[Mesa-users] Missing energy generation in MESA output?

Frank Timmes fxtimmes at gmail.com
Fri Feb 28 01:15:56 UTC 2025 

the energy equation and energy conservation have been a recurring and evolving topic within the source code base and mesa instrument papers - section 8 of mesa iii (paxton et al 2018), in the title and section 3 in mesa iv (paxton et al 2019), and once again in the title and section 5 of mesa vi (jermyn et al 2023). the latter being what i believe represents the current implementation. perhaps they can help shed some light on this situation.

fxt




> On Feb 27, 2025, at 5:35 PM, Farag, Ebraheem via Mesa-users <mesa-users at lists.mesastar.org> wrote:
> 
> Hello Jason,
> 
> (cc'ing mesa-users for this)
> 
> Well, I think I solved the energy issue, but not the dT/dm issue.
> When using T ds/dt, there is a very slight inconsistency with eps_grav.  When I use the eps_grav method, I get practically perfect agreement with the energy equation from the eps_grav profile output.  When I use the dedt method, I have to use T ds/dt, which is the cause of the disparity.
> The new puzzle is why dT/dm gets much better agreement with dP/dm Tbar/Pbar gradT when I use dedt than when I use eps_grav?
> 
> I believe I have an idea of what caused this disagreement you're seeing. ​This is not the most general form of the eps_grav equation. 
> 
> "Δl/Δm = ε_nuc - TdS" 
> 
> rather it should it is:
> 
> "Δl/Δm = ε - dq"
> 
> , where I have chosen to say "ε" to characterize that we have (enuc + eneu_nuc) + eneu_non_nuc, and Tds is replaced with dq. 
> 
> For a purely thermal process dq = Tds, but more formally, dq = Tds + sum[​μi​dX],
> 
> Where the second term includes the heat associated with composition changes due to mixing/ionization/burning. 
> 
> In more complete form,  it is not simply dq = Tds, but rather dq = Tds + sum[partial(e)/partial(yi)] dyi , see equation 58 inhttps://ui.adsabs.harvard.edu/abs/2018ApJS..234...34P/abstract (much more details on eps_grav are here.).
> ----
> 
> I believe this explains why you have a disagreement when adopting the energy equation form and not the eps_grav form. The eps_grav form includes the composition effects, where as you have been assuming eps_grav = -TdS, which is an approximation.
> 
> In the profile columns, i find this comment that warns users of this:
>  "!delta_entropy ! entropy - entropy_start, change during step (does not include effects of diffusion)"
> 
> 
> If we look at the docs, where the eps_grav form is discussed : https://docs.mesastar.org/en/latest/reference/controls.html#eps-grav-form
> 
> We find "The final term reflects the change in internal energy due to changes in composition (at fixed density and temperature) and is referred to in MESA as eps_grav_composition_term. "
> 
> You can actually test turning this term on or off  to see if it restores your agreement with your estimate:
> "include_composition_in_eps_grav = .false.", The default is ".true.".
> 
> The new puzzle is why dT/dm gets much better agreement with dP/dm Tbar/Pbar gradT when I use dedt than when I use eps_grav?
> I'll have to think about this, but if you're plotting the correct quantities then:
> Naively, I would think  this comes from the fact that when adopting the eps_grav form you are explicitly using a derived thermodynamic quantity "S = -∂F/∂T " and then computing an additional derivative "TdS/dt" in your energy equation which can be more sensitive toward any "noise" coming from the EOS and it's precision, and this might propagate into the t-gradient equation. On the other hand, when you adopt the conservative form of the energy equation, you only need "U" (the internal energy) and its direct time derivative, which I believe is a primary variable output from the EOS. Mathematically these approaches should be identical, however numerically they are not.
> 
> In this way, (like a magnifying glass) using the eps_grav form is a way of testing the numerical robustness and consistency of your EOS and its derivatives when placed front and center. I believe you need second order continuity in the eos for this approach to work well:
>  you require discretized form of -eps_grav ~ TdS ~ d(-T1∂F1/∂T1 + T2∂F2/∂T2)/dt  to be not just continuous but smooth, as additional partial derivatives of this quantity are used to construct the Jacobian matrix used by MESA's multi-dimensional newton Raphson scheme. (where i omitted include the eps_grav composition terms here for brevity).
> 
> -EbF
> 
> 
> Well, I think I solved the energy issue, but not the dT/dm issue.
> 
> When using T ds/dt, there is a very slight inconsistency with eps_grav.  When I use the eps_grav method, I get practically perfect agreement with the energy equation from the eps_grav profile output.  When I use the dedt method, I have to use T ds/dt, which is the cause of the disparity.
> 
> The new puzzle is why dT/dm gets much better agreement with dP/dm Tbar/Pbar gradT when I use dedt than when I use eps_grav?
> 
> 
> 
> From: Jason Wright <astrowright at gmail.com <mailto:astrowright at gmail.com>>
> Sent: Thursday, February 27, 2025 2:35:18 PM
> To: Farag, Ebraheem <ebraheem.farag at yale.edu <mailto:ebraheem.farag at yale.edu>>
> Cc: mesa-users <mesa-users at lists.mesastar.org <mailto:mesa-users at lists.mesastar.org>>
> Subject: Re: [Mesa-users] Missing energy generation in MESA output?
>  
> Interesting!
> 
> Implementing eps_grav solved the problem of the energy imbalance in the atmosphere, so that mystery is solved.
> 
> Strangely, where before we had excellent solution for the dT/dm equation, going to the eps_grav formulation created an imbalance in that equation.  I'll investigate more why that might be (have to head out now) but I suppose I must have also triggered an alternative formulation for that equation from the one listed in the first instrument paper somehow.
> 
> It did not change the results in the core, though, as you suspected.  
> 
> I switched to eps_nuc_minus_non_nuc_neu but the difference is very small in my case, (millions of times smaller than the issue I'm having).
> 
> Thanks for your prompt help!
> 
> Jason
> 
> On Thu, Feb 27, 2025 at 1:52 PM Farag, Ebraheem <ebraheem.farag at yale.edu <mailto:ebraheem.farag at yale.edu>> wrote:
> Hi Jason,
> 
> Without thinking too deeply into the information you've shared. 
> 
> (I do see there is an advertised eps_grav parameter in MESA, but it is set to zero everywhere, which is why I'm using T ds/dt.  Is there a way to get that output to "turn on"?)
>  I believe the energy equation in MESA is now -> 'dedt', a conservative form. The older eps_grav form (eq 11 from MESA I) is still an option "eps_grav", see https://docs.mesastar.org/en/latest/reference/controls.html#energy-eqn-option. 
> 
> This might be wrong, but I think when the conservative form is chosen, you admit entropy errors to allow for energy conservation, where as when the original eps_grav form is adopted, you admit energy errors for better thermodynamic consistency. In a perfect world these would be identical, but numerically solving yields this fork in the road. The eps_grac parameter should be on when using the 'eps_grav' form of the energy equation.
> 
> 
> On both problems, I don't just want eps_nuc but rather eps_nuc - non_nuc_neu, where non_nuc_neu are the thermal neutrino losses. This might help with problem 2, unless you already included it. 
> 
> I'm still thinking on problem 1... It could be due to the fact that we use a T-tau relation for low tau, but that's something to check?
> 
> -EbF
> From: Mesa-users <mesa-users-bounces at lists.mesastar.org <mailto:mesa-users-bounces at lists.mesastar.org>> on behalf of Jason Wright via Mesa-users <mesa-users at lists.mesastar.org <mailto:mesa-users at lists.mesastar.org>>
> Sent: Thursday, February 27, 2025 12:27 PM
> To: mesa-users <mesa-users at lists.mesastar.org <mailto:mesa-users at lists.mesastar.org>>
> Subject: [Mesa-users] Missing energy generation in MESA output?
>  
> Hi, all.  Me again teaching stellar structure with MESA.
> 
> One set of our assignments is to test the precision of MESA in computing the equations of stellar structure.  This is a good way for the students to practice coding up physics and exploring which physics matters where in the Sun.  We are using a solar-mass ZAMS star of solar metallicity as our example (from Ed Brown's excellent notes).
> 
> We find that the 1st 3 equations of stellar structure are obeyed extremely well, but we are struggling with the luminosity equation.  
> 
> Δl/Δm = ε_nuc - T
> 
> (Eq. 11 of the first MESA instrument paper).
> 
> Problem 1:
> 
> We get reasonably good agreement for most of the star, but in the outermost cells (m/M > 0.999) there is a big divergence between these quantities.  In the atmosphere of the star, Δl/Δm actually goes very slightly negative, but the reported ε_grav (calculated as -T ds/dt) is still positive there.
> 
> I'm guessing the answer is that I'm not calculating T ds/dt properly.  I'm using the delta_entropy profile output, and log_dt from the history file for that timestep.  Perhaps these do not correspond exactly, or that is not the quantity used in determining Δl/Δm?
> 
> (I do see there is an advertised eps_grav parameter in MESA, but it is set to zero everywhere, which is why I'm using T ds/dt.  Is there a way to get that output to "turn on"?)
> 
> Below:  comparison of the finite difference of Δl/Δm (red, dashed line is negative) and energy generation terms reported by MESA.  
> 
> [Core is on the left, atmosphere on the right.    The logit_10 function "counts nines" near m/M~1 (so 0.9999 is 4) and "counts zeros" near m~0 (so 0.0001 is -4).  ]
> ["eps" here is "net_energy", which does not have the precision to show values below 1 erg/s/g]
> 
> 
> <Screenshot 2025-02-24 at 1.12.43 PM.png>
> 
> 
> Problem 2:
> 
> Another mismatch is more subtle, but suggests I'm missing physics.
> 
> It's hard to see in the above plot, but the discrepancy actually stars at about m/M=0.5 (logit_10(m/M)=0).  If I integrate up (ε_nuc - T ds/dt)dm to compute luminosity and compare it to the actual luminosity, I get about a 0.1% discrepancy with a strange shape.This shape shares some qualities / features of both eps_nuc and eps_grav, but not exactly.
> 
> <Screenshot 2025-02-24 at 1.18.56 PM.png>
> Any ideas what's going on?  Am I misunderstanding something about neutrino losses?
> 
> Happy to share my code if it helps.
> 
> 
> 
> --
>                                                 -
> ---------------
> Jason T Wright
> Professor of Astronomy and Astrophysics
> Director, Penn State Extraterrestrial Intelligence Center
> he/him/his
> https://sites.psu.edu/astrowright/
> https://bsky.app/profile/astrowright.bsky.social
> 
> 
> --
>                                                 -
> ---------------
> Jason T Wright
> Professor of Astronomy and Astrophysics
> Director, Penn State Extraterrestrial Intelligence Center
> he/him/his
> https://sites.psu.edu/astrowright/
> https://bsky.app/profile/astrowright.bsky.social
> 
> 
> --
>                                                 -
> ---------------
> Jason T Wright
> Professor of Astronomy and Astrophysics
> Director, Penn State Extraterrestrial Intelligence Center
> he/him/his
> https://sites.psu.edu/astrowright/
> https://bsky.app/profile/astrowright.bsky.social
> 
> 
> --
>                                                 -
> ---------------
> Jason T Wright
> Professor of Astronomy and Astrophysics
> Director, Penn State Extraterrestrial Intelligence Center
> he/him/his
> https://sites.psu.edu/astrowright/
> https://bsky.app/profile/astrowright.bsky.social
> _______________________________________________
> mesa-users at lists.mesastar.org <mailto:mesa-users at lists.mesastar.org>
> https://lists.mesastar.org/mailman/listinfo/mesa-users

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