Convection vs Radiation at the giant stage of stars

[center o bass]

I just read the following about the evolution of stars:

"When reaching point 2 in the HR-diagram, the radius of the star has
been increasing so much that the surface temperature is close to 2500 K
which is a lower possible limit. When reaching this limit, the dominant
mechanism of energy transport in the star changes from being radiation
to convection. Convection is much more efficient, the energy is released
at a much larger rate and the luminosity increases rapidly."

Two questions: why does convection start to dominate over radiation at 2500K, and why is convection -- in general -- a more efficient way of transporting energy than radiation?

 

[Ken G]

The statement is misleading because the logic is essentially backward-- it seems to claim that red giants are more luminous than dwarfs because they are convective, but in fact they are convective because they are more luminous. So convection does not determine the luminosity of a red giant, but red giants are convective because their luminosity has been determined to be so high that radiative diffusion can't carry it, it must be carried by convection. What's more, convection does not have a particular efficiency at carrying heat, it can carry heat with a wide range of possible efficiencies, whatever it needs to do given the other physics that is actually setting the luminosity. There is a maximum possible efficiency, which is when the gas is convecting at the sound speed, but few stars ever need to be that convective over their whole interior, they just aren't that luminous. Radiation, on the other hand, is limited by the speed that light can diffuse out, which you might think would be very fast given how fast the speed of light is, but the diffusion speed takes the speed of light and divides it by the optical depth, so when the optical depth is huge, it can take a long time to diffuse out.

But convection has a problem-- it only happens when the temperature gradient is steep enough that buoyancy can produce an unstable "turning over" effect, the "rolling boil" effect you see in convecting gas. If the temperature gradient is not steep enough, convection won't happen, and radiation will carry the luminosity of the star. That's where the surface temperature a bit above 2500 K comes in-- when you have that, you get a special kind of opacity where neutral hydrogen picks up a second bound electron, making the "H minus" ion, which is quite good at absorbing light (the extra electron is very weakly bound and easy to knock out of the ion by photon capture). That opacity "bottles up" the radiation, and helps enforce a steep enough temperature gradient where you will get convection. Lower surface temperatures for an ideal gas would be unstable-- rises in T would create more H minus opacity which would absorb more light and raise the temperature. (You can get lower surface temperatures in degenerate material, like brown dwarfs and planets.)

Now, although convection has a higher upper limit on how much luminosity it can carry, it never sets the luminosity of the star, it just carries what luminosity some other process in the star tells it to carry. That "other process" can be of two flavors: outside-in, where the surface layers of the star determine the luminosity and the interior simply provides that luminosity (via convection), or inside-out, where some internal engine determines the luminosity and convection carries that out, and the surface just has to deal with whatever it is. The outside-in case is when you have a protostar that is first forming, which has a history that determines its radius. As mentioned, the surface temperature will always be above 2500 K, typically more like about 4000 K actually, so if we take the surface temperature as known, we can determine the luminosity from the radius of the protostar. The radius is set by the history of whatever stage of contraction the star is currently in, so that's "outside-in" luminosity.

But you are asking about red giants, which have their luminosity determined in a completely different way, they are "inside-out." Their luminosity is determined by the fact that they contain at their centers a ball of degenerate gas, which is a lot like a tiny white dwarf living inside a giant ball of gravitationally bound ideal gas. That is a very special structure, and it gives the star essentially three different pieces-- the white dwarf at the center has a mass that is controlled by the history of adding nuclear burned "ash" to the white dwarf, and that rises with time. Its radius is set by degeneracy physics. Then you have a layer sitting on top of that white dwarf which is an ideal gas, but its temperature is set by the gravity of the white dwarf (via something called the virial theorem). That temperature gets very high as the mass of the white dwarf grows, and indeed it is high enough to have fusion. The rate of fusion in that layer is the internal engine that sets the luminosity of the red giant, and it grows with time simply because the white dwarf mass is growing with time, so the temperature is rising-- and fusion likes high temperature.

Then we come at last to the convective envelope, the third piece of the red giant. This is a rather passive player, it just carries the luminosity set by that internal engine without having essentially any effect on that engine. It must puff out a lot to get down to the necessary cool temperatures to have H minus opacity, and that's why the star is a "giant" (and remember that the luminosity must be carried at the surface by the surface temperature to the fourth power times the radius squared, so if the surface temperature needs to be low to get the convective instability, the radius needs to be huge). So we should say that radiation is too slow to carry the huge luminosity generated by the central engine, and convection will appear and can carry almost any luminosity it needs to, but that convection only constrains the surface temperature-- the luminosity is set by the fusion physics (unlike in main-sequence stars, whose luminosity is set by radiative diffusion), but that's another story). Then the luminosity and surface T get together to determine the radius, and it comes out very large-- hence a red giant.

 

[Drakkith]

Ken, are you saying that radiative diffusion dominates when the temperature gradient is low (and radiation can easily escape), and convection dominates when the temperature gradient is high (and light cannot easily escape)?

 

[Ken G]

Not quite, all heat transport mechanisms prefer high temperature gradients, but radiative diffusion increases smoothly as temperature gradient increases, and convection goes from very low to very high right around a single value of the temperature gradient (the one you get when the star is convective, called the adiabatic lapse rate). So if radiative diffusion would need the temperature gradient to be higher than that to get the luminosity out, which happens in the red giant phase, then the temperature gradient gets "stuck" at the adiabatic lapse rate, and convection takes off, supply essentially any luminosity you need without having much effect on that temperature gradient.

But what I'm really saying is, don't worry about the temperature gradient, it will be whatever it needs to be. What sets the luminosity of the red giant has nothing to do with temperature gradient, it has to do with the fusion rate in a thin shell surrounding the white dwarf-like degenerate helium core. The temperature of that shell is set by the white dwarflike gravity, so that temperature is very high-- way higher than you need for hydrogen fusion, so the hydrogen in that shell fuses like mad. The luminosity is determined by that temperature, and how much mass is in that shell, and that second part is controlled by the need for self-consistency between the fusion rate and the rate that radiation can diffuse out of that shell. Since the shell is narrow, radiative diffusion suffices in that shell-- convection only sets in out outside that shell, to lift all that heat up to the bloated surface of the star. The reason the star is bloated is because that is what the adiabatic lapse rate requires it to be, but we aren't asking why is the star bloated, we are asking why is it so luminous.

So it's all a question of what is determining what, there is always a clear sequence of logic that determines the attributes of any star. In the case of a red giant, that sequence of logic is really quite remarkable-- you have this tiny engine that is maybe 1/1000 or less of the size of the whole red giant, and that tiny engine is controlling its luminosity-- the whole rest of the star is a kind of "convective coat" whose attributes are set simply by the need to carry that huge luminosity.