14.9 SUPERDENSE MATTER

1. 14.9 SUPERDENSE MATTER Throughout this course we have seen that the existence of ‘stable’ astronomical objects has depended on the ability of nature to find a force to compete against the unrelenting attractive force of gravity. During each phase an often temporary solution has been found to maintain the object in a unique configuration. Usually in the process of supporting itself the object has had to expend energy, and when this energy reserve has been exhausted the object has been obliged to shrink to a more compact form. Sometimes violently much of its material has been blown into space, ready to start all over again. white dwarfs and neutron stars seem to be long term cinders. However the question does arise as to how compact can matter become. For the rest of this course we explore the ultimate stages of compactness that are found in nature as revealed to us through astronomical observations. First we must understand what we mean by the term ‘compact’. dr If we fill a sphere with matter at a uniform density r, then the total work done assembling this sphere within its own gravitational field is r r R This is the binding energy of the object and can be defined as a mass defect giving RS is called the Schwarzschild radius, and corresponds to the mass defect being the total mass If we define PHYS3010 - STELLAR EVOLUTION

2. COSMIC STRUCTURES The Universe R < RS 48 40 Galaxies Regime of black holes Nuclei of Active Galaxies 32 Main Sequence Stars 24 White Dwarfs Earth,Moon,Asteroids 16 Neutron Stars 8 Log M (kg) Humans 0 -8 Protons -16 Atoms -24 -8 0 +8 +16 +24 +32 Log R (m) Black Holes. Earlier, in connection with white dwarfs, the red shift of photons emitted from the surface of the star was estimated. If R = RS then the wavelength shift will be infinite and no radiation can emerge. Hence objects having the Schwartzschild radius are called black holes. PHYS3010 - STELLAR EVOLUTION

3. 14.10 THE MASS, SIZE AND DENSITIY OF NEUTRON STARS • A survey of all the objects existing within the Universe in terms of their compactness, as defined by the R/RS ratio above shows that in the context of stellar evolution neutron stars are the most compact objects observed to date. Hence if we want to investigate the ultimate fate of compact matter then it here that we should start looking. We have already discussed much of the relevant background : • White dwarfs are supported by electron degeneracy pressure • The cores of massive stars were left in free-fall after a type II supernova explosion. Neutronisation had taken place. Is there anything to stop it falling? Will neutron degeneracy do the job? The Approximate Density of Neutron Stars The precise forces between nucleons is less well understood than electromagnetic forces and, as a consequence, the neutron degeneracy conditions under such extreme gravitational fields are less well defined. The detailed studies of neutron stars (i.e. masses and radii - the equation of state) will therefore not only be of interest to the astrophysicist, but also may be used as a test of particle physics. Simplistically we may assume that the distances between neutrons are typically so that the density will be The Nominal Maximum Mass No comprehensive equation of state holds for neutron stars, due to the lack of a real understanding of the precise forces involved. Equations similar to the relationships derived for white dwarfs For the non-relativistic and relativistic states respectively are likely to provide a reasonable approximation PHYS3010 - STELLAR EVOLUTION

4. Let us make an estimate of the maximum mass in a simplistic way, similar to the Chandrasekhar limit derived for white dwarfs Now the Fermi energy is If we have N particles so that The gravitational potential well per neutron has a depth We assume mn = mp Thus if the gravitational force is capable of overcoming the support of the degeneracy pressure, the limiting case will be when giving This yields a limiting mass for neutron stars as Similar to the Chandrasekhar limit for white dwarfs PHYS3010 - STELLAR EVOLUTION

5. The Nominal Size of Neutron Stars Since we may estimate the rough size Maximum Temperature of Neutron Stars The condition for degeneracy pressure to dominate is that the Fermi Energy should be greater than the thermal energy of the particles i.e. From our studies of degeneracy pressure, the number density of degenerate particles is giving Thus More Realistic Theoretical Models Unstable neutron drip The above figure shows the gravitational mass vs central density for the case of a model assuming a pure, ideal neutron gas. The stable white dwarf and neutron star configurations are represented by the heavy solid lines. NOTE: Mmax~ 0.7 M0 PHYS3010 - STELLAR EVOLUTION

6. The above figure shows the gravitational mass vs central density for various equations of state. The letters labelling the various curves are defined in Shapiro and Teukolsky. Note that the maximum masses predicted range from about 1.5 to 2.7 M0, and that stable neutron star configurations are predicted for objects with masses from about 0.2 M0. The gravitational mass vs radius for the same set of models. It can be seen that neutron stars will have sizes typically in the range 10 to 15 km, and that as for white dwarfs their sizes are inversely related to their masses. PHYS3010 - STELLAR EVOLUTION

7. r ~ 4 1014 r ~ 2 1017 Outer Crust Inner Crust Core r ~ 1018 Superfluid Neutrons ? 9.7 km 0.6 km 0.3 km 14.11 THE INTERNAL STRUCTURE OF NEUTRON STARS The adjacent figure shows the likely internal structure as determined from a representative model of a 1.4 M0 neutron star. The layering of the internal structure is a direct result of the onset of different regimes in the equation of state as one proceeds to higher densities. The various zones in the models may be described as follows : • The surface layers(r < 109 kg m-3). The strong surface magnetic fields effect the equation of state, making the conductivity high parallel to the magnetic field and negligible in the orthogonal direction • The outer crust (109 < r < 4 1014 kg m-3) The nuclei are extremely close to one another and they form a very stiff (~1017 x steel) body centred Coulomb lattice and exist in b-equilibrium with the relativistic degenerate electron gas. When the energies of the electrons in the degenerate sea around the nuclei are high enough inverse b-decay takes place and results in the production exotic neutron rich nuclei (e.g. 126Fe) which would be unstable in the normal physical environment. • Normally such nuclei would b-decay • i.e. • In the presence of a degenerate sea of energetic (Ee~0.5 MeV) electrons the b-decay process is blocked and in fact the neutron enrichment is caused by the reverse process • The forces involved and the melting point may be gauged if we simplistically relate • kT ~ (1/4pe0)Z2e2/r where r is the distance between the nuclei • This gives T ~ 1010 K as the likely melting point PHYS3010 - STELLAR EVOLUTION

8. Neutron stars Nuclei Minimum Neutron Star Mass This is obtained by setting the adiabatic index to the magic 4/3 value required for radial stability for neutron drip to take place. This yields a minimum mass value This limiting object is expected to have a radius of typically 160 km. • The inner crust (4 1014 < r < 2 1017 kg m-3). This consists of a lattice of neutron-rich nuclei together with free degenerate neutrons and a degenerate relativistic electron gas.. As the density increases, more and more of the nuclei begin to dissolve, and the neutron fluid provides most of the pressure. • The neutron liquid interior. (r > 2 1017 kg m-3) As the density further increases the material of the star contains chiefly a sea of degenerate neutrons with a few electrons and protons remaining. All nuclear structure has vanished. The neutrons form pairs which have extremely weak interactions with other pairs, thus making a superfluid with near zero viscosity • The hyperon core. (r > 3 1018 kg m-3) As the density increases so the Fermi energy of the neutrons increases. When the neutrons have Fermi energies comparable with their rest masses then hyperons and other particles can be created. Many of these are charged and we may expect that any such core will be a solid once more. However, because of our basis lack of full understanding the particle processes any models of the core region (if indeed it can exist) are necessarily even more speculative. If a hyperon core can exist then it is clear that its mass will be directly related to the mass of the neutron star. Low mass neutron stars are unlikely to produce a hyperon core. It is possible to think of a neutron star as a heavy nucleus i.e. } PHYS3010 - STELLAR EVOLUTION

9. 1010 Crust bremsstrahlung Photons 109 T (K) Modified URCA 108 Quarks 107 Pion condensate 106 1 102 104 106 108 t (yr) 14.12 THE COOLING OF NEUTRON STARS - PULSAR GLITCHES It is generally believed that pulsars will be formed with extremely high internal temperatures (T > 1011 K) in the core of the supernova explosion. The predominant cooling mechanism immediately after formation will be due to neutrino emission derived from the variety of particle interactions which take place in such a high temperature - high density environment. The neutrino cooling is very rapid with an initial timescale of seconds. After about a day the temperature drops into the range 109 - 1010 K, i.e. below the melting point of the crust. Photon emission overtakes neutrino emission when the temperature drops to about 108 K, after about 1000 years or more. The relative importance of the various mechanisms is summarised in the adjacent figure. Each curve gives T(t) for each process separately (all except the photons are methods of generating neutrinos) assuming the others are absent. The most effective cooling process at any time will be the one with the lowest T(t). When neutron stars are born we have seen that they are likely to be spinning rapidly with W ~ 104. At these speeds the centrifugal force outwards at the equator will be comparable to the gravitational pull in wards. Now the object will be liquid when it forms and thus take up a surface profile natural to the forces it experiences - i.e. an oblate spheroid. Since neutron stars cool rapidly to the solidification point within ~ 1 day then they will solidify in the natural shape of an object with W ~ 104. As time passes they slow down and their natural shape will converge towards a spheroid. PHYS3010 - STELLAR EVOLUTION

10. SUDDEN CHANGES IN THE SIZE OF NEUTRON STARS - GLITCHES Sudden spin-up period changes have been observed in pulsar periods and are thought to be related to crust quakes instigated by cracks in the rigid crystalline material as it attempts to become more spherical in shape as the neutron star slows down. The accurate measurement of the timing of the pulses enables great sensitivity to be obtained in terms of the measurement of very small changes in radius. Period P Time The moment of inertia is I a MR2so that Since J = IW = Constant Since we can measure period changes at about ten nano-second level we can detect very small changes in the size of the neutron star (at kpc distances!) PHYS3010 - STELLAR EVOLUTION

11. Detailed Study of the W Changes The decrease in W after the glitch exhibits a more complex time structure than is expected from a simple radius change. It can be explained in terms of the viscous coupling between the solid and liquid components within the neutron star. We assume that we have : IS = ISolid IL = ILiquid IT = ITotal Let the ratio of the components be Q = IL/IT W QDW0 DW0 DW0(1 - Q) Time If we have a glitch, all the spin-up is initially taken up by the solid material Finally we will have the entire star spinning together Since If we define the ratio of the liquid moment of inertia to the total moment of inertia by Q = IL/IT PHYS3010 - STELLAR EVOLUTION

12. Then we will have so that Thus at If we assume an exponential decay we obtain a good fit to the observational data by When we look at data from Crab and Vela glitches we find very different values • We can see that the Crab and Vela neutron stars are clearly very different. The Crab must have a lot of liquid and the Vela must be mostly solid. • One scenario is that the Crab is typically one solar mass or slightly more and the Vela a very low mass object, and hence nearly all crust • Alternatively the Vela neutron star could be very massive and close to the upper mass limit. In this case the large amount of solid would be derived from a solid hyperon core • The greater relative change in DW for the Vela supports the latter hypothesis Note the value of t is a measure of the viscous coupling between the liquid and solid components. We can see that the study of glitches provides a powerful probe for the understanding of the internal structures of neutron stars. PHYS3010 - STELLAR EVOLUTION