Plan for next step of mapping the modified RGB profile to the grid

Now that we have been able to generate a modified MESA profile, we must map it to the AstroBear grid. The steps are as follows:

1) Determine the desired cutoff radius and resolution.
2) Solve the modified Lane-Emden equation as explained in the last blog entry, including an iteration over the mass of the gravitation-only particle.
3) Produce an input file for AstroBear with tabulated , , and values.
4) Perform multiple runs for comparison.
5) Assess the level of hydrostatic equilibrium for each run.
Let us consider item (1). For now let us consider a uniform grid. Let the resolution be . The outer radius of the star is , and the cutoff radius is . As in Ohlmann, the softening length for the spline potential of the gravitation-only particle is also . (For the MESA solution will be replaced by the modified Lane-Emden solution for , and for will be determined by re-integrating the hydrostatic equilibrium equation inward using , with equal to the MESA value.) Further, we define , that is, the ratio fo cutoff radius to outer stellar radius. The number of resolution elements over a softening length is given by , where is the grid element size. Finally, let the box size be .

Putting this together, we find

This formula gives the resolution required for a given , , and . Ohlmann+16a states "…we find that a resolution of about 10 cells per softening length is required to ensure energy conservation during the in-spiral." Also, "…in a sphere of five softening lengths of the gravitation-only particle, the maximum cell radius was bound to a tenth of the softening length." This suggests using , so we set . For the RGB star, . To be consistent with our previous simulation, we may choose . As in Ohlmann+16c, we can try , , or . This results in required resolutions , , or , respectively, or, in physical units, , , or . Ohlmann+16a states "The smallest cells near the RG core have a radius of about at the beginning and about at the end of the simulation."

We now consider item (4). It would be useful, initially, to compare the following runs:
a) , .
b) , (i.e. lower resolution).
c) , (i.e. even lower resolution).
d) , , without mass iteration step in item (2), so using a slightly larger particle mass.
e) , , with mass iteration step but without modifying the MESA profile (i.e. skipping item (2) but using the particle mass from model (a)).
f) , (i.e. smaller cutoff radius).
g) , with ambient pressure set to higher value to effectively cut off very outer layer of star where scale height cannot be adequately resolved.

Resolving the scale height

One important additional point is that one should adequately resolve not only the softening length , but also the local pressure scale height . See page 3 of the past presentation ​rgb.pdf for a plot of scale height vs. radius. Modifying the MESA profile for will lead to larger scale heights there, but we must still check whether is large enough. This will likely be most problematic at the stellar surface , where . Here we will not be resolving the scale height so we would expect velocity perturbations to arise. Ohlmann+16c calculates a necessary but not sufficient expression for to ensure that Mach number fluctuations are below a specified level. Assuming the Mach number at , after the first part of a time step (see their Sect. 2.3 and Appendix A for details) one obtains

,

where is the Courant-Friedrichs-Levy constant and is the adiabatic index, so that

.

E.g. for , , and , we obtain . However, Ohlmann+16c somehow obtain the slightly larger value . In any case from this condition we require , and possibly , to avoid Mach number fluctuations greater than . However Ohlmann+16c comments "Even if the resolution requirement is met, other sources of numerical error (e.g. interpolation error, errors from the gravity solver) still introduce spurious velocity fluctuations. Thus, an appropriate relaxation procedure is necessary when mapping stellar models to hydrodynamical grids: the velocity fluctuations have to be damped."

It is important to replot the scale height vs radius plot of ​rgb.pdf for the modified RGB profile, as well as for the case of a higher ambient pressure.

An updated README file with instructions on how to run the code, along with the relevant files, is available ​here.

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