Evolution toward a white dwarf

[virgil1612]

Hello,

I will need to talk to students about the evolution of a solar-like star. We will look at a diagram like the following:

It will be a descriptive presentation, no mathematics. I know pretty well what happens until the formation of the carbon core. The linear upper line happens because the outer shells will be ejected and gradually the core will become visible so of course the temperature strongly increases (what we effectively see is the core). But why this increase in temperature is at constant luminosity, and why do we have that abrupt decrease in luminosity just before the white dwarf stage?

 

[|Glitch|]

Hello,

I will need to talk to students about the evolution of a solar-like star. We will look at a diagram like the following:

It will be a descriptive presentation, no mathematics. I know pretty well what happens until the formation of the carbon core. The linear upper line happens because the outer shells will be ejected and gradually the core will become visible so of course the temperature strongly increases (what we effectively see is the core). But why this increase in temperature is at constant luminosity, and why do we have that abrupt decrease in luminosity just before the white dwarf stage?

As the surface area of the dying star diminishes and the surface temperature increases during its planetary nebula stage, it leaves the bolometric luminosity relatively unchanged.

L = 4πR²σT⁴​

Where:

R = Radius of the star, in meters;
T = Surface temperature of the star, in Kelvin; and
σ = Stefan-Boltzmann Constant, 5.670373(21) × 10⁻⁸ W m⁻² K⁻⁴.​

After the star has exhausted the helium in its inner shell, there is no more energy being generated. As a result, the outward pressure caused by the nuclear reactions stops, and any remaining layers fall back inward. The star shrinks until the collision between electrons in its core provide enough pressure to halt the collapse.

At this point the white dwarf is no longer increasing in temperature, but rather begins its very slow cooling process. It can take billions of years for a white dwarf to cool into a black dwarf due to its high surface temperature and relatively small surface area.

 

[virgil1612]

As the surface area of the dying star diminishes and the surface temperature increases during its planetary nebula stage, it leaves the bolometric luminosity relatively unchanged.

L = 4πR²σT⁴​

Where:

R = Radius of the star, in meters;
T = Surface temperature of the star, in Kelvin; and
σ = Stefan-Boltzmann Constant, 5.670373(21) × 10⁻⁸ W m⁻² K⁻⁴.​

After the star has exhausted the helium in its inner shell, there is no more energy being generated. As a result, the outward pressure caused by the nuclear reactions stops, and any remaining layers fall back inward. The star shrinks until the collision between electrons in its core provide enough pressure to halt the collapse.

At this point the white dwarf is no longer increasing in temperature, but rather begins its very slow cooling process. It can take billions of years for a white dwarf to cool into a black dwarf due to its high surface temperature and relatively small surface area.

OK, let me see if I got it correctly.

On the upper line, the burning He shell is still there, the star gets hotter as the shell gets closer to the surface as material continues to be ejected.

On the left corner point, the fusion in the shell ceases, causing an abrupt fall in luminosity. Without the fusion, the remaining material falls back and concentration increases until degeneracy sets in everywhere. And then it cools.

 

[|Glitch|]

OK, let me see if I got it correctly.

On the upper line, the burning He shell is still there, the star gets hotter as the shell gets closer to the surface as material continues to be ejected.

On the left corner point, the fusion in the shell ceases, causing an abrupt fall in luminosity. Without the fusion, the remaining material falls back and concentration increases until degeneracy sets in everywhere. And then it cools.

Precisely. The luminosity during the planetary nebula phase remains relatively unchanged because the temperature continues to increase even as the size of the star decreases. But once the inner helium layer is consumed and the star has compressed to its smallest point, the luminosity will diminish as the white dwarf cools. That cooling process (from the upper left to the lower left) can take 14 billion years or more.

 

[virgil1612]

Precisely. The luminosity during the planetary nebula phase remains relatively unchanged because the temperature continues to increase even as the size of the star decreases. But once the inner helium layer is consumed and the star has compressed to its smallest point, the luminosity will diminish as the white dwarf cools. That cooling process (from the upper left to the lower left) can take 14 billion years or more.

Just a detail: does the He in the shell burn until it's completely consumed, or the shell stops burning at a certain point, and there is some remaining He falling back on the C core?

 

[rootone]

I guess it reaches a point where the core is mostly Carbon, probably some Oxygen too, outside that some Helium,
but not enough of anything to sustain further fusion.

 

[virgil1612]

I guess it reaches a point where the core is mostly Carbon, probably some Oxygen too, outside that some Helium,
but not enough of anything to sustain further fusion.

Perfect, now I understand. Thank you.
Virgil.

 

[|Glitch|]

Just a detail: does the He in the shell burn until it's completely consumed, or the shell stops burning at a certain point, and there is some remaining He falling back on the C core?

That is dependent on the original mass of the star. A star with about half the mass of the sun is unable to fuse helium, resulting in a helium white dwarf. Stars with an original mass between 0.5 and 8 solar masses can fuse helium into carbon and oxygen, but no further, resulting in a carbon-oxygen white dwarf. Stars with a mass between 8 and 10 solar masses can fuse carbon into magnesium and neon, resulting in an oxygen-magnesium-neon white dwarf.

 

[rootone]

Just an aside here but not completely off topic.
Where does all the Nitrogen come from, like we have in Earth's atmosphere, which is predominantly Nitrogen.
Nitrogen so I have been led to believe is a transitional product in stellar fusion processes.

Why so much Nitrogen?

 

[Ken G]

Why so much Nitrogen?

It's not really that much, if you think about it-- the atmosphere is so low density. But the reason the atmosphere is mostly nitrogen is that something has happened to all the more abundant "volatile" molecules (those that like to be gas) that could be in the atmosphere of Earth:
hydrogen gas escapes
water molecules become liquid and collect in the ocean
carbon dioxide gets dissolved in the ocean and formed into rocks
If that sounds like it requires special circumstances, it does-- notice that neither Mercury, Venus, nor Mars have nitrogen atmosheres. (Titan and Triton do, but for a totally different reason-- everything else freezes!).

 

[Ken G]

But why this increase in temperature is at constant luminosity, and why do we have that abrupt decrease in luminosity just before the white dwarf stage?

Also a point to add-- the reason the luminosity stays constant while the helium shell is burning is that its luminosity is set by the attributes of the core, not the envelope. The envelope is fully convective, and convection is very efficient at accomodating pretty much any luminosity you hand it, even if it is getting peeled away into space so that the stellar surface is getting smaller. As pointed out above, the precipitous drop in luminosity is when the fusion runs out, and the final cooling phase is a white dwarf of constant size, because by then the amount of heat the star will lose is a tiny fraction of the vast internal kinetic energy already in there, hence requiring essentially no adjustment in its structure as it cools. That's why the final cooling phase should follow a contour of constant surface area.

 

[virgil1612]

Also a point to add-- the reason the luminosity stays constant while the helium shell is burning is that its luminosity is set by the attributes of the core, not the envelope. The envelope is fully convective, and convection is very efficient at accomodating pretty much any luminosity you hand it, even if it is getting peeled away into space so that the stellar surface is getting smaller. As pointed out above, the precipitous drop in luminosity is when the fusion runs out, and the final cooling phase is a white dwarf of constant size, because by then the amount of heat the star will lose is a tiny fraction of the vast internal kinetic energy already in there, hence requiring essentially no adjustment in its structure as it cools. That's why the final cooling phase should follow a contour of constant surface area.

Thanks.

 

[|Glitch|]

Just an aside here but not completely off topic.
Where does all the Nitrogen come from, like we have in Earth's atmosphere, which is predominantly Nitrogen.
Nitrogen so I have been led to believe is a transitional product in stellar fusion processes.

Why so much Nitrogen?

With regard to the origins of nitrogen on Earth, it turns out that the nitrogen isotopes in the sun do not match the nitrogen isotopes on Earth. The nitrogen isotopes in the sun are lighter than those on Earth. Also, the nitrogen isotopes found in comets are heavier than those found on Earth. But some of the asteroids do have the same isotope ratios as Earth. Which suggests that the origin of nitrogen on Earth predates the formation of the planets. As to why there is an abundance of nitrogen on Earth, when compared to other planets like Mars and Venus, it is because Earth has plate tectonics and the other planets do not. N₂ is produced where the continental plates converge through volcanic degassing. It is suspected that the nitrogen on Mars and Venus is locked up in aqueous ammonium.

http://www.nature.com/ngeo/journal/v7/n11/full/ngeo2271.html - Nature Geoscience 7, 816–819 (2014), doi:10.1038/ngeo2271 (paid subscription)

Nitrogen is indeed a transitional product of stellar fusion. It is part of the carbon-nitrogen-oxygen (CNO) cycle.

CNO Cycle - Wikipedia

 

[virgil1612]

... the reason the luminosity stays constant while the helium shell is burning is that its luminosity is set by the attributes of the core, not the envelope. The envelope is fully convective, and convection is very efficient at accomodating pretty much any luminosity you hand it.

So, at the upper right point, where the horizontal line starts, the carbon core should have ended its shrinking? If not, any further shrinking of the nucleus would have influenced the luminosity of the shell and the line would no longer be horizontal?

 

[Ken G]

So, at the upper right point, where the horizontal line starts, the carbon core should have ended its shrinking? If not, any further shrinking of the nucleus would have influenced the luminosity of the shell and the line would no longer be horizontal?

Yes, exactly. But note that the timescale for the star to cross the H-R diagram is quite short, there's not much evolution of the core going on there, it's more just the timescale to peel off the envelope once the envelope is inclined to be so peeled. The key point is the nature of the envelope doesn't feed back onto the core properties, the luminosity is set in a very inside-out fashion there-- until the absence of an envelope alters the properties of the burning and shuts it off altogether. I think it must be that the helium itself is eventually peeled off, more so than burnt up, causing the precipitous drop in luminosity.

 

[rrrrr]

On the left corner point, the fusion in the shell ceases, causing an abrupt fall in luminosity. Without the fusion, the remaining material falls back and concentration increases until degeneracy sets in everywhere. And then it cools.

Also, when the helium layer is puffed off the star, some of the light from the star is blocked and the luminosity falls. The rest of the degenerate hydrogen and helium form a layer around the star that prevents what little light inside from coming out.

 

[Ken G]

Also, when the helium layer is puffed off the star, some of the light from the star is blocked and the luminosity falls. The rest of the degenerate hydrogen and helium form a layer around the star that prevents what little light inside from coming out.

Are you sure? It's pretty hard to block light from coming out, as the energy transport times are often shorter than the evolutionary times, so there is a tendency to establish energy balance. Energy balance basically means, whatever energy the core is sending out, finds its way out through the envelope somehow. So when the luminosity drops, it seems most natural to hold the culprit responsible to be the energy generation rate in the core. However, I haven't seen simulations of that short-lived stage of a star's life, so I don't really know.

 

[snorkack]

Note how tremendously the luminosity of Mira Ceti varies! I expect something quite successfully blocks light.