Frontiers and Controversies in Astrophysics

ASTR 160 - Update 2 - Do Black Holes Spin?

Professor Charles Bailyn: Welcome to the second update lecture of Frontiers and Controversies in Astrophysics. These update lectures are designed to bring the information given in this course forward from 2007 to 2012. This particular lecture has to do with the second part of the course in which we discuss black holes. This happens to be my own area of research and so this particular lecture will be perhaps a little more personal than the others.

It’s really a follow-on to some of the stuff I said in lecture 15 of the original series. When I talked about an observation that some colleagues and I made of a black hole in the constellation of Musca, the fly. That set of observations was made in 1992 which is now 20 years ago. Since then, I and my colleagues and competitors have found about two dozen of these things and so it’s become really quite common place to discover black holes in this particular way.

And so what I thought I would do in this particular lecture is talk about what we are doing now, now that what we were doing 20 years ago has become basically straight forward. And one of the things we are engaged in now is studying the spin of black holes. Black holes you know are basically quite simple objects. Unlike other kinds of stars it doesn’t matter what they are made out of, they can be made out of hydrogen or helium or rocks or feathers or whatever you like. It doesn’t actually make any difference. The only thing that matters to a black hole is what its mass is. Actually that’s not quite true. There are three quantities that can make a difference to how a black hole behaves. Perhaps the most important is the mass. But black holes can also have spin and they can also have electric charge. Now electric charge it turns out doesn’t matter very much in the real world because if you had a charged black hole it would pull oppositely-charged particles out of the interstellar medium and it would quickly become neutralized. Opposites attract, you know, so if you had a positively-charged black hole, it would pull all the negatively-charged electrons to it and then it would quickly become neutral. But we do expect that black holes will spin.

Black holes are made when material collapses down into a very small region, and if that material has even the slightest spin, then as it collapses it will start to spin faster and faster. This is the famous figure skater a phenomenon where the skater starts out spinning very slowly, and as he or she pulls her arms in you spin faster and faster and faster; so as the material falls into the black hole the black hole will spin faster and faster and so we do expect black holes to have significant spin.

This spin matters. It turns out the event horizon, which is the boundary between our universe and the region inside the black hole, becomes much more complicated with a spinning black hole than a stationary black hole. We calculated in the course what the so-called Schwarzschild radius of the black hole is that’s the radius of the so-called event horizon. In a spinning black hole, you have not only the event horizon but also a region called the ergosphere. Inside the ergosphere space and time are being pulled around by the spin of the black hole and this is happening at such a rate that objects cannot stay still they have to be pulled around with the space time. Outside the ergosphere you could in principle have an object if you had a strong enough rocket ship or something actually just sort of sitting there in space. Inside the ergosphere that’s simply not possible to do. And it turns out that there are all sorts of consequences of the existence of this ergosphere.

You can extract the spin energy from the black hole and so you can pull energy out of the black hole, which is otherwise very hard to do–a phenomenon first predicted by Roger Penrose therefore known as the Penrose process. One of the features of accreting–mass accreting black holes is that they often shoot out jets perpendicular to where the accretion comes in. These jets can go at a very substantial fraction of the speed of light, relativistic jets, and it is one of the hypotheses about how those jets are organized is if the black hole is spinning fast enough; it kind of winds up space-time into a sort of nozzle, and stuff shoots out the pole.

The whole phenomenon of worm holes, which the science fiction writers are so fond of, has to do with rotating black holes. That’s the thing that allows you to get in close to the event horizon and then come back out. And one of the features of rotating black holes is that they have, or actually, I should say of any black hole–is that there is a innermost stable circular orbit. You can be in a circular orbit around a black hole only so close, and then if you’re a little closer than that, you can’t stay in a stable orbit. You don’t necessarily have to fall in, but you have to be either going in or coming back out. Where that innermost stable circular orbit, or as we call it in the business, the ISCO, where the ISCO is depends on how fast the thing is spinning. And it’s that fact that allows us to try to measure how fast it’s spinning; you look for where the innermost stable circular orbit is, and I’ll come back to exactly how we do that in a minute.

So, as you may recall, the objects that we are looking at are sometimes referred to as “dynamically confirmed black hole candidates.” That’s kind of a joke; we call them that because they have all the properties of a black holes in terms of how they pull other material around–in terms of the dynamics of the material you can see. And what we’re trying to do is to figure out whether they really behave like the black holes as predicted by general relativity.

These things take the form of X-ray emitting binary stars. One of the stars is a black hole, the other is an ordinary star; material falls from the ordinary star into the black hole. As the material falls into the black hole, it heats up, emits large numbers of X-rays, which we can see. And, if you have a situation where you have strong, rapidly varying X-rays, then the only way that that can really happen is an accretion onto a very small compact object. The material goes into a disc, orbits down onto the object, and that inner part of the disc is what emits the X-rays.

And so then you have to determine what the mass of that compact object is, because compact objects with masses greater than three times the mass of the Sun as far as we know can only be black holes. And that was the kind of work that my colleagues and I were doing 20 years ago. By measuring the radial velocity curve in the same way that we do with the planets, you can determine the mass of the compact object, and in so doing, discover whether it’s a black hole or not. And then, there is some evidence from the X-ray behavior of these things as the stuff. If you look at the stuff falling down into the black hole, that it really does cross an event horizon and kind of disappear from the universe because you don’t see the effect of it hitting a surface. So, that’s where we ended up in the lectures in 2007.

And, here’s just a little cartoon courtesy of my colleague Jerry Orosz from San Diego State University showing the dynamically confirmed black hole candidates that we know about in our galaxy at the moment. This is to scale–here’s the Sun and Mercury, just to give you a sense of scale. And I should say the Sun and Mercury are not black holes, they’re just there for the scale. This one here is the X-ray binary in Musca that I talked about in lecture 15, and as you can see, the black hole systems come in various shapes and sizes. Sometimes the companion stars are big giants, sometimes they’re smaller stars more or less like the Sun. Some of these are pointed toward us, some of these are edge on, and so forth. So, this is just a little pictogram describing the systems we know.

So, the trick is to find this innermost stable circular object–circular orbit, to find the ISCO. Inside the ISCO, no stable orbit is possible, and so this disc of accreting gas in orbit around the black hole has to end at the ISCO. And the position of the ISCO depends on the spin. For a non-spinning black hole, the radius of the ISCO–the distance, the closest you can get to the black hole–is three times the Schwarzschild radius. Three times the distance, the size of the event horizon. But, if you have a black hole spinning as fast as it can, then you can have an innermost stable orbit all the way down to one Schwarzschild radius, provided the material is orbiting in the same direction as the black hole. If it’s going the other direction, the ISCO is much bigger, because then it’s trying to fight against the spinning space-time. And so, the position of the inner edge of this accretion disc depends on the spin, in the sense that the further in the disc can go, the more the black hole has to be spinning. It’s also true that the further in it goes, the hotter the material in the disc becomes, because it’s getting heated up by the friction of its orbit; and so as it goes in, it’s getting hotter and hotter and hotter.

And so, here’s a little cartoon of what’s going on. Here’s a non-spinning black hole, you can just barely see where the event horizon is supposed to be. And then there’s an inner edge to this accretion disc here, and you expect the overall light from the accretion disc to be dimmer, cooler, and have less energetic protons emitted, as compared to a spinning black hole where the disc can go all the way down to the event horizon, and therefore, the material can be hotter and brighter, and you would see more energetic photons.

So, here’s the same thing in graphical form. What I’m plotting here is the radius of the innermost stable circular orbit, in units of the gravitational constant times the mass of the black hole over c^2. Two, in these units, is where the Schwarzschild radius is; and this is just plotting against how bright that part of the accretion disc is. For a non-spinning black hole, A is the parameter we use to say how fast it’s spinning. You can see that the end of the disc ends up at six of these units–that is to say, three Schwarzschild radii. But, if it’s spinning close to the maximum–the maximum this parameter can get to be is one–then, it goes all the way down into the middle and it’s much, much brighter. You can also plot the temperature; it gets much, much hotter, and so, you can see that if you observe this disc in its nice gentle decay there, versus this disc, this one will be much, much brighter. And so, the idea is, we go and look at the discs of these things, see how bright they are, and thus, figure out how fast the black hole must be spinning.

So, the first thing you need in order to do this–you need excellent X-ray observations of the disc, so that you can actually measure how bright the radiation coming off of it is. You need to know the size of the system, because a non-spinning black hole that’s big will emit a lot of radiation, whereas a spinning black hole that’s small might emit the same radiation; so you need to know how big it is. And in order to figure out how big it is, you need to know what the mass is because that tells you what the orb–you know the orbital period. If you know the orbital period, and the mass, then you know how far apart the objects are from each other, and you know the size of the accretion disc. You also need to remember you’re measuring all this is units of GM/c^2. G and c^2 are constants of nature, but M is not. And so, in order to understand the scale of the system you’re looking at, you have to have an accurate measurement of the mass.

And so, one of the contributions that I’ve been trying to make with my students here at Yale, is making accurate determinations of the mass of these black holes so that we can do this measurement of the spin. So, determining the mass, as I said, comes from looking at the radial velocity curve, just like we looked at the radial velocity curves of stars with planets going around them. But just from the radial velocity curve, you don’t measure the mass; you measure a quantity called the mass function, which is a lower limit on the mass, not the mass itself. You’ll remember that the whole idea was that the mass needed to be more than three solar masses to prove that it’s a black hole. So, if all you want to do is prove it’s a black hole, it’s fine to just have a lower limit. If the lower limit is four solar masses, then it has to be bigger than that, or you’re happy that you proved it’s a black hole. But if you really want a number, you have to do something more.

So, this is just a review of the mass function. I think I mentioned this is one of the previous lectures. It’s a quantity that’s defined by the orbital period and the amplitude of the radial velocity curve. Here’s the data we acquired on this thing in Musca in 1992. Here’s the point which was taken during the earthquake, and you can see that you can measure the amplitude, and you can measure the orbital period, and so, then you know this quantity, which is strictly less than the mass of the black hole. It has units of the mass, and it’s less than the black hole. And I’ve just put this up to show you how much the quality of the data have improved over the last decade and a half. This is a more recent measurement; you can see how really well these points line up with the sine curve here.

Now, you can demonstrate that this quantity is equal to the mass of the black hole, times a bunch of stuff that is less than one. And that’s why mass of the black hole times a fraction is less than the mass of the black hole. That’s why there’s this inequality; and the key quantity here is the orbital inclination. That is the angle at which we observe the orbit. If you’re looking at the orbit edge on, that’s 90 degrees. If you’re looking at the orbit face-on, if the thing is going this way, then that’s zero degrees. And so, there’s this important term here having to do with how tipped the orbit is relative to your line of sight. There’s also the question of the ratio of the masses, but the mass of the other star divided by the mass of the black hole–typically, the mass of the black hole is much bigger, and so this quantity is close to zero, and then it doesn’t matter. And there are ways, in any case, that you can measure it which I won’t go into here, but the orbital inclination is the key thing that you need to measure.

So, as I say, the radial velocity curve provides the mass functions–let me give you a homework problem for those of you who like homework problems. If you can derive that equation for the mass function, just by starting with Kepler’s laws and conservation of momentum. It took me two and a half pages of algebra to do it. What I’m going to do is flash those two and a half pages of algebra up on the video screen here fast enough so that in real time, you can’t read it. But then, if you go back on the video and stop, you can see it if you want to see what the answer is. And so, here’s my version of deriving that equation: first, there’s these points, then there’s this. And so, if you don’t care about that kind of thing, just ignore it. In any case, the radial velocity curve provides the mass function; to get the mass, we need this orbital inclination, the angle of the orbit. And this comes from the changes of perspective that happen when you observe the companion star during the orbit; these are called ellipsoidal variations. So, I have an ellipsoid here; this is a small souvenir Yale football. And this is more or less the shape of the star. The stars in these systems aren’t circular because of the tidal forces exerted by the presence of the black hole; it stretches the star.

So, now imagine that this ellipsoid is going around in a circle, orbiting the black hole and sort of going around like this. When it’s side onto you, you see more of the star than when it’s end onto you. And so, therefore, you expect to see more light when you’re looking at it this way than when you’re looking at it this way, because you just see more of the star. Then, half an orbit later, when you’re over here, again you see more. And then, when you see the back side, you’ll see less. And so, as this star goes around in its orbit, twice per orbit when you’re looking at the side, you’ll see more light. And twice per orbit, when you’re looking at the end, you’ll see less light.

Now, the interesting thing about this–so, here’s just a representation of this here you’re looking at the side, you’re looking at it end on–but the interesting thing about it is how big this effect is depends on the inclination. Because, if it’s orbiting this way, you see the side the whole time and there’s no change. When it’s orbiting this way, at 90 degrees in inclination, you see the maximum amount of change. And if it’s orbiting in some sort of intermediate phase, the amount of change that you see is going to be determined by what that inclination of the orbit, by what the orientation of the orbit actually is. So, here’s an example: if and when you’re looking at it face-on, and you’re seeing the same side all the time, you see no difference during the course of the orbit. When you’re looking at it edge-on, you see two maxima and two minima in each orbit, and when you’re looking at some intermediate value of the inclination; then, as the thing becomes more and more edge-on, you see a bigger and bigger change.

So, in principle, you could go out and measure the brightness of this thing and figure out–all right, for the amount of brightness difference that you see, which is the best angle of inclination for that star? And then, you could plug it into that equation and figure out the mass of the black hole. So, here’s a typical example: this is data collected by a Yale undergraduate a decade or so ago, and published subsequently. This is different kinds of light; blue light, green, three different kinds of infrared light, and you can see it has this very particular pattern. The line through there is actually a model of the ellipsoidal variation for a particular inclination, in this case 70 degrees, and it agrees really very well.

As I say, this is typical data. What scientists mean by typical data–what they mean is, this is the best data I’ve got. You show the best data you’ve got; you call it–you say, these are your typical results. So, these are typical results; this is actually the best we have. But it’s this kind of data that we acquire in order to figure out exactly what the mass of these things are. And that information is incorporated into this plot of the dynamically confirmed black hole candidate, because you need the exact mass in order to know the size of the system, and in order to know its inclination. You see this kind of grid of an accretion disc is this one pointed at us more, this one is more edge-on, and that’s determined by these ellipsoidal variations, and so we can determine the mass. And, having determined the mass, we can then go on, look at the X-ray data, and try and figure out the spin.

So, one of the other things that’s happened in the past few years, we now have some of these dynamically confirmed black hole candidates that aren’t in our own galaxy, that are in neighboring galaxies in the local group of galaxies, one of which is the large, Magellanic Cloud. As you can see from the southern hemisphere, that’s a sort of small galaxy orbiting our own, and in that is an X-ray source called LMC X-3, which we now know to contain a black hole.

Having discovered that this contained a black hole and having looked at the mass of the black hole–determined the mass of the black hole, my colleague went back into the archival data of X-ray observations of this thing, and figured out how big the inner edge of this–the innermost stable circular orbit, and this is again in units of GM/c^2. The interesting thing here is, it’s always the same answer. In a situation where the brightness of this system–which is measured up here–varies enormously. And where the observations are taken by seven different X-ray telescopes, each with a very different set of characteristics and a very different kind of data that it collects. Nevertheless, you always kind of get the same answer. That’s good, because if the black hole is spinning, it’s going to be spinning at the same rate at any time in the past 20 years, then you would expect the inner edge of the innermost stable circular orbit to always be at the same position.

And so, this kind of data gives us some confidence that the answer is right. It’s also a very interesting answer because it isn’t six, which is what you’d expect from a non-rotating black hole, and it isn’t two, which is what you would expect from a black hole rotating as fast as it could, it’s just a little less than four, and so, it’s an intermediate rotating black hole in this particular case. And so, we’ve now been able to measure half a dozen of these things and figure out how fast they’re rotating. One of the most interesting was a thing with the telephone number GRS1915+105. This particular object is what we call a “microquasar”; it has these very rapid jets being shot out of it, at relativistic speeds. This is a series, I think, of radial observations showing a blob of radio-emitting plasma being ejected from the thing in one direction, another blob being ejected in the other direction. If you figure out how fast those blobs must have been ejected, the answer is like 92% of the speed of light. So, there’s hugely rapid ejection of matter taking place there.

These are the kinds of things we refer to as relativistic jets; and the observations we have are consistent with a very, very rapid spin–almost as fast as a black hole could possibly be spinning. And we like this, because great. One of the hypotheses about how these jets are made is that the spin of the black hole wraps up space-time and ejects material out the kind of funnel that is formed, so we were all happy about that. But, another one of these objects, which also has jets, turns out to have very low spin. So, it’s kind of a puzzle. It didn’t come out exactly the way we were hoping. Furthermore, there’s another method of measuring where the ISCO is that has to do with emission, in particular, from iron atoms in this accretion disc. And, there was a big argument–been a big argument for a while about which method is better. This argument has intensified because within the last couple of years, we’ve actually measured the same object with both methods, and you get different answers. So, this is kind of unfortunate.

At the moment, I believe in the method I just described, but other people believe in another method. And so, it’s kind of one of these big, slightly confusing things. And, I guess that’s the point of this particular lecture. Scientists spend most of their time being confused. I don’t have a good ending to this lecture on spin because we don’t have it right yet. We don’t understand what’s going on; we think we can measure it, but other people measuring it in a different way get a kind of a different answer. And, the answers we get don’t quite line up with everything else that we know. This is characteristic of research in science. If you knew what you were doing, it wouldn’t be research. And so, what has happened in the past five years is, we think we understand this business of measuring the mass of things. We know how to do it, we’ve done it two dozen times, and after you’ve done something two dozen times, the next time you do it – and we still continue to do this where the next time you do it just isn’t as interesting as it was 20 years ago, the very first time it was done. And so, you have to move on to do something else.

Necessarily, the thing that you move on to do something else is confusing and not fully accomplished yet. This is something that confuses people a lot. Because it means that science is always in a state of confusion, but it is–and that’s true–but it is also true that we actually do know something. And so, in the case of the black holes, we’re confused about the spin; but that doesn’t mean we’re confused about the mass, and it doesn’t mean that there’s some question about whether these things really are black holes or not.

If there were still questions about the mass and the nature of the object, we’d still be working on that. But in fact, that’s now more or less settled, so we’ve gone onto something else. This can get you into trouble when there are political issues. We’ve seen this a little bit in the global warming debate, you know? It’s kind of clear that the world is getting hotter, and it’s kind of clear that human activity contributes to that, but then there’s a whole bunch of confusion about the details, and so, people with political motivations can say, look, you guys are all confused! So, why do we have to believe it? And, you could equally well say, you guys are all confused about the spin, so why should we believe these things are black holes anyway?

I like the analogy with crossword puzzles. You know, when you’re doing a crossword puzzle and one corner doesn’t work out, you haven’t figured it out yet, but the rest of it is fine. It might be true that all of the other cross clues that you’ve done are actually wrong and there’s some other solution to each one of those that happens to work out. But, the more cross clues you get, the less likely that is, and you don’t actually spend a lot of time worrying about that because you’re pretty much agreed that the lower left-hand corner is okay. But that stuff somewhere else, out on the edge, that’s still a problem. But the fact that that’s a problem, doesn’t mean that we don’t know anything about anything.

And so, in a way this particular lecture is one of my fables, only this time, I’m telling it about myself, and these fables, as you’ll recall from the main set of lectures, have morals, and here’s the moral of the spin observation: the frontiers of science are always in a state of confusion. If that weren’t the case, they wouldn’t be frontiers. But that doesn’t mean that the central ideas on which this frontier research is based aren’t in any kind of serious doubt. And so, keeping a clear head about what isn’t known and is confusing, and what is known and isn’t confusing, is one of the most difficult things to do, I think, for scientists and non-scientists alike.

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