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How big can a star get? This is a calculation made by one of the original pioneers of modern astronomy, Sir Arthur Eddington. And it’s named after him, the Eddington Limit. Now, astronomers are finding examples of giant black holes early in the Universe, calling into question some of Eddington’s assumptions. Let’s explore this fascinating concept! Why are stars sphere-ish? Why do blackholes not eat everything? Why do pulsating stars pulsate? It all comes down to work done by Eddington at the beginning of the last century, and today we’re going to look back at Eddington’s work and all its applications in modern Astronomy.

Show Notes

• What the Eddington Limit is
• Gravity vs radiation pressure in stars
• Why star growth has an upper limit
• How black holes accrete matter
• Quasars and galaxy-scale feedback
• Evidence for super-Eddington growth
• Why modern observations challenge theory

Transcript

Fraser Cain:

AstronomyCast, Episode 777, The Eddington Limit. Welcome to AstronomyCast, our weekly facts-based journey through the Cosmos, where we help you understand not only what we know, but how we know what we know. I’m Fraser Cain, I’m the publisher of Universe Today.

With me, as always, is Dr. Pamela Gay, a Senior Scientist for the Planetary Science Institute and the Director of Cosmic Quest. Hey Pamela, how are you doing?

Dr. Pamela Gay:

I am doing well. The howling wind is tremendous. If you hear what sounds like ghosts in the background, that is the wind, people.

That is the wind.

Fraser Cain:

Now, as usual, I can’t hear it because we use Zoom to record and the modern technology has made it so, when your dogs are barking, I don’t hear it. And so I’m definitely not going to hear the wind. But yeah, here we are.

Now I’m going to do something kind of mean, but it’s for the best. And that is I’m going to say nice things about you, and you’re just going to have to take it. Which is, you know, I think a lot of people, when they reach out to us about Astronomy Cast and they talk about like our scripts and sort of how we prepare, they don’t realize that there is no script, that this is entirely off of the top of our heads, that I go in and what’s nice is for me, I don’t really even have to prepare at all.

Like I write the intro, which is sort of like my token donation to the cause. And I usually come up with the idea, but, you know, often Pamela has ideas that she wants to express, but then she has to prepare and she doesn’t have to prepare just for whatever is going to be the script. She literally has to prepare for anything that I might ask and I don’t know what I’m going to ask.

So how can she know what I’m going to ask? And so Pamela, you are amazing for being able to gather and prepare so much information. Be ready, sort of on your toes, nimble on your toes to handle whatever comes your way.

And in this case, there’s going to be a lot of information about the Eddington Limit. And I don’t know what I’m going to talk about. You don’t know what I’m going to talk about.

You don’t know what you’re going to talk about. And yet you were always poised and prepared. So amazing job.

Dr. Pamela Gay:

This is a favorite topic. So hopefully I will not disappoint. Great.

Okay.

Fraser Cain:

How big can a star get? This is a calculation made by one of the original pioneers of modern astronomy, Sir Arthur Eddington, and it’s named after him, the Eddington Limit. Now astronomers are finding examples of giant black holes early in the Universe, calling into question some of Eddington’s assumptions.

Let’s explore this fascinating concept. Okay, so the Eddington Limit. What is this calculation that Eddington came up with?

Dr. Pamela Gay:

So back in like circa 1916, that’s when the paper came out. So he’d been working on this. Yeah.

This is fairly modern, but not so modern, because at this point in the history of astronomy, we didn’t actually know how stars worked. Like people had realized they weren’t burning coal, or they hadn’t figured out galaxies existed yet. So yeah, Eddington was trying, like so many other people, to figure out what it was that allowed stars to exist.

And we were at a point in geology and paleontology that we were also realizing, planet is old. And so that meant the star had to be old, that meant stars had to be burning for a long period of time. And they still hadn’t fully figured out all the ins and outs of nuclear fusion.

But Eddington started to propose, okay, so what if we have nuclear something going on in the center of a star?

Fraser Cain:

Not coal.

Dr. Pamela Gay:

Correct.

Fraser Cain:

Correct. As people were perhaps assuming, or would.

Dr. Pamela Gay:

Yeah, it wasn’t any kind of chemical exothermic reaction. They knew that much. It had to be something else.

And we were starting to understand nuclear reactions at this point. And so what he proposed was something was going on, and we were really struggling to figure it out, because electrons were like, no, we won’t allow this. And so they had to figure out electron tunneling and quantum mechanics and stuff like that.

So before we even fully understood the quantum mechanics that would allow the center of a star to do the things it needs to do, Eddington proposed, what if stars are balanced between light pressure pushing outwards and gravity pushing inwards?

Fraser Cain:

And like light pushing?

Dr. Pamela Gay:

Are you mad? It was absolutely amazing. And I mean, it’s more complicated than that.

We have to look at what are the electron pressures involved? What are all the other atomic reactions involved? And we’re still working to figure out the details of stars.

And in coming up with this idea, it was realized, well, shoot, if a star is producing too much light, it’s going to overcome gravity and just blow things apart. If anything is producing too much light, it’s just going to blow things apart. So there is some kind of a limit on how much energy can be presented while at the same time gravity is trying to hold things together.

Fraser Cain:

And this idea, I mean, I think even now, you know, we look at stars, we know they’re in hydrostatic equilibrium and say, well, yeah, it’s the light pressure that is keeping the star from folding in on itself. Explain that idea of just like even the light pressure.

Dr. Pamela Gay:

I love this. And if you ever want to see beautifully, cleanly done maths on this, Chandrasekhar put together a stellar evolution book that was published, I want to say in the 1930s or 40s. And there’s copies of it still floating around.

It’s a little penguin book, Penguin Publishers. And the maths for this is entirely straightforward. It is all algebra.

And so the idea is light has, it doesn’t have mass, but it has energy. And energy in motion has momentum. And so when photons hit things, they transfer momentum.

And so every time you zot something with a photon, it can absorb the photon and it also has a transfer of momentum in the process. And so in the core of a star, we have all of these nuclear reactions taking place. And in the process, photons are being produced.

These photons work their way outwards and they random walk, they’re transferring energy in all directions as they go, which is good because that random walk without any specific direction supports a sphere quite nicely. And because they can escape outwards, you end up with the bulk of the motion on average being outwards balanced against gravity. So in the center, you have this radiation pressure, it goes out, then you have convective zones that are supported through the old fashioned pressure laws and all these different things.

You can just work through the math for each area of the star, figuring out where do these different pressures end up dominating. And this is actually like an undergraduate homework assignment that I still remember both hating doing and really enjoying the fact that I was capable of doing it without getting help.

Fraser Cain:

But that idea, I mean, I think when you think about, say, steam rising or filling up a balloon and you sort of think about the sort of thermodynamic, the movement of the molecules bouncing into each other, that would probably, and people were probably examining this at the time and that was probably their first instinct, well, it’s a giant blob of gas and the gas is hot and here’s how much sort of, you know, entropy is going on. We use that to calculate the star, but no, Eddington said, no, no, it’s the light.

It’s the photons, not just particles bouncing into each other in the way we experience this in steam engines and things like that.

Dr. Pamela Gay:

And what’s wild is it takes all of these things working together. So the light is heating the gas, there’s gas pressure added. So you have light pressure, you have light transferring heat, creating gas pressure.

You have different atomic reactions going on, which are ionizing things and creating an electron pressure. And the classical Eddington didn’t include all the electrons properly. And so we’ve had to modify the equations over the years to better and better represent what’s going on in stars.

It’s super complicated, but Eddington was the first person to really realize what was going on. And this is where having Chandrasekhar coming and being his graduate student made so much sense. Between the two of them, they were able to figure out and Chandrasekhar did surpass his advisor.

And this did lead to a great deal of chaos that we talked about in our episode about Chandrasekhar years ago. But the two of them together were able to explain all the different phases in a star’s life and what keeps stars going the way they’re going.

Fraser Cain:

So, OK, so this is kind of the limit, this hydrostatic equilibrium, but it also sort of defines how quickly a star can accrete mass to grow. Yes. So sort of help me understand this.

Dr. Pamela Gay:

So what you end up with is as stars get more and more massive, it puts more and more pressure on the center of the star, which accelerates the rate at which light is being produced. At a certain point, the light pressure exceeds the gravitational force holding the star together. And so the light pushing outwards starts to push at a rate greater than what gravity can pull inwards.

It’s this balancing of forces that allows a star to start to blow things apart. It’s the slow-mo version of what happens in a supernova. In a supernova, the star’s core stops producing light.

You run out of balance. It collapses violently. In the collapse, you end up with new nuclear reactions going on, releasing energy, and that blasts what’s falling inwards, outwards.

It’s a much slower process or at least a much less violent process that goes on in young stars, in massive stars, as material tries to fall into them. But it’s the exact same physics. And I love the fact that we’re dealing with the same equations, just implemented in different limits for all these different cases.

Fraser Cain:

So then where does this kind of get us in the limits of how big stars can get?

Dr. Pamela Gay:

This is where we keep being like, oh, shoot, our observations don’t match our equations. So we keep finding new ways that stars find to produce things that generate pressure and don’t create pressure. So we think that the limits are somewhere below 200 solar masses.

And I’m going to put it that vaguely because the universe likes to keep going, no, you were wrong. I’m going to make bigger stars. How about this star over here?

Yeah, yeah. And quantum mechanics is incomplete. We know this because particles don’t do what we thought they were supposed to do.

We don’t know the underlying physics to the standard model. We just know the standard model is there may not even be underlying physics, which is super annoying to think about. But at some point below 200 solar masses, you are adding material to a star and it just starts blasting light to the point that it clears the area around it.

Fraser Cain:

And is that sort of separate? Like I sort of I think about the Eddington limit core first, that you’re kind of imagining that you’re you’re adding material to the star. The star is getting hotter both in its core and at its surface.

And like the level of heat is kind of ridiculous. Like say a star like our sun, 15 million at the Kelvin, I think, at the center, while say 5,800 Kelvin at the at the surface. But you take a star like the hottest star, like one of those like 200 times solar masses, and you’re at millions of degrees even on the surface.

I mean, they’re ludicrous. Yeah. And and so if you try to add more material, then this thing is going to you know, it’s going to its age will decrease and it’s going to go through some of its phases of dying almost instantly.

Right. Because it’s just but then I think where you’re getting at next is that not only that, but then you have all of the incredibly intense solar wind that’s coming out, all of the radiation that’s coming off of the star, this is heating up the gas that’s around it. You need cold gas to get a star to form, not hot gas.

But the brightest stars heat up the gas, they make ionized gas. And then that doesn’t want to add to the star. So so in addition to sort of the internal limits of how big a star can get, you also have it sort of interaction with its environment, preventing additional material from falling in.

Dr. Pamela Gay:

And and this is just one of the many super cool things that happens as we we literally live in the realm where we’re looking at the quantum mechanics of how atoms change as a function of temperature, pressure, and everything is balanced between these things. Temperature and pressure defines all these characteristics. And the pressure is coming both from gravity inwards and light outwards.

But some of the energy can go into ionization. And and this starts to lead to really weird things. And and I’m going to take a moment and say variable stars, we have to remember the variable stars.

Because one of the super cool things that Ennikton realized while doing this work is in a star’s atmosphere, you have light going through all these different parts of the star that are at different temperatures, different pressures. And because of this, they have gases at different states. And gases in different states have different optical properties.

And one of the weird ones is helium. So helium one light goes through it. So neutral helium light goes through it.

Helium two, helium with no electrons attached is like I shall not let light pass. And an opacity means that when the photons hit a cloud of fully ionized hydrogen, it is more likely to cause pressure instead of to pass through. It acts like a wall.

So in a star, as it heats up, it hits a point as it’s heating up, heating up, heating up that the helium goes from singly ionized to doubly ionized and it becomes opaque. And it’s like, I’m going to expand instead of getting hotter at this point. And so you have a star’s atmosphere starts expanding instead of heating up the same way.

Now, as gas expands, it cools. So as the star is expanding, it eventually hits the point where it’s like, I’m going to become singly ionized again. And then the light can just pass right back through.

And so now you have a star that doesn’t have the same amount of light pressure because it’s now singly ionized helium and it’s also cooler. And so there’s less light pressure in general. And so it begins to collapse as it collapses.

It heats up and it eventually hits the point where it heats up enough where it starts to expand because it’s heating. But it’s also heating faster than it’s expanding until it hits that doubly ionized helium again. This is the Kappa mechanism, and it’s entirely driven by quantum mechanics playing an extra role with the ionization of helium.

So it’s the little physics like this where things decide instead of cooling, expanding, heating, they’re going to ionize and just do something completely different with all that energy. These are the kinds of things that we keep realizing we’ve left something out of our equations. And this is why stars can be bigger and stuff like that.

Fraser Cain:

You just talked a mini episode about variable stars into this episode.

Dr. Pamela Gay:

I did. I hid variable stars in the episode.

Fraser Cain:

That’s amazing. All right. So now I think we need to kind of pull this all together, which is when you take this law, this limit, and use this to calculate how stars grow in the early on in the early universe.

And then you can kind of apply this to the growth of black holes. You get a certain sort of limit for how big a black hole should be.

Dr. Pamela Gay:

And to be clear, the limit is not the feeding of the black hole. The limit is the accretion disk around the black hole, which acts like a star. And so so you can sneak up on bigger and bigger black holes, but it’s trickstery.

So so the situation that we’re looking at is take black hole. This works for stellar mass black holes. This works for supermassive black holes.

It does not matter what size black hole you have. When the black hole is feeding, because angular momentum is a insert naughty word here, as material attempts to fly in towards that black hole, the angular momentum is like, no, you shall go spiraling around.

Fraser Cain:

Right.

Dr. Pamela Gay:

And so material builds up in a disk of spiraling material that is trying to shed its angular momentum through friction and light and other forces.

Fraser Cain:

And how this works is still a bit of a mystery.

Dr. Pamela Gay:

It’s a good homework equation. Another thing I’ve really enjoyed, but this is a graduate school. Right.

Fraser Cain:

Right. But how this.

Dr. Pamela Gay:

Yeah.

Fraser Cain:

Yeah. I mean, just like how you can get material and how you can get, say, black holes to merge is still a bit, you know, this sort of last part of the momentum is is this is where gravity waves you’re now getting rid of energy through gravity, gravitation instead.

Dr. Pamela Gay:

It’s super cool. Yeah, I love this part of physics. So so you have your black hole.

You have an accretion disk around it. And as the material builds up in the accretion disk, it gets thicker and denser and has super high pressure and temperature and pressure are the two things you need to have that are high in order for nuclear reactions to start occurring.

Fraser Cain:

Right.

Dr. Pamela Gay:

And it’s not identical physics to what’s happening in stars other than like it’s nuclear reactions the same way. But you’re not going to get like the CNO cycle that is is working in the same way as in an accretion disk. It’s slightly different, but also the same physics.

Right. Yeah.

Fraser Cain:

But I guess what your point is, is that this disk around the black hole generates gas is being mushed together and the temperature is increasing and is starting to behave like the interior of a star falls under the Eddington limit. Time to calculate how big these accretion disks can be around various black holes before they’re too hot. They start to blow themselves apart.

The same physics is happening in this situation.

Dr. Pamela Gay:

And with supermassive black holes, the ones in the hearts of galaxies that like to be what we call quasars and active galactic nuclei. But once you start hitting the quasar side of that equation, the black holes can have accretion disks so large that they start generating light pressures that empty out the cores of galaxies.

Fraser Cain:

Right.

Dr. Pamela Gay:

At which point there’s nothing there for the black hole to eat anymore. So it chows down on that accretion disk and then sits there going, I am starving. There is nothing I can do about this.

I have done this to myself.

Fraser Cain:

But so then the math, when you sort of think about it, is like you start at the very beginning. You say, OK, we’ve got the primordial hydrogen and helium. Yeah, we know how big a star can get.

So let’s calculate the bit for the first stars. Great. That tells us the stars.

Let’s say those leave behind black holes. Great. Now we know how big those black holes were.

Now the black holes try to pull in mass.

Dr. Pamela Gay:

And I need to put numbers on this because we are starting to realize that these first stars could have been between a thousand and ten thousand solar masses.

Fraser Cain:

Because now you’re long. I’m trying to follow the standard line here.

Dr. Pamela Gay:

And then obviously it’s not the first stars, the second generation of stars.

Fraser Cain:

But even the first, sure. But even the like, even the first stars, like like you’ve got hot cores, they’re going to, you know, you’re going to reach a limit how big the star can get. Yeah.

You know, maybe it can get bigger because it has less metal that’s poisoning it or whatever. But anyway. So then you get those those first stars die. You get black holes, remnants of black holes, feed the black holes, get accretion discs. You’re limited by how big the black holes can get.

But and so that defines how rapidly this these black holes can add on mass. Their accretion discs get bigger, then they can feed faster. There is a limit.

And astronomers have gone back. They’ve made all these calculations, string them together. They’ve reached the sort of the size of the black hole that you should expect at certain ages of the universe.

And the problem is the ones that we see are too big.

Dr. Pamela Gay:

Yes.

Fraser Cain:

Too massive.

Dr. Pamela Gay:

Yes.

Fraser Cain:

That they broke the rules that at some point they either violated the Eddington limit in terms of stars or they violated the Eddington limit in terms of black holes. But now we’re in a we live in a universe with black holes that at some point took Eddington’s careful calculations, tore them up, stomped on them and said, ha.

Dr. Pamela Gay:

Or they just found another process.

Fraser Cain:

Right. And so now, please, let’s hear. Let’s go through where we think the universe has potentially violated the Eddington limit.

Dr. Pamela Gay:

So and the Eddington limit is is limited to this is what happens when gas is doing the infalling. And it’s because light pressure can push on gas, but light pressure pushing on bigger objects is is going than atoms and gas particles and dust particles is going to have a different kind of effect. So when you start looking at two black holes merging together, the Eddington limit plays a different kind of role.

And also with accretion disks, if you look at the situation of supermassive black hole feeding on the gas and dust around it, feeding on the gas and dust around it empties its area. Now, you merge two galaxies together, you rearrange where all the dust, dust and gas is, and you get to start over. Now, where we start running into problems is we expect all of these things to have time scales.

And the time scales are like, nope, you haven’t had enough time in the universe. And we keep finding this over and over and over. So we need to figure out how to reset the time that it takes for things to happen to be different.

And and this is where like new research just in the past few months, I think time has no meaning. It still has no meaning.

Fraser Cain:

Yeah.

Dr. Pamela Gay:

Is starting to point towards the first generation of stars were far more massive than we had envisioned. And they’re actually starting to come out with, well, if you make it this big, you get this chemical ratio. And we actually see nebulae filled with that chemical ratio exactly as expected at less than a billion years of of the universe being in existence.

Fraser Cain:

Right. And so it might be that if you have that first star, just hydrogen, helium, no metals. Right.

They’re able to get much bigger.

Dr. Pamela Gay:

We know that is true. And it’s true because you don’t have all the additional lines that electrons can go into to change how light is held back. You end up with with a lot more of this helium being it’s opaque, self-allowing stars just get big and hot and stuff like that.

Fraser Cain:

And like you mentioned, some recent research and there has been examples of observations that have been made with X-ray observatories and things like that, where they’re literally watching black holes feed at super Eddington rates.

Dr. Pamela Gay:

And that we’re still trying to figure out what what did we miss? And this is where we’re starting to realize, oh, shoot, you have to include electrons in ways that we didn’t originally. You have to include what is the chemical constituency of the accretion disk in ways that we hadn’t thought of before.

Fraser Cain:

Magnetic fields.

Dr. Pamela Gay:

Yeah. So you have to balance every single force. You have to balance every single quantum reaction and trying to figure out what did we forget.

This is where creativity is a part of science that we don’t acknowledge nearly enough because it’s it’s one thing to go through and do our homework where we’re like, let’s just worry about what hydrogen and helium are doing. They’re the bulk of the universe. And then you start realizing, OK, so to explain stars like our sun, you have to start including heavier atoms.

Otherwise, it doesn’t work at the mass it’s at. OK. And so we’re getting more and more complex in a lot of cases.

But the computer power and also the creativity of the person running all of the maths sometimes means we just don’t think of things or our computers aren’t powerful enough for us to include all those things. Both factors are at play. There are a lot of times where you’re like, shoot, if I run this the way I have it written, it’s going to take four months.

So let’s simplify the equations. And then there’s also the human willingness of, well, I could figure out all of those atomic quantum mechanic electrons bouncing around doing their thing. And I do not want to.

So I’m going to simplify. I’m going to do this in one dimension instead of three dimensions. I’m there’s so many ways that we simplify things because that makes the math doable.

And there’s so many ways that we simplify things because we don’t have computational power yet.

Fraser Cain:

Yeah, but you sort of hold that the most complicated field of science is magneto hydrodynamics, plasma dynamics.

Dr. Pamela Gay:

Yeah, yeah. And that’s exactly what this is.

Fraser Cain:

And this is one of those problems that you have. You have magnetic fields. You have magnetic fields, you have plasma, you have moving fluids, charged particles moving like a fluid in this environment, and it is one of the most complicated things. And then it also brings aspects of general relativity and also brings on a whole bunch of quantum mechanics.

What I love about this, though, is that we, it’s kind of like, you know, when you’re like doing homework assignments, you’ve got some complicated problem that you’re trying to do.

Dr. Pamela Gay:

Yeah.

Fraser Cain:

And you do your math, but you know the answer. You go and you look at the answer key, but it just gives you one number. It just says, 45 meters per second.

And you go back in your calculation, you got 31 meters per second. You’re like, how, where did I go wrong? And then you examine every single part of the calculation to get you to go, like, I know what the answer has to be.

And so I have to sort of revisit all of my assumptions and try and figure this out. The universe has told us what the reality is. The Eddington limit works very well most of the time, and yet we live in this universe that is slightly different.

And so it’s those assumptions somewhere that we’re off track and that, but you, but you know, you’re not just like completely moving in an area where you have no idea where you’re going. So it has structure.

Dr. Pamela Gay:

What’s so amazing about this is when Eddington first did this, when Chandrasekhar expanded on this, when I did it as a homework assignment, we were looking at a line through a star from the center to the surface, looking for all of the places where changes in pressure and temperature changed what physics was dominant. So in the core, you have nuclear reactions. At what point as you move away from the core, does the pressure and temperature hit a limit where you switch from one mode to the next?

It’s a straight line calculation through a star and it’s good enough. As we now look at accretion disks around supermassive black holes, we’re still largely trying to figure out how to do it by taking a cut through that disk, looking both up and down and also center outwards. So now it’s two dimensions.

We’re still simplifying and now because of all the rotations and because of everything else, it’s no longer something you can do with pen and paper. We have gone from 1916 working on a chalkboard to 2020s working on a supercomputer and it’s the exact same physics. We’re just changing where we’re applying it.

Fraser Cain:

Yeah. Well, it’s a fascinating concept and, you know, I think we’re going to see a lot of work and thanks to James Webb and other big observatories, we’re making a lot of progress. So stay tuned for maybe someone coming up with the answer.

Thanks, Pamela.

Dr. Pamela Gay:

Thank you, Fraser. And thank you so much to everyone out there on Patreon who allows us to keep the show going. Rich is able to make us sound good.

Aviva is able to keep the website updated. Everything works because of you. This week, I would like to thank the following $10 and up patrons.

Alex Cohen, Andrew Palestra, Arctic Fox, Boré Andro-Lovesville, Benjamin Davies, Boogie Net, Brian Kilby, Kami Rassian, Cooper, David, Davius Rosetta, Don Mundus, Elliot Walker, Father Prax, Frank Stewart, Gerhard Schweitzer, Gordon Dewis, Hal McKinney, James Signovich, Jean-Baptiste Lemontier, Jim McGean, Joanne Mulvey, John M, JP Sullivan, Katie Byrne, Kimberly Rake, Larry Dzat, Lou Zeeland, Mark Phillips, Matt Rucker, Michael Prashada, Michelle Cullen, Name, Olga, Paul Jarman, Philip Grant, R.J. Basque, Ron Thorson, Sam Brooks and his mom, Scott Bieber, Subhana, Stephen Coffey, The Big Squish Squash, Tiffany Rogers, Tricor, Wanderer M101, and Zach Coquindal. Thank you all so very much.

Fraser Cain:

All right. Thanks, everyone. And we will see you all next week.

Dr. Pamela Gay:

Bye-bye, everyone.

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