How fast do they eat?

The popular notion of a black hole `sucking in everything' from its surroundings is only correct near the event horizon, and even then, only if the angular momentum of the infalling matter isn't too great. Far away from the black hole, the external gravitational field is identical to that of any other spherical body having the same mass. Therefore, a particle can orbit around a black hole in accordance with Newtonian dynamics, just as it would around any other star.What could unravel this pattern of going round and round in circles (or indeed ellipses) and pave the way for more exotic behaviour? The answer is that there is invariably more than one particle orbiting the black hole. The richness of the astrophysical phenomena we observe arises because there is a lot of matter orbiting around a black hole and this matter can interact with itself.What is more, gravity isn't the only law of physics that must be obeyed: so too must the law of conservation of angular momentum. Applying these laws to the bulk quantities of matter that may be attracted towards the black hole gives rise to remarkable observable phenomena, good examples of which are found in the case of exotic objects known as quasars. Quasars are objects at the centres of galaxies having a supermassive black hole at their very heart which, because of its effect on nearby matter,can cause it to outshine the collective light from all the stars in one of those galaxies, across all parts of the electromagnetic spectrum.We shall meet quasars, and other examples of `active galaxies', in Chapter 8, together with scaled-down counterparts of these called microquasars whose black holes are orders of magnitude less massive than those inside quasars. For now let's get back to thinking about the matter around a black hole.

As we have noted, you cannot directly observe an isolated black hole because it simply won't emit light; you can only detect a black hole by its interactions with other material. Any matter falling towards a black hole gains kinetic energy and by turbulence, that is to say swirling against other infallingmatter doing a similar thing, becomes hot. This heating ionizes the atoms leading to the emission of electromagnetic radiation. Thus, it is the interaction of the black hole on the nearby matter that leads to radiation being emitted from the vicinity of the black hole, rather than direct radiation from the black hole itself.

Black holes are not aloof, non-interacting entities in space. Their gravitational fields attract all matter, whether nearby gas or stars,towards them. Because gravitational attraction increases strongly with proximity, stars are ripped apart if they are unfortunate enough to have a close encounter with a black hole; an example is pictured in Figure 15. A certain fraction of the attracted matter will be entirely swallowed or accreted by the black hole. Matter doesn't just accelerate into the black hole whooshing through the event horizon. Rather, there is something of an elaborate courtship ritual as the gravitationally-attractedmatter draws near the black hole. Very often it is found that a particular geometry characterizes accreting matter: that of a disc. If the gravitational field were spherically symmetric, the black hole would play no role in determining the plane within which the gas would settle to form an accretion disc-the disc plane would be determined by the nature of the gas flow far from the black hole. If, however, the black hole has spin, accreted gas will eventually settle into the plane perpendicular to its spin axis, regardless of how it flows at large radii. If there is any rotation at all in the attracted matter,then this must be thought of in terms of the conservation of angular momentum that we met in Chapter 3 when we considered the rotation of material that ultimately collapsed to form a spinning black hole. The rotation means that the matter will be following (fairly circular but actually) spiralling-in orbits as it loses energy. Close to the black hole, the Lense-Thirring effect that we met in Chapter 3 means that at small radii the accretion disc may become aligned with the equatorial plane of the spinning black hole. (In this context, this effect is known as the Bardeen-Petterson effect.)

15. Artist's impression of an accretion disc (fromwhich a jet is shownto emanate-see Chapter 8) and a donor star which is being rippedapart by the gravitational tidal forces from the black hole which is atthe centre of the accretion disc.

If gas is a significant component of the collapsing matter then gas atoms can collide with other gas particles on their own orbits and these collisions result in electrons in those atoms being excited to higher energy states. When these electrons fall back to lower energy states they release photons whose energies are precisely the difference between the higher energy level of the electron and the lower energy level it has fallen to. The release of photons means that radiative energy leaves the collapsing gas cloud and so this loses energy.While energy is released in these processes, bulk angular momentum is not. Because angular momentum remains in the system, the coalescingmatter continues to rotate in whatever plane conserves the direction of the original net angular momentum.Thus, the attractedmatter will invariably form an accretion disc:a rather long-lived holding pattern formaterial orbiting the black hole. Depending on just how close to the black hole the orbiting material can get, the matter can get so hot that the radiation emitted from the accretion disc actually comprises X-ray photons,corresponding to high temperatures of ten million degrees (it doesn'tmatter too much whether the Kelvin or Celsius temperature scale is being used when the temperatures are quite this hot!).

A simple analysis of some familiar equations fromNewtonian physics shows that the gravitational energy release for a given amount of infalling mass depends on the ratio of its mass multiplied by that of the black hole it is spiralling towards, and how close to the black hole the infalling mass gets. For a given mass of attractor such as a black hole, the closer the infalling mass approaches it, the greater the gravitational potential energy released as can be seen in the cartoon in Figure 16. The energy that is available to be radiated out is the difference between the energy the infalling mass has far away before it is accelerated(calculated using Einstein's famous formula E = mc2, where E is energy, mis mass, and c is the speed of light) and the energy it has at the innermost stable circular orbit of the black hole.

Although fusion holds great hope as a future source of energy for Earth, it can only yield at most 0.7% of the available `E = mc2' energy. In contrast, significantlymore of the available rest mass can be released as energy from accreting material, via electromagnetic or other radiation. Quite how close to a black hole the accretingmaterial can get depends, as described in Chapter 4,on how fast the black hole is spinning. If the black hole is spinning fast, the holding pattern of the material can be orbiting much closer in, on much smaller orbits. In fact, accretion of mass onto a spinning black hole is the most efficient way known of using mass to get energy. This process is thought to be the mechanism by which quasars are fuelled. Quasars are the sites of the most powerful sustained energy release in the Universe and are discussed further in Chapter 8.

16. Diagram showing how the potential energy of a mass (a testparticle) decreases with decreasing distance to a black hole.

I've already mentioned there is an equivalence between mass and energy and for a Schwarzschild (non-rotating) black hole, an amount of energy equivalent to 6% of its original mass could in principle be liberated, and that Roy Kerr's solutions to the Einstein field equations show that the last stable circular orbit has a much smaller radius fromthe spinning black hole thanwould a non-rotating black hole of the same mass. In principle, vastlymore rotational energy can be extracted from a Kerr black hole, but only if the infalling matter is orbiting in the same sense as the black hole itself. If matter is orbiting in the opposite direction to the way the black hole is spinning, i.e. it is on a retrograde orbit, then not quite 4% of the rest energy could be released as electromagnetic radiation. If, however, the matter infalling towards a maximally spinning black hole were orbiting in the same sense as the black hole were spinning, then in principle a remarkable 42% of the rest energy could be released as radiation, if the matter could lose sufficient angular momentum that it could orbit the black hole as close as the innermost stable prograde circular orbit.

How fast do they eat?

The accretion rate of the black hole at the centre of our Galaxy,in Sagittarius A*, whose discovery we met in Chapter 6, is100-millionth of the mass of the Sun per year. This doesn't sound very much until you realize that this corresponds to an appetite of300 Earth masses per year. To account for the typical, immense luminosities of quasars, matter-infall rates amounting to a few times the mass of our Sun each year are required. To account for the typical luminosities of the smaller-scale microquasars that we shall also meet in Chapter 8, the required matter-infall rates might be one millionth of this value.

Another context in which a similar energy extraction process may be taking place is in gamma-ray bursts, usually referred to as GRBs. These are sudden flashes of intense beams of gamma rays that seem to be associated with violent explosions in distant galaxies. They were first observed by US satellites in the late 1960s and the received signals were initially suspected to be from Soviet nuclear weapons.

Given the ubiquity of matter spiralling into a black hole via a disc,physicistsfind it helpful tomake simple and instructive calculations to get a handle on themagnitudes of some of the important physical quantities: if ' one considers a spherical geometry rather than a disc geometry then some interesting limits emerge. A particularly illustrative example comes from the world of stars, which are much better approximations to spheres of plasma than are accretion discs. Sir Arthur Eddington pointed out that the radiation released by the excited electrons colliding with other ions in the hot gas of a star will exert a radiation pressure on any matter that it subsequently intercepts. Photons can `scatter' (which simplymeans`give energy and momentum to') electrons contained in the hot ionized plasma within the interior of a star. This outward pressure is communicated via electrostatic forces (the electrically-charged analogue of the gravitational force) to the positively charged ions such as the nuclei of hydrogen (also known as protons) and the nuclei of helium and other heavier elements that are present.

In the case of a star, the net radiation heads radially outwards and this resulting outward radiation pressure acts oppositely to the gravitational force that pulls matter inward towards the centre.For the more-or-less spherical geometry of a star, there is a maximum limit to the amount of outward radiation pressure before it overwhelms the inward gravitational pull and the star simply blows itself apart. This maximum radiation pressure is known as the Eddington limit. Higher radiation pressure inevitably follows from higher luminosity of radiation, and the luminosity of an object can be estimated from its brightness if we know the distance to the object. Therefore, with certain simplifying assumptions including approximating an accretion disc to a sphere, the amount of radiation pressure inside an object can be inferred. This simple method is sometimes used to make an indicative estimate of the mass of the black hole: from the observed luminosity of the radiation to emerge from the surrounding plasma, if it is deemed to be at the maximal limiting value of the `Eddington luminosity' (above which higher luminosity would give sufficiently high radiation pressure that it would exceed the gravity from the mass within and hence blow itself apart) then the mass can be estimated.

This Eddington luminosity can be thought of in terms of a maximum rate at which matter can accrete, for suitable assumptions about how efficient the process of accretion is. This gives a quantity called the Eddington rate which (for the assumed efficiency) is a maximal value. There are ways of breaking this particular maximum limit, not the least of which is the rejection of the assumption of spherical symmetry (this is fine for a star but manifestly doesn't apply to the disc-geometry of accretion discs that we need to consider in order to understand how black holes grow).

How to measure the speed of rotation within an accretion disc

Because of advances in astronomical technology it is now possible to measure the speed at which material is orbiting a black hole,at least for examples that are relatively close to Earth. One of the big challenges is that it is difficult to obtain information on a sufficiently fine angular scale. The spatial resolution required needs to be at least one hundred, if not one thousand, times finer than that routinely obtained by optical telescopes. In principle, the route to achieving finer resolution with a telescope would be to observe at shorter wavelengths and to build a larger telescope, in particular to reduce the ratio of the wavelength of observation to the diameter of the telescope being used. Unfortunately, the latter gets hideously expensive very quickly while the former takes the usual visible observing wavelengths into the ultra-violet regime, to which the atmosphere of the Earth is rather opaque. The route to achieving a smaller ratio of observing wavelength to telescope diameter is, counter-intuitively, to observe at radio wavelengths(much longer wavelengths than either visible or ultra-violet) for which the atmosphere and ionosphere are usually transparent, and to take the telescope diameter to be most of the Earth's diameter.

There are a few technical issues about this approach which need a little discussion: it turns out that thanks to some very useful mathematics developed by the French mathematician Jean Baptiste Joseph Fourier, it is possible to recover much of the signal that a full telescope aperture would observe, even if the actual collecting area only exists in a sparse subset of the full aperture that one would ideally prefer. If the signals from discrete antennas(each looking like an individual telescope-see Figure 17 showing the Very Long Baseline Array, known as the VLBA) are correlated together, it is possible to reconstruct images of small regions of the sky that have detail as fine as that which would be obtained if an Earth-sized telescope could have been fully built. Just to give an idea of how fine this resolution is, suppose that I was standing on top of the Empire State Building in New York, and you were in San Francisco.With this amount of resolution you would be able to resolve detail that is separated by the size of my little finger nail.

17. Artist's impression of the Very Long Baseline Array (VLBA) ofantennas that collectively give images with a resolution equal to thatwhich would be obtained by a telescope with an aperture a significantfraction of the diameter of the Earth.

18. The VLBA hasmeasured the distribution of discretemasersorbiting within the accretion disc of the galaxy NGC 4258 (also knownasMessier 106) around its central black hole whose mass is 40 milliontimes the mass of our Sun.

(I am glossing over the fact that the Earth is a sphere so there is no direct line of sight between San Francisco and the Empire State Building, but you get the idea.) This means that with instruments like the VLBA we can see individual features less than a light-month apart in other galaxies.

High resolution across an image in a spatial sense, and high resolution in a spectral sense (meaning that one can discern very precisely what the wavelengths of particular features are in a spectrum) is a very powerful combination.Making use of the Doppler effect, a team led by JimMoran of Harvard University used the VLBA to make observations of the accretion disc surrounding the central black hole of a nearby galaxy known as NGC 4258. They measured the variation in wavelength of a particular spectroscopic signal (a `watermaser' ) across the rotating accretion disc and used this redshifting and blueshifting, as the masing matter moved towards and away from the Earth, to detect the variation in the speed with which matter at a given distance orbits around the black hole. These exquisitely beautiful data confirmthat thematter orbits around the black hole just as Kepler's laws would describe, and these orbits are depicted in Figure 18.

Swirling matter

In the innermost stable orbit of a black hole whose mass is100 million times the mass of our Sun, the angular momentum is over 10,000 times smaller than the angular momentum of matter orbiting in a typical galaxy. It is clear that for matter to be accreted by the black hole, this requires the removal of the vast majority of this angular momentum, and this is accomplished by processes within the accretion disc. The orbits in an accretion disc may be regarded as a good approximation to circular although in fact they are subtly and gradually spiralling in. Kepler's laws say that the matter orbiting on the smaller radii will be moving faster than the matter on slightly larger orbits. This differential rotation allows a black hole to absorb the plasma that comprises the accretion disc:the rapidly rotating inner orbits friction burn against the neighbouring material on orbits with slightly larger radii. This difference in velocity will mean that the matter on slightly larger orbits will, by viscous turbulence effects, be dragged along a little faster and so correspondingly the matter on inner orbits will be slightly slowed. Therefore, because orbital motion has increased further out, angular momentum has been transferred to the outer material from the inner material, heating as it does so.

Overall, angular momentum is conserved, and the inner material can systematically lose angular momentum, making it more likely to be swallowed by the black hole. Note that if a blob of orbiting matter has too much angular momentum, it will stay further away from the centre of mass about which it is orbiting: it would be moving too fast to get any closer.What kind of viscous effects might be relevant to the plasma within an accretion disc?Inter-atomic viscosity can be small in this situation-the gaseous plasma of which the accretion disc is comprised is very far removed from the consistency of treacle. In fact, magnetic fields may be very important in transferring angular momentum out of accreting inflow.Where do the magnetic fields come from? The plasma in an accretion disc is very hot, and so the atoms are partially ionized into electrons and positively charged nucleons.Therefore, there are flows of charged particles and moving charges produce magnetic fields, as described by the equations of James Clerk Maxwell. Once even very weak magnetic fields exist, they can be stretched and amplified by differential rotation and modified by the turbulence of the plasma, up to levels at which they can give the required viscosity. This is the basis of what is known as the magnetorotational instability. The importance of this mechanism in this context was first realized by Steve Balbus and John Hawley in the early 1990s when working at the University of Virginia.

By viscous turbulence and probably other means, plasma can eventually lose angular momentum and orbit at smaller radii closer to the black hole. Once the gaseous plasma reaches the innermost stable orbit, no more friction is needed for it to slip down into the black hole, after which it will never be seen again,but it will have augmented the mass and spin of the black hole.

What do accretion discs look like,and how hot are they?

We have seen that viscous and turbulence effects play a significant role in removing angular momentum from the orbiting material so that it can orbit more closely to the black hole and be swallowed by it. A consequence of the viscous action, however, is that the bulk orbital spiralling motion gets converted into random thermal motion and hence the matter heats up. The greater the random thermal motion of matter, the more heat energy it has and the higher its temperature. As mentioned in Chapter 5, wherever there is heat, there will be thermal electromagnetic radiation.Every body emits thermal radiation, unless it is at absolute zero.

Such heating is what is responsible for the highly luminous radiation we observe from accretion discs. For the accretion discs that surround the supermassive black holes that are at the hearts of quasars, the characteristic size of an accretion disc is a billion kilometres and the bulk of the radiation from these accretion discs is in the optical and the ultra-violet region of the spectrum. For the accretion discs that surround the vastly less massive black holes in the so-called microquasars (that are discussed in Chapter 8),the accretion discs are a million times smaller in extent and the radiation is dominated by X-rays. The more massive a black hole is, the larger the innermost stable circular orbit is and hence the cooler the surrounding accretion disc will be.

The maximum temperature in an accretion disc around a supermassive black hole 100 times the mass of our Sun will be around 1million Kelvin while for a disc around a stellar-mass black hole, it can be up to a factor of 100 higher.

How do you measure how fast a black hole is spinning?

Given you can't actually directly see black holes, you can't see them spinning either. But there are nonetheless two main routes to measuring how fast a black hole is spinning. As discussed in Chapter 4, when black holes spin very fast, it is possible for matter to be in stable orbit around the black hole much closer in than would be possible if they were not spinning. It turns out that matter in these very tight orbits is heated by strong turbulent and viscous effects as it swirls in, and this immense heat can lead to X-rays being emitted, depending on how close to the black hole the matter has swirled in before being swallowed up. General relativity predicts that the shape of the spectral lines is affected by the distance the emittingmatter is from the black hole (arising from the gravitational redshift) in a way that has a characteristic signature. This signature arises from fluorescing iron atoms within this matter and the method of extracting information from X-ray light was pioneered by Andrew Fabian of Cambridge University.

These are challenging measurements to interpret, because of many different factors, such as the inclination of the accretion disc with respect to Earth, and indeed the nature of wind and outflowing matter from the surface of the accretion disc, in the vicinity of (along our line of sight to) its inner rim whose characteristics hold the key to unlock information about the black hole that is otherwise inaccessible. Other methods for measuring the spin of stellar mass black holes involvemeasuring a significant range of the X-ray spectrum and accounting for the different temperatures of the inner regions of the accretion disc (which are hotter) and the regions further out (that become gradually cooler).It is possible to estimate from the shape of the X-ray spectrum the inclination of the disc and from the highest temperature(assuming you know the mass of the black hole, and its distance from Earth) to infer how far from the black hole the innermost material is orbiting. Analogous methods to measure the spin of supermassive black holes at the hearts of quasars are being developed by Christine Done at Durham University. How close that matter is able to orbit (before being swallowed by the black hole) tells you how fast the black hole itself must be spinning.

Black holes are very messy eaters

It transpires that only a fraction (estimated to be 10%, though it can be very significantly higher) of the matter that gets attracted in towards a black hole gets as far as the event horizon and actually gets swallowed. Chapter 8 considers what happens to the matter infalling towards a black hole that doesn't actually get swallowed within the event horizon. From across the accretion disc itself, matter can blow off as a wind; from within the innermost radii of the accretion disc very rapid jets of plasma squirt out at speeds that are really quite close to the speed of light.As Chapter 8 shows, what doesn't get eaten by the black hole gets spun out and spat out rather spectacularly.