Can a star with thousands of solar masses be formed?

[nicolauslamsiu]

Can a extremely massive star ( eg. thousands solar masses) be formed?

 

[Drakkith]

I believe the upper limit is on the order of several hundred solar masses, which can only be reached when the forming star has an extremely small amount of elements heavier than helium. These conditions generally only existed in the very earlier universe when the first stars, population III stars, were formed. Of course, there is some uncertainty regarding the upper limit. Extremely massive stars, beyond 500-1000 solar masses, could be formed from the merging of two or more stars, though this would be an extremely rare event since stars would need to be both extremely massive (hundreds of solar masses) and need to be in some kind of decaying orbit or be interacting with a multi-body system in such a way to cause the merger.

 

[Matterwave]

We have not observed any stars >~200 solar masses, so any stars more massive than that are purely theoretical. Stars in the 1000's of solar mass range will be getting quite close to the Eddington limit I think.

 

[Ken G]

They would encounter the Eddington limit, but that's only a limit on the luminosity given the mass, it's not a limit on the mass.

 

[Matterwave]

They would encounter the Eddington limit, but that's only a limit on the luminosity given the mass, it's not a limit on the mass.

Right, but unless you can somehow suppress the luminosity, the Eddington limit says that the high luminosity of these very massive stars will start to blow itself apart. I suppose that doesn't mean they can't form though, just that after they form, they are liable to shed their own mass until they are much smaller.

 

[Ken G]

Actually, to be precise, the Eddington limit limits the luminosity given the mass, it won't make the star fly apart. It just makes the star adjust its internal structure so it doesn't blow itself apart-- it imposes a limit on the luminosity, it does not impose a limit on the mass. What does impose a limit on the maximum mass must have something to do with radiation pressure, that's true, but it's not well known, and it's not the Eddington limit. It should have something to do with how many metals are present, because they add to the opacity, but the Eddington limit does not refer to metals.

 

[Matterwave]

Actually, to be precise, the Eddington limit limits the luminosity given the mass, it won't make the star fly apart. It just makes the star adjust its internal structure so it doesn't blow itself apart-- it imposes a limit on the luminosity, it does not impose a limit on the mass. What does impose a limit on the maximum mass must have something to do with radiation pressure, that's true, but it's not well known, and it's not the Eddington limit. It should have something to do with how many metals are present, because they add to the opacity, but the Eddington limit does not refer to metals.

What kind of internal adjustments can the star make to reduce it's luminosity? I am not aware of any...

 

[Ken G]

The adjustment is what happens in high-mass stars-- the internal temperature is not as high as it would be if the Eddington limit was not being approached. Let me explain.

The luminosity is set by the time it takes to dump the luminous energy content. The luminous energy content is proportional to temperature to the 4th power times radius cubed, and the time it takes to diffuse out (for Thomson opacity, let's say) is proportional to the radius times the optical depth. Put that all together, and the luminosity is proportional to the temperature to the fourth power and radius to the fourth power, divided by mass. Now the temperature and the radius are not independent, and if gas pressure dominates, then the temperature is proportional to the mass divided by the radius, that's called the "virial theorem." So you get that the luminosity is proportional to mass cubed, and the temperature and radius fall out, they don't matter. This isn't exactly right, because we took a constant opacity, but it's not bad as long as gas pressure dominates.

However, gas pressure cannot dominate for the high-mass stars, because if the luminosity continues to scale like mass cubed as mass increases, it will eventually push the star apart as you say. But that doesn't happen, because the star won't push itself apart, it will simply readjust its structure such that the luminosity does not scale like mass cubed any more. This happens when radiation pressure dominates over gas pressure, which changes the virial theorem to something completely different-- it makes it so that temperature to the fourth power (radiative energy density) is proportional to mass squared over radius to the fourth power (that's what energy density will be in the virial theorem). So when radiation pressure dominates, you get temperature is proportional to mass to the one half times radius to the minus 1, not mass over radius. This makes a huge difference-- now the luminosity scales like two less powers of mass, so it is proportional to mass not mass cubed, That's how the star avoids violating the Eddington limit, its internal temperature is not as high for a given mass and radius as it would be if it had to use gas pressure to balance gravity. Or if temperature and mass are regarded as given, then it is the radius that will be less than if gas pressure had to balance gravity. Smaller radius reduces the radiative diffusion rate, which reduces the luminosity and keeps it from violating the Eddington limit.

 

[Calion]

As others have said, the Eddington limit seems to impose a maximum initial mass of around 150 solar masses. That being said, it is possible for more massive stars to form via mergers. The most massive currently known star, R136a1, is thought to be around 265 solar masses and probably formed through stellar mergers in the center of a dense cluster.

There are some theories that the first stars in the universe could have been more massive, with masses between 100 and 1000 solar masses. Since the first stars were virtually free of heavy elements (since they are formed in stars), they would have had lower opacity that could have allowed a higher Eddington limit. However, these stars have not been observed, so their initial masses are unknown.

 

[Ken G]

If mergers can create main-sequence stars of 265 solar masses, you can be sure the Eddington limit does not prevent stars like that from existing in the first place, because the merger would need to fight that limit the same way the initial formation of the star would-- had it really been a mass limit. But it isn't-- as I pointed out above, the Eddington limit does not have anything to do with metal abundances, it is a limit that refers to free electron opacity only. What's more, as you know, stars of much higher mass are expected in the first generation of stars, which had no metals. So whatever limit the presence of metals are imposing on maximum stellar masses, it simply is not the Eddington limit. This is because the Eddington limit is a limit on the luminosity, given the mass-- not a limit on the mass. Indeed, some authors prefer the superior label "Eddington luminosity", because the Eddington limit means that a star of high mass will adjust its structure to bring its luminosity down below that limit on luminosity. Whatever determines the maximum mass should indeed be something that relates to radiation pressure-- but it's not the Eddington limit. We need a different term for it, because the Eddington luminosity is already an important concept for these other reasons.

 

[Matterwave]

The adjustment is what happens in high-mass stars-- the internal temperature is not as high as it would be if the Eddington limit was not being approached. Let me explain.

The luminosity is set by the time it takes to dump the luminous energy content. The luminous energy content is proportional to temperature to the 4th power times radius cubed, and the time it takes to diffuse out (for Thomson opacity, let's say) is proportional to the radius times the optical depth. Put that all together, and the luminosity is proportional to the temperature to the fourth power and radius to the fourth power, divided by mass. Now the temperature and the radius are not independent, and if gas pressure dominates, then the temperature is proportional to the mass divided by the radius, that's called the "virial theorem." So you get that the luminosity is proportional to mass cubed, and the temperature and radius fall out, they don't matter. This isn't exactly right, because we took a constant opacity, but it's not bad as long as gas pressure dominates.

However, gas pressure cannot dominate for the high-mass stars, because if the luminosity continues to scale like mass cubed as mass increases, it will eventually push the star apart as you say. But that doesn't happen, because the star won't push itself apart, it will simply readjust its structure such that the luminosity does not scale like mass cubed any more. This happens when radiation pressure dominates over gas pressure, which changes the virial theorem to something completely different-- it makes it so that temperature to the fourth power (radiative energy density) is proportional to mass squared over radius to the fourth power (that's what energy density will be in the virial theorem). So when radiation pressure dominates, you get temperature is proportional to mass to the one half times radius to the minus 1, not mass over radius. This makes a huge difference-- now the luminosity scales like two less powers of mass, so it is proportional to mass not mass cubed, That's how the star avoids violating the Eddington limit, its internal temperature is not as high for a given mass and radius as it would be if it had to use gas pressure to balance gravity. Or if temperature and mass are regarded as given, then it is the radius that will be less than if gas pressure had to balance gravity. Smaller radius reduces the radiative diffusion rate, which reduces the luminosity and keeps it from violating the Eddington limit.

It's been too long since I have done the calculation to find $L\propto M^3$ so I can't even remember if the derivation depended on gas pressure or not, so I'll have to take your word for it. But IIRC the relation should be more like $L\propto M^{3.8}$? :)

 

[Ken G]

It does require gas pressure, so it doesn't hold near the Eddington limit. You're right that the empirical relation when we stay away from the Eddington limit is more like M to the 3.5 than M to the 3, but that's because the full empirical relation extends to low-mass stars that start to get different physics, like extra opacity as the stars get cooler. It's best to keep it as simple as possible if our goal is conceptual understanding-- we have the simulations if we really want to do it accurately.