Electron-positron energy deposition near neutron and quark stars

Electron-positron energy deposition rate from neutrino pair annihilation in the equatorial plane of rapidly rotating neutron and quark stars

Z. Kovács, K. S. Cheng and T. Harko
Department of Physics and Center for Theoretical and Computational Physics, The University of Hong Kong,
Pok Fu Lam Road, Hong Kong, Hong Kong SAR, P. R. China
E-mail: zkovacs@mpifr-bonn.mpg.deE-mail: hrspksc@hkucc.hku.hkE-mail: harko@hkucc.hku.hk
Abstract

The neutrino-antineutrino annihilation into electron-positron pairs near the surface of compact general relativistic stars could play an important role in supernova explosions, neutron star collapse, or for close neutron star binaries near their last stable orbit. General relativistic effects increase the energy deposition rates due to the annihilation process. We investigate the deposition of energy and momentum due to the annihilations of neutrinos and antineutrinos in the equatorial plane of the rapidly rotating neutron and quark stars, respectively. We analyze the influence of general relativistic effects, and we obtain the general relativistic corrections to the energy and momentum deposition rates for arbitrary stationary and axisymmetric space-times. We obtain the energy and momentum deposition rates for several classes of rapidly rotating neutron stars, described by different equations of state of the neutron matter, and for quark stars, described by the MIT bag model equation of state and in the CFL (Color-Flavor-Locked) phase, respectively. Compared to the Newtonian calculations, rotation and general relativistic effects increase the total annihilation rate measured by an observer at infinity. The differences in the equations of state for neutron and quark matter also have important effects on the spatial distribution of the energy deposition rate by neutrino-antineutrino annihilation.

keywords:
neutrinos: dense matter – equation of state: stars: rotation: relativity.
pagerange: Electron-positron energy deposition rate from neutrino pair annihilation in the equatorial plane of rapidly rotating neutron and quark starsReferencespubyear: 2002pagerange: Electron-positron energy deposition rate from neutrino pair annihilation in the equatorial plane of rapidly rotating neutron and quark starsReferencespubyear: 2002

Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11

1 Introduction

Since the pioneering works of Cooperstein et al. (1986), Cooperstein et al. (1987), and Goodman et al. (1987), the energy deposition rate from the neutrino annihilation reaction has been intensively studied. This reaction is of considerable importance for Type II supernova dynamics, neutron star collapse, or for close neutron star binaries near their last stable orbit. Neutrino-antineutrino annihilation into electrons and positrons can deposit more than ergs above the neutrino-sphere of a type II supernova (Go87, ). This energy deposition, together with neutrino-baryon capture, significantly increases the neutrino heating in the envelope via the so-called delayed shock mechanism (BeWi85, ; Be90, ). For large the energy deposition rate is proportional to , where is the distance from the center of the neutrino-sphere. The initial estimations of the neutrino annihilation reaction efficiencies were based on a Newtonian approach, by assuming that , where is the gravitational mass of the star, and is the distance scale. However, for a full understanding of the effects of the neutrino annihilation in strong gravitational fields, general relativistic effects must be taken into account (SaWi99, ). For a static neutron star, by adopting for the description of the gravitational field the Schwarzschild metric, the efficiency of the process is enhanced over the Newtonian values up to a factor of more than in the regime applicable to Type II supernovae, and by up to a factor of for collapsing neutron stars (SaWi99, ). The neutrino pair annihilation rate into electron pairs between two neutron stars in a binary system was calculated by Salmonson & Wilson (2001). A closed formula for the energy deposition rate at any point between the stars was obtained, where each neutrino of a pair derives from each star, and this result was compared with that in which all neutrinos derive from a single neutron star. An approximate generalization of this formula was also given to include the relativistic effects of gravity. The interstar neutrino annihilation is a significant contributor to the energy deposition between heated neutron star binaries.

The neutrino-antineutrino annihilation into electrons and positrons is an important candidate to explain the energy source of the gamma ray bursts (GRBs) (Pa90, ; MeRe92, ; RuJa98, ; RuJa99, ; AsIw02, ). The semi-analytical study of the gravitational effects on neutrino pair annihilation near the neutrinosphere and around the thin accretion disk were considered in Asano & Fukuyama (2000), by assuming that the accretion disk is isothermal, and that the gravitational field is dominated by the Schwarzschild black hole. General relativistic effects were studied only near the rotation axis. The energy deposition rate is enhanced by the effect of orbital bending toward the center. However, the effects of the redshift and gravitational trapping of the deposited energy reduce the effective energy of the gamma-ray burst’s source. Although each effect is substantial, the effects partly cancel one another. As a result, the gravitational effects do not substantially change the energy deposition rate for either the spherically symmetric case or the disk case (AsFu00, ). Using idealized models of the accretion disk, Asano & Fukuyama (2001) investigated the relativistic effects on the energy deposition rate via neutrino pair annihilation near the rotation axis of a Kerr black hole, by assuming that the neutrinos are emitted from the accretion disk. The bending of neutrino trajectories and the redshift due to the disk rotation and gravitation were also taken into consideration. The Kerr parameter, , affects not only behavior of the neutrinos, but also the inner radius of the accretion disk. When the deposition energy is mainly contributed by the neutrinos coming from the central part, the redshift effect becomes dominant as a becomes large, and the energy deposition rate is reduced compared with that neglecting the relativistic effects. On the other hand, for a small , the bending effect becomes dominant and makes the energy increase by factor of 2, compared with that which neglects the relativistic effects (AsFu01, ).

The effect of the inclusion of the slow rotation of the star in the general relativistic treatment of the neutrino-antineutrino annihilation into electron positron pairs was considered in Prasanna & Goswami (2002). It was shown that the inclusion of the rotation results in a reduction in the heating rate, as compared to the no rotation case. The energy-momentum deposition rate (MDR) from the collisions above a rotating black hole/thin accretion disk system was calculated by Miller et al. (2003), by imaging the accretion disk at a specified observer using the full geodesic equations, and calculating the cumulative MDR from the scattering of all pairs of neutrinos and antineutrinos arriving at the observer. The dominant contribution to the MDR comes from near the surface of the disk with a tilt of approximately in the direction of the disk’s rotation. The MDR at large radii is directed outward in a conic section centered around the symmetry axis and is larger by a factor of 10-20 than the on-axis values. There is also a linear dependence of the MDR on the black hole angular momentum. The deposition of energy and momentum, due to the annihilation of neutrinos and antineutrinos in the vicinity of steady, axisymmetric accretion tori around stellar-mass black holes was investigated in Birkl et al. (2007). The influence of general relativistic effects were analyzed in combination with different neutrinosphere properties and spatial distribution of the energy deposition rate. Assuming axial symmetry, the annihilation rate 4-vector was numerically computed. The local neutrino distribution was constructed by ray-tracing neutrino trajectories in a Kerr space-time, using null geodesics. Different shapes of the neutrinospheres, spheres, thin disks, and thick accretion tori were studied, whose structure ranges from idealized tori to equilibrium non-selfgravitating matter distributions. Compared to Newtonian calculations, general relativistic effects increase the total annihilation rate measured by an observer at infinity by a factor of two when the neutrinosphere is a thin disk, but the increase is only 25% for toroidal and spherical neutrinospheres. Thin disk models yield the highest energy deposition rates for neutrino-antineutrino annihilation, and spherical neutrinospheres the lowest ones, independently of whether general relativistic effects are included. General relativity and rotation cause important differences in the spatial distribution of the energy deposition rate by neutrino and antineutrino -annihilation (Bi07, ). The study of the structure of neutron star disks based on the two-region (i.e., inner and outer) disk scenario was performed by Zhang & Dai (2009), who calculated the neutrino annihilation luminosity from the disk in various cases. The effects of the viscosity parameter , energy parameter (measuring the neutrino cooling efficiency of the inner disk), and outflow strength on the structure of the entire disk, as well as the effect of emission from the neutron star surface boundary emission on the total neutrino annihilation rate were investigated. An outflow from the disk decreases the density and pressure, but increases the thickness of the disk. Moreover, compared with the black hole disk, the neutrino annihilation luminosity above the neutron star disk is higher, and the neutrino emission from the boundary layer could increase the neutrino annihilation luminosity by about one order of magnitude higher than the disk without boundary emission. The neutron star disk with the advection-dominated inner disk could produce the highest neutrino luminosity, while the disk with an outflow has the lowest(ZhDa09, ). A detailed general relativistic calculation of the neutrino path for a general metric describing a rotating star was studied in Mallick & Majumder (2009). The neutrino path was calculated along the equatorial and polar plane. The expression for the minimum photosphere radius was obtained and matched with the Schwarzschild limit. The minimum photosphere radius was calculated for stars with two different equations of state, each rotating with two different velocities. The results show that the minimum photosphere radius for the hadronic star is much greater than for the quark star, and that the minimum photosphere radius increases as the rotational velocity of the star decreases. The minimum photosphere radius along the polar plane is larger than that along the equatorial plane. The estimate of the energy deposition rate for neutrino pair annihilation for the neutrinos coming from the equatorial plane of a rotating neutron star was calculated along the rotation axis in Bhattacharyya et al. (2009), by using the Cook-Shapiro-Teukolsky metric. The neutrino trajectories, and hence the neutrino emitted from the disk, are affected by the redshift due to the disk rotation and gravitation. The energy deposition rate is very sensitive to the value of the temperature, and its variation along the disk. The rotation of the star has a negative effect on the energy deposition rate, it decreases with increase in rotational velocity. The standard model for Type II supernovae explosions, confirmed by the detection of neutrinos emitted during the supernova explosion, predicts the formation of a compact object, usually assumed to be a neutron star (Cha09, ). However, the newly formed neutron star at the center of SN 1987A may undergo a phase transition after the neutrino trapping timescale ( s). Consequently the compact remnant of SN 1987A may be a strange quark star, which has a softer equation of state than that of neutron star matter (Cha09, ). Such a phase transition can induce stellar collapse and result in large amplitude stellar oscillations. A three-dimensional Newtonian hydrodynamic code was used to study the time evolution of the temperature and density at the neutrinosphere. Extremely intense pulsating neutrino fluxes, with submillisecond period and with neutrino energy (greater than 30 MeV), can be emitted because the oscillations of the temperature and density are out of phase almost . The dynamical evolution of a phase-transition-induced collapse of a neutron star to a hybrid star, which consists of a mixture of hadronic matter and strange quark matter, was studied in Cheng et al. (2009). It was found that both the temperature and the density at the neutrinosphere are oscillating with acoustic frequency. Consequently, extremely intense, pulsating neutrino/antineutrino fluxes will be emitted periodically. Since the energy and density of neutrinos at the peaks of the pulsating fluxes are much higher than the non-oscillating case, the electron/positron pair creation rate can be enhanced dramatically. Some mass layers on the stellar surface can be ejected, by absorbing energy of neutrinos and pairs. These mass ejecta can be further accelerated to relativistic speeds by absorbing electron/positron pairs, created by the neutrino and antineutrino annihilation outside the stellar surface.

It is the purpose of the present paper to consider a comparative systematic study of the neutrino-antineutrino annihilation process around rapidly rotating neutron and strange stars, respectively, and to obtain the basic physical parameters characterizing this process (the electron-positron energy deposition rate per unit volume and unit time, and the total emitted power, respectively), by taking into account the full general relativistic corrections. In order to obtain the electron-positron energy deposition rate for various types of neutron and quark stars we generalize the relativistic description of the neutrino-antineutrino annihilation process to the case of arbitrary stationary and axisymmetric geometries. To compute the electron-positron energy deposition rate, the metric outside the rotating general relativistic stars must be determined. In the present study we study the equilibrium configurations of the rotating neutron and quark stars by using the RNS code, as introduced in SteFr95 , and discussed in detail in Sterev . This code was used for the study of different models of rotating neutron stars in No98 and for the study of the rapidly rotating strange stars (Ste99 ). The software provides the metric potentials for various types of compact rotating general relativistic objects, which can be used to obtain the electron-positron energy deposition rate in the equatorial plane of rapidly rotating neutron and quark stars.

The present paper is organized as follows. In Section 2 we present the basic formalism for the calculation of the electron-positron energy deposition rate from neutrino-antineutrino annihilation. The general relativistic corrections to this process are obtained in Section 3. The equations of state of dense neutron and quark matter used in the present study are presented in Section 4. In Section 5 we obtain the electron-positron energy deposition rates in the equatorial plane of the considered classes of neutron and quark stars. We discuss and conclude our results in Section 6.

2 Neutrino pair annihilation

A considerable amount of energy can be released by the neutrino pair annihilation process in the regions close to the so called neutrino-sphere, with radius , at which the mean free path of the neutrino is equal to the radius itself (BeWi85, ).

The energy deposition rate per unit volume is given by

(1)

for any point , where and are the number densities in the momentum spaces with the momenta and , and , , and are the 3-velocities and the energy of the colliding neutrino-antineutrino pairs, respectively Go87 . The cross-section of the collision is denoted by .

By applying the decompositions and , with the solid angle vector pointing in the direction of , and with the assumption that the neutrino-sphere emits particles isotropically, the integral in Eq. (1) can be separated into an energy integral and an angular part. After evaluating the energy integral for fermions, for a spherically symmetric geometry the energy deposition rate per unit volume can be represented as

(2)

where is the neutrino temperature, is the Riemann function, , with , and , respectively, and with the angular part of the energy deposition rate given by

(3)

The radial momentum density transported into the plasma from the colliding pairs can be written as (SaWi99, )

(4)

where

(5)

with the unit vector normal to the stellar surface.

Since the integrals in Eqs. (3) and (5) depend only on the radial coordinate , by virtue of the symmetry, the angular part and the function can be given by the analytic formulae

(6)

and

(7)

where , and is the maximal angle between and (Go87, ; SaWi99, ). In order to determine the quantity , one need to use the equations of motion of the neutrinos radiated by the stellar matter, and propagating outside the neutrino-sphere. The equations of motion in the Newtonian case, or the geodesic equations in the general relativistic case fully determine as a function of the radial coordinate. Therefore, the properties of the gravitational potential produce an imprint on the energy deposition rate calculated in the region close to the neutrino-sphere. Salmonson & Wilson (1999) extended the calculations of Goodman et al. (1987) to the case of the Schwarzschild geometry, and compared their results to those of the Newtonian case.

Although rotating configurations of neutron stars break the spherical symmetry of the space-time, one can still carry out an analysis similar to the static case if the study is restricted to the investigation of the annihilation of neutrino and antineutrino pairs propagating in the equatorial plane of the rotating star. By eliminating the angular dependence from the equations, a formalism similar to the spherically symmetric case can be used to calculate the energy deposition rate in the equatorial plane. The obtained result is not equal to the total deposition rate of the high energy electron positron pairs created in the annihilation process. However, this quantity can still be applied in the comparison of the neutrino and antineutrino annihilation energy deposition rate for different models of rotating neutron and quark stars, or in studying the general effects of the rotation of the stellar object on this process.

3 General relativistic effects on the electron-positron energy deposition rate

The metric of a stationary and axisymmetric geometry is given in the general form by

(8)

For this metric the null-geodesics equations in the equatorial plane are

(9)
(10)
(11)

where is the energy, and is the angular momentum of the particles propagating along the null-geodesics. Then the parametric equation for can be written as

(12)

where the impact parameter is defined as .

For the metric given by Eq. (8), the locally non-rotating frame (LNRF) (in which the wordlines of the freely falling observers are constant, = constant and constant, respectively, with ) has the basis of one-forms (BPT72, )

Since the velocity measured in the LNRF is given by , the angle between the particle trajectory and the tangent vector to the circular orbit with the radial coordinate can be written as

From this expression we obtain for the equation

(13)

By inserting Eqs. (9) and (10) into the definition of , we obtain for the expression

which can be substituted into Eq. (13) to give

Then the derivative can be eliminated from the parametric equation (12) and we obtain

This result gives a second order algebraic equation for ,

(14)

which can be solved to give the impact parameter as

(15)

A particular system of coordinates that is used in the study of the general-relativistic rotating configurations is the quasi-isotropic coordinate system , in which the line element can be represented as (SteFr95, ; Sterev, )

(16)

where , , and the angular velocity of the stellar fluid relative to the local inertial frame are all functions of the quasi-isotropic radial coordinate and of the polar angle .

If for neutron stars the metric (8) is given in an isotropic coordinate system in the form (16), then the second order algebraic equation  (14) for can be written in terms of the metric functions and as

For the impact parameter, corresponding to , we obtain Ca07

From the parametric equation Eq. (12) we obtain the deflection angle of the particle trajectory for a given as

(17)

In this equation measures the change in the angle between the source and the observer, for a photon emitted at the radial coordinate , and observed at the radial coordinate . This equation can also be given in the equatorial plane in terms of the metric functions and , respectively, appearing in the line element Eq. (16) Ca05

In the equatorial plane of black holes the photon radius is defined as the innermost boundary of circular orbits below which massless particles with are gravitationally bound (BPT72, ). For static black holes the photon radius is , and for rotating black holes it reduces to , as the spin parameter of the black hole approaches unity. This orbit may exist for ultra-compact stars as well. For static stars with a stellar radius less than there is always such a ”photon sphere”, whereas, depending on the geometry of the space-time, the rotating stars can also have a photon radius at both lower and higher radii MaMa09 . On the other hand, very massive rotating quark stars in the Color-Flavor-Locked phase can reach masses higher than the equilibrium limit for static stars (), of the same order as the stellar mass black holes, and thus they can also have a photon radius Ko09 .

If the integral in Eq. (17) diverges to infinity (or tends to zero), the null particles are rotating around the central objects in circular orbits. In this case , and the algebraic equation Eq. (14) for reduces to , with the solution

(18)

For any value of satisfying Eq. (18), the integral (17) is divergent, and the null particles have circular orbits. In Eq. (18) the impact parameter in the equatorial plane is given as a function of the radial coordinate only.

In the case of the ultracompact static stars with radii less than the (local) maximum of the function is located at , providing the photon radius. If , then is a monotonically decreasing function, without a local maximum. For rotating compact stars, the function provides the same criterion for the existence of the potential barrier: for massless particles the equatorial orbit where attains its local maximum defines the innermost boundary of circular orbits. Even if this value is less than the equatorial radius , neutrinos are still free to propagate along orbits lying on the photon radius.

By assuming that the mean free path of the neutrinos is equal to or less than the photon radius, Salmonson & Wilson (1999) identified the photon and the neutrino spheres with each other. Accordingly, we will also consider the orbit at the photon radius as the minimal radius where the annihilation process should still be taken into account, provided it is outside the star. The contribution of the electron-positron pairs formed inside the star to the deposition rate is neglected, because of their complicated interactions with the neutron and quark matter.

Since the impact parameter measured at infinity is constant along the trajectory of any null particle, the neutrinos propagating from a point at the photon sphere radius (or the stellar surface ) with the angle , will reach another point with radial coordinate with the angle

(19)

Eq. (19) allows to express in the analytic expression of the angular part of the energy, given by Eq. (6), as

(20)

For the line element given by Eq. (16), Eq. (20) has the form

(21)

Thus the function can be represented in the general form

(22)

In quasi-isotropic coordinates we have

The neutrino temperature at the radius can be expressed in terms of the temperature of the neutrino stream at the neutrino-sphere radius , by taking into account the gravitational redshift. The redshift formula for is the same as for the photon energy,

(24)

For the observed luminosity of the neutrino annihilation the redshift relation is given by

(25)

since for isolated gravitating systems, such as rotating stars, the spacetime is asymptotically flat. Here the neutrino luminosity at the neutrino-sphere is

(26)

where is the radiation constant. In this formula the curvature radius is used to obtain the total area of the spherical surface through which the neutrino radiation is emitted. If we insert Eq. (20), describing the path-bending of the neutrinos, and the redshift formulae Eqs.  (24)-(26) into the decomposed expression Eq. (2) of , we can calculate the effects of the gravitational potential on the deposition rate in the equatorial plane:

(27)

The proportionality factor, omitted from Eq. (27), is the same as the one in the Newtonian case, which will be used in the following to normalize the deposition rate for the general relativistic case. Eq. (27) describes the energy deposition rate in pairs from the neutrino-antineutrino annihilation process at radius in the equatorial plane above the neutron or quark star photon sphere radius , and with the neutrino luminosity observed at infinity . This relation can be also given in the quasi-isotropic coordinate system in terms of the metric functions of the line element Eq. (16),

(28)

4 Equations of state and stellar models

In order to obtain a consistent and realistic physical description of the rotating general relativistic neutron and quark stars, as a first step we have to adopt the equations of state for the dense neutron and quark matter, respectively. In the present study we consider the following equations of state for neutron and quark matter:

1) Akmal-Pandharipande-Ravenhall (APR) EOS (Ak98, ). EOS APR has been obtained by using the variational chain summation methods and the Argonne two-nucleon interaction. Boost corrections to the two-nucleon interaction, which give the leading relativistic effect of order , as well as three-nucleon interactions, are also included in the nuclear Hamiltonian. The density range is from g/cm to g/cm. The maximum mass limit in the static case for this EOS is . We join this equation of state to the composite BBP (g/cm) (Ba71a, ) - BPS ( g/cm g/cm) (Ba71b, ) - FMT ( g/cm) (Fe49, ) equations of state, respectively.

2) Douchin-Haensel (DH) EOS (DoHa01, ). EOS DH is an equation of state of the neutron star matter, describing both the neutron star crust and the liquid core. It is based on the effective nuclear interaction SLy of the Skyrme type, which is particularly suitable for the application to the calculation of the properties of very neutron rich matter. The structure of the crust, and its EOS, is calculated in the zero temperature approximation, and under the assumption of the ground state composition. The EOS of the liquid core is calculated assuming (minimal) composition. The density range is from g/cm to g/cm. The minimum and maximum masses of the static neutron stars for this EOS are and , respectively.

3) Shen-Toki-Oyamatsu-Sumiyoshi (STOS) EOS (Shen, ). The STOS equation of state of nuclear matter is obtained by using the relativistic mean field theory with nonlinear and terms in a wide density and temperature range, with various proton fractions. The EOS was specifically designed for the use of supernova simulation and for the neutron star calculations. The Thomas-Fermi approximation is used to describe inhomogeneous matter, where heavy nuclei are formed together with free nucleon gas. We consider the STOS EOS for several temperatures, namely , and MeV, respectively. The temperature is mentioned for each STOS equation of state, so that, for example, STOS 0 represents the STOS EOS for . For the proton fraction we chose the value in order to avoid the negative pressure regime for low baryon mass densities.

4) Relativistic Mean Field (RMF) equations of state with isovector scalar mean field corresponding to the -meson- RMF soft and RMF stiff EOS (Kubis, ). While the -meson mean field vanishes in symmetric nuclear matter, it can influence properties of asymmetric nuclear matter in neutron stars. The Relativistic mean field contribution due to the -field to the nuclear symmetry energy is negative. The energy per particle of neutron matter is then larger at high densities than the one with no -field included. Also, the proton fraction of -stable matter increases. Splitting of proton and neutron effective masses due to the -field can affect transport properties of neutron star matter. The equations of state can be parameterized by the coupling parameters , , and , where and are the masses of the respective mesons, and and are the coefficients in the potential energy of the -field. The soft RMF EOS is parameterized by fm, fm, and , while the stiff RMF EOS is parameterized by fm, fm, and , respectively.

5) Baldo-Bombaci-Burgio (BBB) EOS (baldo, ). The BBB EOS is an EOS for asymmetric nuclear matter, derived from the Brueckner-Bethe-Goldstone many-body theory with explicit three-body forces. Two EOS’s are obtained, one corresponding to the Argonne AV14 (BBBAV14), and the other to the Paris two-body nuclear force (BBBParis), implemented by the Urbana model for the three-body force. The maximum static mass configurations are and when the AV14 and Paris interactions are used, respectively. The onset of direct Urca processes occurs at densities fm for the AV14 potential and fm for the Paris potential. The comparison with other microscopic models for the EOS shows noticeable differences. The density range is from g/cm to g/cm.

6) Bag model equation of state for quark matter (Q) EOS (It70, ; Bo71, ; Wi84, ; Ch98, ). For the description of the quark matter we adopt first a simple phenomenological description, based on the MIT bag model equation of state, in which the pressure is related to the energy density by

(29)

where is the difference between the energy density of the perturbative and non-perturbative QCD vacuum (the bag constant), with the value g/cm.

7) It is generally agreed today that the color-flavor-locked (CFL) state is likely to be the ground state of matter, at least for asymptotic densities, and even if the quark masses are unequal (cfl1, ; cfl2, ; HoLu04, ; cfl3, ). Moreover, the equal number of flavors is enforced by symmetry, and electrons are absent, since the mixture is automatically neutral. By assuming that the mass of the quark is not large as compared to the chemical potential , the thermodynamical potential of the quark matter in CFL phase can be approximated as (LuHo02, )

(30)

where is the gap energy. With the use of this expression the pressure of the quark matter in the CFL phase can be obtained as an explicit function of the energy density in the form (LuHo02, )

(31)

where

(32)

and . In the following the value of the gap energy considered in each case will be also mentioned for the CFL equation of state, so that, for example, CFL200 represents the CFL EOS with . For the bag constant we adopt the value g/cm, while for the mass of the strange quark we take the value MeV.

The pressure-density relation is presented for the considered equations of state in Fig. 1.

Figure 1: Pressure as a function of density (in a logarithmic scale) for the equations of state DH, RMF soft, RMF stiff, STOS 0. STOS 0.5, STOS 1, BBBAV14, BBBParis, APR, Q, CFL150, and CFL300, respectively.

To calculate the equilibrium configurations of the rotating neutron and quark stars with the EOS’s presented here we use the RNS code, as introduced in Stergioulas & Friedman (1995), and discussed in details in Stergioulas (2003). This code was used for the study of different models of rotating neutron stars (No98, ), and for the study of the rapidly rotating strange stars (Ste99, ). The RNS code produces the metric functions in a quasi-spheroidal coordinate system, as functions of the parameter , where is the equatorial radius of the star, which we have converted into Schwarzschild-type coordinates according to the equation .

5 Electron-positron energy deposition rate in the equatorial plane of rapidly rotating neutron and quark stars

To demonstrate the existence and the location of the photon radius for neutron and quark stars in the static and the rotating cases, respectively, in Fig. 2 we present as a function of the Schwarzschild coordinate radius (normalized to the mass of the star). For the static neutron star (modeled, for example, by the STOS 0.5 EOS), the exterior geometry is always the Schwarzschild one, and the curve connects to the curve of the static black hole at a radius somewhat lower than . Then, the curve has a local minimum at , similarly to the case of the function , corresponding to the Schwarzschild case (LaLi, ). Therefore, the star has a photon radius at . This is not the case for the static quark star (with an EOS of the CFL 300 type), which is not so compact, and its curve reaches the Schwarzschild black hole curve at a somewhat higher radius than . The radial dependence of for rotating stars has also these two features: the metric potentials of the neutron star with RMF stiff type EOS and the Q and CFL type quark stars give local minima for , whereas the neutron star with STOS 0.5 type EOS is not compact enough to have a photon radius (see Table 4). We note that the photon radii of the rotating stellar objects are located at higher radii than , which is opposite to the behavior of the rotating black holes, where the photon radii approach as the black hole spins up.

Figure 2: The impact parameter as a function of the Schwarzschild coordinate for static black holes, and static and rotating neutron and quark stars, respectively.

The deposition rate of the total energy of the pairs generated in the neutrino annihilation is usually characterized by the integral of over the spatial proper volume of the neutrino stream. The integral of in the radial direction, measuring the total amount of energy converted form neutrinos to electron-positron pairs at all radii , is then given by

(33)

For a spherically symmetric geometry the integration over the angular coordinates and is a straightforward computation. This is not the case in the axially symmetric case, where the -dependence of the metric as well as the nature of the null trajectories is much more complicated. In order to avoid this problem, we restrict the study of the electron-positron energy deposition rate to neutrino pairs moving in the equatorial plane only. Instead of the quantity defined by Eq. (33), we consider its derivative with respect to in the equatorial plane,

(34)

, and therefore its -derivative, is neither an observable, nor a Lorentz-invariant quantity SaWi99 , but can be used to measure the total amount of local energy deposited via the neutrino pair annihilation outside the neutrino-sphere. The derivative , evaluated at allows one to describe the energy deposition rate only in the equatorial plane, but its value normalized to the Newtonian case can be applied to compare the energy deposition rates in different gravitational potentials. Since the quantity given by Eq. (34) has a simple form in the Newtonian case, the ratio of the general relativistic case to the Newtonian model is given by

(35)

Here is the deposition rate calculated by taking into account the general relativistic effects, whereas is the deposition rate for the simple Newtonian case, without taking into account the bending of the neutrino path and the redshift in the neutrino temperature. We will use the ratio given by Eq. (35) to compare the electron-positron energy deposition rates in the space-times of neutron and quark stars, with different equations of state.

Figure 3: The electron-positron energy deposition rate in the equatorial plane around rotating neutron and quark stars, with the same total mass , and same rotational velocity .

In Fig. 3 we present the ratio given by Eq. (35) as a function of the neutrino-sphere radius, measured in curvature coordinates, for each neutron and quark star model previously described. The total mass of the central objects is set to , and the rotational frequency of the stars is about . For this configuration the physical parameters of the stars are shown in Table 1.

EOS DH RMF stiff STOS 0 STOS 0.5 STOS 1 BBBAV14 BBBParis APR Q CFL150
1.29 0.57 0.369 0.383 0.40 2.15 1.70 1.225 0.931 0.71
1.81 1.80 1.85 1.80 1.79 1.80 1.80 1.80 1.79 1.80
2.05 2.00 2.01 1.95 1.93 2.08 2.07 2.11 2.09 2.09
12.01 15.79 21.03 22.84 22.72 10.57 10.98 10.99 11.79 12.36
4.99 5.00 4.90 4.71 4.45 5.01 5.00 5.00 4.79 5.00
11.16 7.97 5.37 4.56 4.54 13.96 13.19 13.23 11.97 11.28
3.43 8.93 14.91 12.25 9.83 2.35 2.66 3.41 4.45 5.64
1.15 2.00 2.95 2.49 2.15 0.95 1.00 1.12 1.28 1.50
2.03 3.52 5.30 4.65 4.25 1.66 1.76 1.98 2.35 2.63
8.54 43.82 106.53 80.44 61.22 4.42 5.43 7.87 1.30 1.91
6.85 0.00 0.00 0.00 0.00 3.19 2.69 0.00 0.00 0.00
7.48 -3.40 7.91 3.87 2.27 8.14 7.90 -2.11 0.00 0.00
5.85 4.52 4.10 4.07 4.08 6.67 6.34 5.86 5.27 5.00
9.10 12.85 18.00 19.97 19.91 7.64 8.06 8.05 8.86 9.40
0.88 0.72 0.54 0.53 0.58 0.92 0.91 0.90 0.87 0.84
Table 1: Physical parameters of the compact stars with total mass and rotational frequency . Here is the central density, is the gravitational mass, is the rest mass, is the circumferential radius at the equator, is the angular velocity, is the angular velocity of a particle in circular orbit at the equator, is the rotational-gravitational energy ratio, is the angular momentum, is the moment of inertia, gives the mass quadrupole moment so that , is the height from the surface of the last stable co-rotating circular orbit in the equatorial plane, is the height from surface of the last stable counter-rotating circular orbit in the equatorial plane, is the ratio of the central value of the potential to , is the coordinate equatorial radius, and is the axes ratio (polar to equatorial), respectively.

From the analysis of the compactness of these stars, that is, of the ratio of the total mass to the equatorial radius, we see that the mass stars are not compact enough to have a photon or a neutrino-sphere. In other words, the impact parameter given by Eq. (18) is a monotonic function of the radial coordinate, without any minimum. For comparison we also plotted the static case with the same total mass, which is essentially the same for any type of the equation of state describing the properties of the dense star matter, since the exterior metric of the static stars is described by the Schwarzschild metric. The energy deposition rate for this case coincides with the versus plot given by (SaWi99, ), since the multiplicative factor coming from the integration of over is unity for the Schwarzschild space-time. From Fig. 3 it can be immediately seen how the energy deposition rate of the electron-positron pairs is enhanced by the rotation of the central object. It is also clear that the measure of enhancement depends on the type of the equation of state used to describe the dense neutron and quark stars. In the rotating case, the increase in the ratio of the energy deposition rates, given by Eq. (35), is the smallest, as compared to the static case, for the BBBAV14 and BBBParis type EOS’s. The neutron stars with DH and APR type EOS’s produce roughly the same ratios, which are somewhat greater than the BBB type neutron star pair production rates. For the quark stars, described by the Q and the CFL type EOS’s, we obtain even higher deposition rates. Although none of these types of central objects possesses neutrino-spheres, they are still compact enough to have equatorial radii less than . The equatorial radii of the neutron stars with the RMF and STOS type EOS’s are much larger than those of the previous group: for the RMF type EOS, and or even more for the STOS type stars, depending on the temperature of these stellar configurations. This means that although they give the highest deposition rates, with the STOS Mev type star having the maximal value, the energy released by the neutrino pair annihilation outside the star in the equatorial plane is smaller than in the case of the first group. This result might be not true in the regions close to the poles of the star, if the axis ratios would be considerably smaller for the second group as compared to the axis ratios of the first group. In the case of the STOS type EOS is between 0.5 and 0.6, which is indeed much smaller than 0.85-0.9, the average values of the axis ratios for the first group. This means the difference between the polar radii of the fist group, and the STOS type neutron star, is smaller than the difference between their equatorial radii, and the lower boundary of the integral of over the radial coordinate are closer to each other. In this case, the integrated deposition rate can be higher for the neutron star with STOS type EOS than the one with the other types of equations of state for neutron stars and quark stars.

However, in this framework one should be very cautious with any statement on the physical processes located far from the equatorial plane, since the neutrinos reaching the region close to the pole of the rotating stars have impact parameters rather different from the neutrinos moving in the equatorial plane, and the formalism applied for the latter cannot be extended straightforwardly to the motion outside the equatorial plane.

EOS RMF stiff STOS 1.0 Q CFL 300
2.10 0.640000 1.60000 0.394500
2.79986 2.8098 2.80017 2.80677
3.28120 3.19101 3.34023 3.28947
14.8969 19.6359 15.4159 18.0389
9.78914 5.90127 10.0016 5.96099
10.4349 6.99013 10.4087 8.35765
1.30916 1.09688 2.00292 1.50674
5.49201 5.05616 6.64450 6.02253
4.93060 7.52988 5.83855 8.87919
5.68507 8.48428 9.63523 13.4904
1.93604 0.306693 - 0.00000
19.6015 15.1710 21.4442 -
8.23642 6.2819 7.57682 5.75394
9.99343 14.9859 10.2897 13.1041
0.590234 0.640000 0.502344 0.620000
Table 2: Physical parameters of the compact stars with total mass and rotational frequency and .

Next, we consider the electron-positron energy deposition rate for a more massive group of stars, with the total mass set to . Although this value of the mass is smaller than the theoretical stability mass limit of 3 Solar masses for ultra-compact objects, not all types of the EOS’s considered here do have configurations of such high masses, even in the rapidly rotating case. We have obtained solutions to the field equations for this mass regime only for the quark stars, for the RMF, and for the STOS type neutron star EOS’s, respectively. For the other equations of state we did not even find a compatible rotation frequency in the high mass regime. For the stable massive configurations the angular velocity varied up to