## a solar model...

 Quote by Orion1 Which equation definition is correct?
Neither or both. The difference is much smaller than the other errors that you have in the model so it doesn't matter which one you choose.

If you are doing high precision work it may matter, but in that situation the person that hands you the values of mu that you are working with will tell you what units those things are in. Astronomers will usually quote mu in terms of the mass of hydrogen, whereas I think people in other fields will quote in terms of atomic mass units.

 Quote by Orion1 [The Tolman–Oppenheimer–Volkoff equation under General Relativity should be used for the Equation of State for hydrostatic equilibrium for stellar modeling?
A bad idea unless you have something in your doctoral dissertation that requires it.

The problem is that if you treat hydrostatic equilibrium with GR, you'll have to treat everything with GR. If your pressure equations are GR, and your radiation equations are Newtonian you end up with a big, big mess that's worse than if everything was Newtonian.

Intuitively you can think about this by imagining a black hole. When a black hole forms, the stuff inside of the event horizon stops radiating, and are treating pressure with GR and radiation Newtonian, you'll get a big mess.

One thing that you should realize that the TOV doesn't work if you have a radiating body. If the thing you are trying to study is radiating, then the T01 component of the stress-energy tensor is non-zero, and you have to rederive everything.

There is a "standard" way of bolting on GR to stellar codes.

What you do is to run an newtonian code and then set up a "GR time correction" factor and a "GR volume correction" factor that corrects for everything. If you do this for any sort of stellar code, you'll find that both correction factors are so close to one that it doesn't matter.
 Also there are two "philosophies" of stellar modelling. There is the "cartoon" approach which were leave out everything accept for the important parts so that you have something that you can understand. The thing about doing things via cartoon is that it helps a lot in understanding how things work, but you aren't going to get anything that is close to something matches the sun. You can also try a photo realistic approach in which you dump everything into a computer. You get realistic output, but there is a loss of insight. Something that would be interesting would be try to create the models in python or mathematica.
 I note the advice from the Science Advisors and provide my current 'cartoon' model 'college trial' for core pressure, for the benefit of insight... Experimentally determined parameters: Total solar radius: $$R_{\odot} = 6.955 \cdot 10^8 \; \text{m}$$ Total solar core density: $$\rho_c = 1.622 \cdot 10^5 \; \frac{\text{kg}}{\text{m}^3}$$ Stellar density function: $$\rho(r) = \rho_c \left( 1 - \alpha \left( \frac{r}{R_{\odot}} \right)^{\delta} \right)^{\gamma} \; \; \; \delta > 0 \;, \gamma > 0 \;, \alpha > 0$$ Integration via substitution (ref. 2): $$\frac{dP(r)}{dr} = - \frac{4 \pi G \rho(r)}{r^2} \int_0^{r} r^2 \rho(r) dr$$ Equation of State for hydrostatic equilibrium: $$\frac{dP(r)}{dr} = - \frac{4 \pi G \rho_c^2 r}{3} \left( 1 - \alpha \left( \frac{r}{R_{\odot}} \right)^{\delta} \right)^{\gamma} \cdot _2F_1 \left( - \gamma, \frac{3}{\delta}, \frac{3 + \delta}{\delta}, \alpha \left( \frac{r}{R_{\odot}} \right)^{\delta} \right) \; \; \; \; \; \; \alpha = 1 \; \; \; \delta = 2 \; \; \; \gamma = 1$$ Total stellar core pressure integration: $$P_c = \int_0^{R_{\odot}} \left( \frac{dP(r)}{dr} \right) dr$$ Orion1 solar model core pressure: $$\boxed{P_c = -7.115 \cdot 10^{17} \; \frac{\text{N}}{\text{m}^2}}$$ Standard Solar Model core pressure (SSM): $$P_c = -2.477 \cdot 10^{16} \; \frac{\text{N}}{\text{m}^2}$$ Known problems with 'cartoon' modeling approach: 1. The model is static and does not evolve. 2. The model only uses experimentally determined parameters. 3. There is a functional discontinuity in the SSM, ZAMS, and Lane-Emden (Eddington) models on the radiative-convective shell boundary. 4. Newtonian laws are incompatible with General Relativity. Reference: Sun - Wikipedia a solar model - Orion1 #12 Equations of stellar structure - Wikipedia Analytic Models For The Mechanical Structure Of The Solar Core - Sidney A. Bludman, Dallas C. Kennedy Hypergeometric2F1 function - Wolfram NASA - Sun Fact Sheet
 Would adding a positive differential radiation pressure that opposes negative differential gravitational pressure to the Newtonian Equation of State for hydrostatic equilibrium produce a more accurate stellar core pressure reading? $$\frac{dP(r)}{dr} = \frac{dP_g(r)}{dr} + \frac{dP_r(r)}{dr}$$ Stellar Equation of State for hydrostatic equilibrium: $$\frac{dP_g(r)}{dr} = - G \frac{m(r) \rho(r)}{r^2}$$ What would the equation be for stellar interior differential radiation pressure? Reference: Radiation pressure - Wikipedia

 You can also try a photo realistic approach in which you dump everything into a computer. You get realistic output, but there is a loss of insight. Something that would be interesting would be try to create the models in python or mathematica.

My Equation of State for hydrostatic equilibrium equation seems sophisticated enough to use a photo realistic approach to modulate the constants α, δ, γ using a realistic output computer data dump into Mathematica of the mass and density functions for the Standard Solar Model (SSM) at present time.

Would the constants α, δ, γ become stellar evolution functions of time using such an approach?

Is Mesa star capable of doing this?

Reference:
Stellar structure - Equations of stellar structure - Wikipedia
Mesa star - stellar evolution code
 Mentor This thread has turned into one person talking to himself.

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