6

IV. NEUTRINO EMISSIVITY AND OPACITY

In a non-equilibrium situation where a temperature gradient exists in a substance, energy is transported not only by means of thermal conduction, as discussed in the last section, but also by radiation. Parameters of radiative transfer intrinsic to the matter system are its emissivity and opacity. There are two major forms of radiation. In one form, the radiative energy is transmitted by photons and in the other by neutrinos.

In a dense matter system whose constituent electrons are highly degenerate (i.e., when the electron Fermi energy is high compared with the thermal energy kBT), the degenerate electrons cannot avail themselves as effective scatterers for the passage of energy carriers created by the thermal gradient; and the thermal conductivity is correspondingly high. Therefore, energy transport is far more effective through heat conduction than it would be for radiative transfer. For this reason, the problem of photon emissivity and opacity in dense matter receives very little attention. Most astrophysical studies of photon emissivity and opacity are performed for relatively low-density substances, such as those forming the interiors of luminous stars. Since our subject is dense matter, these parameters will not be discussed here. Interested readers are referred to astrophysical texts suggested in the Bibliography. More relevant to dense matter physics is the topic of neutrino emissivity and opacity.

A. Neutrino Emissivity

The reaction described by Eq. (37) is a sample reaction in which a neutrino is produced. Indeed, neutrinos are produced throughout the second density domain whenever neutronization occurs, and in these processes neutrinos are produced even at zero temperature. At high temperature there are other reactions that are more effective in neutrino production. Reactions involving neutrinos belong to a class of interaction called the weak interaction. The coupling constant for the weak interaction is much smaller than that of the electromagnetic interaction. Neutrino interaction rates are in general slower than comparable photon interaction rates by a factor of at least 1020. Neutrinos can pass through thick layers of substance and experience no interaction. For example, the neutrino mean free path through matter with density similar to our sun is about a billion solar radii. Hence, once they are produced in a star they are lost into space, and they serve as an efficient cooling mechanism for hot and dense stellar objects.

During a supernova process, the collapse of the stellar core raises the core temperature to as high as 1011 K. It quickly cools to a temperature of 109 K through neutrino emission as the core stablizes into a neutron star. Neutrino emission dominates photon emission until the temperature drops to 108 K. It is estimated that neutrino cooling dominates for at least the first few thousand years of a neutron star after its formation, and therefore the neutrino cooling mechanisms deserve attention. We are primarily interested in those neutrino emission mechanisms that supersede photon emission under similar conditions. They determine therefore the cooling rate of neutron stars in the early period, and they also play a major role in the dynamics of a supernova process.

An important neutrino production process is the modified Urca reactions which involve neutrons, protons, and electrons:

             (53a)

             (53b)

where n and n` denote neutrino and anti-neutrino, respectively. In the following we shall not distinguish neutrinos from anti-neutrinos and address them collectively as neutrinos. These two reactions are very nearly the inverse of each other, and when they are occurring at equal rates, the numbers of neutrons, protons, and electrons in the system is unaltered, while neutrinos are being produced continuously. Urca is the name of a casino in Rio de Janeiro. Early pioneers of neutrino physics saw a parallel between nature's way of extracting energy from the stellar systems and the casino's way of extracting money from its customers, and they named the reactions after the casino. Reactions (53a,b) are modifications of the original Urca reactions by adding an extra neutron to the reaction. This increases the energy range over which neutrinos may be produced and thus improves the production rate.

Our current understanding of the weak interaction theory is provided by the Weinberg-Salam-Glashow theory. Even though most of the neutrino reactions to be discussed in this section have never been verified under laboratory conditions, they are nevertheless believed to be correct, and quantitative estimates of their reaction rates are reliable. When reaction (53a) occurs, a transition is made from a state of two neutrons to a state of a neutron, proton, electron and neutrino. The theory evaluates the transition probability from an initial state of two neutrons that occupy quantum states of definte momenta to a final state of four particles of definite momenta. Whenever a transition is made, a neutrino of a specific energy is produced. The total neutrino energy emitted from the system per unit time is found by summing the neutrino energies of all allowed transitions multiplied by their respective transition probabilities. The neutrino emissivity is the total neutrino energy emitted per unit time per unit volume of the substance.

Consider a neutron matter system in the density range of 1014-1015 g/cm3, that is composed mainly of neutrons with a small admixture (about 1%) of protons and electrons. Quantum states occupied by the particles are given by the Fermi-Dirac distribution fj of Eq. (32). Note that the temperature dependence of the final result comes from the temperature factor in the distribution. Designating the initial-state neutrons by subscripts 1 and 2, the final-state neutron by 3, and the proton, electron, and neutrino by p, e, and n, respectively, we find that the emissivity of the neutron matter system is given by

                                 (54)

where the summations over quantum states are represented by integrations over d3nj = V(d3pj/h3), en = c|pn| is the neutrino energy, and W(i -> f) is the transition probability from initial state i to final state f per unit time and per unit volume of the system. It has the following structure

             (55)

where pi and pf denote the total momenta of the initial and final states, respectively, Ei and Ef the total energies of the initial and final states, respectively, and H(i -> f) the matrix element of the Hamiltonian describing the interaction; the summation symbol S indicates the summations over all spin orientations of the particles. The mathematical delta-functions are here to insure that only energy-momentum-conserving initial and final states are included in this evaluation. In Eq. (54), the initial-state particles are assigned distributions fj while the final-state particles are assigned distributions (1 - fj) because the particles produced in the reaction must be excluded from states that are already occupied, and therefore they must take up states that are not occupied, which are expressed by (1 - fj). The chemical potentials in the distributions fj determine the particle numbers in the system. They are related by m1 = m2 = m3, and mp = me, while in most cases un = 0. The quantity un will be different from zero only in circumstance where the neutrino opacity is so high that the neutrinos are trapped momentarily and become partially degenerate.

If the neutrons and protons in the system are assumed to be non-interacting ,the emissivity due the reaction (53a) can be easily evaluated and the result may be expressed conveniently as

                 (56)

where r0 = 2.8 x 1014 g/cm3 is the density of nuclear matter, and the symbol T9 stands for temperature in units of 109 K. Here the emissivity (in units of ergs per cubic centimeter per second) is evaluated for a dense system of neutron matter whose proton and electron contents are determined by the ground state requirement under reaction Eq. (37). It is expressed in this form because neutron matter exists only with densities comparable to nuclear matter density, and T9 is a typical temperature scale for neutron star and supernova problems.

An interesting point to note is that this emissivity is given by eight powers of temperature. This comes about for the following reason. For the range of temperature considered, the thermal energy is still small compared with the degenerate or Fermi energy eF, of the fermions in the system (except neutrinos), and most of the easily accessible quantum states are already occupied, leaving only a small fraction of quantum states in each species contributing to the reaction (of the order of kBT/eF per species). Since there are two fermion species in the initial state and three (not counting neutrinos) in the final state, a factor of T5 is introduced. The allowed neutrino states are restricted only by energy conservation, and their number is given by the integration over d3nn d(Ei - Ef) which is proportional to the square of the neutrino energy en2. This, together with the neutrino energy term in Eq. (54), gives en3. Since en must be related to kBT, which is the only energy variable in the problem, all together they give the T8 dependence to the expression. The above deduction has general applicability, and if there were one less fermion in both the initial and final states, then the emissivity from such a process should be proportional to T6.

The emissivity due to reaction (53b), EUrca(b), can be shown to be of comparable magnitude to that evaluated above, and the total emissivity due to the modified Urca process is simply twice that of Eq. (56):

                         (57)

At high density, muons would appear alongside electrons, and neutrino production reactions similar to those of Eq. (53a) and (53b) become operable, but with the electrons there replaced by muons. Similar results are obtained, but muons do not appear in dense matter until its density exceeds 8 x 1014 g/cm3, and the emissivity due to the additional muon processes is only a minor correction.

A more significant consideration is the inclusion of nuclear interaction, which has been neglected in the above evaluation. Nuclear interaction appears in the evaluation of the matrix element H(i -> f). When nuclear interaction is included, the total emissivity due to modified Urca process is changed to

                             (58)

which is a factor of six higher than that evaluated without the inclusion of nuclear interaction.

Other important neutrino production mechanisms in neutron matter are

                                                               (59a)

                                                                (59b)

in each of which a pair of neutrinos is produced as the nucleons scatter from each other. Neutrino emissivities for these processes are evaluated to be

                                         (60a)

                                         (60b)

where the subscript nn denotes reaction (59a) and np denotes (59b). These emissivities are evaluated with nuclear interaction taken into consideration. The processes are, however, not as effective as the modified Urca processes in neutrino production.

If neutron matter turns superfluid after its temperature falls below the critical temperature Tc, then the neutrino production rates evaluated above must be reduced. Tc should be in the range of 109-1010 K. Superfluidity is explained by the fact that an energy gap appears in the energy spectrum of the particle, just above its Fermi energy. What that means is that a group of quantum states whose energies fall in the energy gap are excluded from the system. Normally, in an inelastic scattering process, the neutrons or protons are scattered into these states; but since the states are absent, they must be excited into much higher energy states, and thus scattering becomes difficult and less likely to occur. The consequence is that neutrons and protons may move about relatively freely without being impeded by scatterings. This is the explanation of superfluidity. By the same token, the above neutrino production mechanisms that depend on the scattering of neutrons and protons are similarly reduced. Qualitatively, if superfluidity occurs in both the neutron and proton components, EUrca and Enp are reduced by a factor of exp[-(Dn + Dp)/kBT], where Dn and Dp are the widths of the energy gaps for neutron and proton superfluidity, respectively. For Enn, the reduction is exp[-2Dn/kBT]. If superfluidity occurs in just one component, the reductions are obtained by setting the energy gap of the normal component in the above expressions to zero.

In the case of pion condensation in neutron matter, interactions involving pions and nucleons also modify the neutrino production rate. Some estimates have shown that the neutrino production rates thus modified can be significant. However, due to the uncertainty in our present understanding of the pion condensation problem, no emissivity for this situation will be quoted here.

There are also neutrino production mechanisms not involving the direct interaction of two nucleons, such as the following

    1. Pair annihilation

                                             (61a)      

    2. Plasmon decay

                     (61b)

    3. Photoannihilation

                    (61c)

    4. Bremsstrahlung

          (61d)

    5. Neutronization

                 (61e)

where e+ denotes a positron, plasmon a photon propagating inside a plasma, and (Z,A) nucleus of atomic number Z and mass number A. A photon in free space cannot decay into a neutrino pair, because energy and momentum cannot be conserved simultaneously in the process. However, when a photon propagates inside a plasma, its relation between energy and momentum is changed in such a way that the decay becomes possible. Quanta of the electromagnetic wave in a plasma are called plasmons. The neutrino production rates for these processes have been found to be relatively insignificant at typical neutron star densities and temperatures and will not be listed here.

B. Neutrino Opacity

When a radiation beam of intensity I(erg/cm2-sec) is incident on a substance of density r(g/cm3), the amount of energy absorbed from the beam per unit volume per unit time E (erg/cm3-sec) is proportional to the opacity k = E/rI. The opacity is expressed in units of cm2/g. Each neutrino emission process has an inverse process corresponding to absorption. In addition to absorption, scattering can also impede the passage of neutrinos through the medium. Both absorption and scattering contribute to the opacity. Some of the more important processes are listed below

    1. Scattering by neutrons                 

                        (62a)      

    2. Scattering by protons                     

                        (62b)

    3. Scattering by electrons

                        (62c)

    4. Scattering by nuclei             

                           (62d)

    5. Absorption by nucleons

                        (62e)

    6. Absorpion by nuclei                       

                      (62f)

Similar processes occur for anti-neutrinos, which shall not be displayed here.

For each of these reactions, a reaction cross section is evaluated from the Weinbery-Salam-Glashow theory of weak interactions. The cross section represents an area effective in obstructing the incident beam of neutrinos. The opacity may be expressed in terms of the reaction cross sections as follows:

                              (63)

where nj denotes the number density of the particles that react with the neutrino and sj their reaction cross sections, and the summation S is over all the reactions listed above. Often, neutrino opacity is expressed by a neutrino mean free path, which is defined as

                                               (64)

Among these reactions, the contribution of reaction (62f) to the opacity of dense matter is quite negligible because it entails the production of electrons; and since electrons in the system are already highly degenerate, it is difficult to accommodate the newly produced electrons.

The most important reaction in this regard is reaction (62d) where the neutrino is scattered by the nuclei. This is the result of coherent scattering, in which all nucleons in a nucleus participate as a single entity in the process. The cross section of a coherent process involving A nucleons is proportional to A2 times the cross section of scattering from a single nucleon. This reaction therefore dominates all others when the matter system is composed of giant nuclei, which is the case when matter density is below nuclear density, that is, before nuclei dissolve into neutron matter.

For neutron matter the important reactions are scatterings by neutrons, protons, and electrons as indicated by reactions (62a), (62b), and (62c). Neutrinos are scattered elastically by the nucleons and nuclei since the scatterers are massive. The neutrinos may change directions after scattering but do not lose their energies to the scatterers. They lose energy only if they are scattered by electrons. Electron scattering is therefore an important process in lowering the energies of the high-energy neutrinos, bringing them into thermal equilibrium with all neutrinos should the neutrinos be trapped in the system for a duration long enough for this to happen. Even though neutrinos interact very weakly and are therefore very difficult to confine, neutrino trapping is in fact believed to occur at the moment when the a collapsing stellar core reaches the point of rebound initiating the explosive supernova process. Therefore a great deal of attention has been given to the problem of neutrino opacity and the issue of neutrino trapping.

The cross sections for these processes in the reference frame of the matter system are evaluated to be as follows:

1. neutrino-electron scattering,

     (65)

2. neutrino-nucleon scattering,

    (66)

3. neutrino-nucleus scattering,

   (67)

where s0 = 1.76 x 10-44 cm2 is a typical weak interaction cross section, en the neutrino energy, and eF the Fermi energy of the electrons. The total neutrino opacity of the substance is given by

         (68)

According to the Weinberg-Salam-Glashow theory, neutrinos also interact with quarks, and therefore dense quark matter also emits and absorbs neutrinos. However, since our understanding of quark matter is still far from complete, no results related to neutrino emissivity and opacity in quark matter will be quoted at this time.

Bibliography

Chandrasekhar, S. (1939). "An Introduction to the Study of Stellar Structure," Dover, New York. 

Clayton, D. D. (1968). "Principles of Stellar Evolution and Nucleosynthesis," McGraw-Hill, New York. 

Gasiorowicz, S. (1979). "The Structure of Matter: A Survey of Modern Physics," Addison-Wesley, Reading, Massachusetts. 

Leung, Y. C. (1984). "Physics of Dense Matter," Science Press, Beijing, China. 

Schwarzschild, M. (1958). "Structure and Evolution of the Stars," Dover, New York. 

Shapiro, S. L., and Teukolsky, S. A. "Black Holes, White Dwarfs, and Neutron Stars, The Physicsof Compact Objects," Wiley, New YOrk. 

Zeldovich, Ya. B., and Novikov, I. D. (1971). "Relativistic Astrophysics," Chicago University Press, Chicago.