Chapter 6 How do you weigh a black hole?

  The Sun, the planets that orbit around it, together with dwarf planets (of which Pluto is the most famous example), asteroids,and comets collectively comprise the Solar System. The Solar System itself orbits within the disc of our Galaxy around its centre of mass at the Galactic Centre. The speed atwhich our Solar System travels around its circular path through the Galactic disc is about 7 km/s, and to complete an entire circuit around the Galactic Centre will take a couple of hundred million years. In addition to this orbital motion, the whole Solar System moves perpendicular to the Galactic plane. The kind ofmotion it exhibits is well known to physicists as simple harmonic motion with the restoring force, which pulls our Solar System back towards the equilibrium position of the plane of the Galaxy, coming from the gravitational pull of the stars and gas that comprise the Galactic disc. At the moment, we are about 45 light-years above this equilibrium point. In about 21 million years from now the Solar System will be at its extreme point 320 light-years above the Galactic plane. 43 million years after that, the Solar System will be back in the mid-plane of the Galaxy.When the Solar System lies in the centre of the Galactic plane then, the Earth will suffer maximum exposure to the cosmic rays that are whizzing around in the plane of the Galaxy, trapped along lines of magnetic field,and travelling around them on some kind of a cross between a helter-skelter and a tramline. There have been speculations that the Sun's motion through the Galactic plane could have been responsible for the mass exinction of dinosaurs. But this kind of speculation is hard to verify or refute because the timescales for this orbital motion are of course rather tricky for human observers, who don't tend to live longer than one century. This is a common problem in observational astronomy when we want to follow some process that changes on timescales much longer than the few centuries over which we've been making astronomical observations of any reasonable accuracy and thoroughness.

  There are, however, orbital motions within the Galaxy that are significantly easier to measure, at least in the sense that the relevant timescales are commensurate with the attention spans of humans and their telescopes. Of particular interest in the context of black holes are the orbital motions of the stars in the innermost regions of theMilkyWay, that appears in a part of the sky known as SagittariusA*. Looking into this region, most easily seen from the southern hemisphere, one is looking towards the very centre of our own Galaxy, 27,000 light-years away from us. This is a particularly densely populated region of space, which leads us to two problems when we want to study the Galactic Centre. The first is that there is a relatively high space density of stars and the second is that there is lots of dust.

  The first problem means you need to use a measurement technique that enables high resolution imaging, i.e. fine details can be separated from one another in the way that a telephoto lens gives finer detail on a given camera than a wide-angle lens does.Just using a larger telescope is invariably insufficient for this,but there are various techniques developed for untangling the turbulence in Earth's atmosphere through which we inevitably view all celestial objects, unless we put the telescope on a satellite above the atmosphere. Of particular importance is a technique known as adaptive optics. This technique corrects for atmospheric variations by observing the blurring of a bright star (called a guide star) and deforming the primary mirror of the telescope to cancel out this varying blurring. When a bright star isn't available in the part of sky that is of interest, a powerful collimated laser beam can be shone up to excite atoms in the atmosphere and the atmospheric corrections derived from that.

  The second issue, the presence of vast quantities of dust towards the Galactic Centre, is problematic because it is hard to see optical light through dust, just as it is hard for ultra-violet light from the Sun to penetrate through the opacity of a sunhat. The solution to this problem is that one needs to observe at infra-red wavelengths rather than visible wavelengths.

  How to measure the mass of the black hole at the Galactic Centre

  Such infra-red observations have been championed by two groups,one led by Andrea Ghez in California and one led by Reinhard Genzel in Germany. The work of both teams independently provides a wonderfully clear measurement of the mass at the centre of the Galaxy. Figure 14 shows the data from Andrea Ghez and her team. Over the last few years they have made repeated observations right into the very heart of the Galactic Centre and watched how the stars have moved since the last time they observed them. Because the spectral types of these stars are known, their masses are known. Year by year, as the orbital path of each of these stars becomes apparent, the dynamical equations(known as Kepler's laws, the same laws that govern the motion of the planets around our Sun) enable Ghez and her team to solve for each orbit independently and deduce the mass of the `dark' region that is at the common focus of all these orbits. These independent solutions determine the mass of this dark region rather well. It is now known to be just over 4 million times the mass of our Sun,within a region whose radius is no more than 6 light-hours.Because the object is dark but extraordinarily massive, the only conclusion is that there is a mammoth black hole at the centre of our Galaxy.

  

  14. Figure showing the successive positions of stars that orbit around the central black hole in our Milky Way.

  There is no reason to believe that our Galaxy, theMilkyWay, is unique in having a black hole at its centre. On the contrary, it is strongly suspected that all galaxies may well have a black hole at their centres, at least the more massive ones. The reason for this is because of a seemingly fundamental relationship, discovered by John Magorrian, then at the University of Durham, and co-workers, between the mass of a black hole at the centre of a galaxy and the mass of the galaxy itself. Of course the business of measuring the mass of a black hole and the mass of a galaxy is tricky. The technique that works so beautifully at the centre of our Galaxy cannot be applied to external galaxies because they are simply too far away.

  The masses of the central black holes at the hearts of elliptical galaxies exceed a million times the mass of our Sun and indeed extend up to and beyond a billion times the mass of our Sun. For this reason, they are often termed supermassive black holes.

  Despite the difficulties in measuring the masses of black holes and the masses of galaxies, it has been found for a wide range of different galaxies that the mass of the central black hole scales with the mass of its host galaxy. This is thought to suggest that both the central black hole and the galaxy itself grew and evolved together across cosmic time.

  Many black holes throughout the Galactic disc

  Besides the single, central supermassive black hole at the heart of a galaxy, there are thought to bemillions of black holes distributed throughout the extent of each galaxy, and these are believed to have formed in a very different way from the galactic-central ones which grow by gradual accretion of infalling matter. These stellar mass black holes are formerly massive stars, once shining very brightly, with fusion powering away inside them keeping them very hot and pressurized, and crucially able to resist gravitational collapse.When their nuclear fuel is all used up, there is no longer any radiation pressure to hold up the star, and nothing to balance the inward force of gravity. For a star with a similar mass as our Sun, the collapse under gravity ultimately results in a compact object known as a white dwarf. The word compact has special meaning in astrophysics and connotes that the matter is dense in a way that is utterly distinct from normal matter. By the standards of normal matter, white dwarfs are compact because the matter has been extremely compressed. This matter is ionized, meaning that all the electrons are separate from their parent nuclei, yet cold (normally matter is only ionized at high temperature). The pressure that withstands the persistent inward gravitational pull arises from the electrons refusing to be compressed into too confined a region (a consequence of the Heisenberg uncertainty principle); the technical name for this effect is `electron degeneracy pressure'. Had the collapsing star,when it had used up all its fuel, been more massive, then the gravitational infall would have been greater still and the electrons and their counterpart protons would have fused together to form neutrons. These can form a much more compact object than a white dwarf-a neutron star.

  But, if we are interested in black holes, then we must turn to stars which are considerably more massive than those which go on to produce white dwarfs or even neutron stars. A star above thismass will be very luminous while its fuel lasts and nuclear fusion can be sustained. Once all the fuel is used up, it's game over for the star and the lights will switch off. The star is now sufficiently massive that the gravitational force can overwhelm even the strong neutron degeneracy pressure and so the collapse is so powerful that even this pressure cannot balance gravity and the collapse leads inexorably to a black hole. The collapse of a massive star is often accompanied by the explosion of a spectacular supernova remnant, leaving a black hole as the only remnant at the original location of the progenitor star. In such explosions many elements,particularly those heavier than iron, are synthesized.

  The first black hole to be securely identified from a determination of the masses of the two stars in a binary star system is called V404 Cyg. Jorge Casares and Phil Charles and their co-workers observed the orbits of the two stars very carefully and inferred from their analysis that this binary pair includes a compact object having a mass at least six times greater than the mass of our Sun,and is thus a black hole. (Its mass was later found to be twelve times the mass of the Sun.)

  It is possible to make plausible estimates of the numbers of stars in galaxies and their masses.We can then estimate the number of『stellar-mass' black holes in our Galaxy by considering how many massive stars would have formed early enough in its history to have evolved sufficiently by now to use up all their nuclear fuel via fusion. Even if only a very small proportion of stars in our Galaxy go on to form black holes, with more than 1011 objects in theMilky Way that still gives us a lot of black holes.

  How can one measure the masses of these black holes that pervade galaxies? In fact for some stellar-remnant black holes, the technique is dynamically very similar to that used for the black hole at the centre of our Galaxy. The reason for this is that a very significant fraction of stars in our Galaxy, and therefore most probably in other galaxies also, come in pairs that formed binary star systems. It is easy to surmise how this might come about:gravitational forces are attractive and many two-body orbits are stable, so once two stars encounter one another and become gravitationally bound together, they are likely to remain so. For a binary system, if we can measure the time taken for the stars to do a complete loop around one another, a time known as the orbital period, and if we know the distance between them, then we are well on the way to finding their masses. If the compact object is in orbit around a normal (fusion fuelled) star of known spectral type and therefore known mass, then the mass of the compact star is straightforward to derive. If a compact object such as a black hole is a singleton and not in a binary, then the lack of dynamical informationmeans that there is no means of inferring its mass and or indeed of determining that it is a black hole. The smallest black hole that we can measure is a few times the mass of our Sun,but the heaviest stellar-mass black holes can exceed a hundred times the mass of our Sun.

  The measurement of the mass of a black hole, given modern day technology, is tractable although it still requires a good measure of patience and tenacity. Given thatmass is one of essentially only two fundamental physical properties of a black hole such studies get us half-way to characterizing it! However, measuring the spin of a black hole is harder, and in Chapter 7 I describe the heroic efforts that are needed to try and do this.