7.4: Von Zeipel's Theorem and Eddington-Sweet Circulation Currents

For a solution to exist for the structure of a distorted star, we had to invoke mass motions in the star itself. This result was essentially obtained by von Zeipel³ in the middle 1920s. At that time, the source of stellar energy was unknown, and von Zeipel set about to place constraints on the energy generation within a distorted star and in so doing produced one of the most misunderstood theorems of stellar astrophysics. The theorem is essentially a proof by contradiction that stars cannot simultaneously satisfy radiative and hydrostatic equilibrium if the stars are distorted. The normal version of the theorem is given for rigidly rotating stars and this is the version quoted by Eddington⁴. However, in the original publication, the version developed for rigid rotation is followed immediately by a version appropriate for tidally distorted stars5. Thus, clearly the theorem results from the induced distortion itself and is independent of the details that produce the distortion. We describe the version for rotation here, but keep in mind that is it the distortion that is important, not the mechanism by which that distortion is generated.

a. Von Zeipel's Theorem

As originally stated by von Zeipel³ in 1924, this theorem says that for a rigidly rotating star in hydrostatic and radiative equilibrium, the rate of energy generation is given by \[\epsilon=\text { (const) }\left(1-\frac{\omega^2}{2 \pi G \rho}\right)\label{7.3.1}\]

In light of what we now know about stars, this is an absurd result, because it requires that the energy generation rate become negative near the surface as the density goes to zero. As is the case when any theorem yields an absurd result, one must challenge the assumptions. To see where the trouble is likely to be, let us sketch von Zeipel's argument.

The equation of hydrostatic equilibrium \[\nabla P=\rho \nabla \Phi\label{7.3.2}\]

indicates that the potential gradient is related to the pressure gradient by the scalar density \(\rho\). Hence, both vectors point in the same direction, and we can describe the change in pressure as a proportional change in potential so that \[d P=\rho d \Phi\label{7.3.3}\]

From this it is clear, that the pressure must be constant on a level surface where dΦ = 0. This is equivalent to saying that the pressure can be written as a function of the potential Φ alone. If the pressure is a function of Φ alone, then the scalar \(\rho\), relating the potential and pressure gradients, must also be a function of Φ alone. Or \[\rho=\frac{d P(\Phi)}{d \Phi}=\rho(\Phi)\label{7.3.4}\]

As long as the chemical composition µ is constant or at least not varying over an equipotential (level) surface, the ideal-gas law guarantees that the temperature will also be a function of Φ alone: \[T=\frac{P(\Phi) \mu m_h}{k \rho(\Phi)}=T(\Phi)\label{7.3.5}\]

This is what we stated in Section 7.2 to be Poincare's theorem.

Now the radiative temperature gradient which arises from radiative equilibrium requires that \[\vec{F}=\frac{4 a c T^3}{3 \bar{\kappa} \rho} \hat{n}=-\left(\frac{c}{\bar{\kappa} \rho}\right) \nabla\left(\frac{a T^4}{3}\right)=-\frac{c}{\bar{\kappa} \rho} \frac{d P_r}{d n} \hat{n}\label{7.3.6}\]

which, expressed in terms of the potential gradient, becomes \[\vec{F}=-\frac{c}{\bar{\kappa} \rho} \frac{d P_r}{d \Phi} \frac{d \Phi}{d n} \hat{n}=-\frac{c}{\bar{\kappa} \rho} \frac{d P_r}{d \Phi} \nabla \Phi\label{7.3.7}\]

However, since \(\bar{\kappa}\), \(\rho\), and T are all state variables or functions of them, they are all functions of Φ alone and \[\vec{F}=f(\Phi) \nabla \Phi\label{7.3.8}\]

But ∇Φ is just the local gravity, and it is most certainly not a function of the potential alone or constant on level surfaces. Indeed, for a critically rotating star, the gravity varies from the mass gravity at the pole to zero at the equator, where the mass gravity is balanced by the centripetal acceleration. Thus, equation \ref{7.3.8} basically says that in the presence of the radiative temperature gradient \[|\vec{F}|=\text { (const) }|\vec{g}|\label{7.3.9}\]

which is sometimes known as von Zeipel's law of gravity darkening.

If we further consider radiative equilibrium in the absence of mass motions, we can write \[\nabla \cdot \vec{F}=\rho \epsilon=\nabla \cdot[f(\Phi) \nabla \Phi]=f(\Phi) \nabla^2 \Phi+\nabla f(\Phi) \cdot \nabla \Phi\label{7.3.10}\]

For a star in rigid rotation, ∇²Φ will depend on only the density and some constants and so will be a function of Φ alone. The left-hand side of equation \ref{7.3.10} will depend on only the state variables and must also be a function of Φ alone. But, again, the gravity ∇Φ is not a function of Φ alone, so \[\nabla f(\Phi)=0\label{7.3.11}\]

Therefore, evaluating ∇²Φ by means of equations \ref{7.1.10} and \ref{7.2.1}, we get \[\epsilon=\text { (const) } \frac{\nabla^2 \Phi}{\rho}=\text { (const) }\left(1-\frac{\omega^2}{2 \pi G \rho}\right)\label{7.3.12}\]

The absurdity of equation \ref{7.3.12} results primarily from the assumption that the effects of mass motions are not present in equation \ref{7.3.10}. The addition of mass motions removes the exclusive dependence of radiative equilibrium on the potential and the remainder of the argument falls apart releasing the constraint on ε. The gradient of f(Φ) is no longer zero and allows for the variation of ε with radius that we know must exist. However, as we shall see, small amounts of energy are all that is required to be carried by the currents of the mass motions. Thus the radiative gradient will still be basically the temperature gradient that is operative in the radiative zones of the star. The result given in equation \ref{7.3.9} will still be largely correct, and we may expect the radiative flux to be redistributed in accordance with the local value of the gravity. Therefore, particularly for the rapidly rotating upper main sequence stars with radiative envelopes, we may expect that their surface will not be uniformly bright, but will become darker with decreasing local gravity. While it is true that the conditions of radiative equilibrium become rather different in the stellar atmosphere as the photons begin to escape into outer space, the thickness of the atmosphere compared to the depth of the radiative envelope is minuscule and whatever variation of radiative flux has been established at the base of the atmosphere will be largely reflected in the flux emerging from the star. So von Zeipel's theorem, while telling us nothing about the energy generation within the star, does tell us quite a lot about the manner in which the radiation leaves the star.

b. Eddington-Sweet Circulation Currents

We have seen that radiative and hydrostatic equilibrium cannot be simultaneously satisfied in a distorted star and that the failure of these conditions results in the mass motion of material carrying energy to make up the deficit produced by the departure from spherical geometry. That the energy transfer is accomplished by means of the physical motion of material seems ensured. There simply is no other mechanism to effect the transfer. Radiation has been accounted for, conduction is ineffective and the environment is stable against classical convection. These arguments persuaded Eddington⁴ (p. 286) to suggest the existence of such currents which were later quantified by Sweet⁶. Let us now estimate the speed of these currents and determine the amount of energy they may carry. The currents will be quite slow since, even in the most distorted of stars the local departure of the energy flux from spherical symmetry is quite small. Even so, any mass motion could be important if it transports material throughout the star on a nuclear time scale. The possibility would then exist for a resupply of nuclear fuel, and that could upset some of our stellar evolution calculations.

Conservation of Energy and Circulation Velocity The distortion of a star will force a departure from radiative equilibrium and a change in the divergence of the radiative flux from that expected for spherical stars. We argued earlier that the change in the divergence will be brought about by the additional nonradial transport of energy by mass motions, as expressed by the second term on the right-hand side of equation \ref{7.2.6}. Thus, to estimate the velocity of those motions, we must estimate the entropy gradient that distortion will establish.

From thermodynamics remember that the entropy can be expressed in terms of the state variables of an ideal gas as \[S=C_p \ln T-n R \ln P+S_0\label{7.3.13}\]

Therefore, the general energy source term in equations \ref{7.1.1b} can be written as \[T \frac{\partial S}{\partial t}=C_P \frac{\partial T}{\partial t}-\frac{\partial P}{\partial t}\label{7.3.14}\]

and the entropy gradient of equation \ref{7.2.6} becomes \[T \nabla S=C_P \nabla T-\nabla P\label{7.3.15}\]

Now the temperature and pressure gradients are both normal to equipotential surfaces, so the vector nature of equation \ref{7.3.15} is unimportant and it must hold for the magnitude of the individual terms. Therefore, \[T|\nabla S|=C_P \rho|\nabla P|\left(\frac{1}{\rho} \frac{|\nabla T|}{|\nabla P|}-\frac{1}{\rho C_P}\right)\label{7.3.16}\]

Equation \ref{7.3.16}, when combined with the ideal-gas law and the fact that the temperature and pressure gradients point in the same direction, enables equation \ref{7.3.15} to be written as \[T|\nabla S|=C_P \rho\left(\frac{T}{P}\right)|\nabla P|\left(\frac{P}{\rho T} \frac{d T}{d P}-\frac{P}{C_P \rho T}\right)\label{7.3.17}\]

Now the adiabatic gradient can be expressed as \[\left.\frac{d T}{d P}\right|_{\mathrm{ad}}=\frac{T}{P(n+1)}=\frac{P /(k T)}{\rho C_P}\label{7.3.18}\]

Since for the zeroth-order values of these gradients \[\left(\frac{d T}{d P}\right)_0=\frac{(d T / d r)_0}{(d P / d r)_0}\label{7.3.19}\]

we can write the zeroth-order value for the entropy gradient as \[T|\nabla S|_0=\left[|\nabla P|\left(\frac{d T}{d P}\right)_{\mathrm{ad}}^{-1}\left(\frac{d T}{d P}-\left.\frac{d T}{d P}\right|_{\mathrm{ad}}\right)\right]_0=\left(|\nabla P| \frac{\Delta \nabla T}{\left.\nabla T\right|_{\mathrm{ad}}}\right)_0\label{7.3.20}\]

In the equilibrium model, there are no mass motions; the velocity in equation \ref{7.2.6} is already a first-order term and so to estimate its value we need only keep zeroth-order terms in the entropy gradient. The zeroth-order pressure gradient is just \[T|\nabla S|_0=\left[|\nabla P|\left(\frac{d T}{d P}\right)_{\mathrm{ad}}^{-1}\left(\frac{d T}{d P}-\left.\frac{d T}{d P}\right|_{\mathrm{ad}}\right)\right]_0=\left(|\nabla P| \frac{\Delta \nabla T}{\left.\nabla T\right|_{\mathrm{ad}}}\right)_0\label{7.3.21}\]

Combining this with equations \ref{7.2.6} and \ref{7.3.20}, we can write the perturbed equation for energy conservation as \[\begin{aligned}
\nabla \cdot \vec{F} & =\rho \epsilon-\vec{g}_0 \cdot \vec{v}_2 \rho_0(r) \frac{(\Delta \nabla T)_0}{\left(\left.\nabla T\right|_{\mathrm{ad}}\right)_0} \\
& =\rho_0 \epsilon_0-g_0 v_{2,0}(r) \rho_0(r) \frac{(\Delta \nabla T)_0}{\left(\left.\nabla T\right|_{\mathrm{ad}}\right)_0}
\end{aligned}\label{7.3.22}\]

Now, from von Zeipel's gravity darkening law [equations \ref{7.3.6} and \ref{7.3.7}] we have \[\vec{F}=-\left(\frac{4 a c T^3}{3 \bar{\kappa} \rho} \frac{d T}{d \Phi}\right) \vec{g}\label{7.3.23}\]

which means that we can write the divergence of the flux as \[\begin{aligned}
\nabla \cdot \vec{F} & =\nabla \Phi \cdot \frac{d}{d \Phi}\left(-\frac{4 a c T^3}{3 \bar{\kappa} \rho} \frac{d T}{d \Phi} \nabla \Phi\right) \\
& =-\nabla \Phi \cdot \nabla \Phi\left[\frac{d}{d \Phi}\left(\frac{4 a c T^3}{3 \bar{\kappa} \rho} \frac{d T}{d \Phi}\right)\right]-\frac{4 a c T^3}{3 \bar{\kappa} \rho} \frac{d T}{d \Phi} \nabla \cdot \nabla \Phi
\end{aligned}\label{7.3.24}\]

But, since the radiative flux and gravity are vectors pointing in the same direction, \[\nabla \cdot \vec{F}=|\vec{g}|^2\left[\frac{d(F / g)}{d \Phi}+\frac{F}{g} \cdot \nabla^2 \Phi\right]\label{7.3.25}\]

For rotation we can obtain the generalized potential from equations \ref{7.1.1a} and \ref{7.1.10}. Expressing the rotational potential in cylindrical coordinates, we get \[\nabla^2 \Phi=4 \pi G \rho-s^{-1} \frac{\partial}{\partial s}\left[s \frac{\partial}{\partial s}\left(\frac{1}{2} \omega^2 s^2\right)\right]=4 \pi G \rho-s^{-1} \frac{\partial}{\partial s}\left(\omega^2 s^2\right)\label{7.3.26}\]

Equation \ref{7.3.24} for the perturbed flux divergence can be broken into its perturbed components so that \[\begin{aligned}
& \nabla \cdot \vec{F}_0=\frac{d\left(F_0 / g_0\right)}{d \Phi} g_0^2+\frac{F_0}{g_0}\left\{4 \pi G \rho-\left[\frac{1}{s} \frac{d}{d s}\left(\omega^2 s^2\right)\right]_0\right\} \\
& \nabla \cdot \vec{F}_2=\frac{d\left(F_0 / g_0\right)}{d \Phi} 2 g_0 g_2-\frac{F_0}{g_0}\left[\frac{1}{s} \frac{d}{d s}\left(\omega^2 s^2\right)\right]_2
\end{aligned}\label{7.3.27}\]

Since the zeroth-order flux-to-gravity ratio can be written as \[\frac{F_0}{g_0}=\frac{L(r) /\left(4 \pi r^2\right)}{G M(r) / r^2}=\frac{L(r)}{4 \pi G M(r)}\label{7.3.28}\]

its derivative with respect to the generalized potential is \[\frac{d\left(F_0 / g_0\right)}{d \Phi}=-\frac{1}{g_0^2} \frac{\rho_0 L(r)}{M(r)}\label{7.3.29}\]

The luminosity can be written in terms of an average energy generation ratee and M(r) so that \[L(r)=\int_0^r 4 \pi r^2 \rho \epsilon d r \equiv \bar{\epsilon} M(r)\label{7.3.30}\]

which yields \[\nabla \cdot \vec{F}_2=-\bar{\epsilon} \rho_0 \frac{2 g_2}{g_0}-\frac{L(r)}{4 \pi G M(r)}\left[\frac{1}{s} \frac{d}{d s}\left(\omega^2 s^2\right)\right]_2\label{7.3.31}\]

If we further assume that the distortion is small so that \(\mathrm{g}_2 / \mathrm{g}_0 << 1\), then equation \ref{7.3.31} can be combined with equation \ref{7.3.22} to give the velocity for the induced circulation currents as \[v_c \equiv v_{2,0}(r)=\frac{\left(\left.\nabla T\right|_{\mathrm{ad}}\right)_0}{(\Delta \nabla T)_0} \frac{L(r)}{4 \pi g_0 \rho G M(r)}\left[s^{-1} \frac{d\left(\omega^2 s^2\right)}{d s}\right]_2\label{7.3.32}\]

Eddington-Sweet Time Scale and Mixing If we take reasonable values for the parameters in equation \ref{7.3.32}, namely, \[g_0=\frac{G M}{R^2} \quad L(r)=L \quad \omega=\text { const }=w\left(\frac{8 G M}{27 R^3}\right)^{1 / 2}\label{7.3.33}\]

then we can rewrite the circulation velocity as \[v_c=\frac{\left(\left.\nabla T\right|_{\mathrm{ad}}\right)_0}{(\Delta \nabla T)_0} \frac{\bar{\rho}}{\rho} \frac{16 R^2 L}{81 G M^2} w^2\label{7.3.34}\]

Here we have introduced the fractional angular rotational velocity w, which is just ω normalized by the critical angular velocity, ω_c, where the effective equatorial gravity is zero for a centrally condensed star (Roche model). In addition, if we introduce a time scale such as the Kelvin-Helmholtz time scale [equation \ref{3.2.11}], we can further reduce the expression for the circulation velocity to some dimensionless ratios multiplied by \(\mathrm{R} / \tau_{\mathrm{K}-\mathrm{H}}\) and get \[v_c \approx \frac{\left(\left.\nabla T\right|_{\mathrm{ad}}\right)_0}{(\Delta \nabla T)_0} \frac{\bar{\rho}}{\rho} \frac{16 w^2}{135} \frac{R}{\tau_{\mathrm{K}-\mathrm{H}}}\label{7.3.35}\]

If we introduce the Eddington-Sweet time scale as the time required for the circulation currents to carry the material a distance R, then \[\tau_{\mathrm{E}-\mathrm{S}}=\tau_{\mathrm{K}-\mathrm{H}} \frac{135}{16 w^2} \frac{\rho}{\bar{\rho}} \frac{(\Delta \nabla T)_0}{\left(\left.\nabla T\right|_{\mathrm{ad}}\right)_0}\label{7.3.36}\]

The sun is a rather slowly rotating star, and if the angular velocity of the core is displayed on the surface, there is little rotational distortion, so the Eddington-Sweet currents should be small. Certainly the core density will exceed the mean density, and we have already indicated that the radiative core of the sun is barely stable against convection. Thus, the following values for the solar radiative core should provide fair estimates of the internal conditions necessary for the evaluation of the circulation currents: \[w_{\odot}^2 \sim 10^{-5} \quad \frac{\rho(\text { core })}{\bar{\rho}}>1 \quad \frac{(\Delta \nabla T)_0}{\left(\left.\nabla T\right|_{\mathrm{ad}}\right)_0} \sim 1\label{7.3.37}\]

These values, when substituted into equation \ref{7.3.35}, yield \[\tau_{\mathrm{E}-\mathrm{S}} \geq 10^5 \tau_{\mathrm{K}-\mathrm{H}}>\tau_n\label{7.3.38}\]

So the material in the core takes much longer than the nuclear time scale to circulate, and we would not expect the core of the sun, or solar-type stars that are slowly rotating to be mixed. So the stellar evolution scenarios we developed for lower main sequence stars in Chapter 5 remain intact.

The situation is less clear for upper main sequence stars. Here the envelope has a density is lower than the mean density, is in radiative equilibrium, and in danger of supplying fuel to the core. In addition, many of these stars rotate very rapidly so that we might expect much larger circulation current velocities. Reasonable values of the parameters in equation \ref{7.3.35} for rapidly rotating B stars are \[w_B^2 \approx 1 \quad \frac{\rho(\mathrm{env})}{\bar{\rho}}<1 \quad \frac{(\Delta \nabla T)_0}{\left(\left.\nabla T\right|_{\mathrm{ad}}\right)_0} \sim 1\label{7.3.39}\]

which lead to \[\tau_{\mathrm{E}-\mathrm{S}}(B \text { star }) \sim \tau_{\mathrm{K}-\mathrm{H}}<\tau_n\label{7.3.40}\]

On the basis of this analysis we would have to conclude that there is an excellent possibility that rapidly rotating stars on the upper main sequence may be mixed thoroughly throughout and their main sequence life times may be prolonged. However, we have not dealt with the formation of the helium core itself and the effects caused by the change in chemical composition.