Neutron Stars

If the mass of a star is large enough, the density of matter may grow even larger than in normal white dwarfs. The equation of state of a classical degenerate electron gas then has to be replaced with the corresponding rel-ativistic formula. In this case decreasing the radius of the star no longer helps in resisting the gravitational attraction. Equilibrium is possible only for one particular value of the mass, the Chandrasekhar mass MCh, already introduced in Sect. 11.5. The value of MCh is

about 1.4 M0, which is thus the upper limit to the mass of a white dwarf. If the mass of the star is larger than MCh, gravity overwhelms the pressure and the star will rapidly contract towards higher densities. The final stable state reached after this collapse will be a neutron star (Fig. 14.2). On the other hand, if the mass is smaller than MCh, the pressure dominates. The star will then expand until the density is small enough to allow an equilibrium state with a less relativistic equation of state.

When a massive star reaches the end of its evolution and explodes as a supernova, the simultaneous collapse of its core will not necessarily stop at the density of a white dwarf. If the mass of the collapsing core is larger than the Chandrasekhar mass (> 1.4 Me), the collapse continues to a neutron star.

An important particle reaction during the final stages of stellar evolution is the URCA process, which was put forward by Schonberg and Gamow in the 1940's and which produces a large neutrino emission without otherwise affecting the composition of matter. (The URCA process was invented in Rio de Janeiro and named after a local casino. Apparently money disappeared at URCA just as energy disappeared from stellar interiors in the form of neutrinos. It is claimed that the casino was closed by the authorities when this similarity became known.) The URCA process consists of the reactions

(Z, A) + e-^ (Z - 1, A) + Ve, (Z - 1, A) ^ (Z, A) + e- + ve, where Z is the number of protons in a nucleus; A the mass number; e- an electron; and ve and ve the electron neutrino and antineutrino. When the electron gas is degenerate, the latter reaction is suppressed by the Pauli exclusion principle. In consequence the protons in the nuclei are transformed into neutrons. As the number of neutrons in the nuclei grows, their binding energies decrease. At densities of about 4 x 1014 kg/m3 the neutrons begin to leak out of the nucleus, and at 1017 kg/m3 the nuclei disappear altogether. Matter then consists of a neutron "porridge", mixed with about 0.5% electrons and protons.

Neutron stars are supported against gravity by the pressure of the degenerate neutron gas, just as white dwarfs are supported by electron pressure. The equation of state is the same, except that the electron mass is replaced by the neutron mass, and that the mean molecular weight is defined with respect to the number of free neutrons. Since the gas consists almost entirely of neutrons, the mean molecular weight is approximately one.

The typical diameters of neutron stars are about 10 km. Unlike ordinary stars they have a well-defined solid surface. The atmosphere above it is a few centimetres thick. The upper crust is a metallic solid with the density growing rapidly inwards. Most of the star is a neutron superfluid, and in the centre, where the density exceeds 1018 kg/m3, there may be a solid nucleus of heavier particles (hyperons), or of quark matter, where the quarks that normally constitute neutrons have become unconfined.

A neutron star formed in the explosion and collapse of a supernova will initially rotate rapidly, because its angular momentum is unchanged while its radius is much smaller than before. In a few hours the star will settle in a flattened equilibrium, rotating several hundred times per second. The initial magnetic field of the neutron star will also be compressed in the collapse, so that there will be a strong field coupling the star

\

Solid \\

^ Neutrons as

matter U

superfluid

1017 1018 kg/m3

Hyperons 1

Core

Mantle

Crust

Surface 107 kg/m3

Core

Mantle

Crust

Fig. 14.2. The structure of a neutron star. The crust is rigid solid material and the mantle a freely streaming superfluid

to the surrounding material. The angular momentum of the neutron star is steadily decreased by the emission of electromagnetic radiation, neutrinos, cosmic ray particles and possibly gravitational radiation. Thus the angular velocity decreases. The rotation can also break the star into several separate objects. They will eventually recombine when the energy of the system is reduced. In some cases the stars can remain separated, resulting e. g. in a binary neutron star.

The theory of neutron stars was developed in the 1930's, but the first observations were not made until the 1960's. At that time the pulsars, anew type of rapidly pulsating radio sources, were discovered and identified as neutron stars. In the 1970's neutron stars were also seen as members of X-ray binaries, such as X-ray pulsars and X-ray bursters (Fig. 14.9) (see Sect. 14.4).

Pulsars. The pulsars were discovered in 1967, when Anthony Hewish and Jocelyn Bell in Cambridge, England, detected sharp, regular radio pulses coming from the sky. Since then about 1500 pulsars have been discovered (Fig. 14.4). Their periods range from 0.0016 s (for the pulsar 1937 + 214) up to 20 minutes.

In addition to the steady slowing down of the rotation, sometimes small sudden jumps in the period are observed. These might be a sign of rapid mass movements in the neutron star crust ("starquakes") or in its surroundings.

The origin of the radio pulses can be understood if the magnetic field is tilted at an angle of 45° — 90c with respect to the rotation axis. There will then be a magnetosphere around the star, where the particles are tied to the magnetic field and rotate with it (Fig. 14.3). At a certain distance from the star, the speed of rotation approaches the speed of light. The rapidly moving particles will there emit radiation in a narrow cone in their direction of motion. As the star rotates, this cone sweeps around like a lighthouse beam, and is seen as rapid pulses. At the same time relativistic particles stream out from the neutron star.

The best-known pulsar is located in the Crab nebula (Fig. 14.5). This small nebula in the constellation Taurus was noted by the French astronomer Charles Messier in the middle of the 18 th century and became the first object in the Messier catalogue, M1. The Crab nebula was found to be a strong radio source in 1948 and an X-ray source in 1964. The pulsar was discovered in

Fig. 14.3. The magnetic field around a rotating neutron star carries plasma with it. At a certain distance the speed of the plasma approaches the speed of light. At this distance the radiating regions S emit radiation in a narrow forward beam. Radiation from the point P hits the observer located in the direction of the arrow. (Drawing from Smith, F.G. (1977): Pulsars (Cambridge University Press, Cambridge) p. 189)

Fig. 14.3. The magnetic field around a rotating neutron star carries plasma with it. At a certain distance the speed of the plasma approaches the speed of light. At this distance the radiating regions S emit radiation in a narrow forward beam. Radiation from the point P hits the observer located in the direction of the arrow. (Drawing from Smith, F.G. (1977): Pulsars (Cambridge University Press, Cambridge) p. 189)

Fig. 14.4. Consecutive radio pulses at 408 MHz from two pulsars. To the left PSR 1642-03 and to the right PSR 1133+16. Observations made at Jodrell Bank. (Picture from Smith, F.G. (1977): Pulsars (Cambridge University Press, Cambridge) pp. 93, 95)

Fig. 14.4. Consecutive radio pulses at 408 MHz from two pulsars. To the left PSR 1642-03 and to the right PSR 1133+16. Observations made at Jodrell Bank. (Picture from Smith, F.G. (1977): Pulsars (Cambridge University Press, Cambridge) pp. 93, 95)

Fig. 14.5. A time-sequence of the pulsation of the Crab pulsar in visible light. The pictures were taken once about every millisecond; the period of the pulsar is about 33 milliseconds. (Photos N.A. Sharp/NOAO/AURA/NSF)

1968. In the following year it was observed optically and was also found to be an X-ray emitter.

Neutron stars are difficult to study optically, since their luminosity in the visible region is very small (typically about 10-6 LQ). For instance the Vela pulsar has been observed at a visual magnitude of about 25. In the radio region, it is a very strong pulsating source.

A few pulsars have been discovered in binary systems; the first one, PSR1913+16, in 1974. In 1993 Joseph Taylor and Russell Hulse were awarded the Nobel prize for the detection and studies of this pulsar. The pulsar orbits about a companion, presumably another neutron star, with the orbital eccentricity 0.6 and the period 8 hours. The observed period of the pulses is altered by the Doppler effect, and this allows one to determine the velocity curve of the pulsar. These observations can be made very accurately, and it has therefore been possible to follow the changes in the orbital elements of the system over a period of several years. For example, the periastron direction of the binary pulsar has been found to rotate about 4° per year. This phenomenon can be explained by means of the general theory of relativity; in the solar system, the corresponding rotation (the minor fraction of the rotation not explained by the

Newtonian mechanics) of the perihelion of Mercury is 43 arc seconds per century.

The binary pulsar PSR 1913+16 has also provided the first strong evidence for the existence of gravitational waves. During the time of observation the orbital period of the system has steadily decreased. This shows that the system is losing orbital energy at a rate that agrees exactly with that predicted by the general theory of relativity. The lost energy is radiated as gravitational waves.

Magnetars. The energy emitted by common pulsars has its origin in the slowing down of their rotation. In some neutron stars, the magnetars, the magnetic field is so strong that the energy released in the decay of the field is the main source of energy. Whereas in ordinary pulsars the magnetic field is typically 108 T, in magnetars a typical value may be 109-10n T.

Magnetars were first invoked as an explanation of the soft gamma repeaters (SGR), X-ray stars that irregularly emit bright, short (0.1s) repeating flashes of low-energy gamma rays. Later a second class of mysterious objects, the anomalous X-ray pulsars (AXP), were identified as magnetars. AXP are slowly rotating pulsars, with a

rotation period of 6 to 12 seconds. Despite this they are bright X-ray sources, which can be understood if their energy is of magnetic origin.

It is thought that magnetars are the remnants of stars that were more massive and rapidly rotating than those giving rise to ordinary pulsars, although the details are still subject to debate. A magnetar first appears as a SGR. During this phase, which only lasts about 10,000 years, the very strong magnetic field is slowing down the rate of rotation. At the same time the field is drifting with respect to the neutron star crust. This causes shifts in the crust structure, leading to powerful magnetic flares and the observed outbursts. After about 10,000 years the rotation has slowed down so much that the outbursts cease, leaving the neutron star observable as an AXP.

Gamma ray bursts. For a long time the gamma ray bursts (GRB), very short and sharp gamma ray pulses first discovered in 1973, remained a mystery. Unlike the much less common SGR, the GRB never recurred, and they had no optical or X-ray counterparts. A first major advance was made when satellite observations with the Compton Gamma Ray Observatory showed that the gamma ray bursts are almost uniformly distributed in the sky, unlike the known neutron stars.

The nature of the gamma ray bursts is now becoming clear thanks to dedicated observing programmes that have used burst detections by gamma and X-ray satellites such as Beppo-SAX and, in particular, Swift rapidly to look for afterglows of the GRB at optical wavelengths. The detection of these afterglows has made it possible to determine distances to the bursts and their location in their host galaxies (see Fig. 14.6).

It has become clear that there are at least two kinds of bursts, with the self-descriptive names long soft bursts, and short hard bursts. The long soft gamma ray bursts, lasting longer than 2 seconds, have now been convincingly shown to be produced in the explosions of massive stars at the end of their life, specifically supernovae of types Ib and Ic (Sect. 13.3). Only a small fraction of all type Ibc SNe give rise to GRB. The explosions that produce GRB have been called hypernovae, and are among the brightest objects in the Universe. A gamma ray burst observed in late 2005 took place when the Universe was only 900 million years old, making it one of the most distant objects ever observed. The con-

Fig. 14.6. The location of the peculiar type Ibc supernova SN 1998bw at redshift z = 0.0085 is in the circle to the lower left. This was also the position of the faint gamma-ray burst GRB 980425, the first GRB to be connected with a supernova. The circle on the upper right marks an ultraluminous X-ray source. (C.Kouveliotou et al. 2004, ApJ 608, 872, Fig. 1)

Fig. 14.6. The location of the peculiar type Ibc supernova SN 1998bw at redshift z = 0.0085 is in the circle to the lower left. This was also the position of the faint gamma-ray burst GRB 980425, the first GRB to be connected with a supernova. The circle on the upper right marks an ultraluminous X-ray source. (C.Kouveliotou et al. 2004, ApJ 608, 872, Fig. 1)

Fig. 14.7. Some pulsars shine brightly in gamma-rays. In the center the Crab pulsar and on the upper left the gamma source Geminga, which was identified in 1992 to be the nearest pulsar with a distance of about 100 pc from the Sun. (Photo by Compton Gamma Ray Observatory)

200 190 180 Galactic Longitude

Fig. 14.7. Some pulsars shine brightly in gamma-rays. In the center the Crab pulsar and on the upper left the gamma source Geminga, which was identified in 1992 to be the nearest pulsar with a distance of about 100 pc from the Sun. (Photo by Compton Gamma Ray Observatory)

Fig. 14.8. The pulses of the X-ray pulsar Hercules X1 have the period 1.24 s. The best-fitting curve has been superimposed on the observations. (Tananbaum, H. et al. (1972): Astrophys. J. (Lett.) 174, L143)

ditions required for hypernova explosions are still not certain.

The nature of the systems giving rise to short gamma ray bursts, lasting less than 2 seconds, have been more difficult to ascertain. The most popular theory has been that they are produced in compact binary systems consisting of two neutron stars or a neutron star and a black hole. These systems lose energy by gravitational radiation, and eventually the two components should merge, producing a burst of gamma radiation. This theory has

Fig. 14.9. The variations of the rapid X-ray burster MXB 1730-335. An 100 second interval is marked in the diagram. (Lewin, W.H.G. (1977): Ann. N.Y. Acad. Sci. 302, 310)

now received strong support when the afterglow of a few short bursts has been detected in the outer parts of their host galaxies. Since the stars in these regions are all old and no longer give rise to core-collapse supernovae, the neutron star merger hypothesis appears most likely. However, it is still also possible that some of the short bursts are exceptionally bright magnetar flares.

* The Radius of White Dwarfs and Neutron Stars

The mass of a white dwarf or a neutron star determines its radius. This follows from the equation of hydrostatic equilibrium and from the pressure-density relation for a degenerate gas. Using the hydrostatic equilibrium equation (10.1)

one can estimate the average pressure P:

Here we have used p a M/R3. Thus the pressure obeys

In the nonrelativistic case, the pressure of a degenerate electron gas is given by (10.16):

and hence

By combining (14.1) and (14.2) we obtain

Thus the smaller the radius of a white dwarf is, the larger its mass will be. If the density becomes so large that the relativistic equation of state (10.17) has to be used, the expression for the pressure is

As the star contracts, the pressure grows at the same rate as demanded by the condition for hydrostatic support (14.1). Once contraction has begun, it can only stop when the state of matter changes: the electrons and protons combine into neutrons. Only a star that is massive enough can give rise to a relativistic degenerate pressure.

The neutrons are fermions, just like the electrons. They obey the Pauli exclusion principle, and the degenerate neutron gas pressure is obtained from an expression analogous to (14.2):

Pn a

mn^n where mn is the neutron mass and the molecular weight per free neutron. Correspondingly, the radius of a neutron star is given by

If a white dwarf consists purely of helium, /xe = 2; for a neutron star, « 1. If a white dwarf and a neutron star have the same mass, the ratio of their radii is

Rwd = (Mns \1/3 (^n\5/3 m Rns V Mwd J \Me/ me ( 1 \5/3

Thus the radius of a neutron star is about 1/600 of that of a white dwarf. Typically Rns is about 10 km.

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