Since β alone determines how relativistic the star is, the constant p (or A) simply gives an overall dimensional scaling to the problem. It can be given any desired value by an appropriate choice of the unit for distance. It is β, therefore, whose variation produces nontrivial changes in the structure of the model. The lower limit on β is, as we remarked above, zero. The upper limit comes from the causality requirement, Eq. (10.57), and the observation that Eqs. (10.62) and (10.63) imply

This range spans a spectrum of physically reasonable models from the Newtonian (β ≈ 0) to the very relativistic (surface redshift 0.22).

10.7 Re a l i s t i c s t a r s a n d g ra v i t a t i o n a l co l l a p s e

Buchdahl’s theorem

We have seen in the previous section that there are no uniform-density stars with radii smaller than (9/4)M, because to support them in a static configuration requires pressures larger than infinite! This is in fact true of any stellar model, and is known as Buchdahl’s theorem (Buchdahl 1959). Suppose we manage to construct a star in equilibrium with a radius R = 9M/4, and then give it a (spherically symmetric) inward push. It has no choice but to collapse inwards: it cannot reach a static state again. But during its collapse, the metric outside it is just the Schwarzschild metric. What it leaves, then, is the vacuum Schwarzschild geometry outside. This is the metric of a black hole, and we will study it in detail in the next chapter. First we look at some causes of gravitational collapse.

Formation of ste l lar-mass black holes

Any realistic appraisal of the chances of forming a black hole must begin with an understanding of the way stars evolve. We give a brief summary here. See the bibliography in § 10.8 for books that cover the subject in detail.

An ordinary star like our Sun derives its luminosity from nuclear reactions taking place deep in its core, mainly the conversion of hydrogen to helium. Because a star is always radiating energy, it needs the nuclear reactions to replace that energy in order to remain static. The Sun will burn hydrogen in a fairly steady way for some 1010 years. A more massive star, whose core has to be denser and hotter to support the greater mass, may

remain steady only for a million years, because the nuclear reaction rates are very sensitive to temperature and density. Astronomers have a name for such steady stars: they call them ‘main sequence stars’ because they all fall in a fairly narrow band when we plot their surface temperatures against luminosity: the luminosity and temperature of a normal star are determined mainly by its mass.

However, when the original supply of hydrogen in the core is converted to helium, this energy source turns off, and the core of the star begins to shrink as it gradually radiates its stored energy away. This shrinking compresses and actually heats the core! Interestingly, this means that self-gravitating systems have negative specific heat: as they lose energy, they get hotter. Such systems are thermodynamically unstable, and the result is that every star will eventually either collapse to a black hole or be held up by nonthermal forces, like the quantum-mechanical ones we discuss below.

Eventually the temperature in the shrinking core gets high enough to ignite another reaction, which converts helium into carbon and oxygen, releasing more energy. Because of the temperature sensitivity of the nuclear reactions, the luminosity of the star increases dramatically. In order to cope with this new energy flux, the outer layers of the star have to expand, and the star acquires a kind of ‘core-halo’ structure. Its surface area is typically so large that it cools below the surface temperature of the Sun, despite the immense temperatures inside. Such a star is called a red giant, because its lower surface temperature makes it radiate more energy in the red part of the spectrum. The large luminosity of red giants causes many of them gradually to blow away large fractions of their original material, reducing their total mass. Stars showing such strong stellar winds form spectacular ‘planetary nebulae’, a favorite subject of astronomical photographs.

Eventually the star exhausts its helium as well, and what happens next depends very much on what mass it has left at this point. It may just begin to cool off and contract to form a small-mass white dwarf, supported forever by quantum-mechanical pressure (see below). Or if its core has a higher temperature, it may then go through phases of turning carbon into silicon, and silicon into iron. Eventually, however, every star must run out of energy, since 56Fe is the most stable of all nuclei – any reaction converting iron into something else absorbs energy rather than releasing it. The subsequent evolution of the star depends mainly on four things: the star’s mass, rotation, magnetic field, and chemical composition.

First consider slowly rotating stars, for which rotation is an insignificant factor in their structure. A star of the Sun’s mass or smaller will find itself evolving smoothly to a state in which it is called a white dwarf. This is a star whose pressure comes not from thermal effects but from quantum mechanical ones, which we discuss below. The point about relatively low-mass stars like our Sun is that they don’t have strong enough gravitational fields to overwhelm these quantum effects or to cause rapid contraction earlier on in their history. A higher-mass star will also evolve smoothly through the hydrogen-burning main-sequence phase, but what happens after that is still not completely understood. It is even more complicated if the star is in a close binary system, where it may pass considerable mass to its companion as it evolves off the main sequence and into a red giant. If a star loses enough mass, through a wind or to a companion, then its subsequent evolution may be quiet, like that expected for our Sun. But it seems that not all stars follow this route. At some point

in the nuclear cycle, the quantum-mechanical pressure in the core of a sufficiently massive star can no longer support its weight, and the core collapse. If the star is not too massive (perhaps an initial mass of up to 15–20 M+), then the strong nuclear repulsion forces may be able to stop the collapse when the mean density reaches the density of an atomic nucleus; the infalling matter then ‘bounces’ and is expelled in a spectacular supernova explosion of Type II. (But the physics of this bounce, and even the fraction of collapses that experience it, is not yet – in 2008 – well understood.) The compact core is left behind as a neutron star, which we will study below. If the original star is even more massive than this, then computer simulations suggest that the collapse cannot be reversed, and the result is a black hole, perhaps accompanied by some kind of explosion, maybe a burst of gamma rays (Woosley and Bloom 2006). These so-called ‘stellar-mass’ black holes may have masses anywhere from 5 to 60 M+, depending on the progenitor star. A number of such black holes have been identified in X-ray binary systems in our Galaxy, as we will discuss in the next chapter.

This picture can be substantially altered by rotation and magnetic fields, and this is the subject of much current research. Rotation may induce currents that change the mainsequence evolution by mixing inner and outer layers of the star. In the collapse phase, rotation becomes extremely important if angular momentum is conserved by the collapsing core. But substantial magnetic fields may allow transfer of angular momentum from the core to the rest of the star, permitting a more spherical collapse.

The composition of the star is also a key issue. Most stars formed today belong to what astronomers call Population I, and have relative element abundances similar to that of the Sun: they form from gas clouds that have been mixed with matter containing a whole spectrum of elements that were created by previous generations of stars and in previous supernova explosions. However, the very first generation of stars (perversely called Population III) were composed of pure hydrogen and helium, the only elements created in the Big Bang in any abundance (see Ch. 12). These stars may have been much more massive (hundreds of solar masses), and they may have evolved very rapidly to the point of gravitational collapse, leading perhaps to a population of what astronomers call intermediate-mass black holes, from perhaps 100 M+ upwards to several thousand solar masses (see Miller and Colbert 2004).

Intermediate mass black holes are given this name because their masses lie between those of stellar-mass black holes and those of the supermassive black holes that astronomers have discovered in the centers of most ordinary galaxies. We will discuss these in more detail in the next chapter.

Quantum mechanical pressure

We shall now give an elementary discussion of the forces that support white dwarfs and neutron stars. Consider an electron in a box of volume V. Because of the Heisenberg uncertainty principle, its momentum is uncertain by an amount of the order of

where h is Planck’s constant. If its momentum has magnitude between p and p + dp, it is in a region of momentum space of volume 4π p2 dp. The number of ‘cells’ in this region of volume p is

Since it is impossible to define the momentum of the electron more precisely than p, this is the number of possible momentum states with momentum between p and p + dp in a box of volume V. Now, electrons are Fermi particles, which means that they have the remarkable property that no two of them can occupy exactly the same state. (This is the basic reason for the variety of the periodic table and the solidity and relative impermeability

states, which is then the maximum number of electrons that can have momenta between p and p + dp in a box of volume V.

Now suppose we cool off a gas of electrons as far as possible, which means reducing each electron’s momentum as far as possible. If there is a total of N electrons, then they are as cold as possible when they fill all the momentum states from p = 0 to some upper limit

The number pf is called the Fermi momentum. Notice that it depends only on the number of particles per unit volume, not on their masses.

Each electron has mass m and energy E = (p2 + m2)1/2. Therefore the total energy

The pressure can be found from Eq. (4.22) with Q set to zero, since we are dealing with a closed system: