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BE turnover updates/discussion
The time interval we chose to test stability of the BE sphere in the light case was 5 sound crossing times of the SPHERE. Perhaps if we had known better at that time, we would have chose some time-scale for the ambient. But, we were interested in the stability of the sphere, so it seemed reasonable to choose that timescale. 5 sound crossing times = 1.1 for the sphere. It is different for the ambient in the different cases, given the change in ambient sound speed.
In looking for the turn-over, I found the 1/50 case did not collapse in t=1.1 (which is 5 tsc for the BE SPHERE), but the 1/45 case did.
A closer look at the timescales operating in the AMBIENT show why.
For the ambient, tff=1.6, slightly longer than the simulation time of t=1.1. Perhaps the ambient did not have time enough to collapse sufficiently so to trigger collapse onto the sphere. Running the simulation out longer showed the ambient to collapse far enough down to trigger collapse of the sphere. I found it only needed a little longer, and the BE sphere collapsed fully to a sink particle by t=1.4.
A movie (http://www.pas.rochester.edu/~erica/50LineOutLonger.gif) shows some interesting behavior of the ambient in this small time period between t=1.1 and 1.4. It is also of interest that the BE sphere collapse is triggered on a timescale similar to the timescale of the ambient's gravitational infall. Does this trend hold up in the different cases?
In checking the tsims compared to the tffs, I see that the trend holds for jeans stable mediums, but that as the medium becomes jeans unstable, the simulation time becomes much shorter than the free-fall time of the ambient. Maybe this is because the time-scale for the jeans unstable medium is more accurately given by some faster timescale?
Why did 1/45 (the seeming turn-over point) collapse in the time interval t=1.1? A look at the table, http://www.pas.rochester.edu/~erica/BEcalcs.pdf, shows that for the ambient, tff=1.4, a little shorter than that of the 1/50 case..
I think this table shows some interesting trends. Also, it is interesting to consider the timescale of collapse for the BE sphere once the BE sphere goes unstable; is this a constant throughout the different runs? That is, say we take the collapse time to be the time it takes the unstable BP case's sphere to collapse (~.2), do all spheres collapse in t=0.2 (i.e. it takes t=tfinal-0.2 time for the sphere to go unstable). It is interesting that this timescale for the BP case is on order the sound crossing time for the sphere, as well as the average free fall time of the sphere. Other authors have suggested a more correct dynamical timescale is given by the thermal radius/the sound speed. When I calculated this however, it was was way off from the simulation time. I am not sure why.
As for the Jeans length of the ambient (calculated at t=0), the table shows that when the jeans length is ~2-10 smaller than the box size we see strong compression waves. When the jeans length is comparable to the box size, as in the 1/10 case, we see a turn-over in the type of collapse, changing from a strong compression wave early phase, to a redistribution of matter. I am curious if in a case like the 1/10, if I make the box 2-10x larger than the jeans length, will the collapse be qualitatively different.
Lastly, we talked about increasing the size of the 1/50 box so that it is strongly jeans unstable. On first thought, I would expect ambient collapse to happen on timescales given by tff, regardless of the size of the box. This seems justified given the 1/50 case is "jeans stable", but still collapses, and on a timescale specified by the density of the ambient. However, given the jeans stability of the ambient seems tied with the generation of strong compression waves, I am inclined to think that the timescales may change from a jeans unstable to stable medium. This is at odds with the assumption that within our box, we can approximate the collapse of the ambient as a uniform sphere with collapse time on order the free fall time.
Another point we have talked about before that I am not entirely sure of is this, given the matter in the box is finite and contained inside the box (i.e. we are not supplying a continuous flow of matter into the box at the boundary) might the ambient material see the edge of the sphere as a wall and build up on it in some way as to move toward a new HSE? In contrast, if we were continuously supplying new matter into the box, it seems this wouldn't happen because the matter will always be falling down to the BE sphere, and will eventually trigger collapse.
Maybe this equilibration of the ambient would be possible for a very stable BE sphere configuration..?
So in summary, there seem to be 4 points here I am making,
- How is the Jeans stability of the ambient tied with physical characteristics of the BE sphere collapse, such as early phase type, i.e. compression wave vs. matter redistribution
- How is the Jeans stability of the ambient tied with the timescale of the ambient collapse
- What is the timescale the BE sphere collapse operates on
- Can the ambient move to a new HSE with the sphere?
Comments
I am running both the 1/50 and 1/10 cases with a box that is from 0-90 in x,y,z with the BE sphere quadrant at 0 (total length is 180~6lambdaj for the 1/10 case and ~lambdaj for the 1/50 case) with 643 base resolution + 7 levels. This is to test if 1) the 1/50 case has a different collapse profile, and 2) if the 1/10 case will have compression wave dominated infall onto the BE sphere.
The 1/10 runs are done, here are movies of them, the first is in the BIG box (~100 long), the 2nd is in the original box. If you open both of these at the same time, you will see the collapse is the SAME in the 2 cases. This rules out that the medium collapses faster (and stronger) in a highly jeans unstable case.
http://www.pas.rochester.edu/~erica/lineoutrho10bigger.gif
http://www.pas.rochester.edu/~erica/lineoutrho10.gif
This ⇒ the ambient collapses on ~ its free fall time, the shorter the free fall time, the stronger the induced compression waves, and the faster the sphere turns over into an unstable regime. The longer the free fall time, the longer it takes the sphere to become unstable (slight and slow build up of ambient material to accumulate inside of the sphere). It is a coincidence the tsim~tff.
The results for 1/50 are very different than 1/10. I see an effect on increasing the box length on the collapse time in the 1/50 case (http://www.pas.rochester.edu/~erica/rho50bigger.gif and http://www.pas.rochester.edu/~erica/rho50longer.gif), where I did not in the 1/10 case. This must mean the velocity profile is strongly different in the smaller vs bigger boxes for the 1/50 case. Plots to come..
The stability case does not collapse when I run it for a time ~ tff (tfinal=2.6) for the ambient. Here is a movie - http://www.pas.rochester.edu/~erica/stabilitylonger.gif
However, the stability case does collapse when I run it in a bigger box-
http://www.pas.rochester.edu/~erica/stabilitylongerbigger.gif
In reference to comment 3, I am saying it is a coincidence that 1/50 case tsim~tff