THE NUCLEAR PHYSICS OF NEUTRON STARS

J. M. IRVINE

Department of Theoretical Physics, The University, Manchester, Ml3 9PL

CONTENTS

1. INTR~OUC~ON 201 1.1. Historical review 201 1.2. stellar evolution 202 1.3. Neutron stars as pulsars 203 1.4. Stellar stability and neutron-star masses 207

2. NEUTRON STAR COOLING 211 2.1. Introduction 211 2.2. Neutrino opacities 211

3. PULSES 214 4. THE ATMOSPHERE 217 5. THE NEUTRON STAR CRUST 218

5.1 The nuclear lattice 218 5.2. Neutron drip 220

6. THE NUCLEON FLUID 222 6.1. Beta-stable nucleon matter 222 6.2. Nucleon supertluidity 224 6.3. Post-glitch relaxation times 226

7. SUPER NUCLEAR DENSITIES 227 7.1. The baryonic soup 227 7.2. Pion condensation 229 7.3. Quark matter 232

8. SIZES AND MASS= OF NEUTRON STARS 232 REFERENCES 236

1. INTRODUCTION

1.1. Historical review

In 1932, when news of the discovery of the neutron reached Copenhagen, Landau, who was visiting Neils Bohrs at the time, speculated upon the possible existence of a final state of stellar evolution in which a star, having exhausted all possible sources of free energy and apparently doomed to collapse for ever under the inexorable force of gravity, might be sustained against this collapse by the Fermi pressure of the neutrons.“’

Landau’s speculation was immediately taken up by the astronomers Zwicky and Baade”) who suggested that neutron stars might be formed at the centre of supernovae explosions. By

202 J. M. Irvine

1939 the pioneering work of Tolman et al. (3) had led to the first estimates of stable neutron star masses and radii.

For most of the next three decades neutron stars remained a curiosity outside the mainstream of astrophysics. The reason for this is not hard to find; how was one to see these neutron stars? All estimates indicated that the maximum mass of a stable neutron star would be of the order of a solar mass and that in the neutron star configuration such a star would have a radius of a few kilometres. Having exhausted its sources of free energy the neutron star would cool extremely rapidly and at temperatures of less than lo6 Kit would be impossible to detect the black-body radiation from such an object with the technology currently available.

The situation changed dramatically in 1967 with the discovery by HewishC4’ and his collaborators of the first pulsar and with the almost immediate suggestion by Gold”’ that pulsars might be rapidly rotating magnetic neutron stars. The suggestion that pulsars are neutron stars is now nearly universally accepted in astronomical circles and it is the view that we shall adopt in this article.

In the remainder of this introduction we shall review the place of neutron stars in stellar evolution, examine the conditions of stellar stability with a view to predicting masses of stable neutron-star configurations. We shall describe some of the observed characteristics of pulsars and the interpretation of these observations in terms of the properties of neutron stars. In the following sections we shall discuss a number of features of neutron-star physics where nuclear physics makes an essential contribution to our understanding of the phenomena.

For readers wishing to learn more about pulsars there is an excellent account in the recent book by F. G. Smith@’ while a wider account of neutron-star physics is given in the monograph by J. M. Irvine.“’

1.2. Stellar evolution

All massive objects in the universe are held together by the force of gravity.7 When an object has exhausted its sources of free energy and cooled down so that the thermal pressures are negligible, opposition to gravitational collapse may arise from the short-range repulsive forces between atoms, the Fermi pressure of the electrons, nuclear forces or the Fermi pressure of the nucleons. If the interatomic forces are sufficient to halt the collapse then the end-point of the evolution is a planet. If the collapse is halted by the electron Fermi pressure the end point is a black dwarf star (i.e. a cold white dwarf). Stars which are stabilized by nuclear forces and nucleon Fermi pressures we shall refer to as neutron stars even although, as we shall see, they may not be composed primarily of neutrons as was originally envisaged by Landau. If none of these forces are sufficient to stabilize the star then it is predicted that the end-point of the evolution will be a black-hole.

In a main sequence star like the sun the principal obstacle to gravitational collapse is thermal pressure. The high temperatures are generated by the exothermic release of energy in the fusion of light elements to form heavier elements and by the compression of the stellar matter by the gravitational field. As the elements get heavier the coulomb repulsion between the nuclei becomes greater and higher and higher pressures are required to raise the temperature to the point where the next fusion process can begin. The heaviest element that

t An exception would he small meteorites which are held together by the ordinary forces of solid-state physics.

The Nuclear Physics of Neutron Stars 203

can be created depends critically on the mass of the star since this dictates the maximum gravitational pressure. The most stable nucleus is 56Fe so that even in the most massive stars once the fusion chain has reached 56Fe there is no further source of free energy in the fusion process. The star will then start to collapse. Heat will be generated at the expense of this gravitational collapse and the temperature may rise temporarily but eventually this heat will be radiated away from the star and the collapse will continue. As the stellar material is compressed Fermi pressures will rise. The Fermi pressure PF for a non-relativistic gas is proportional to n/m 5’3(8) where n is the number density and m is the mass of the fermions. Thus the Fermi pressure rises most rapidly for the lightest fermions, i.e. the electrons.? At densities - 10’ gem- 3 the electron Fermi pressure is sufficient to support a star of around solar mass. At these densities the electron Fermi energy is much greater than either the thermal energy or the binding energy to the nuclei so that we have a highly relativistic free electron gas (P,an4/3, see ref. 8). The bare nuclei predominantly 56Fe will minimize their coulomb interaction energy by forming a lattice. In the centre of the star there will be traces of heavier elements like cobalt, nickel, copper and zinc produced by overshoot in the fusion chain, while in the surface there will be lighter elements like chromium and manganese where the pressure has not been sufficient to complete the fusion chain. The cold black-dwarf star is surely the ultimate in super metals-a stainless-steel, chromium-plated sphere populated by a free, degenerate electron gas.

Thereis a critical maximum mass for the.black-dwarf configuration. If the central density of the star is too high then so also is the electron density and Fermi energy. The electron capture cross-section rises rapidly with rising electron density and energy. As the electrons are captured neutrinos are produced which are radiated away from the star; the electron density is reduced and hence the Fermi pressure falls and gravitational collapse can begin again; as the electrons are captured by the nuclei the charge number on the nuclei falls and the nuclear lattice structure is weakened. Eventually the extremely neutron-rich nuclei will merge together to form a beta-stable nucleon fluid. At densities - 1015 gcme3 the nucleon Fermi energy is sufficient to stabilize a star of solar mass. This then is the prototype neutron star. In fact, this final stage of evolution from a white dwarf to the neutron-star phase is extremely rapid and almost certainly an explosive event with the neutron star being created as the ember of a supernova explosion. The most widely studied supernova is the one which gave rise to the Crab nebula and it is gratifying to note that at the centre of the nebula is the Crab pulsar.

1.3. Neutron stars as pulsars

We have noted above that typical densities for neutron stars are - 1015 gcme3 to be compared with the saturation density of ordinary atomic nuclei - 2.8 x lOi g cmp3. A solar mass star at a density of - 1015 g crnm3 would have a radius of only - 10 km and would generate a gravitational field at its surface sufficient to bind an electron gravitationally to the star with a potential energy of order 10 keV, i.e. a thousand times more tightly bound than it is in the ordinary hydrogen atom. In the neutron-star phase the dominant cooling mechanism is not the loss of radiant electromagnetic energy but through neutrino radiation. Hence the rate of cooling is dictated by the rate of neutrino production and the rate at which

t Neutrinos will be radiated away from the star and hence will form an extremely low density Fermi gas in the stable stellar configuration. If the collapse is extremely rapid neutrino pressures can become very great and may even lead to an explosive situation as in a supernova.

204 J. M. Irvine

neutrinos escape from the star, i.e. the opacity of neutron star matter to neutrinos. As we shall see in more detail later all estimates predict that neutron stars will cool very rapidly and within a few years of formation temperatures in excess of lo6 K are not expected. At such temperatures the huge gravitational field leads to extremely compressed atmospheres. We can obtain an extremely rough estimate of the scale height h of the atmosphere in terms of the temperature T and the surface gravitational acceleration g by assuming an isothermal exponential atmosphere whence

h=kT/mg (1.1)

where m is the mass of’the atmospheric molecules. For monatomic hydrogen we obtain a scale height - 5 cm.

As a spinning star is compressed conservation of angular momentum requires that it spin faster. The sum has a rotational period of - 25 days and a radius of lo9 m, thus compressed into the neutron-star configuration with a radius of 104m it will rotate with a period of - 20 msec. The periods of the observed pulsars lie in the range 30 msec to 4 sec.

Similarly, conservation of magnetic flux requires that the surface magnetic field increases inversely with the surface area of the star. The sun has a mean magnetic field of 10M3 T and hence in the neutron star configuration would have a surface field - 10’ T. Thus the spinning neutron star behaves like a huge, rapidly rotating magnetic dipole. Such a star would generate a surface electric quadrupole field of - 10’4Vm-‘.

These huge electromagnetic fields have a number of effects: first, any atmosphere will be strongly compressed by the polymerizing effect of the large surface magnetic field. The electric field will draw charges from the surface of the star into the magnetosphere where they will be accelerated along tightly curved orbits resulting in synchrotron radiation. It is the lighthouse-like effect of this radiation (see Fig. 1) which is thought to give rise to the pulsar

FIG. 1. A magnetic neutron star rotating about an axis which is not the magnetic axis will generate an electric quadrupole field and radiate electromagnetic energy (see equation (1.2)). The electric field will draw charges from the star’s surface which will move along tightly curved lines of force producing synchrotron radiation and acting as a primary source of pulsar signals.

The Nuclear Physics of Neutron Stars 205

signals. However, the process is not a simple one-stage event as described above since this fails to describe the detailed nature of the pulsar signals. All pulsars are observed at radio frequencies, but some of the faster pulsars have been observed in the optical and even X-ray ranges. Indeed, the Crab pulsar has been observed to pulse with the same period, to within a millisecond, over 39 octaves ranging from radio frequencies N 100 MHz to gamma quanta with energies in excess of 10 MeV (lOI MHz). A rotating magnetic dipole will radiate electromagnetic energy at a rate

d&e, 32x2 A2 sin’ u -- dt= 3c3 r4 (1.2)

where JZ is the magnetic moment, c( is the angle between the magnetic axis and the rotation axis and r is the rotational period. For typical neutron star parameters this corresponds to a source of strength N 1O32 W. Circumstantial evidence for the view that pulsars are rapidly rotating magnetic neutron stars comes from the observation that the expansion of the Crab nebula is accelerating implying a continuing source of energy driving the expansion. Measurements of the rate of expansion indicate the need for a source of at least m 103r W. Further evidence for the presence of large electromagnetic fields comes from the observation of high-energy X-ray sources within the nebular material.

Pulsar periods are extremely stable being steady to N 1 part in 10’ which makes them comparable with the best quartz crystal clocks. However, there is a general slowing down of pulsars. If we assume that the rate of loss of rotational kinetic energy

(1.3)

where .# is the moment of inertia of the neutron star N 103’ kg m2 is compensation for the electromagnetic radiation loss of equation (1.2) then we arrive at a slowing down rate

8 _4Z2 sin2 a i=3

C392 (1.4)

For typical neutron-star parameters the slowing-down rate predicted by equation (1.4) is consistent with pulsar observations. Differentiating equation (1.4) we obtain the result

. .

+=-1 (1.5)

which is independent of specific neutron-star parameters and is a test of the basic model of pulsars as rotating magnetic stars. Due to the extreme stability of pulsar signals accurate measurements of 5 are extremely difficult and to date approximate measurements have only been made in the case of the Crab pulsar where the relationship of equation (1.5) appears to be satisfied within the observational uncertainties of N 20 %.

Measurements of i are complicated by the observation that the fastest (and most rapidly slowing) pulsars are observed to glitch, i.e. at intervals of the order of years, the gradual steady lengthening of the pulsar period is interrupted by a sudden speeding up of the pulsar signal. The relaxation time for a glitch is on the order of days and during this time the pulsar signal may be quite noisy with both spin ups and spin downs occurring before the pulsar settles down to a regular pattern of gradually increasing pulse period (see Fig. 2) once more. The most plausible explanation of the glitch phenomena is that it results from a starquake

206 J. M. Irvine

and just as earthquakes can tell us a great deal about the earth’s interior so we may hope that glitches can tell us something about the interior of the neutron star.

0.0692095 r

#’

ii , , ,

.Y

B / , 8 ,

,

.i o.06g20go -/ L

B 2

0.0692065 I JOll Feb . MOr AV 1969

FIG. 2. The Vela “glitch” of 1969.

If there is a starquake this implies that some portion of the star is a solid capable of sustaining shear. Since the star is rapidly rotating it is unlikely to be spherical, but instead will adopt an equilibrium oblate deformed shape. As the star slows down this shape will no longer be in equilibrium and elastic energy will build up in the solid. Eventually the solid will acquire a critical shear and will crack, adjusting itself to a new and less-deformed equilibrium shape. When this happens the moment of inertia of the star will suddenly be reduced and in order to conserve angular momentum the star will spin up (glitch). Observations of glitches suggest that for typical neutron-star parameters a change in the eccentricity of an oblate spheroid by one part in w lo6 would suffice to account for the glitch phenomena. The period between glitches and the regular slowing-down rate of the pulsar in the interglitch phase will contain information about the amount of solid matter in the star and its elastic moduli. The relaxation time following a starquake should be governed by the velocity of sound and the size of the star. At a radius of lo4 m and with estimates of the velocity of sound in neutron star matter of 1O’m set-r we would predict relaxation times of N 1O-3 secincontradiction to the observed relaxation times of the order of days. This discrepancy remains as long as the bulk of the star is either solid or a normal fluid. However, it is known from the rotational properties of finite nuclei and in particular the rotational spectra of heavy nuclei”‘) that a nucleon fluid at densities near to the nuclear saturation density behaves like a superfluid. Hence, if the solid portion of the star represents a small fraction of the total mass of the star (say a crust) and the bulk of the matter is in the form of a nucleon superfluid, then the reduced viscosity of the superfluid will slow down the time that the information about the adjustment of the solid can be transferred to the bulk of the star. Thus observations of glitch relaxation times may yield information about the amount of the star that is in a superfluid phase. The noise in the glitch relaxation phenomena presumably contains additional information about secondary effects, e.g. the pinning of superfluid vortex lines to the crust, etc.

The Nuclear Physics of Neutron Stars

1.4. Stellar stability and neutron-star masses

207

The condition of hydrodynamic equilibrium for an element of stellar material (see Fig. 3) is that the pressure gradient should balance the gravitational forces, i.e. in Newtonian gravitational theory

dP GWr)P(r) z=- r2 (1.6)

where M’(r) is the mass within the radius r

M’(r) =4n s

’ p(r’)r’2 dr’. (1.7) 0

General relativistic corrections to equation (1.6) are of order 5i?/R where R is the actual radius of the star and R is the Schwarzschild radius

.9I = 2MG/c2 (1.8)

where the mass of the star M= M’(R). For a star of solar mass the Schwarzschild radius is N 3 x 10’ m and hence in the case of a neutron star general relativistic effects can be N 30 %. The relativistic generalization of the equilibrium condition (1.6) is (l I)

dP _ r - GCMr) +4Wr3/c21Cd~) + PWlc21 , dr r(r - 2GM’(r)/c’) (1.9)

Thus given an equation of state for the pressure in terms of the density P= P(p) for stellar matter we can substitute into equation (1.9) to obtain a first-order differential equation for

FIG. 3. An element of stellar material in equilibrium with the pressure gradient balancing the gravitational forces.

208 J. M. Irvine

Central density

FIG. 4. In regions where the mass of the equilibrium neutron-star configuration is an increasing function of the central density we have stable equilibrium, where it is a decreasing function of central density we have unstable equilibrium.

the density as a function of radius. Given any central density pe this equation can be integrated out to the point at which the density vanishes thus defining the radius of the star. Integrating the density distribution out to this radius yields a predicted mass for the star. The essential point to note is that given a central density the equation of state implies a unique mass and radius for the equilibrium stellar configuration. While all such configurations are guaranteed to be in equilibrium they are not all necessarily stable.

Consider a star of mass M(p,) in a region where A4 increases as the central density pe increases. If such a star was to be compressed slightly it would require a central density which could only be held in equilibrium by a larger mass star which would in turn generate a stronger gravitational field. The weaker gravitational field of our star would thus not be strong enough to maintain the compression (see Fig. 4). Similarly, if we were to expand the star this would reduce the central density which would now correspond to that for the equilibrium configuration of a lower mass star with a weaker gravitational field. Hence the greater gravitational field of our test star would oppose the expansion. Thus in a region where A4 increases with pe we have stable equilibrium. Similar arguments show that in a region where the mass of the star is a decreasing function of the central density we have unstable equilibrium and such configurations will not appear in nature. It follows that maximum and minimum masses of stable stars of various configurations are given by the turning-points in the mass versus central density curve.

The reader is reminded that the equation of state for a system in thermodynamic equilibrium is

(1.10)

and for systems at absolute zero temperature the Helmholtz free energy F becomes the same as the internal energy E and equation (1.10) can then be written in terms of the particle number density n and energy per particle E = nE

p+ (1.11)

The Nuclear Physics of Neutron Stars 209

If we are dealing with a multicomponent system each of density ni there is a partial pressure Pi from each component. The equation of state not only yields the pressure it also tells us the nature of matter at a given density p

P=Cnimi, (1.12)

i.e. at each density p matter will distribute itself amongst the components i and will adopt such phases as will minimize the free energy. Thus each phase transition is associated with the lowering of the free energy at a given density and hence a softening of the equation of state.

There are also limits on the possible equation of state imposed by causality. The velocity of sound c, is defined to be

(1.13)

and this must not exceed the velocity of light in uucuo c. There is an even more stringent restriction on the equation of state imposed by the condition that the stress-energy tensor have a positive trace, i.e.

P<& (1.14) which implies

c,<---c. >

(1.15)

We note also that wherever aP/ap -c 0 the velocity of sound is undefined corresponding to regions of instability where phase boundaries will be established.

The simplest equation of state is that for a perfect Fermi gas. For a non-relativistic gas the energy per particle is

&=$EF=

and hence the equation of state is

h2 3 z/3 pF=- - 0 20m R

$13

’

For an extremely relativistic Fermi gas we have

hc 3n l/3 E=PFC=l T 0

whence the pressure becomes

(1.16)

(1.17)

(1.18)

(1.19)

increasing with the $ power of the density rather than the 3 power. It is this relativistic softening of the equation of state which is essential if the inequality (1.15) is not to be violated.

In Fig. 5 we plot the predicted masses of equilibrium stellar configurations as a function of the central density assuming a perfect Fermi gas equation of state for electrons and neutrons.

210 J. M. Irvine

1.0

0.75

r” 2 0.5

0.25

I I I I I 13 14 15 16 17

Log central density (g cm?

FIG. 5. Equilibrium stellar masses as a function of central density. At low densities it is assumed that the star is composed of pure s6Fe and is supported by an electron Fermi gas pressure. At high densities a pure neutron gas is assumed.

There are two maximum masses predicted; one for black dwarfs where a central density of - 3 x lo8 g cm- 3 yields a star of the Chandrasekhar mass

~3 x 1030 kg, (1.20)

corresponding to the maximum mass that can be supported by electron Fermi pressure. This limit is rather well defined since equation of state at this density is quite well known. There is a second maximum at 10’5-10’6gcm-3 of -0.5 M, corresponding to the maximum mass

I.0 -

Black dwarfs

I I I I I I IO 100 1000 10000

Radius ( km)

FIG. 6. Predicted stellar masses as a function of radius with the same assumptions as Fig. 5.

The Nuclear Physics of Neutron Stars 211

neutron star which can be supported by the non-interacting neutron Fermi gas pressure alone. This limit is not well defined, as we shall see, due to uncertainties in the equation of state at this density. In Fig. 6 we plot the corresponding masses as a function of the predicted radius of the equilibrium stellar configurations. We note that the regions of equilibrium correspond to regions where the radius is a decreasing function of mass.

2. NEUTRON STAR COOLING

2.1. Introduction

The dominant sources of free-energy loss are through radiation from the star of photons and neutrinos. The major source of photons is from electron bremsstrahlung and if we assign to the electrons a temperature T, then we will have a photon luminosity for the star of

L,(Te)=4xaR2T,4 (2.1)

corresponding to a black-body radiation spectrum. The failure to observe such a spectrum of radiation from any pulsar suggests that T, 5 lo6 K.

The explanation of long post-glitch relaxation times in terms of neutron superfluidity requires that the neutron temperature be very much less than the neutron fluid l-point N 10’ K (see Section 6.3).

As we shall see at temperatures above * lo6 K the cooling is dominated by neutrino radiation processes. cl ‘) The three dominant neutrino-production processes are:

(a) Neutrino pair bremsstrahlung by nucleons:

N+N-+iv+N+v+iC

(b) The neutrino Urea process:

(2.2)

(c) Pionic absorption:

N+N+e-+N+N+v. (2.3)

rr+N-+N+e+V, N+v+S. (2.4)

Together with such lesser reactions as neutrino pair production, collective electron plasma excitations, photo neutrino reactions, pair annihilation and neutrino synchrotron loss, etc.

The relative importance of these neutrino-production processes depends on the environ- ment in which they take place. The efficiency of pionic absorption as a neutrino-producing reaction means that this will dominate in any region of the star where free pions are present,(“) i.e. in the core of a star containing a pion condensate (see Section 7).

Having produced neutrinos their efficiency as cooling agents will be governed by how fast they can get out of the star.

2.2. Neutrino opacities

Prior to the discovery of weak neutral currents in 1974 it was confidently asserted that the mean free path of neutrinos in neutron-star matter was very much greater than the radius of a neutron star. The argument went as follows: as long as neutrino reactions with baryons were

212 J. M. Irvine

associated with conventional beta decay any neutrino reaction required a change in baryon charge. Typical neutrino energies are expected to be less than N 1 MeV and then reactions of the type

v+N-+N+e (2.5)

are forbidden by conservation of energy and momentum. Hence it was argued that the dominant source of neutrino energy loss was through inelastic neutrino-electron scattering. For neutrinos of energy E, propagating in a degenerate, highly relativistic electron gas with Fermi energy .+>>E, the neutrino-electron scattering cross-section is

gve N (2 x 10~44)(~,/m,cZ)2(~,/~~)Cm2 (2.6)

where the scale N 10-44cm2 is given by the observed inverse beta-decay cross-section. For antineutrinos the cross-section of equation (2.6) should be multiplied by 3. The mean free path for electron-neutrinos is then

IZv=[b,,n,j-l (2.7)

where the electron number density n, is given by the Fermi energy sF and vice versa. Using the extreme relativistic equation of state for the electrons [equation (1.18)] we find that the mean free path is approximately given by

1, = (5 x lC13)(p,/p)4/3 f km Y

(2.8)

where p0 is the nuclear saturation density -2.8 x 1014gcm-3 and the neutrino energy is measured in MeV. Hence we see that the electron-neutrino mean free path N lo3 times greater than the neutron-star radius. The muon-neutrino mean free path is even greater because of the relatively much lower muon number density.

The suggestion by Weinberg and Salam (14) of the existence of neutral currents allows a whole new class of neutrino-baryon scattering processes in which there is no change in the baryon charge. In particular we can now have inelastic neutrino-neutron scattering.

For non-relativistic neutrons we have for the neutron-neutrino Fermi part of the neutral current coupling

(2.9)

where g is the weak coupling constant of ordinary beta-decay and n, is the neutron number density. Thus we expect a contribution to the neutrino opacity from the scattering of neutrinos off of neutron-density fluctuations in the stellar medium. The differential cross- section for scattering with a momentum transfer q through an angle 6’ is

(2.10)

where h,(q) is the Fourier transform of the neutron-density fluctuation. Such fluctuations may arise from thermal fluctuations in the neutron fluid or from the local clustering of neutrons into nuclei. In the former case the mean square density fluctuations can be calculated from the equation of state

lc%z.(q)I* = kTn,2 VlK, (2.11)

The Nuclear Physics of Neutron Stars 213

where Kr is the isothermal bulk modulus

K--VaP T- 0 av T’ For a degenerate Fermi gas of neutrons we find

KT 3X 10’ A= 2 0 I wkrn

(2.12)

(2.13)

where the neutrino energy E, and the thermal energy kTare measured in MeV, n, is neutrons fm-3andKTisinMeVfm-3.

The neutrino mean free path depends on the equation of state for the stellar material through the isothermal bulk modulus KT. For interacting neutron matter at nuclear saturation-densities estimates of the isothermal bulk modulus from calculated equations of state have yielded

K,lrnz N 250 MeV fm3 (2.14)

and assuming E, 1: kT this implies that the mean free path of neutrinos falls below ,G 1 km only for temperatures in excess of 1Ol2 K. The mean free path may be greatly reduced if the equation of state softens as it does whenever there is a phase transition, i.e. KT is undefined at a phase transition, where quantum fluctuations may become important (critical opales- cence). Phase transitions depend critically on the interactions in the system and hence estimates of neutrino opacities which ignore the interactions, e.g. neutron Fermi gas models, may be wrong by many orders of magnitude.

We can place limits on the total opacity due to any softening of the equation of state. Suppose that over a layer of thickness 6r the nuclear density changes by an amount 6p. Then the pressure change

6P = KTSp/p (2.15)

must be balanced by the gravitational forces in order to preserve hydrodynamic stability, thus

K

T = WWp2mN dr

r2 sp’

The number NA of mean free paths in such a layer is thus

(2.16)

(2.17)

Combining equations (2.11) to (2.17) we can conclude that NA can be as large as N 20. This is, however, a theoretical upper limit and all realistic calculations suggest a considerably lower neutrino opacity.

Now we consider the second possible source of inhomogeneity in the baryon density, i.e. the clustering of the nucleons to form nuclei. The relevant equation is (2.10) where now &I, represents the density distribution of nucleons in the nucleus with a statistical factor for the local neutron to proton ratio. At neutrino energies less than 1OOMeV the neutrino wavelength is much greater than the size of the nucleus which we can thus treat as a point yielding coherent neutrino-nucleus scattering so that equation (2.10) becomes

214 J. M. Irvine

da --=&(l +cos8)A2 d(cos 8)

(2.18)

where a2 - ,, ,.. 0.8 + 0.4 (see ref. 15). At a density p z 10’ 2 g cmm3 and with nuclei of mass number A N 50 the neutrino’s mean free path is - 100 m, i.e. of the order of the thickness of the crust (see Section 5.1). As we shall see in Section 5.2. following the onset of the neutron-drip region at the base of the crust very exotic nuclei with extremely large mass numbers may be favoured (e.g. ref. 16) so that the neutrino mean free path could easily be reduced to - 1 m. If the neutrino pressure were sufficient to blow these exotic nuclei out of the star (perhaps in conjunction with a supernova) we would have a possible source for superheavy elements.

Bearing in mind the uncertainties over the neutrino opacities discussed above we shall now assume that the opacity is low, i.e. the mean free path is much greater than the size of the neutron star. We shall also consider the star to be primarily composed of a uniform, quasi- free, hadronic fluid. The matrix elements of the weak interaction hamiltonian for the dominant neutrino-producing processes are then easily calculated and first-order per- turbation theory plus simple thermodynamics yields for the neutrino luminosity the results

L~msstrah’ung - (104r MeV set- 1)(M/M,)(p0/p)3Ti, (2.19)

Ly~(10~~ MeVsec-‘)(M/M,)(p,/p)3Ti (2.20)

and

L:-(lOsl MeVsec-‘)(n,/n,)(M/M,)T,6 (2.21)

where il4, is the Chandrasekhar mass of equation (1.20), p. is the nuclear saturation density and the subscript 9 on the temperatures implies that they are measured in units of IO9 K. Thus if pions are present in sufficient numbers in neutron star matter we should expect that neutrino production following pionic absorption will be the dominant cooling mechanism. In the absence of pions, say, in a low-mass neutron star where the central density is not high enough to lead to pion condensation the neutrino-Urea process should dominate.

Given the luminosity L we define a cooling time t

t= s

” de/r(E) (2.22) Ei

for the star to cool from an initial temperature Ti = Ei,k to a final temperature T, = .c,.,~. Here r(E) is the average luminosity over the internal energy interval E to E+dE where E =nE. Assuming a star of approximately Chandrasekhar mass an initial temperature of lo9 K and that the cooling is dominated by pionic absorption we find that the star will have cooled to a final temperature 5 lo4 K on the order of days. Assuming that the Urea process dominates the cooling time is longer but still much less than lo3 years.

3. PULSES

In trying to explain the source of pulsar signals we have to account for the fact that there are no pulsars with periods longer than a few seconds or shorter than a few tens of milliseconds, that all the pulsars are seen at radio frequencies but that some of the faster pulsars are also seen in the optical and X-ray regions.

The Nuclear Physics of Neutron Stars 215

A magnetic dipole moment _.M will produce a surface magnetic field B

BdZJR? (3.1)

B is assumed to be N 10*-lO’” Tfor typical neutron stars. If the star is rotating in a vacuum with angular frequency w we would expect a surface electric quadrupole field to be generated

E@B. c

(3.2)

For the Crab pulsar this would yield a surface field strength N 10’ 2 V cm- ‘. Depending on the nature of the magnetosphere the actual surface electric field may be much less than this, i.e.

Es = EE,O (3.3)

where E = 1 for no magnetosphere and E = 0 for an exactly co-rotating magnetosphere. The field emission electron current is

j-1000E~exp{-2Wx10’S/EoB}ACm-2 (3.4)

where W is in MeV, Es in Vcm-‘, o in set-’ and B in tesla. For the Crab pulsar a typical prediction would be j- lo6 Acmv2. Note that the current decreases rapidly as the pulsar slows down. This current follows the strongly curved magnetic field lines and hence gives rise to synchrotron radiation. The radiation is focused in a cone emanating from the magnetic poles of the star. As the star rotates it will produce a lighthouse-like pulsed signal (see Fig. 1). While this is probably a correct picture of the primary source of the pulses in this simple form it does not meet the criteria laid down at the beginning of this section. There is no consensus view as to the correct explanation of the source of the pulses and so we will content ourselves by outlining qualitatively how the simple model discussed above may be refined in order to account for the observations.

\

/

@

3

FIG. 7. The magnetic field lines for a neutron star rotating about an axis perpendicular to the magnetic axis.

216 J. M. Irvine

The magnetic field lines which emanate from the magnetic polar regions and reach the velocity-of-light cylinder are open lines and if a particle tied to them is not to exceed the velocity of light they must be twisted as illustrated in Fig. 7. The radius of curvature p of the magnetic field line at a radius r is

pm (rc/o)1/2 (3.5)

and the radius of the velocity of light cylinder R, is

R, = c/w. (3.6)

A relativistic particle of charge e and mass m with energy E >>m,, c2 moving along an orbit with radius of curvature p will radiate the power spectrum

&,=gf(g.gJ3 v<vo

where the limiting frequency v. is

lc & 3 vo=---

2np 2 [ 1

(3.7)

(3.8)

Primary charges leaving the polar regions will be accelerated along the magnetic field lines and radiate gamma rays tangentially. These gamma rays, if they are energetic enough, will give rise to electron-positron pair production. These secondary charges will be accelerated in opposite directions and will in turn give rise to more gamma rays which will lead to more pair production and thus a cascade will be established (Fig. 8).

FIG. 8. A highly relativistic primary electron following the curved magnetic field lines radiates high-energy y-rays which subsequently produce electron-positron pairs and so on until a cascade of charged sheets is produced.

If the primary charges are electrons their low mass means that they will rapidly be accelerated to relativistic velocities and their maximum energy is likely to be radiation reaction limited, i.e. they will reach an energy at which the rate at which they are radiating away energy is equal to the rate at which they are gaining it from the electric field. The secondary pair production leads to the formation of charged sheets moving out along the magnetic field lines. The high-frequency components of the resulting radiation field will suffer attenuation due to synchrotron self-absorption and only the radio frequencies will

The Nuclear Physics of Neutron Stars 217

survive. This mechanism for producing radio-frequency pulses requires that the primary electrons produce y-rays of sufficient energy to yield pair production. This means that the limiting frequency of equation (3.8) must be such that

hv, > 2m,c2, (3.9)

this implies that the rotational period should be

(3.10)

If the energy E is radiation-reaction limited this implies r S 5 set as is indeed observed.

Now we consider what happens if the primary charges are protons. The much greater mass means a lower rate of acceleration and in fact it is unlikely that their energies will ever reach the radiation-reaction limit. It is certainly clear that the pair cascade will occur at a much greater distance from the star and that the spacing between the charged sheets will be much larger. In this situation the magnetic field strength is weaker and the synchrotron self- absorption peak is at much lower frequencies, i.e. the radio frequencies will be absorbed while optical and X-ray frequencies may escape. The critical rotational frequency for proton- induced pair production is much higher because of the greater proton mass and we obtain an upper limit for the rotational period of - 0.05 sec.

An extreme lower limit on the rotational period is provided by the maximum rotational frequency consistent with stellar equilibrium, i.e.

(3.11)

or

w&&W, (3.12)

which leads to periods z 2 0.001 set below this we will have mass shedding by the star.

4. THE ATMOSPHERE

We saw in Section 1 that the most primitive estimates led to a maximum scale height for a neutron-star atmosphere of a few centimetres [see equation (l.l)]. In this region the dominant effects are provided by the huge electromagnetic fields.

All the charged particles become tied to the magnetic field lines and all the spins of the charged particles are completely polarized. An electron in a uniform magnetic field of strength - 10’ Twill spiral about the magnetic field lines in a spiral of radius of order one- hundredth of a Bohr radius. Inserting a nucleus will trap the electron in a one-dimensional coulomb field so that the resultant atom is cylindrical in nature and of length a tenth of a Bohr radius. These cylindrical atoms will generate huge electric quadrupole fields and hence the atoms will interact strongly so that the atmosphere will polymerize and form a polymeric solid with its strands parallel to the magnetic field lines, i.e. the Van Allen belts condense onto the surface of the star.

From the dimensions of the atoms given above we see that in the close-packed solid configuration we expect a typical density of - 10’ gem - 3. This extremely anisotropic solid

218 J. M. Irvine

has a Young’s modulus a million times greater than steel, it is a perfect conductor along its strands and has a band gap of N 10 KeV opposing conduction at right angles to the magnetic field lines. The nearest analogy to this material on earth is a high field type II superconductor.

The properties of this atmosphere are extremely important for a discussion of the coupling of the magnetosphere to the body of the star; however, the depth of the atmosphere is so small (N 1 cm) that it has negligible consequences for the structural and mechanical properties of the star.

There is relatively little nuclear physics input to a description of the atmosphere and so we shall elaborate no further in this article. Readers may find more details in the works of Ruderman”‘* la) and Irvine.“’

5. THE NEUTRON STAR CRUST

5.1. The nuclear lattice

Below the atmosphere the first region of the star is the surface crust and in this section we shall briefly review the properties of the crust in the density range lo6 5 p 6 2 x 10” g cmm3.

The energy per nucleon in this region is minimized by clustering the nucleons together to form nuclei. In this density range the spacing between the nuclei R,,

RL = (~o/~)“~Riv, (5.1)

is much greater than the radius of the nuclei R,. Hence the interactions between the nuclei are purely coulombic and are not sufficiently strong to favour the formation of nuclei far from the normal line of beta stability.

The Fermi energy of electrons at these densities are k MeV, thus the electrons are extremely relativistic and their Fermi energies are much greater than their binding energies to nuclei near the normal line of beta stability. To a good approximation we can consider the electrons to form a free highly relativistic degenerate Fermi gas.

The nuclei will arrange themselves within the electron gas so as to minimize the internal energy density and thus will form a lattice. The crust in fact will have the same type of composition as that discussed for the prototype black-dwarf star matter in Section 1.2.

There are thus in the crust region three contributions to the internal energy density:

(a) The electron gas energy. The energy density of a degenerate Fermi gas of electrons is

1 se=?IZhJ

s 6 J@ 2c2+m,Zc4)p2dp

=& (&(Pz +mzc2)3’2 - $rn,Z c2 [-&& + m,Z c’)~/’

with the number density of electrons given by

87~ 4=~PZ.

(5.2)

(5.3)

The Nuclear Physics of Neutron Stars 219

Equation (5.2) has the expected low-density limit

se-+-m,c2 (5.4) &’ 0

and the familiar high density limit

(5.5)

This extreme relativistic limit is sufficient for most of our needs, especially for densities in excess of w lo9 gcmm3.

(b) The lattice energy. This is most simply calculated in the Wigner-Seitz approxi- mation.“” We divide the lattice into unit cells with a single nucleus at the centre of each cell and then approximate each cell by a sphere. Since each cell is electrically neutral we can, to a first approximation, neglect the interaction between cells. The coulomb energy of each cell is estimated by assuming that the Z electrons uniformly fill a sphere of radius rL such that

+7rr$z,= 1, (5.6)

where nN is the number density of nuclei. The total lattice energy is then the coulomb energy of the cell less the coulomb self-energy of the nucleus, i.e.

Ed= -ky(l -$(r’)/rz), (5.7) L

where (r2) is the mean square radius of the nuclear charge distribution and in the simplest approximation is

(r2) -&jA2/3, (5.8)

where A is the mass number of the nucleus and r. N 1.2 fm.

(c) The nuclear energy. At these low densities (5 10” g cm-j) we have argued that we only have nuclei close to the line of beta stability, in this event it is adequate to take the energy of the nuclei sN from the nuclear data tables or their parameterizations in the form of semi empirical mass formulae.

Thus the total energy density in the crust region is

&bnN, 4 Z;p)=~ehh+~L(A~ z, nNbN+ &NtA, an,. (5.9)

In order to predict the equilibrium configuration of matter the internal energy is minimized with respect to the degrees of freedom n,, nN, A, Z at each fixed density p. Since charge neutrality requires

n, = ZnN (5.10)

and the density is given by

p =&/c2 (5.11)

there are in fact only two independent degrees of freedom. Given the equilibrium internal energy density the equation of state is given by equation (1.11) and hence a pressure versus density curve can be calculated.

220 J. M. Irvine

5.2. Neutron drip

As the density rises so does the electron Fermi energy and there may also be adjustments to the lattice structure, but at first the coulomb interactions are not strong enough to change the nuclear composition and the dominant nuclear species will be 56Fe. Eventually, however, the coulomb interaction between the nuclei become strong enough to favour nuclear con- figurations which are not in the normal valley of beta stability and at higher densities we shall gradually see the appearance of more and more neutron-rich nuclei. At this point the calculation of the equation of state becomes complicated by the fact that we can no longer take the nuclear contribution to the energy from the nuclear data tables. The nuclear contributions must either be deduced from extrapolations of the semi empirical mass formulae or calculated ab initio from some nuclear model.

As the matter is squeezed to higher and higher densities the Fermi energies of the nucleons will rise. For nucleons bound in nuclei these Fermi energies are initially negative. As the density increases the neutron Fermi energy approaches zero. At the point when it reaches zero it becomes energetically favourable to create neutrons in a gas outside the nuclei and this is heralded by a first-order phase transition defining the onset of the neutron drip region at a density -2 x 10” gcmW3.

Note that there is no correspondingproton drip because the energy of a proton in the space between the nuclei is raised by its long-range coulomb interaction with the lattice nuclei. This coulomb energy rises rapidly enough with the compression of the lattice to prevent any proton drip from occurring.

The calculation of the equation of state now proceeds as it did in the outer crust region except that the nuclear contribution must be calculated from an ab initio calculation, e.g. a HartreeFock calculation with Skyrme forces. (*O) Such calculations would yield nuclear energies which depend not only on the mass and charge numbers A and 2 but also on the proton and neutron distributions which we shall describe by the labels 5, and <,. As the nuclear charge distribution starts to fill an appreciable volume of the unit cell we will have to replace the approximation (5.8) with a more careful numerical calculation of (r*(<,)). The other new feature of the equation of state is the contribution from a non-relativistic neutron gas. The contribution of an interacting neutron gas may be calculated by a Thomas-Fermi or Brueckner-Hartree-Fock approach” ‘* i4) or by the variational approaches of Owen et ~1.‘~~’ or the hypernetted chain technique. (**) The calculation of the contribution from the nuclei is now further complicated by the fact that these nuclei are immersed in a neutron gas which acts rather like a detergent on oil drops by reducing the surface energy of the nuclei so that the energy of the nuclei depends not only on the inter&parameters A, Z, cP, r,, but also on the external neutron-gas number density.

The expression (5.9) for the internal energy density in the neutron-drip region now reads

+&~(A~Z~rnr~p,hh~ (5.12)

where the charge neutrality constraint (5.10) and density condition (5.11) are still operative. In ordinary nuclei the neutron and proton distributions are approximately proportional;

however, in the neutron-drip region the presence of the neutron gas outside the nuclei draws out the proton distribution and the charge radius grows faster than the mass radius. There is, of course, a continual problem of how exactly to define the nuclear radius in such a situation

The Nuclear Physics of Neutron Stars 221

Log density (g cmm3)

FIG. 9. The neutron-star matter equation of state calculated by Baym et al. (23*24) for the outer regions of the star.

where there is a continuous neutron background and no clear-cut zero to the nucleon density.

The constrained variation of b(p) then yields the equation of state and indicate equilibrium values of n, = 0 below densities p - 2 x 10’ ’ g cm - 3 corresponding to no neutron drip below this density and a first-order phase transition, i.e. discontinuity in the slope of the pressure-density curve at this density marking the onset of neutron drip. In Fig. 9 we plot the calculated equation of state for this region obtained by Baym et u1.(23*24)

As the density increases the nuclei continue to get more and more neutron rich and in Fig. 10 we plot the predicted nuclear species as a function of density obtained by Baym et al. The proton charge distributions extend further and further outside the neutron distributions and

Log density (g cm-s)

FIG. 10. Nuclear species predicted by Baym ef al. ‘23*24) for the outer regions of a neutron star. The solid line is the charge number Z, the broken line the mass number A.

222 J. M. Irvine

steadily the proportion of nucleons that are in the neutron gas to those in the nuclei increases. Finally, at N lOi gem- 3 the nuclei dissolve smoothly into a uniform, beta-stable, very neutron-rich nucleon fluid. There is, of course, throughout a relativistic electron gas background maintaining charge neutrality.

6. THE NUCLEON FLUID

6.1. Beta-stable nucleon matter

The equation of state for a normal, degenerate nucleon fluid can be calculated by one of the standard methods. However, recently (25) doubts have been cast on the results of Brueckner-Bethe-Goldstone” or 14) calculations. Reliable Monte Carlo@@ and Fermion Hypernetted Chain’22’ results are not yet available for systems involving realistic nucleon- nucleon interactions due to difficulties associated with the non-central components of the force. We shall base our comments on the results of the lowest order constrained variational calculations of Owen el~~L’~r’ as applied to beta stable nucleon matter’27’ using the Reid soft core interaction.‘28’

The requirement that the neutron star matter be beta-stable, i.e. in thermal equilibrium with respect to the reaction

P+e++n+v,,

is

P”-PL,=L

(6.1)

(6.2) where pi is the chemical potential (here Fermi energy plus rest mass energy) for the ith species of particle. Charge neutrality requires that

np=ne. (6.3)

Subject to these constraints the internal energy is calculated and the neutron to proton ratio (the only degree of freedom) is varied to find the equilibrium composition of matter at each density. In Fig. 11 the results of such a calculation are presented. We see that as the density increases from a low value the proton abundance falls extremely rapidly. However, when the density reaches -ipo the strong neutron-proton component of the Reid interaction temporarily results in a slight rise in the proton to neutron ratio and at a density -5 x 10’4gcm-3 it reaches a maximum of -7 %. In Fig. 12 we plot the corresponding

equation of state and compare it with that calculated for a pure neutron fluid. The equations of state are very similar apart from the slight softening of the beta-stable-fluid case due to the attractive nature of the neutron-proton interaction in the density range 3-5 x 1014gcm-3.

Before we proceed further there is an important point to note about the validity of the equation presented in Fig. 12. If the lowest order constrained variational calculations are repeated for nuclear matter with a phenomenological interaction designed to fit the two nucleon data, like the Reid potential, we find (21) that the saturation point occurs at 2CL 23 MeV/N and at a density corresponding to a Fermi momentum of 1.7-l .9 fm- ’ compared with the semi-empirical mass formulae of N 16 MeV/N at 1.35 fm- ‘. This overbinding of nuclear matter can either be due to a fault in the N-Ninteraction or an incorrect treatment of the physics of the many-nucleon problem. We believe that it is the latter and is associated

The Nuclear Physics of Neutron Stars 223

20

15

s

t IO \ N

5

L

Log density (g cm-33

FIG. 11. et al.‘27

The proton-to-neutron ratio as a function of density in a beta-stable nucleon fluid as calculated by Bishop

250 r

5 IO 15 20 25

Density (10’4q cmm3)

FIG 12. The equation of state for a beta-stable nucleon fluid (solid curve) and neutron matter (dashed curve) as calculated by Bishop et ~1.“~’

with effects in the many body system which are not present in the two-nucleon problem. Phenomenological interactions like the Reid soft-core force attempt to represent all possible contributions to the free two-body scattering amplitude. In particular, they include the contribution illustrated in Fig. 13. The intermediate nucleon and N* (1234) states would be summed over completely in the generation of a free two-body interaction. However, when this interaction takes place inside nucleon matter it is modified in two respects.(29) First, the

224 J. M. Irvine

R*P ___________

_-- _______. “*P

N

N”

N

FIG. 13. N-N scattering via an intermediate state containing a nucleon and a N*(1234).

intermediate nucleon state must lie outside the background nucleon Fermi sea (Pauli effect) and secondly, the intermediate N and N* will both feel a mean field due to the surrounding nucleons (dispersion effect). The amplitude of Fig. 13 contributes greatly to the attractive portion of the N-N interaction, hence without the Pauli and dispersion corrections the phenomenological interactions should overestimate the binding energy of dense nucleon matter. Green and Niskanen’30) have estimated that the effect of allowing for the Pauli and dispersion effect in nuclear matter leads to a density-dependent reduction in the binding energy per particle

Owen et ~l.‘~” have noted that when this is taken together with their lowest-order constrained variational calculations the nuclear saturation point is moved to N 15 MeV at k, N 1.4 which is in satisfactory agreement with the semi-empirical mass formulae fits. When similar considerations are applied to neutron matter Green and Haapakoski propose to reduce the contribution to the 1So’31’ channel by a factor N (1 - 4n,‘.4) which represents a very considerable stiffening of the neutron-matter equation of state.‘27’ These considerations have not been applied to beta stable-nucleon matter as yet, but there can be no doubt that the resulting equation of state will be very much stiffer than that indicated in Fig. 12.

6.2. Nucleon superfluidity

Calculations of the superfluidity of nucleon fluids follow directly the usual Bardeen- Cooper-Schriefer approach. (32) Superfluidity clearly plays an important role in the transport properties of the fluid, but does not contribute significantly to the calculation of pressure gradients from the equation of state. We shall only briefly present the formalism for s-wave pairing, a discussion of p-wave pairing may be found in the literature.‘33’ 34)

The Nuclear Physics of Neutron Stars 225

At absolute zero temperature the ground state of a Fermi fluid will be superfluid provided that the energy gap

A=GxukvI, (6.5) k

is non-zero, where UC + vi = 1 and vi is the occupation probability for the single-particle state of momentum k.

?$=+{I -(&k-&F)[(&k-&F)2+A2] -1’2} (6.6)

with G an average s-wave pairing matrix element of the two-body residual interaction

G= (k, -klVlk’, -k’). (6.7)

In Fig. 14 we illustrate the form of v,” and we see that the effect of the interaction is to smear out the Fermi surface and that the sum in equation (6.5) only receives contributions from a range of width 20 about the Fermi surface. Assuming a constant density of single-particle states q over the interval .+ - w to aF + o we can solve equations (6.5) and (6.6) to obtain

A,,=2wexp[-l/G?] (6.8)

At finite temperatures there is an additional smearing of the single-particle occupations provided by the Fermi distribution function

SF(a)= {expC(a--EF(WkTl+ l>-‘, (6.9)

whence we find that the gap vanishes at a critical temperature T,(a)

kTC=0.57A,, (6.10)

and for kTc A0 we have

A(T)=A 1-J 0{ fF)exp C- AJkTl} (6.11)

while at T= T,

A(T)-3.06kT,J(l -T/T,). (6.12)

2w I *

l r ck

FIG. 14. The occupation probability of v: for a single-particle state of momentum k in a Fermi fluid at zero temperature containing an s-wave pairing interaction.

226 J. M. Irvine

P

2

%,-

1

= .z .- L

0 2-

I I I I I 4 6 12 16

Density (lO”g cmm3)

FIG. 15. The calculated neutron superfluid critical temperature as a function of density for a system with a Skyrme type interaction.‘“”

The fraction of normal fluid to superfluid at a total density p

P = Pnormal +hprlluid

for low temperatures Tc T, is (6.13)

ev C - AolKl. (6.14)

In Fig. 15 we plot the calculated neutron superfluid critical temperature and we see that overmostoftherange 10’2gcm-3 to 1015gcm - 3 T, >> T (assuming an ambient temperature TS lo6 K). It must be remembered that in this density range the Fermi temperature TF is Z 10’ 2 K so that the fluid is really extremely cold.

At lower densities (5 1012 gcme3) the neutron fluid is not superfluid, then as the density increases so does the neutron Fermi energy and the density of states at the Fermi surface. At N 10’ 3 gem-’ the strong ‘So neutron-neutron interaction induces a BCS’32’ type of pairing correlation resulting in the transition to a superfluid. As the density rises still further so also does the neutron Fermi momentum. At -250 MeV the ‘So phase shift changes sign and this is reflected in the disappearance of the s-wave pairing neutron superfluid phase at -3 x lOi gcmw3. As the energy increases so do the 3P2 phase shifts overtaking the ‘So phase shifts at N 170 MeV and the sign of these phase shifts remains the same up to more than 500MeV.Thusat -3x 1014gcm-3 it is predicted that P-wave correlations set in leading to an anisotropic superfluid.

The proton to neutron abundance is so low throughout this region (Fig. 11) that we never reach a sufficiently high proton density to yield a proton-superconducting phase transition.

6.3. Post-glitch relaxation times

We consider an extremely simple model containing two components, one which is conducting and is assumed to rotate at the characteristic pulsar frequency o. The other

The Nuclear Physics of Neutron Stars 227

component is neutral and couples only weakly to the charged components and is characterized by a relaxation time z. If the neutral component is a superfluid all the circulation is confined to vortex cores and the spin up of the supertluid occurs when angular momentum is transferred to these vortex cores. The total coupling is proportional to the number of neutrons in the vortex core which depends on the size of the vortices and their density times the probability of a quasi-particle excitation.‘35’

The neutron excitations are quantized in a plane perpendicular to the vortex core axis, hence

AE=h2/2m,(f2 zrr2A2/4,, (6.15)

where 5 is the superfluid coherence length

5 = hv,/aA (6.16)

with or the Fermi velocity [er = em&]. The quasiparticle excitation probability is governed by a Boltzmann factor

exp ( - n2A2/4.z,kT) ‘Y 10-14. (6.17)

Assuming that the coupling between the charged components and the superfluid is the direct magnetic coupling of the electrons to the neutrons we find from the linearized Boltzmann transport equation that the relaxation time is inversely proportional to the coupling.‘36’ i.e.

_=,, 2!,2a2g2g 1 “*kT T ‘R24 *E;

kexp[-~2A2/4~,kT-J. (6.18)

Here iV, is the number of vortex lines per unit area so that N,(‘/R’ is the fraction of the superfluid in the vortex cores since the radius of a vortex is -5, a is the fine structure constant and gn is the neutron g-factor while EF is the electron Fermi energy.

For a superfluid energy gap - 2 MeV and an electron Fermi energy - 100 MeV we obtain a relaxation time on the order of a year. Younger stars we might expect to have higher temperatures and hence smaller pairing gaps leading to shorter relaxation times while older colder neutron stars would have longer relaxation times. If we can use the rate of pulsing as a measure of the age of the pulsar then this interpretation of the phenomena is consistent with the observations.

7. SUPER NUCLEAR DENSITIES

7.1. The baryonlc soup

We now consider the nucleon fluid of Section 7.1 at densities greater than the nuclear saturation density p,-,=2.8 x 1014gcm-3.

At a density - 3 x 1Ol4 g cm- 3 the electron Fermi energy reaches 105 MeV (the rest mass of the muon) and a new degree of freedom is possible with the system being in thermal equilibrium under the reaction

p+/J*n+v,. (7.1)

228 J. M. Irvine

As the density continues to increase a Fermi sea of condition (6.2) is extended to become

~“-~(p=/Q?=~Lr

while charge neutrality now requires

ng=ne+np.

muons develops. The equilibrium

(7.2)

(7.3)

Note that these muons are stable, unlike free muons, and the condition (7.2) prohibits the normal decay mode

~+e+V~+v,. (7.4)

This suggests that as the Fermi energies continue to rise there will be a series of thresholds at which new species of particle will arise.

An early statistical bootstrap approach was made to this problem, i.e. it is assumed that the hadrons are themselves composed of hadrons the interactions between which generate a self-consistent observable free hadronic spectrum. The whole system is assumed to be in statistical equilibrium. We begin by guessing a density of states qin(E) for the particles of total energy E and use this to calculate a density of states Q,,,(E), we then iterate until the self- consistency condition of the bootstrap hypothesis is satisfied

tlin(&) = &8dE) (7.5)

Consider, for example, the number of states of one particle with spin degeneracy g in a box of volume V with momentum between p and p + dp, it is g Vdp/h’. Generalized to n independent particles with total energy E it is

(7.6)

where the momentum delta function and the missing factor V/h3 correspond to taking the centre of mass at rest. Further generalized to an assumed density of states vi”(s) for-the bootstrap hypotheis we have I

(7.7)

where the l/n! term prevents double counting. In writing down equation (7.7) we have ignored the Pauli principle and interactions between the particles. Ignoring the Pauli principle may not be a serious deficiency in the high-energy distribution tail for a hot baryon system, e.g. for predicting production rates in a violent high-energy reaction, but it is certain to cause problems for extremely low-temperature degenerate-neutron star cores.

In Fig. 16 we present the results of a simple statistical bootstrap calculation of particle abundances in a neutron star core. These predictions are extremely uncertain due to the omission of the Pauli principle and any baryonic interactions. It is also uncertain because until one has a reliable model for the baryons the number of baryonic states, their masses and coupling constants must be very uncertain. In the next section we shall see how careful consideration of known strong interactions radically changes these predictions.

The Nuclear Physics of Neutron Stars 229

I6

Log density (g cmS3)

FIG. 16. Abundances of particles predicted by a statistical bootstrap calculation as a function of neutron-star- matter density. All particles up to a mass of 1.7 GeV were included in the analysis.

7.2. PIon condensation

If we return to the discussion which we used to describe the appearance of muons, we would be led to conclude that the next landmark in the equation of state might be reached when pn - pp = 138 MeV, the rest mass of the pion. The system would now reach thermal equilibrium with respect to the reactions

n++p+n- (7.8)

and real pions would spontaneously appear in the system. Like the muons these pions would be dynamically stable with the usual decay channels

IL- + p+ylr, e+?,, etc., (7.9)

closed by the extended equilibrium condition

~“-~p=&=rU~=l&. (7.10)

The pionic fluid will differ from the electron and muon gases in that the pions being bosons will form a Bose condensate rather than a degenerate Fermi gas. Also the pions being hadrons, unlike the electrons and muons, will interact strongly with the nucleon background to acquire a self-energy (effective mass) significantly smaller than the free pion mass. Several calculations’37*38’3g) have indicated that the strong pion-nucleon p-wave interaction reduces the effective mass of the pions to N 110 MeV so that the appearance of pion condensation follows extremely rapidly on the heels of the appearance of the muons. The p- wave 71--N interaction has the non-relativistic form N k, . eN where k, is the pionic momentum and a, is the nucleon spin and hence, in order to take advantage of this

230 J. M. Irvine

interaction, the pions must condense into a non-zero momentum state. Thus we have a new element in our nucleon fluid, a charged Bose condensate of non-zero momentum.

This phase transition to a pion condensate is very similar to a paramagnetic- ferromagnetic phase transition where the ferromagnetic phase corresponds to a condensate of spin waves. The pion condensate corresponds to the onset of long-range spin-isospin ordering of the nucleons induced by the tensor force (propagated by OPEP, i.e. the same form as a dipole-dipole interaction modified by a charge exchange coupling) and the pions play the role of spin waves in this lattice except that instead of a simple spin-flip disturbance propagating through the lattice we now have a spin and isospin wave both spin and charge flips propagating together.

The charge neutrality condition is now extended to be

np=ne+np+nrr. (7.11)

As the density continues to rise the relative abundance of electrons and muons continues to fall as they become more and more energetically expensive due to their ever-rising Fermi energies. More and more of the charge balance is maintained by a growing amplitude of the energy-inexpensive pion condensate. The relatively low energy cost of pions reverses the trend to a greater and greater relative neutron abundance and by the time we reach N lOI gcmm3 we have virtually symmetric nuclear matter containing no significant numbers of electrons and muons and where the proton-charge density is compensated for by the pion-condensate amplitude.‘37’

In this region we also see the appearance of the next component of neutron-star matter, the N*( 1234 MeV) nucleon resonance. This is a p-wave resonance in the 71-N system with spin 3/2 and isospin 3/2 and hence each N* momentum state is sixteen-fold degenerate compared with the four-fold degeneracy of each N state. Thus as the density increases the Fermi energy of the nucleons has to increase faster than that for the N*‘s. This, together with a reduction in the N* self-energy due to its interaction with the surrounding nucleon Fermi sea, leads to the spontaneous appearance of real N*‘s in the fluid.

The charge neutrality condition is now

2n$+ +n&+n,=n,+n,+n,+n~. (7.12)

and the equilibrium conditions are

~“-Clp=C(e=&=LI=L&*, &* =HY ,U;*=p,, /&+=2&-/L”. (7.13)

In Fig. 17 we plot the calculated abundances of hadronsC3” as a function of density. A comparison with Fig. 16 indicates how different the predictions are when interactions and the Pauli principle are taken into account.

Qualitatively it is easy to see what the effect of pion condensation will be on the equation of state. Before the onset of the condensate the energy will be rising according to the equation of state of Fig. 12. Then at high density where the symmetrization of nuclear matter is complete we will have transferred to the nuclear matter equation of state (see Fig. 18). In the intermediate densities we have a classic form of the equation of state in the presence of a phase transition as typified by the Van der Waal’s equation of state. The region where &/an is negative corresponds to negative pressure and is unstable. In thermal equilibrium, then, we would make the usual Maxwell construction and predict a phase boundary. In this case on the low-density side of the boundary we would have a beta-stable-nucleon fluid containing no pion condensate, while on the high-density side we would have symmetric nuclear matter

The Nuclear Physics of Neutron Stars * 231

FIG. 17. Abundances of baryons predicted in an interacting hadronic fluid allowing for pion condensation.@‘)

,-

I-

FIG. 18. Equation of state for nucleon matter allowing for a pion condensate. The two solid curves represent the neutron matter and nuclear matter equations of state. The shaded region represents the uncertainty allowed in the equation of state of pion condensed matter.

with the proton-charge density balanced by a pion condensate. The density discontinuity across the phase boundary is not inconsiderable. Our estimates(37’ would indicate a sharp rise in the density by a factor - 3 at the phase boundary.

232 J. M. Irvine

7.3. Quark matter

The success of models in which hadrons are composed of weakly interacting quarks and yet the apparent absence of free quarks has led to the development of quark-confinement models in which the quarks interact with one another so as to be almost free when close together but interact strongly at large distances, thus inhibiting the escape of quarks from the interior of the hadrons.

In the most successful of such bug models the spin 4 quarks have a whole new series of quantum numbers: flavoured quarks can be up, down or strange, they can be coloured red, blue or green and heavy quarks may exist with charm, top and bottom. These quarks interact with one another through the exchange of members of a family of eight massless coloured gluons. The lowest mass baryons and their resonances, i.e. N, C, A, Z and A, etc., together with the low mass mesons, i.e. rr, K, etc., can be constructed out of up, down and strange quarks. Theoretical predictions suggest that the masses of the up and down quarks may be small (N 0 MeV) while the strange quark is likely to be somewhat heavier (N 300 MeV). The remaining quarks are expected to have masses k 1 GeV. Hence we do not expect the heavier mass quark states to be populated below m 10” gem- 3. This we shall see shortly is very close to the limit of the central density of the heaviest possible stable-neutron-star configuration.

At nuclear density the low mass quarks are going to be highly relativistic and the simplest model for the internal energy of the quark Fermi seas would be that for the highly relativistic, degenerate Fermi gas [equation (1.19)]. The masses of the quarks can then be fixed by requiring that the internal energy has a minimum, subject to baryon number conservation, at a density of nQz2.15fme3 corresponding to three quarks per nucleon with a nucleon root mean square radius of 0.8 fm, and with a saturation value of 940 MeV, corresponding to the rest mass of the nucleon.

The situation is somewhat analogous to that at the interface of the neutron-star crust and the nucleon-fluid core. There we had at lower densities nuclei composed of nucleons which, when they were compressed sufficiently so that nucleons lost their identity with a given nucleus, dissolved into a nucleon fluid. Here we have nucleons (and perhaps pions and N*‘s) all composed of quarks which when sufficiently compressed dissolve into a quark fluid which in this case should be a degenerate extremely relativistic very weakly interacting Fermi gas. Since the quark bags have a root mean square radius of w 0.8 fm we expect this transition to occur at 3-4 times the nuclear saturation density, i.e. N 10’ 5 g cmW3.

The formation of the quark matter need not destroy the pion condensate since long-range correlations in the quark matter may persist. One suggestion is that the quark matter may well be a ferromagnetic superfluid with pairs of quarks condensing out into spin 1 states.

8. SIZES AND MASSES OF NEUTRON STARS

In Sections 5,6 and 7 of this short review we have tried to indicate how the equation of state of neutron-star matter may be calculated for what we hope are the three major regions of the star. There is, of course, considerable scope for variations in the results of such calculations depending on the many-body techniques employed, the interactions used, etc. This uncertainty is particularly acute at super nuclear densities. Armed with these cautionary

The Nuclear Physics of Neutron Stars

Quark matter

233

Leg density ( g cm-9

FIG. 19. An equation of state for cold high-density matter.

comments we present the results of some calculations which may act as an indicator to the nature of a likely scenario.

In Fig. 19 we present an overall view of the equation of state for neutron-star matter. Note that this is presented on a logarithmic scale so that features such as the density discontinuity at the pioncondensate phase boundary appear suppressed.

In Fig. 20 we present a plot of the mass of the equilibrium stellar configuration as a function of the star’s central density. This is arrived at by taking the equation of state of Fig. 19, inserting it in equation (1.9) to obtain a numerical solution. We see that there is a predicted maximum stable stellar mass of order three solar masses and that for this situation

Log density (g cm-31

FIG. 20. The calculated stable stellar mass as a function of central density for the equation of state of Fig. 20.

234 J. M. Irvine

IO

i

I I I J IO loo

Radius ( km)

FIG. 21. The predicted mass versus radius curve for the stable neutron-star configurations of Fig. 20.

the central density is - lOi gem- 3. Statistical bootstrap calculations lead to a softer equation of state with maximum central densities - 10” gcmm3 and maximum stable- neutron-star masses smaller than a solar mass. Models which do not have a central quark- matter core and which employ particularly repulsive short-range hadronic interactions can yield harder equations of state and higher central densities supporting more massive stars.

In Fig. 21 we plot the radius of stable-neutron stars as a function of their mass for stars whose matter obeys the equation of state represented in Fig. 19.

Finally, in Fig. 22, we plot the profiles for three stable-neutron-star configurations corresponding to three possible masses from Fig. 20. We see that in the extremely small mass configuration, where the radius is largest, most of the star is composed of crust material reminiscent of a white dwarf star and there is a relatively small core of highly neutron-rich nucleon fluid. In the high mass configuration where the radius is smallest the crust and the nucleon fluids both form relatively thin skins covering a large quark-matter core. In other models this central core region might be composed of various hadronic mixtures, but in no case would the bulk of the star conform to Landau’s original view of neutron-star matter. Only at the intermediate mass neutron star is a situation achieved where the bulk of the star might be described as a neutron fluid.

From the discussion that has gone before we might then expect that extremely light neutron stars would suffer particularly violent glitches and that these would be separated by relatively long periods of time and that relaxation following a glitch would be particularly rapid. The very massive stars would suffer relatively frequent glitches which might be of such a minor nature as to go undetected. The relaxation time following one of these minor glitches would to some extent depend on whether the core of the star is superfluid or not, but the relatively high abundance of charged hadronic states would certainly shorten this time. Again only in the intermediate mass region would the description of the long relaxation time following a moderate glitch, as observed, for example, in the Crab and Vela pulsars, be applicable.

The Nuclear Physics of Neutron Stars 235

Log radius (km)

FIG. 22. Neutron-star cross-sections for three stable-mass configurations.

The whole discussion of the masses of neutron stars has, up to this point, ignored the possibility that the neutron star is probably rotating extremely rapidly. Intuitively we would expect the effect of the rotations to lead to more massive neutron stars where, in addition to the pressure gradients derived from the equation of state, centrifugal forces will exist to oppose gravitational collapse.

From the radii presented in Fig. 21 and the observed pulsar frequencies we may deduce, at least for the visible neutron stars, that the maximum velocity of a point on the stellar equator may be as great as one-tenth the velocity of light. If the core of the neutron star is a superfluid then on average the neutral superfluid component will be rotating slightly more rapidly than the charged components, i.e. presumably the sources of the pulse periods.

Unfortunately, the calculation of the effects of the rotation are not as simply performed as they would be if we were dealing with the Newtonian stability condition [equation (1.6)]. Within the framework of the relativistic condition [equation (1.9)] a perturbation approach has been developed by Hartle and ThorneC4’* ‘) and we estimate that including such effects the maximum stable-neutron-star mass might be at most three times greater than that indicated in Fig. 20.

What would be extremely useful would be some independent measurement of the mass of an observed glitching pulsar so that a judgement could be made about the possible consistency of the neutron-star predictions which we have made.

We have clearly only touched the tip of an iceberg in our brief discussion of neutron-star physics, but we hope that the nuclear physicist might have gained an insight into a whole new realm of physics to which his knowledge is essential and that the astrophysicist might have uncovered new questions that he might ask of the nuclear physicist in his attempts to explain the fascinating phenomena associated with pulsars.

J. M. Irvine

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