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Part II
Advanced Topics and Appendices

Lecture 35
Hydrostatic and Thermal Equilibrium of the Sun

Overview

This appendix is the stellar analogue of several themes that recur throughout these notes. Gravity replaces magnetic curvature as the confining force, nuclear burning replaces external or Ohmic heating, and radiative or convective transport replaces the closures used in laboratory plasmas. The same four questions organize the problem: what sets equilibrium, what sets temperature, how is energy transported, and which gradients are regulated by marginal stability? The static Sun is also the branch from which the Parker-wind problem later departs; Lecture 8 begins by showing why a hot corona cannot remain hydrostatic all the way to infinity.

Historical Perspective

The modern stellar-structure problem was assembled in stages. Kelvin and Helmholtz emphasized gravitational binding as the global energy reservoir of a star. Eddington made clear that a star is a self-gravitating gas sphere whose interior temperature is fixed primarily by gravity rather than by chemistry. Schwarzschild identified convective instability as an entropy-ordering problem, and Chandrasekhar later recast stellar stability in the same variational language that became so powerful in plasma physics. This appendix does not attempt a precision solar model. Its purpose is narrower and more useful for these notes: to isolate the simplest 1-D structure equations and show how hydrostatic balance, radiative diffusion, nuclear heating, and convective marginality fit together.

35.1 Global Scales and Hydrostatic Support

Basic solar parameters. The Sun has mass, radius, and luminosity

\[M_\odot \simeq 2\times 10^{30}\,\mathrm {kg}, \qquad R_\odot \simeq 7\times 10^{8}\,\mathrm {m}, \qquad L_\odot \simeq 3.8\times 10^{26}\,\mathrm {W}.\]
Treating the Sun as an ionized hydrogenic plasma, the mean number density is of order
\[\bar n \sim \frac {M_\odot }{m_p V_\odot } \sim 8\times 10^{29}\,\mathrm {m^{-3}},\]
while central values are much larger,
\[n(0)\sim 10^{32}\,\mathrm {m^{-3}}, \qquad T(0)\sim 1.5\,\mathrm {keV}, \qquad p(0)\sim 10^{11}\,\mathrm {atm}.\]
These are the scales one expects for a self-gravitating plasma compressed by its own weight.

Hydrostatic balance. In the absence of dynamically important magnetic fields, force balance reduces to hydrostatic equilibrium,

\[\frac {dp}{dr} = -\rho (r)\frac {G M(r)}{r^2}, \tag{A.4}\]
with enclosed mass satisfying
\[\frac {dM}{dr} = 4\pi r^2 \rho (r). \tag{A.5}\]
Equation (A.4) is the spherical version of the hydrostatic balance used in Lecture 18; the difference is that the gravitational field is now generated self-consistently by the same mass distribution whose pressure it compresses.

Pressure scale height. A useful local measure of the stratification is the pressure scale height,

\[H_p \equiv -\frac {p}{dp/dr} = \frac {p}{\rho g}, \qquad g(r)=\frac {GM(r)}{r^2}.\]
For an ideal gas,
\[p=\frac {\rho }{\mu m_p}k_B T,\]
so that
\[H_p = \frac {k_B T}{\mu m_p g}. \tag{A.8}\]
Thus a temperature of order \(k_B T\sim \mu m_p g H_p\) is required just to support a scale height \(H_p\) against gravity.

Tutorial

Virial estimate of the solar temperature scale. For a self-gravitating sphere the virial theorem gives

\[2K+U=0,\]
where \(K\) is the total thermal energy and \(U\) the gravitational potential energy. If we approximate the Sun as a sphere of total mass \(M_\odot \), radius \(R_\odot \), and mean molecular weight \(\mu \), then
\[K \sim \frac {3}{2}\frac {M_\odot }{\mu m_p}k_B \bar T, \qquad U \sim -\frac {3}{5}\frac {G M_\odot ^2}{R_\odot }.\]
Using \(2K+U=0\) gives the mean thermal scale
\[k_B \bar T \sim \frac {\mu m_p}{5}\frac {G M_\odot }{R_\odot }. \tag{A.11}\]
This is only an average temperature. A centrally condensed star has a significantly hotter core than this global estimate, which is why the central solar temperature reaches the \(10^7\,\mathrm {K}\) or keV range. The key lesson is conceptual: the basic thermal scale of a star is set by gravity.

Negative heat capacity. Self-gravitating systems behave opposite to ordinary laboratory gases. If the star radiates energy and \(U\) becomes more negative, the virial theorem forces \(K\) to increase. In that sense a self-gravitating system can heat up as it loses energy. This is one reason why gravitational confinement is both effective and subtle.

35.2 Energy Generation and Transport

Nuclear power. The solar luminosity is generated mainly by the proton–proton chain,

\[4p + 2e^- \rightarrow {}^4\mathrm {He} + 2\nu _e + 26.7\,\mathrm {MeV}.\]
At the level of stellar structure, the convenient quantity is the heating rate per unit mass, \(\varepsilon (\rho ,T,X_i)\), so that the luminosity equation is
\[\frac {dL}{dr} = 4\pi r^2 \rho \,\varepsilon (\rho ,T,X_i). \tag{A.13}\]
This is equivalent to writing the local fusion power as a reaction rate times the energy release per reaction. The essential point is that \(\varepsilon \) depends very steeply on temperature, which gives stars a powerful self-regulation mechanism.

Radiative diffusion. In the deep solar interior, energy is carried mainly by photons diffusing through an optically thick plasma. Using the Rosseland mean opacity \(\kappa \) (per unit mass), the radiative flux is

\[F_{\rm rad} = -\frac {4acT^3}{3\kappa \rho }\frac {dT}{dr}, \tag{A.14}\]
where \(a=4\sigma _{\rm SB}/c\) is the radiation constant. Since
\[L(r)=4\pi r^2 F_{\rm rad},\]
we obtain
\[\frac {dT}{dr} = -\frac {3\kappa \rho L(r)}{16\pi a c\,r^2 T^3}. \tag{A.16}\]
This is the standard stellar analogue of a heat-flux closure in transport theory: the luminosity fixes the temperature gradient required to carry that heat.

Tutorial

Deriving the radiative temperature gradient \(\nabla _{\rm rad}\). Define the logarithmic temperature gradient

\[\nabla \equiv \frac {d\ln T}{d\ln p}.\]
To obtain the radiative value, divide Eq. (A.16) by the hydrostatic relation Eq. (A.4):
\[\frac {dT}{dp} = \frac {\dfrac {dT}{dr}}{\dfrac {dp}{dr}} = \frac {-\dfrac {3\kappa \rho L}{16\pi a c\,r^2 T^3}}{-\rho GM/r^2} = \frac {3\kappa L}{16\pi a c\,G M\,T^3}.\]
Multiplying by \(p/T\) gives
\[\nabla _{\rm rad} = \frac {p}{T}\frac {dT}{dp} = \frac {3\kappa L p}{16\pi a c\,G M\,T^4}. \tag{A.19}\]
This quantity answers a practical question: if radiation alone is responsible for energy transport, how steep must the temperature gradient be in order to carry the local luminosity \(L(r)\)? When the required gradient is too steep, the stratification becomes convectively unstable and the star switches to a different transport channel.

Why stars self-regulate. Suppose the core temperature rises slightly. Then the fusion power increases, the luminosity grows, and the required transport gradient steepens. That modifies the density and temperature profiles, which feeds back on the burning rate. The Sun is not static because nuclear physics is weak; it is static because gravity, transport, and burning form a tightly coupled feedback loop.

35.3 Convective Stability and Entropy

Entropy proxy. As emphasized in Lecture 18, convective stability is an entropy-ordering problem. For an ideal gas,

\[s=s_0+c_v\ln \!\left (\frac {p}{\rho ^\gamma }\right ),\]
so the quantity
\[K\equiv \frac {p}{\rho ^\gamma }\]
is a monotonic proxy for entropy. Stable stratification requires entropy to increase outward,
\[\frac {ds}{dr}>0 \qquad \Longleftrightarrow \qquad \frac {d}{dr}\ln \!\left (\frac {p}{\rho ^\gamma }\right )>0.\]
This is the spherical version of the Schwarzschild logic developed earlier in the plane-parallel setting.

Schwarzschild criterion in \(\nabla \) form. Using the ideal-gas relation \(p\propto \rho T\), one has

\[d\ln \rho =d\ln p-d\ln T=(1-\nabla )\,d\ln p.\]
Hence
\[\begin{aligned}d\ln \!\left (\frac {p}{\rho ^\gamma }\right ) &= d\ln p-\gamma \,d\ln \rho \nonumber \\ &= \left [1-\gamma (1-\nabla )\right ]d\ln p \nonumber \\ &= \gamma \left (\nabla -\nabla _{\rm ad}\right )d\ln p,\end{aligned}\]

where

\[\nabla _{\rm ad}=\frac {\gamma -1}{\gamma }. \tag{A.25}\]
Since pressure decreases outward, \(d\ln p/dr<0\), entropy increasing outward is equivalent to
\[\boxed {\nabla <\nabla _{\rm ad}.} \tag{A.26}\]
Thus the radiative equilibrium is stable only if the gradient it requires satisfies
\[\nabla _{\rm rad}<\nabla _{\rm ad}.\]
If instead \(\nabla _{\rm rad}>\nabla _{\rm ad}\), radiative transport alone would demand an entropy profile that overturns under buoyancy, and convection sets in.

Marginality. Convection tends to relax the stratification toward nearly adiabatic conditions,

\[\nabla \approx \nabla _{\rm ad}.\]
This is closely analogous to the way interchange or ballooning turbulence tends to regulate pressure gradients in magnetically confined plasmas. The mechanism differs, but the logic of marginality is the same.

Caution

This appendix deliberately stays at the level of a static, spherically symmetric, weakly magnetized star. Real solar structure also involves composition gradients, partial ionization, mixing-length convection, differential rotation, magnetic fields, and a non-hydrostatic corona. The goal here is not fidelity to every solar detail. It is to isolate the cleanest equilibrium and transport logic in a self-gravitating plasma.

35.4 A Minimal 1-D Stellar Structure Model

A standard spherically symmetric stellar model is obtained by solving a coupled first-order system for the radial profiles of enclosed mass \(M(r)\), luminosity \(L(r)\), pressure \(p(r)\), and temperature \(T(r)\). The independent variable is radius \(r\in [0,R_\star ]\).

Unknowns. \[ y(r) \equiv \big (M(r),\,L(r),\,p(r),\,T(r)\big ). \]

Closure relations. The density is obtained from an equation of state,

\[\rho =\rho (p,T,X_i),\]
where \(X_i\) denotes composition. The heating rate and opacity are then written as
\[\varepsilon = \varepsilon \!\big (\rho ,T,X_i\big ), \qquad \kappa = \kappa \!\big (\rho ,T,X_i\big ).\]
Once these are specified, the structure equations close.

Structure equations. The stellar-structure system is

\[\begin{aligned}\frac {dM}{dr} &= 4\pi r^2 \rho , \\ \frac {dp}{dr} &= -\rho \,\frac {G M}{r^2}, \\ \frac {dL}{dr} &= 4\pi r^2 \rho \,\varepsilon (\rho ,T,X_i), \\ \frac {dT}{dr} &= \nabla \,\frac {T}{p}\,\frac {dp}{dr}.\end{aligned} \tag{A.31}\]

In a radiative region,

\[\nabla =\nabla _{\rm rad} = \frac {3\kappa L p}{16\pi a c\,G M\,T^4}, \tag{A.35}\]
while in a convective region one often approximates
\[\nabla \approx \nabla _{\rm ad}=\frac {\gamma -1}{\gamma }.\]
A simple closure model is therefore
\[\nabla = \min \!\big (\nabla _{\rm rad},\,\nabla _{\rm ad}\big ), \tag{A.37}\]
that is, radiative where stable and adiabatic where convection sets in.

Boundary conditions. At the center, regularity requires

\[M(0)=0, \qquad L(0)=0, \tag{A.38}\]
while the central pressure and temperature,
\[p(0)=p_c, \qquad T(0)=T_c, \tag{A.39}\]
serve as shooting parameters. For numerical work it is convenient to start at a small radius \(r=r_0\ll R_\star \) using the series expansions
\[\begin{aligned}M(r_0) &\simeq \frac {4\pi }{3}\rho _c r_0^3,\\ L(r_0) &\simeq \frac {4\pi }{3}\rho _c \varepsilon _c r_0^3,\\ p(r_0) &\simeq p_c - \frac {2\pi }{3}G\rho _c^2 r_0^2,\\ T(r_0) &\simeq T_c + \left .\frac {dT}{dr}\right |_{r_0} r_0,\end{aligned}\]

with \(\rho _c=\rho (p_c,T_c)\) and \(\varepsilon _c=\varepsilon (\rho _c,T_c)\).

At the surface one imposes global targets such as

\[M(R_\star )=M_\star , \qquad L(R_\star )=L_\star , \tag{A.44}\]
possibly together with a photospheric condition,
\[T(R_\star )=T_{\rm eff}, \qquad L_\star = 4\pi R_\star ^2 \sigma _{\rm SB} T_{\rm eff}^4. \tag{A.45}\]
In the simplest model, the surface pressure is taken to be negligible compared with the interior pressure.

Shooting idea. Given target values \((M_\star ,L_\star )\), one chooses trial central values \((p_c,T_c)\), integrates outward, and iterates on \((p_c,T_c)\) until the surface conditions are met. This is the same shooting logic used repeatedly in plasma equilibrium and eigenvalue problems.

Takeaways

A static star is governed by the same kind of intertwined logic that appears throughout MHD: a force balance equation, a source equation, a transport law, and a marginal-stability condition. For the Sun those are hydrostatic equilibrium, nuclear burning, radiative or convective energy transport, and the Schwarzschild criterion. Gravity changes the details, but not the structure of the reasoning.

35.5 Homework: A Minimal Stellar Structure Model (pp burning + Kramers opacity)

Goal. Build a simple 1-D stellar model by solving the coupled ODEs for hydrostatic equilibrium, mass conservation, luminosity generation, and radiative energy transport. Use simplified closures for pp-chain burning and opacity, and identify where the model becomes convectively unstable by comparing \(\nabla _{\rm rad}\) to \(\nabla _{\rm ad}\).

Structure equations. Solve for \(M(r),\,p(r),\,L(r),\,T(r)\) on \(r\in [0,R_\star ]\):

\[\begin{aligned}\frac {dM}{dr} &= 4\pi r^2 \rho , \\ \frac {dp}{dr} &= -\rho \,\frac {G M}{r^2}, \\ \frac {dL}{dr} &= 4\pi r^2 \rho \,\varepsilon (\rho ,T), \\ \frac {dT}{dr} &= -\frac {3\kappa (\rho ,T)\rho L}{16\pi a c\,T^3\,r^2},\end{aligned} \tag{A.46}\]

where \(a=4\sigma _{\rm SB}/c\) is the radiation constant. Assume an ideal-gas equation of state,

\[p=\frac {\rho }{\mu m_p}k_B T, \qquad \text {so}\qquad \rho =\frac {\mu m_p}{k_B}\frac {p}{T}, \tag{A.50}\]
with mean molecular weight \(\mu =0.6\).

Simplified microphysics. Use the parameterizations

\[\varepsilon (\rho ,T)=\varepsilon _0\,\rho \,T^4, \tag{A.51}\]
for pp burning and
\[\kappa (\rho ,T)=\kappa _0\,\rho \,T^{-7/2}, \tag{A.52}\]
for Kramers opacity. The constants \(\varepsilon _0\) and \(\kappa _0\) are to be calibrated.

Boundary conditions and numerical start. At the center,

\[M(0)=0, \qquad L(0)=0, \qquad p(0)=p_c, \qquad T(0)=T_c, \tag{A.53}\]
where \(p_c\) and \(T_c\) are shooting parameters. Start the integration at a small radius \(r=r_0\ll R_\star \) using
\[\begin{aligned}\rho _c &= \frac {\mu m_p}{k_B}\frac {p_c}{T_c},\\ M(r_0) &\simeq \frac {4\pi }{3}\rho _c r_0^3,\\ L(r_0) &\simeq \frac {4\pi }{3}\rho _c \varepsilon (\rho _c,T_c) r_0^3,\\ p(r_0) &\simeq p_c-\frac {2\pi }{3}G\rho _c^2 r_0^2,\\ T(r_0) &\simeq T_c + \left .\frac {dT}{dr}\right |_{r_0} r_0.\end{aligned}\]

At the surface, impose

\[M(R_\star )=M_\star , \qquad L(R_\star )=L_\star , \tag{A.59}\]
and optionally define the effective temperature through
\[L_\star = 4\pi R_\star ^2 \sigma _{\rm SB} T_{\rm eff}^4. \tag{A.60}\]

Tasks.

1.
Non-dimensionalization. Introduce dimensionless variables \[ \tilde r = \frac {r}{R_\star },\qquad \tilde M=\frac {M}{M_\star },\qquad \tilde L=\frac {L}{L_\star },\qquad \tilde p=\frac {p}{p_\star },\qquad \tilde T=\frac {T}{T_\star }, \] and choose convenient scales \(p_\star \) and \(T_\star \). Rewrite Eqs. (A.46)–(A.49) in dimensionless form and identify the dimensionless control parameters.
2.
Calibration of \(\varepsilon _0\) and \(\kappa _0\). Choose \(R_\star =R_\odot \) and \(M_\star =M_\odot \). Pick trial values for \((p_c,T_c)\) and then tune \(\varepsilon _0\) and \(\kappa _0\) so that the integrated model yields \(L(R_\odot )\approx L_\odot \) and a plausible core temperature \(T_c\sim (1\text {--}2)\,\mathrm {keV}\).
3.
Shooting for \((p_c,T_c)\). Implement a shooting method and iterate \((p_c,T_c)\) until the surface constraints (A.59) are satisfied at \(R_\odot \). Report the resulting profiles \(M(r)\), \(L(r)\), \(T(r)\), \(\rho (r)\), and \(p(r)\).
4.
Convective stability check. Compute the radiative gradient in the standard form
\[\nabla _{\rm rad} = \frac {3\kappa L p}{16\pi a c\,G M\,T^4}, \tag{A.61}\]
and compare it with
\[\nabla _{\rm ad}=\frac {\gamma -1}{\gamma }, \qquad \gamma =\frac 53.\]
Identify radii where \(\nabla _{\rm rad}>\nabla _{\rm ad}\). How does the location of the convectively unstable region depend on \(\kappa _0\)?
5.
Virial check. Estimate the gravitational binding energy
\[U\sim -\int _0^{R_\star } \frac {G M(r)}{r}\,dM,\]
and the thermal energy
\[K\sim \int _0^{R_\star } \frac {3}{2}p\,dV.\]
Verify that the result is consistent with the virial theorem, \(K\approx -U/2\), to order unity.

Deliverables. Submit (i) plots of \(\rho (r)\), \(T(r)\), \(M(r)\), and \(L(r)\); (ii) your fitted \(\varepsilon _0\) and \(\kappa _0\); (iii) the radius range that is convectively unstable according to \(\nabla _{\rm rad}>\nabla _{\rm ad}\); and (iv) a short paragraph explaining what sets the core temperature and why the Sun can remain in long-lived equilibrium.

Optional extensions.