Convection in Stars: The Role of Mass and Structure in Fuel Consumption

[Xforce]

As most people know, massive stars such as blue giant of Wolf-Rayet stars are short lived,only for a few million years.well tiny stars such as red dwarfs seems to be immortal by human’s prospective, and indeed they can live up to a trillion years! Most people thought it was only caused by the huge fuel consumption of the massive stars. But it turns out small stars have a convection between the outer layer and the core,and all the hydrogen in the star can be used as fuel. Stars more massive than 0.5 solar masses have a radiation zone, where radiation passes by.
In a medium sized star like sun, the convection zone is between the core and the surface and the core is a radiation zone. The convection zone is away from the core, means that the hydrogen between the surface and core can not be used as fuel. In a planetary nebula there is a lot of hydrogen. In a massive star, the convection zone is in the core and the radiation zone is between surface and core. Because large stars (more than 8 solar masses)can fuse multiple elements until iron-56, so the fuel in the core should be convected from the inner core and outer core. But turns out there are still layers of different elements in a dying core of a large star. Why is this happening and why stars with different masses have different structures?

 

[phyzguy]

I has to do with the density, pressure and temperature gradients in the star, which depend on the mass of the star. Here are a couple of links that might help:

https://en.wikipedia.org/wiki/Schwarzschild_criterion
http://www.ucolick.org/~woosley/ay112-14/lectures/lecture8.4x.pdf

 

[Ken G]

Basically, if you set up the internal structure of a main-sequence star (the kind that fuse hydrogen in their core, which you are mostly talking about) and assume the energy transport is entirely by radiation (i.e., by diffusion of light), you start from the core temperature of hydrogen fusion, and start moving outward in your model. As you do, the temperature drops (to carry the heat outward), but the pressure drops even more (as there is less and less weight bearing down as you go out), so the density drops a great deal. Eventually you get to a place where light can start to escape into free space. However, low-mass stars are higher density stars, and you think you have a solution. But then you encounter two problems.

The temperature drop occurs over the scale of the photon "mean-free-path," meaning that if the density is higher, the temperature must drop faster with distance in order to carry the radiation outward. This implies that low-mass main-sequence stars present you with a problem-- they are high-density objects, so the mean-free-paths are short, and the temperature drops quite quickly. If you assume all is radiative, you find that the temperature drops to zero before you reach the place where light can escape to free space! This is not allowed, it means the radiative assumption requires the temperature drop to be too steep, and some other heat transport mechanism must take over before you get to zero temperature. That other mechanism requires a slower drop in temperature with distance, so you can reach the surface at a reasonable temperature, albeit somewhat cool for a star. That other mechanism is convection, and it sets in when it is needed to prevent a lower-mass star from requiring too steep of a temperature drop.

But you also encounter a problem in high-mass main-sequence stars. These have low density, so the mean-free-paths are much longer, and there is no problem with the temperature dropping too much-- you get to the surface and the temperature is still high, and all is radiative, and this is why those stars are hot at their surfaces. But there is a problem deep down in the core, because light is leaking out of these large low-density stars so quickly that the fusion rate is regulated to be quite high. To get such high fusion rates requires CNO cycle fusion of hydrogen, rather than p-p type, and CNO cycle fusion is incredibly temperature sensitive. This means the fusion is much faster very close to the center where the temperature is highest, rather than a little farther out. But there is very little mass close to the center, so the gravity is quite weak there. This presents a problem, because you have a large amount of radiation being created, and very little gravity, so radiation pressure on the atoms gets higher than the gravitational effects on those atoms, and the atoms begin to stream outward. They can't get very far because of the huge weight above them, but they also are not happy to stay put, so they become very stirred up and dynamical, and convection sets in-- but this time in the core.

 

[snorkack]

Basically, if you set up the internal structure of a main-sequence star (the kind that fuse hydrogen in their core, which you are mostly talking about) and assume the energy transport is entirely by radiation (i.e., by diffusion of light), you start from the core temperature of hydrogen fusion, and start moving outward in your model. As you do, the temperature drops (to carry the heat outward), but the pressure drops even more (as there is less and less weight bearing down as you go out), so the density drops a great deal. Eventually you get to a place where light can start to escape into free space. However, low-mass stars are higher density stars, and you think you have a solution. But then you encounter two problems.

The temperature drop occurs over the scale of the photon "mean-free-path," meaning that if the density is higher, the temperature must drop faster with distance in order to carry the radiation outward. This implies that low-mass main-sequence stars present you with a problem-- they are high-density objects, so the mean-free-paths are short, and the temperature drops quite quickly. If you assume all is radiative, you find that the temperature drops to zero before you reach the place where light can escape to free space! This is not allowed, it means the radiative assumption requires the temperature drop to be too steep, and some other heat transport mechanism must take over before you get to zero temperature. That other mechanism requires a slower drop in temperature with distance, so you can reach the surface at a reasonable temperature, albeit somewhat cool for a star. That other mechanism is convection, and it sets in when it is needed to prevent a lower-mass star from requiring too steep of a temperature drop.

I think it is wrong approach, though not mistake.
Look at the analogy of Sun.
The lower temperature outer regions of Sun are convective because they are poorly conductive - the conductivity of gas increases a lot moving away from ionization temperature.
As you go inside, Sun has tachocline because at higher temperature the increasingly ionized gas gets more transparent and increasingly capable of conduction. At some point the conduction short-circuits the convection and causes the interior to become stagnant at below adiabatic gradient.
Lower mass stars have lower interior temperatures, bigger convective shell and smaller radiative core. I have seen the mass below which a star has no radiative core at all and convection all the way to centre quoted as 0,25 solar, not 0,5, though.

 

[Ken G]

There are certainly many ways to look at the situation, and what one focuses on as the "key" element can be somewhat subjective. I stand by my answer, but I agree that other issues like changes in opacity with temperature are also important details in the full story. So the way I would put it is, I answered why convection zones in core and envelope are somewhat inevitable given the basic physics of a star, but where they actually appear when you quantify things do include all the important effects you mention.

 

[Nik_2213]

IIRC, the above three configurations are complicated by elemental mix, such that later generations of stars have subtly different innards and process ratios.

Also, you have fun effects such as tidal stirring by siblings, radiative heating, mass transfer unto 'blue straggler' rather than supernova etc etc...

 

[anorlunda]

'blue straggler'

Thanks @Nik_2213. You expanded my vocabulary today.

 

[PAllen]

tidal stirring by siblings..

I think a lot of this was going on in my family growing up ...

 

[snorkack]

There are certainly many ways to look at the situation, and what one focuses on as the "key" element can be somewhat subjective. I stand by my answer, but I agree that other issues like changes in opacity with temperature are also important details in the full story. So the way I would put it is, I answered why convection zones in core and envelope are somewhat inevitable given the basic physics of a star, but where they actually appear when you quantify things do include all the important effects you mention.

But the thing is, there is a basic physical reason for convection zones in envelope - increase of opacity at low temperatures.
Yet envelope convection is not inevitable.
Stars more massive than Sun have less and soon no convective envelope.
Sun´s envelope is convective between 2 000 000 K level and 6000 K level.
Compare Sirius.
1,71 times the radius and 25,4 times the mass. Nearly 9 times the heat flux per area.
What makes Sirius´ envelope so transparent compared to Sun, that all heat is carried by radiation and no convection?

 

[Ken G]

But the thing is, there is a basic physical reason for convection zones in envelope - increase of opacity at low temperatures.
Yet envelope convection is not inevitable.

My point was, it pretty much is. If all stars had a constant opacity per gram everywhere, we'd still get that low mass stars have convective envelopes and high-mass stars have convective cores. So that's what I'm stressing as the crucial physics, it's there independent of any opacity variations so that must be the most important thing to understand. Then you look closer, and you find there are significant opacity variations like you are talking about, and the domains where you find convection get altered as a result.

What makes Sirius´ envelope so transparent compared to Sun, that all heat is carried by radiation and no convection?

It stems in large part from its lower column mass, leading to lower total optical depth. If you fix the core temperature, the virial radius scales roughly proportional to mass, so the column density (and optical depth at fixed opacity per gram) scales roughly inversely proportional to mass.

 

[snorkack]

My point was, it pretty much is. If all stars had a constant opacity per gram everywhere, we'd still get that low mass stars have convective envelopes and high-mass stars have convective cores. So that's what I'm stressing as the crucial physics, it's there independent of any opacity variations so that must be the most important thing to understand. Then you look closer, and you find there are significant opacity variations like you are talking about, and the domains where you find convection get altered as a result. It stems in large part from its lower column mass, leading to lower total optical depth. If you fix the core temperature, the virial radius scales roughly proportional to mass, so the column density (and optical depth at fixed opacity per gram) scales roughly inversely proportional to mass.

Look at Sirius again.
1,71 times the radius of Sun means about 2,9 times the area of Sun.
2,06 times the mass of Sun means column density surface to centre is just about 1,4 times less than Sun.
The heat flux per area is nearly 9 times bigger. If it is just radiation/conduction, should require correspondingly bigger conductivity.
Precisely what enables Sirius´ envelope, between 10 000 K level and 2 000 000 K level, to pass through so much more radiation without convecting? The basic physics of increased opacity per gram at low temperatures because of bound-bound transitions, bound-free transitions and free-free transitions (inverse braking radiation) should still apply!

 

[Ken G]

Again, if you look at the details of where convection appears along the main sequence, and how suddenly it appears, you will need to look at the details of just how it appears and what feedback mechanisms there are, as you are saying. However, that is not the question I'm answering, because it's not the question I'm seeing. I'm answering why, if you look at a mass distribution of main-sequence stars, you should expect to see at the high-mass end core convection due to dynamical instability in the weak-gravity core, which gives way at lower masses to envelope convection to keep the temperature from dropping to zero before the surface of the star is reached. At lower and lower masses, the problem with the radiative temperature gradient gets so bad that convection must appear sooner and sooner. That's the guts of why we see what we see, but yes, the details of when we see it and how suddenly the transition occurs do involve a lot more opacity and ionization physics, so it's not an either/or, it's a question of how deeply into it do you want to get.

 

[Ken G]

Also, I now see there is a second question being asked, which is why do we get "onion skin" models of massive star interiors if the core is convective. The answer to that is the "core" is defined as only the central part of the "onion", it's where the hottest fusion is occurring. Between the layers of the "onion," we have shell fusion, but those regions are narrow and are not mixing the shells. The core is convective because the gravity is weak there yet there is a steep radiation pressure gradient, so the situation is dynamically unstable.

 

[snorkack]

A reason why massive star interiors are convective is that while at low temperatures, opacity drops steeply with temperature as the atoms are ionized and inverse braking radiation also decreases, at higher temperatures the opacity reaches a floor of Thomson scattering. Therefore the concentrated heat flux from massive star core CNO cycle is liable to cause convection.
But this requires availability of protium to fuse. When protium is exhausted, the helium core, hot as it is, will become stagnant.

 

[Xforce]

What about red giants? They have depleted their hydrogen fuel in the core, start to fuse helium and expand their size over 1 million times. By the time helium is ignited it creates a nuclear explosion called a helium flash. In a red giant the core is about the same size as earth, but contains half of the mass of the whole star. The core is covered by a layer of thin, translucent plasma that could readily exceed the size of Earth orbit. How does the convection mechanisms work in a red giants or red super-giants?

 

[Ken G]

A reason why massive star interiors are convective is that while at low temperatures, opacity drops steeply with temperature as the atoms are ionized and inverse braking radiation also decreases, at higher temperatures the opacity reaches a floor of Thomson scattering.

The opacity environment in a convective core isn't all that much different from what is in the Sun, because most ions are highly stripped in both cases, and we are talking about a fairly small temperature difference between the solar core and convective cores. It's true that the lower density of convective cores also lowers the opacity, but it's not by a large amount, I can't think that's really the key issue.

Therefore the concentrated heat flux from massive star core CNO cycle is liable to cause convection.

It's the super-high temperature sensitivity of CNO cycle fusion that would seem to be the dominant cause, though there are some opacity differences in detail.

 

[Ken G]

What about red giants? They have depleted their hydrogen fuel in the core, start to fuse helium and expand their size over 1 million times. By the time helium is ignited it creates a nuclear explosion called a helium flash. In a red giant the core is about the same size as earth, but contains half of the mass of the whole star. The core is covered by a layer of thin, translucent plasma that could readily exceed the size of Earth orbit. How does the convection mechanisms work in a red giants or red super-giants?

It's similar to what is happening early in a star's life. Stars start out on the "Hayashi track", which is an opacity induced limit on how cold their surface temperatures can actually get. The interiors are entirely convective, and the surface temperature controls the luminosity (given that the radius is controlled by the age as the star contracts), but convection is so vastly effective at carrying heat that it is almost unaffected by whatever luminosity it needs to support. In the case of a red giant, the luminosity is not set by the surface temperature, it is set by the shell fusion in the deep interior, so when the surface temperature hits the Hayashi limit (around 3000 K or so), what gets determined is the radius of the envelope-- whatever is necessary to carry the fusion rate. But that feeds back on the shell fusion, because the radius controls the pressure in the shell by controlling the weight of the envelope. Red giants are fascinating systems!

Red supergiants are also mostly convective. There what you have are two sides of the "Hertsprung gap," which basically means stars can be rather torn between two possible interior solutions, one that is Hayashi-like and dominated by convection, and the other Henyey-like and dominated by radiative diffusion. One way to understand why young stars ultimately leave the Hayashi track is that as they shrink at nearly constant surface temperature, their luminosity must drop, until at some point they find they can actually support a higher luminosity by radiative diffusion than by convection-- not because convection is less effictive (it is spectacularly effective), but because it is simply not called on to carry much energy flux due to the rather "red" surface temperature it generally implies. When radiation can support a higher surface temperature, it takes over, which happens for the higher-mass stars. But the point is, both these solutions are in some sense allowed, so the star can get rather confused about which one it is supposed to have. Sudden resolution of that "confusion" is what causes the jump across the "Hertsprung gap", and creates many of the red supergiants when the star decides it should be mostly convective and have a low surface temperature and large size, rather than mostly radiative with a high surface temperature and smaller size.