4.6: Convective Stability

a. Efficiency of Transport Mechanisms

We calculated the fluxes that can be transported by radiation, convection, and conduction, and we found that they produce rather different temperature gradients. However, we have seen from the integral theorems that the central temperature is set largely by the mass of the star, and in Chapter 3 we learned that the energy produced by nuclear processes will be a strong function of that temperature. Thus, virtually all the energy will be produced near the center and, in steady state, must make its way to the surface. In general, it will do this in the most efficient manner possible. That is, the mode of energy transport will be that which produces the smallest temperature gradient and also the greatest luminosity. In short, the star will choose among the methods available to it and select that which allows it to leak away its energy as fast as possible.

To carry enough energy to support the luminosity of the sun, conductive transport would require an immense temperature gradient. This is another way of saying that conduction is not important in the transport of energy. Convection will produce a temperature gradient which is nearly the adiabatic gradient and is fully capable of carrying all the energy necessary to sustain the solar luminosity. If we compare the radiative temperature gradient given in equation \ref{4.2.14}, and the adiabatic gradient as given in equation \ref{4.3.19}, we get \[\frac{(\nabla T)_{\mathrm{rad}}}{(\nabla T)_{\mathrm{ad}}}=\frac{15 \bar{\kappa}(r) \rho(r) L(r) k}{32 \pi a c \mu m_h G M(r) T^3(r)} \sim 1\label{4.5.1}\]

From such an estimate the dominance of one mechanism over another is not obvious. Could both methods compete roughly equally? Or is it more likely that one method will prevail in part of the star, while the remainder will be the domain of the other. We have continually suggested that the latter is the case, and now we shall see the reason for this assertion.

b. Schwarzschild Stability Criterion

For convection to play any role whatever, convective elements must be formed, and the conditions must be such that the elements will rise and fall. The statistical distribution law says that particles exist with the full range of velocities, and it would be remarkable if the particles were so uniformly distributed that any given volume had exactly the same number of particles of each velocity. This would be a very special particle distribution and not at all a random one. A random distribution would require that on some scale some volumes have more high-speed particles than others and hence can be considered to be hotter. In fact, an entire spectrum of such volumes will exist and can be viewed as perturbations to the mean temperature. Thus, the first of our conditions for convective transport will always be met. Temperature fluctuations will always exist. But will they result in elements that move? In developing an expression for the adiabatic gradient, we assumed that the convective element will expand adiabatically and so do no work on the surrounding medium. This is certainly the most efficient way the element can move, and it cannot be exactly met in practice. To move, the element must displace the material ahead of it. There must be some "viscous" drag on the element requiring the element to do "work" on the surrounding medium. So the adiabatic expansion of a convective element is clearly the "best it can do" in getting from one place to another. Let us see if we can quantify this argument.

Let us assume that the gas is an ideal gas, and for the reasons mentioned above we assume that the element will behave adiabatically. Under these conditions we know that, the element will follow a polytropic equation of state, namely, \[P \propto \rho^\gamma \quad \gamma=\frac{5}{3}\label{4.5.2}\]

Now consider a volume element which is displaced upward and has state variables denoted with an asterisk, while the surrounding values are simply P, T, and ρ (see Figure 4.2).

If \(\rho_2^* \geq \rho_2\) then the element will sink or will not have risen in the first place. Initially, we require the conditions at point 1 to be the same (we are displacing the element in an ad hoc manner). Thus, \[P_1^*=P_1 \quad T_1^*=T_1 \quad \rho_1^*=\rho_1\label{4.5.3}\]

Adiabatic expansion of the element requires that pressure equilibrium be maintained throughout the displacement, so \[P_2^*=P_2\label{4.5.4}\]

We may express the conditions at point 2 in terms of a Taylor series and the conditions at point 1 so that \[\begin{aligned}
P_2^*=P_2 & =P_1+\frac{d P}{d r} d r+\cdots+ \\
\rho_2 & =\rho_1+\frac{d \rho}{d r} d r+\cdots+
\end{aligned}\label{4.5.5}\]

Figure 4.2 shows a schematic representation of a convective element with state variables denoted by * and surrounded by an ambient medium characterized by state variables P,T, and ρ. The element is initially at position 1 and is displaced through a distance dr to position 2.

Using the equation of state, we may write \[\begin{aligned}
\rho_2^*=(\text {const})\left(P_2^*\right)^{1 / \gamma}=(\text {const})\left(P_2\right)^{1 / \gamma} & =\rho_1\left(1+\frac{1}{P_1} \frac{d P}{d r} d r+\cdots+\right)^{1 / \gamma} \\
& \approx \rho_1\left(1+\frac{1}{\gamma P_1} \frac{d P}{d r} d r+\cdots+\right)
\end{aligned}\label{4.5.6}\]

If we take \(\rho_2^* \geq \rho_2\) to be a condition for stability (i.e., the element will return to its initial position if displaced), then equations \ref{4.5.5}, and \ref{4.5.6} require that \[\frac{1}{\gamma P} \frac{d P}{d r} \geq \frac{1}{\rho} \frac{d \rho}{d r}\label{4.5.7}\]

The ideal-gas law requires that \[\frac{1}{P} \frac{d P}{d r}=\frac{1}{\rho} \frac{d \rho}{d r}+\frac{1}{T} \frac{d T}{d r}\label{4.5.8}\]

which can be used to replace the density gradient in inequality (4.5.7) to get \[\left(1-\frac{1}{\gamma}\right) \frac{T}{P} \frac{d P}{d r} \leq \frac{d T}{d r}\label{4.5.9}\]

Dividing by dT/dr, we obtain the Schwarzschild stability criterion for a polytropic gas \[\frac{d \ln P}{d \ln T} \geq \frac{\gamma}{\gamma-1}=n+1\label{4.5.10}\]

which for a monatomic gas with a \(\gamma=5 / 3\) is just \[\frac{d \ln P}{d \ln T} \geq 2.5\label{4.5.11}\]

Thus, if the logarithmic derivative of pressure with respect to temperature is greater than or equal to 2.5, convection will not occur. In other words, if the actual temperature gradient is less than the adiabatic gradient, convection will not occur. This, then, is our means for deciding whether convection or radiation will be the dominant mode of energy transport. Should radiation be able to transport the energy with a temperature gradient less than the adiabatic gradient, no energy will be carried by convection, for the gas is stable against the thermal perturbations which must exist. However, if this is not the case, convection will be established; and it is so very efficient that it is capable of carrying all the energy with a temperature gradient that is just slightly super-adiabatic. For most of stellar structure, we may regard energy transport as being bimodal; either radiation or convection will transport the energy, with the decision being made by equation \ref{4.5.10}. The Schwarzschild stability criterion has been shown to be quite general and will hold under the most varied of conditions, including those stars where general relativity must be included to describe their structure.