the fictitious bounding sphere of R=15Rbe, material seems to be supplied from the external domain, and then this spherical region (1<r<15) seems to collapse homologously (shrinking and increasing density everywhere) for most of the sim. Speeds however, have a more complicated behavior. Outside of the sphere of r=15, speeds remain close to 0, but immediately within this sphere they continue to increase and move inward, becoming hypersonic near the spherical (r=15) boundary early on.

Here are some lineouts I made at a 45-degree angle—

BP

These plots show a very different scenario. The ambient remains quiescent in this case; the density of the ambient does not change and the radial velocity remains close to 0 out there. The only motions we see is inside of the BE sphere itself, as it undergoes classic outside-in collapse. The free fall time for the ambient in these cases is tff=40Myr, or t=2.38>>tsim.

Ambient 1/3

This case is closest to the Matched case. Here we see the effects of the Poisson boundary conditions in a similar way that we did above in the Matched case but not as exaggerated. That is, the velocity increases near the spherical boundary, but not as drastically. Near the end of the simulation, when the speeds have approached vrad/cs~4, we very quickly see a build up of density in the corresponding ambient region. I wonder if this time corresponds to the time we see a 'turn-over' in the solution from compression wave to classic collapse? The free-fall time of ambient in this case is tff = 7 Myr, or t=0.44>tsim=0.28.

Ambient 1/10

Here the ambient remains quiescent most of the way through the simulation, as the BE sphere begins to collapse. We see only very late in the sim, after the collapse is well underway, a transonic flow begin to develop in the ambient medium, closer in to the sphere than the other cases. This is interesting, both the diminished strength of the flow speed and the closer proximity of the flow to the sphere, i.e. further away from the spherical boundary than in previous cases. This flow doesn't quite reach the BE sphere surface by the end of the simulation. The free-fall time of the ambient in this case is tff = 14 Myr, or in computational units, tff=0.7869 > tsim = 0.4.

Ambient 1/30

This case is nearest the stability case. The freefall time of the ambient is 24 Myr, and in computational time, t=1.4, which exceeds the time of the simulation, tsim=0.7 (see above table). The fact that we do not see any radial motions develop in the ambient medium over the course of the simulation is in agreement with the tff >> tsim. The only motions we see develop is inside of the sphere as it undergoes collapse.

Stability Case

The freefall time of the ambient is tff=43 Myr, or t=2.5>>tsim.

Remarks

Tsim is the final time of the simulations and is defined by the time when a sink particle forms from the collapse, accept in the stability case where a sink did not form, and tsim is taken to be 5 crossing times for the sphere. All of these cases have ambient freefall times that exceed the simulation time, yet some drastically more than others. This may be a diagnostic for which ambients are more conducive to homologous collapse than others.

Does Ambient Collapse?	Case	tff	tsim	tsc
Yes	Matched	0.25	0.216
Yes	1/3	0.44	0.28
?	1/10	0.79	0.4
?	1/30	1.4	0.7
No	BP	2.38	0.192