White dwarfs

From our derivation we should expect this to be the order of magnitude of the maximum mass supportable by a relativistic electron gas, when most of the gravity comes from a cold nonrelativistic gas of nuclei. This is called the Chandrasekhar limit, and a more precise calculation, based on integrating Eq. (10.82) more carefully, puts it at M ≈ 1.3 M+. Any star more massive than this cannot be supported by electron pressure, and so cannot be a white dwarf. In fact, the upper mass limit is marginally smaller, occuring at central densities of about 1010 kg m−3, and is caused by the instability described next.

Neutron stars

If the material is compressed further than that characteristic of a white dwarf (which has ρ 1010 kg m−3), it happens that the kinetic energy of electrons gets so large that if they combine with a proton to form a neutron, energy can be released, carried away by a neutrino. So compression results in the loss of electrons from the gas which is providing pressure: pressure does not build up rapidly enough, and the star is unstable. There are no stable stars again until central densities reach the region of 1016 kg m−3. By this density, essentially all the electrons are united with the protons to form a gas of almost pure neutrons. These are also Fermi particles, so they obey exactly the same quantum equation of state as we derived for electrons, Eqs. (10.78) and (10.79). The differences between a neutron star and a white dwarf are two: firstly, a much higher density is required to push pf up to the typical momentum of a neutron, which is more massive than an electron; and secondly, the total energy density is now provided by the neutrons themselves: there is no extra gas of ions providing most of the self-gravitation. So, here, the total equation of state at high compression is Eq. (10.79):

Unfortunately, there exist no simple arguments giving an upper mass in this case, since the fully relativistic structure equations (Eqs. 10.30 and 10.39) must be used.

In fact, neutron star matter is the most complex and fascinating state of matter that astronomers have yet discovered. The dense degenerate gas of neutrons appears to be superfluid, despite the very high temperatures (106 K or higher) that we find inside neutron stars. The neutrons are in chemical equilibrium with a much less dense gas of protons and electrons, and the protons may exhibit superconductivity! These properties probably are important for understanding why neutron stars have developed such strong magnetic fields that do not align with their rotation axes, but the connection is not understood. At the center of a neutron stars the density may be so large that the neutrons dissolve into essentially free quarks. It has proved extremely difficult for nuclear theorists to compute an equation of state for nuclear matter under these conditions. The physical conditions are out of reach of laboratory experiment, and the short range and complexity of the nuclear force (the strong interaction) require physicists to make one or another approximation and assumption in order to arrive at an equation of state. There are thus dozens of proposed equations of state, all leading to stars with very different properties from one another (Lattimer and Prakash 2000). Different equations of state predict different relations between the mass and radius of a neutron star, and also vastly different maximum masses for neutron stars, ranging from about 1.5 M+ to perhaps 2.5 M+. If it were possible to measure both the mass and the radius of one neutron star, much of this uncertainty would be resolved. Alternatively, if we had the masses of a large enough sample of neutron stars, enough to give us confidence that we were observing the maximum mass, then that would help as well.

Neutron stars are observed primarily as pulsars (as mentioned in the previous chapter), although some non-pulsating neutron stars are known through their X-ray and gamma-ray emission. Although the connection between stellar evolution and the formation of neutron stars is not fully understood, pulsar studies make it clear that neutron stars are often created in supernova explosions. Astronomers have been able to associate a number of pulsars with supernova remnants, including one of the youngest known pulsars, PSR B0531+21, in the center of the Crab nebula. Studies of the motion of pulsars suggest that they get strong ‘kicks’ when they are born, with typical velocities of 400 to 1000 km s−1. This compares with the orbital speed of the Sun around the center of the Galaxy, which is about 200 km s−1, and the typical random speeds of stars relative to one another, which is some tens of km s−1. The kick must result from some kind of asymmetry in the gravitational collapse and subsequent initial explosion.

From the point of view of understanding neutron star structure, the most interesting pulsars are those in binary systems, where the orbital dynamics allow astronomers to measure or at least to place limits on their masses. Many accurately measured masses are now known, and remarkably they cluster around 1.4 M+ (Lorimer 2008, Stairs 2003); however, some stars seem to have masses as high as 2 M+ or more. Neutron stars in some binary systems become X-ray sources, in the same way as for black holes: gas falls on to them from their companions. This leads to the stars being spun up to very high rotation rates. The fastest pulsar known is PSR J1748-2446ad, which spins at 716 Hz!

Rotation can, in principle, considerably increase the upper limit on stellar masses (Stergioulas 2003), at least until rotation-induced relativistic instabilities set in (Friedman and Schutz 1978, Andersson et al 1999, Kokkotas and Schmidt 1999). Realistically, this probably doesn’t allow more than a factor of 1.5 in mass.

10.8 F u r t h e r re a d i n g

Our construction of the spherical coordinate system is similar to that in most other texts, but it is not particularly systematic, nor is it clear how to generalize the method to other symmetries. Group theory affords a more systematic approach. See Stefani et al. (2003).

It is clear that the computation of Riemann and Einstein tensors of metrics more complicated than the simple static spherical metric will be time-consuming. Modern computer-algebra systems, like Maple or Mathematica, have packages that can do this automatically. The most elegant way to do these computations by hand is the Cartan approach, described in Misner et al. (1973).

A full discussion of spherical stellar structure may be found in Shapiro and Teukolsky (1983). In deriving stellar solutions we demanded continuity of g00 and grr across the surface of the star. A full discussion of the correct ‘junction conditions’ across a surface of discontinuity is in Misner et al. (1973).

There are other exact compressible solutions for stars in the literature. See Stefani et al. (2003).

The evolution of stars through the different stages of nuclear burning is a huge subject, on which astrophysicists have made great progress with the help of computer simulations. But there are still many open questions, particularly concerning the evolution of stars in binary systems, where interaction with the companion becomes an issue. Good references include Hansen et al. (2004) and Tayler (1994).

A more rigorous derivation of the equation of state of a Fermi gas may be found in quantum mechanics texts. See Chandrasekhar (1957) for a full derivation of his limit on white dwarf masses. See also Shapiro and Teukolsky (1983).

The instability which leads to the absence of stars intermediate in central density between white dwarfs and neutron stars is discussed in Harrison et al. (1965), Shapiro and Teukolsky (1983), and Zel’dovich and Novikov (1971). For a review of the fascinating story of pulsars, see Lyne and Smith (1998) or Lorimer and Kramer (2004).

It is believed that the main route to formation of black holes is the collapse of the core of a massive star, which leads to a supernova of Type II. Such supernovae may be sources of gravitational radiation. The more massive collapse events may lead to so-called hypernovae and gamma-ray bursts. A good introduction to contemporary research is in the collection by Höflich et al. (2004). For an up-to-date view, the reader should consult recent proceedings of the Texas Symposium in Relativistic Astrophysics, a series of conferences that takes place every two years. Other types of supernovae arise from white-dwarf collapse and do not lead to neutron stars. We will discuss the mechanism underlying supernovae of Type Ia in Ch. 12 on cosmology.

There are many resources for neutron stars and black holes on the web, including websites about X-ray observations. Readers can find popular-style articles on the Einstein Online website:

http://www.einstein-online.info/en/.