Mapping a modified RGB profile to the grid: first results

As explained in the previous blog entry, the next step was to map the modified RGB profile to the AstroBear grid. In this blog entry I report the initial results.

From the statement in Ohlmann+16c that it is necessary to resolve the softening length by about 10 grid cells, we had estimated a minimum resolution of about or cells, for the case , with the outer radius of the star. For the case , the minimum resolution would be or cells.

Another constraint to satisfy is to resolve the scale height . We estimated that the scale height needs to be resolved by at least few grid cells, to avoid unreasonably large fluctuations in the Mach number. Near to the surface of the star , becomes very very small, and, moreover, suffers a discontinuity at , where the star transitions suddenly to the ambient medium. So resolving the scale height at the surface is numerically unfeasible (we return to this issue below). The scale height may also be too small to resolve near the transition radius . A plot of the pressure scale height vs. radius for the RGB profile, showing both original MESA and modified profiles, is available ​here for the case and ​here for the case . Note that the unphysical behaviour of near the transition radius and noise in for are probably consequences of how the IDL routine used to differentiate the pressure profile handles the different sampling in radius for (obtained using modified Lane-Emden) and (MESA). This is not expected to cause a problem in AstroBear, which interpolates the inputted profile for the AstroBear grid. For , the minimum value near is at , whereas for , the minimum value near is at . With a resolution of , we resolve by at least a few cells for , which may be sufficient. For , and resolution, is only marginally resolved by cell, which is apparently insufficient (see results below).

We now present plots for the density from each of the following simulations:

A) Fiducial run: resolution ( resolution), modified RGB profile with , box size cm . ​3d, ​1d
B) Low res run: resolution ( resolution) but otherwise same as (A). ​3d, ​1d
C) High res run: resolution ( resolution) but otherwise same as (A) ( was attempted but protections caused the simulation to terminate). ​3d, ​1d
D) No iteration over core mass run: , so not fully self-consistent (see previous blogs), otherwise same as (A). ​3d, ​1d
E) Small softening length run: instead of , otherwise same as (A). ​3d, ​1d
F) No spline run: Spline softening of potential not employed in AstroBear, otherwise same as (A). ​3d, ​1d
G) Original MESA profile but with spline potential run: No modification to profile, but otherwise same as (A). ​3d, ​1d
H) Direct MESA run: No modification to profile, nor is spline softening employed, otherwise same as (A). ​3d, ​1d

I also made side-by-side movies of the fiducial run (on the left) and an alternative run (on the right), for easy comparison:
i) (A) and (B) ​3d, ​1d
ii) (A) and © ​3d, ​1d
iii) (A) and (D) ​3d, ​1d
iv) (A) and (E) ​3d, ​1d

Discussion
From the above plots we can conclude:
I) The fiducial run (A) is quite stable at the center, as expected, but there is still a slight drop in the central density during the run. This might be alleviated by using higher resolution, but run ©, with slightly higher resolution, shows only a marginal increase in stability.
II) In all the runs, the SURFACE of the star becomes unstable on the local dynamical time, as expected.
III) A resolution of is needed for approximate stability at the center, as expected.
IV) The run (E) is NOT quite stable at the center when using the same resolution as for the fiducial model. This is not surprising because both and are smaller in this model, so not as well-resolved.
V) The iteration procedure used for the particle mass to ensure that is equal to the MESA value makes a small but noticeable difference to the stability and final profile of the star (see the 1d plot comparison in (iii) above). If this step is not included, the density profile develops an unphysical kink at .

Next steps
Some logical next steps are:
1) Figure out a viable method for reducing and smoothing out the pressure gradient at the surface. This will involve specifying an ambient value and/or an extension of the profiles for . This involves coming up with some different methods, testing them, and comparing the results.
2) Once (1) is implemented, protections should no longer be a problem, so higher resolution (e.g. ) could be tested.
3) It will then be useful to make other plots to study, e.g. the degree to which hydrostatic equilibrium is being satisfied, or the local Mach number of any residual motions.
4) If the star is reasonably stable after these steps, then AMR can be implemented and tested.
5) If damping of residual motions is still needed, it can first be implemented as in Ohlmann+16c.

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