Week 5-- Interior of the Sun
Up until now we have mostly talked about the Sun as if it was all one thing, a ball with radius R and mass M, and applied global force-balance and energy-transfer constraints (in the form of the virial theorem and the radiative diffusion rate, respectively). This is useful for a basic understanding, but of course there is also local structure within the Sun, and both force balance and energy transfer are processes that also occurring locally at any point in the Sun. For example, when we look at the luminosity L (rate of energy emitted per time) of the Sun, and apply the Stefan-Boltmann law over the surface area of the Sun, we can calculate the surface T needed to emit L. It comes out about 6000 K, much less than the roughly 14 million K in the solar core. Hence the interior of the Sun exhibits a strong temperature gradient, which is necessary for radiative diffusion to occur, but which gets so steep near the surface that convection sets in. To understand why, we must look closer at the processes of convection and radiative diffusion, and in particular, how they relate to the local temperature gradient.

I. Radiative Diffusion as a Type of Conduction
Radiative diffusion induces a T gradient, because the radiation needs to go primarily outward, and it would not have a preferred direction if the T was uniform everywhere. This is very similar to the way heat conduction works, where a hot surface next to a cool one has heat go from hot to cool (because there are more ways for that to happen, yes?). What's more, the rate of heat transfer (per unit area) is generally proportional to the T gradient (a rule originally discovered by Newton), so a given T gradient can be associated with a given radiative flux density. Since we can estimate the luminosity from the radiative diffusion rate, and we know the characteristic area through which it is being transferred, we can estimate the radiative flux density, and infer the necessary T gradient. This is the T gradient we expect to find in the interior of a radiatively diffusing star, which is true over most of the interior of the Sun.

There is an important caveat here. The rate of "conduction" of the radiative flux does not depend just on the T gradient, but also on a coefficient of the T gradient, called the "thermal conductivity." This is similar to how metal generally conducts heat better than wood, for example, the free electrons in metal imply a higher thermal conductivity. In the case of radiation, however, the thermal conductivity depends on the density of photons, rather than of free electrons, and the density of photons is much higher at higher T (when the "coveted energy" is higher, more energy is dumped into each oscillating mode of the radiation field to make more photons, and also, photons of higher energy can be produced easily). This means at the lower T near the surface of the Sun, the radiative "conduction" is much less efficient, so a steeper T gradient is needed to carry the radiative flux due to the low thermal conductivity at low T. To compensate for the low conductivity, the T gradient itself must get steeper and steeper at lower and lower T, until eventually it becomes so steep that the gas becomes convectively unstable, for reasons we can explore next. This is the region called the convection zone, where it is not radiative diffusion that carries the solar luminosity, but rather, the rising of hot gas and the falling of cool gas.

II. Convection
Convection is the process by which higher T gas at the same pressure P as its surroundings will have a lower mass density than its immediate surroundings, which causes it to rise because gravitational potential energy is released when this gas changes places with the gas above it. Hence convection is associated with the expression "hot air rises." This can lead to a continuous process of hot fluid rising and cooler fluid falling, like when boiling water is heated from below on a stove. Since force balance is always achieved very quickly, we can generally assume that the pressure structure is established much more quickly than energy transport can occur, and this implies that we can take the pressure gradient as externally fixed by the need to hold up the weight of the overlying gas. Then we consider what happens to the temperature of the gas as it moves around. We normally assume the energy transport is slow, so as the gas rises and falls, it expands and contracts adiabatically. The result of moving gas parcels up to lower P or down to higher P, while letting them expand and contract adiabatically, defines the "adiabatic temperature gradient".

Now to determine whether or not the gas will undergo convection, we simply compare the adiabatic temperature gradient induced by convection with the radiatively diffusing temperature gradient induced by radiative conduction, and whichever is less steep will be the one that occurs. The reason the less steep one wins out is that if the radiatively conductive T gradient was steeper, then any gas parcel we pull from that gradient and raise it up a little (just by chance), it will find, by virtue of its adiabatic expansion, that it is warmer than its surroundings (since its surroundings are ruled by the assumed-steeper radiative conduction T gradient). But if it is warmer than its surroundings, but at the same P, it will be less dense, which means it will be buoyant, and will hence continue to rise. This is an instability-- any small perturbation leads to greater and greater perturbation, the situation is not sustainable and the T gradient will be replaced by the less-steep adiabatic T gradient as convective motions prevail.

On the other hand, if the adiabatic T gradient is more steep than the radiative conductive gradient, the opposite happens-- now the gas that was perturbed to rise finds itself, after adiabatic expansion to the new lower P, cooler than its surroundings, so denser, so it sinks back down-- which is a restoring effect, not an instability. All this means that in any region of the star where the adiabatic T gradient is less steep than the radiative conduction gradient (and the latter is getting very steep near the surface where T is low and radiative conduction is inefficient), convection will set in, and the T gradient will be supplanted by the adiabatic T gradient. In such a layer, it is as though a single gas parcel were being moved to all the different radii within that layer, and whatever T it has based on its adiabatic expansion and compression to the surrounding P, that is the T found at that radius. The whole layer is like the life story of any single one of its parcels as it moves all over the layer, keeping the P gradient required by force balance.