[Mesa-users] TOV approximation, other_cgrav

[Bill Paxton]

As usual, it is necessary (if painful) to check that the results have “converged” in the sense that the response to changes in timesteps or resolution are small compared to to the uncertainties you want to study.    i.e., the “small” change from cell centered to face centered may just be reflecting that neigher case is actually converged.

b



> On Nov 26, 2020, at 11:34 AM, Nicholas Herrington <up813350 at myport.ac.uk> wrote:
> 
> Hi Bill,
> 
> Apologies for not explaining my use case, I am a PhD student and I am running a series of supermassive star models with very rapid accretion rates. I have one model accreting at 1 Msun/yr and this reaches the onset of collapse at ~ 230,000 Msun, with the corrections I previously stated (https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.2236W/abstract <https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.2236W/abstract> eqn 11, §2), which uses s% P(k) and s% rho(k) in the other_cgrav routine. When I switch this to the face centred variables, rho = 0.5d0*(s% rho(k-1) + s% rho(k)) and P = 0.5d0*(s% P(k-1) + s% P(k)) within your other_cgrav (same corrections), this model reaches the onset of collapse at ~380,000 Msun. So I'm trying to figure out the difference in final masses based on using cell centred or face centred pressure and density. I will try to look into the origin of the corrections in the paper I linked also. 
> 
> Thanks for your response,
> Nick
> 
>> On 26 Nov 2020, at 19:00, Bill Paxton <paxton at kitp.ucsb.edu <mailto:paxton at kitp.ucsb.edu>> wrote:
>> 
>> 
>>> On Nov 26, 2020, at 10:51 AM, Nicholas Herrington via Mesa-users <mesa-users at lists.mesastar.org <mailto:mesa-users at lists.mesastar.org>> wrote:
>>> 
>>> Hi,
>>> 
>>> From a previous thread related to modelling a star with a neutron star in its core, I saw Bill provided a other_cgrav routine for the TOV approximation. I have been using the other_cgrav routine to implement the TOV approximation and looking at the other_cgrav.inc provided, I wanted to ask what is the significance of using face centred pressure and density instead of cell centred (I used cell centred variables in my version).
>> 
>> i have no idea what is or is not significant for your application!   But it is a fact of mesa that the momentum equation that uses cgrav applies to cells faces where r and m and cgrav are defined.   Pressures are defined at cell “centers” (actually they are cell average values, but the distinction is blurred in mesa).  That’s convenient for the momentum equation since it uses the pressure difference across the face.
>> 
>> 
>> 
>>> Additionally the version of the TOV corrections to cgrav in the script follows Gnew = G*f1*f2/f3, would you be able to explain where this comes from? I am not knowledgeable in the solutions to arrive at G, I have been following the correction from https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.2236W/abstract <https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.2236W/abstract> (equation 11 in section 2).
>> 
>> i’m no expert on GR or TOV either, and that’s why this is in a user-accessible hook so you can fix it!
>> 
>> the comments in other_cgrav.inc in src directories for the current, pre-release version, include comments that summarize my entire (lack of) understanding of this issue.
>> 
>>       ! use the Tolman–Oppenheimer–Volkoff (TOV) equation. 
>>       ! See first equation in https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation <https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation>.  
>>       ! want to replace -G*m/r^2 by -G*m/r^2*(1 + P/(rho c^2))(1 + 4 pi r^3 P /(m c^2))/(1 - 2 G m/(r c^2))
>> 
>> Those factors in ()’s that multiply gravity in TOV in the comment simply multiply standard_cgrav in the code.
>> 	    G = standard_cgrav
>>             f1 = 1d0 + P/(rho*clight**2)
>>             f2 = 1d0 + 4d0*pi*r**3*P/(m*clight**2)
>>             f3 = 1d0 - 2d0*G*m/(r*clight**2)
>>             s% cgrav(k) = G*f1*f2/f3
>> 
>> -bill
>> 
>> 
>> 
>> 
>> 
>> 
>> 
>>> 
>>> Thanks
>>> Nick
>>> 
>>> _______________________________________________
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>>> 
>> 
> 

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