[Mesa-users] Doubt regarding the boundary conditions for a selection of the optical depth at the surface (tau_surf) greater that the base value (tau_base, corresponding to the photosphere).

Wed Jun 5 09:57:51 UTC 2024

Dear MESA community,

As context of the science problem I am interested in, I am using MESA 23.05.1 to create supermassive stars (masses around 1000 Msun, with accretion rates ~10-2 Msun/yr). During the evolution of such stars, the solver encounters numerical difficulties related to weird values of densities or temperatures at the surface of the star. From feedback of other mesa users and also from comments in this mailing list, I tested the option of setting up the surface optical depth (tau_surf) to larger values with respect to the base value at the photosphere (tau_base = 2/3,
      for my selection of atmosphere:
            atm_option = 'T_tau'
            atm_T_tau_relation = 'Eddington'
            atm_T_tau_opacity = 'fixed'
), achieving this by changing the value of tau_factor, to avoid any numerical difficulties in the surface. I also mention that I am using the following MLT options (no MLT++ or MLT++ local):
            MLT_option = 'Mihalas'
           okay_to_reduce_gradT_excess = .false.
(For more details regarding on why I am recurring to change the position of my boundary deeper in the star see the Appendix below)

For a particular case, if tau_factor = 100 the pressure at the surface of that star is two orders of magnitude lower than expected (by comparing with the stellar model with tau_factor = 1). From the MESA code I see that the P(tau) relation used to set the pressure at the surface is:

      P(tau) = (tau_surf * g / kappa) * (1 + Pextra_factor * (kappa / tau_surf) * (L / M) / (6 * pi * c * G))

From previous messages in this mailing list, it is advised to set high values of Pextra_factor if tau_factor is much greater than one as this "can be numerically helpful". In order to understand that and make a physically motivated decision on the value of Pextra_factor, I compared the pressure from the stellar model and the P(tau)  relation for the models of tau_factor = 1 and tau_factor = 100 (at least for simplicity here I only show those two models):

[cid:b17f3b5f-98b8-4393-9cf3-be0e5bdc8a27]
      Fig.1.- Pressure vs. optical depth (tau): The thick green line represents the pressure of the profile for a stellar model with 1000 Msun and tau_factor = 1. The thin green line for the radiation pressure from the same model. The dashed green line is the radiation pressure term from the P(tau) relation computed as if the boundary of the star is set for a given optical depth ( (g) * (L / M) / (6 * pi * c * G)). The dashed mint line is the total pressure coming from the P(tau) relation for the boundary conditions. The blue solid line is the pressure of the profile for a stellar model with 1000 Msun and tau_factor = 100.

From Fig. 1 we can see that the pressure at the boundary of the model with tau_factor = 100 is consistent with the pressure from the P(tau) relation of the model with tau_factor = 1 at the same optical depth. But such pressure is below the actual pressure of the stellar model with tau_factor = 1 at tau = 100 * 2/3.

As well, in order to use the values coming from the profile sf the model with thau_factor = 1, I show the value of tau_effective as function of tau. Where tau_effective is the value of the optical depth in order to have a pressure from P(tau) equal to the local pressure at a given optical depth.

[cid:8c20186e-d07c-44b3-beea-22c9fa1ba3c1]
      Fig. 2.- Tau_effective vs. tau from the profile for a stellar model with 1000 Msun and tau_factor = 1.

From Fig. 2 it shows that selecting tau_factor = 100 (the same as tau_surf = 100 * 2/3)  the pressure computed from the boundary condition will be lower than the expected one for the stellar model. One option is to set a value of Pextra_factor to make both pressures equal, the correspondent value of Pextra_factor in orter to have P(tau) = P_local is shown in the next figure.

[cid:2804b402-dcd6-4d91-90d4-39d3496aa4c8]
      Fig. 3.- Pextra_factor vs. tau such that P(tau) = P_star(tau) for a stellar model with 1000 Msun and tau_factor = 1. The green line represents positive values while the red negative ones.

From this experiment I get that Pextra_factor should set to ~800 to have a consistent value of the pressure at the boundary with tau_factor. = 100. But my concern is regarding to the validity of this value as the star evolves. Is there another approach to have self consistent boundary conditions when tau_factor is much greater than one? From the information I have gathered it seems that there is no way of doing that without compromising the physical accuracy of the model.

Please let me know if there is another approach to this problem, it will be much appreciated.

Kind regards,
Jaime.

____

Appendix. Why I need to set the boundary of my stellar models at larger optical depth?

The aim of building the supermassive stars is to take such models, inject a given amount of energy using the extra_heat quantity and explore the evolution of the model. While doing this, the stars expand, until the surface reaches de blending zone between the FreeEOS and Ideal EOS. Weird values of T, P and rho at that point force the solver to retry the calculation until it crashes due to reaching the minimum timestep limit. By using the EOS plotter module, I see that the boundary of my stars are around a region where FreeEOS and the Ideal one does not agree very well on the value of Pgas for the metallicity I am considering (Z = 1e-4), see image below (the limits for the color map where changed in order to appreciate the values of Pgas better around the location of the stellar boundary, marked with a red dot). For other quantities that are computed in the EOS module the transition seems smooth). Then, setting the boundary at a larger optical depth will allow me to have a more compact model and avoid reaching such region in the EOS.

[cid:2783ead6-35d4-43b0-8e41-aa596c2c4fb9]

[cid:a7d87526-1732-4b13-a08c-c8c7d5d102a4]

Jaime ROMAN GARZA

Ph.D. candidate – Département d'Astronomie

______________________________________________________________________________________________________________________________________________________________________________________________________________________________________

GW sources | Stars, formation and evolution

Observatoire de Genève | 51 Chemin Pegasi - 1290 - Versoix - CH

• +41 22 379 24 17                 • : Jaime.RomanGarza at unige.ch<mailto:Jaime.RomanGarza at unige.ch>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <https://lists.mesastar.org/pipermail/mesa-users/attachments/20240605/d24edcd7/attachment.htm>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: Unknown-5.png
Type: image/png
Size: 41474 bytes
Desc: Unknown-5.png
URL: <https://lists.mesastar.org/pipermail/mesa-users/attachments/20240605/d24edcd7/attachment.png>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: Unknown.png
Type: image/png
Size: 29683 bytes
Desc: Unknown.png
URL: <https://lists.mesastar.org/pipermail/mesa-users/attachments/20240605/d24edcd7/attachment-0001.png>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: Unknown-2.png
Type: image/png
Size: 25338 bytes
Desc: Unknown-2.png
URL: <https://lists.mesastar.org/pipermail/mesa-users/attachments/20240605/d24edcd7/attachment-0002.png>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: eos_plotter_lnPgas_lowlimit.png
Type: image/png
Size: 77684 bytes
Desc: eos_plotter_lnPgas_lowlimit.png
URL: <https://lists.mesastar.org/pipermail/mesa-users/attachments/20240605/d24edcd7/attachment-0003.png>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: Outlook-ea2my4so.png
Type: image/png
Size: 28771 bytes
Desc: Outlook-ea2my4so.png
URL: <https://lists.mesastar.org/pipermail/mesa-users/attachments/20240605/d24edcd7/attachment-0004.png>