Emissivities of Neutrinos in Neutron Star Cores

VOLUME 76, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 18 MARCH 1996

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cores, protona self-ges are

1994

Emissivities of Neutrinos in Neutron Star Cores

Ø. Elgarøy, L. Engvik, and E. OsnesDepartment of Physics, University of Oslo, N-0316 Oslo, Norway

F. V. De Blasio, M. Hjorth-Jensen, and G. LazzariECT , European Centre for Theoretical Studies in Nuclear Physics and Related Areas, Trento, I

(Received 24 October 1995; revised manuscript received 17 January 1996)

In this work we consider various URCA processes relevant for neutrino emissions in theregion of neutron stars. The parameters which enter these processes, such as effective massepairing gaps, proton fractions, and electron and muon chemical potentials, are calculated inconsistent Brueckner-Hartree-Fock approach forb-stable nuclear matter. Modern meson-exchanpotential models are employed. We find that the emissivities from modified URCA processereduced due to the presence of superconducting protons at densities below0.4 fm23.

PACS numbers: 26.60.+c, 74.25.Bt

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The thermal evolution of a neutron star may proviinformation about the interiors of the star, and in receyears much effort has been devoted to measuring neustar temperatures, especially with the Einstein Observaand ROSAT, see, e.g., Ref. [1]. The main cooling mecanism in the early life of a neutron star is believed tothrough neutrino emissions in the core of the neutron s[2,3]. The most powerful energy losses are expected togiven by the so-called direct URCA mechanism

n °! p 1 l 1 nl , p 1 l °! n 1 nl , (1)

as discussed recently by several authors [2,4,5]. Tlabel l refers to the leptons considered here, electronsmuons. However, in the outer cores of massive neutstars and in the cores of not too massive neutron starsfM ,

s1.3 2 1.4dMØg, the direct URCA process with electronis allowed at densities where the momentum conservakn

F , kpF 1 ke

F is fulfilled. (Throughout this work wewill reserve lettersn, p, e, and m to neutrons, protonselectrons, and muons, respectively. Also, we will seth c 1.) This happens only at densitiesr several times thenuclear matter saturation densityr0 0.16 fm23.

Thus, for a long time the dominant processes for ntrino emission have been the so-called modified URCprocesses first discussed by Chiu and Salpeter [6]which the two reactions

n 1 n °! p 1 n 1 l 1 nl ,

p 1 n 1 l °! n 1 n 1 nl (2)

occur in equal numbers. These reactions are just the uprocesses of neutronb decay, and electron and muocapture on protons of Eq. (1), with the addition of an exbystander neutron. They produce neutrino-antineutrpairs, but leave the composition of matter constantaverage. Equation (2) is referred to as the neutron braof the modified URCA process. Another branch is tproton branch

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aonche

n 1 p °! p 1 p 1 l 1 nl ,

p 1 p 1 l °! n 1 p 1 nl , (3)

pointed out by Itoh and Tsuneto [7] and recently reaalyzed by Yakovlev and Levenfish [8]. The latter athors showed that this process is as efficient as Eq.In addition, one also has the possibility of neutrino-pbremsstrahlung, processes with baryons more masthan the nucleon participating, such as isobars or hypons [4,5], or neutrino emission from more exotic statsuch as pion and kaon condensates [9,10] or quark mter [11,12]. Actually, Prakashet al. [4] showed that thehyperon direct URCA processes gave a considerable ctribution to the emissivity, without invoking exotic stateor the large proton fractions needed in Eq. (1).

The aim of this Letter is to give an estimate of the prcesses of Eqs. (2) and (3) at densities corresponding toouter core of massive neutron stars or the core of notmassive neutron stars, performing self-consistent nonrtivistic Brueckner-Hartree-Fock (BHF) calculations forb-stable nuclear matter. We impose the relevant equilibriconditions, with and without muons, employing modemeson-exchange potential models for the nucleon-nuclinteraction.

In addition, we study the role of superconducting prtons in the core of the star by calculating the pairing gfor protons in the1S0 state. The proton superconductivity reduces considerably the energy losses in the abreactions [8], and may have important consequencesthe cooling of young neutron stars. The final outcomeour calculations yields then the equation of state, effectmassesmp for protons and neutrons, their correspondiFermi momentakF , the Fermi momenta for electrons anmuons, and the proton pairing gap. These ingredientster into the calculation of the various reaction rates.

Here we focus on a nonrelativistic calculational prcedure, since variables like the nonrelativistic effectimasses enter to the determination of the modified URprocesses and the solution of the pairing gap discus

© 1996 The American Physical Society


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below. However, when discussing the equation of st(EOS) inb-stable matter, we also include results fromrelativistic Dirac-Brueckner-Hartree-Fock (DBHF). Thcalculational procedure for the DBHF calculation is dicussed in Ref. [13].

First, we solve self-consistently the BHF equationsthe single-particle energies, using aG matrix definedthrough the Bethe-Brueckner-Goldstone equation as

G V 1 V fQysv 2 H0dgG , (4)

whereV is the nucleon-nucleon potential,Q is the Paulioperator which prevents scattering into intermediate staprohibited by the Pauli principle,H0 is the unperturbedHamiltonian acting on the intermediate states, andv isthe so-called starting energy, the unperturbed energy ofinteracting states. Methods to solve this equation areviewed in Ref. [14]. The single-particle energies for staki (i encompasses all relevant quantum numbers like mmentum, isospin projection, spin, etc.) in nuclear mter are assumed to have the simple quadratic form´ki k2

i y2mp 1 di, wheremp is the effective mass. The termmp andd, the latter being an effective single-particle ptential related to theG matrix, are obtained through thself-consistent BHF procedure. The so-called model-spBHF method for the single-particle spectrum has beused, see, e.g., Ref. [14], with a cutoffkM 3.0 fm21.This self-consistency scheme consists of choosing aquate initial values of the effective mass andd. The ob-tainedG matrix is in turn used to obtain new values formp

andd for protons and neutrons. This procedure continuuntil these parameters vary little. The BHF equationssolved for different proton fractions, using the formalisof Refs. [14,15]. The conditions ofb equilibrium requirethat mn mp 1 me, wheremi is the chemical potentiaand that charge is conserved throughrp re, whereri

is the particle number density in fm23 for particle i. Ifmuons are present, we also need to satisfy charge convation rp re 1 rm and energy conservation througme mm. The nucleon-nucleon potential is defined bthe parameters of the meson-exchange potential modethe Bonn group, version A in Table A.2 of Ref. [16].

From the derived effective BHF and DBHF interations, one can in turn calculate the energy per nucleonfunction of the total density [14]. This is plotted in Fig.with and without muons together with the relevant protfractions. We also compare our results with those forb-stable matter with muons and electrons of Wiringa, Fiand Fabrocini [17], namely, those obtained with the Ubana potential UV14 including three-body forces (UVIIAs can be seen from this figure, muons yield contributioto the energy per particle at densities around0.15 fm23

for both the BHF and DBHF approaches. The preseof muons has only little effect on the energy per parcle and the resulting maximum masses for neutron stas discussed in Ref. [18]. The proton fractions chandramatically, however, at high densities with muons, athe DBHF muon results allow the direct URCA proce

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FIG. 1. Energy per baryonEyA and proton fractions asfunctions of the total baryonic density. See text for furthdetails. WFF88 refers to the results from Ref. [17].

of Eq. (1) with electrons at densities greater thanr 0.5 fm23, while that with muons starts atr 0.6 fm23.The corresponding densities for the nonrelativistic BHcalculation arer 0.62 and0.7 fm23. These are in turncomparable to the results of Wiringa, Fiks, and Fabroc[17], also based on a nonrelativistic approach. Howevthese authors include nonrelativistic three-body forcesincluded in our BHF calculation. These three-body forcyield an additional repulsion, and a stiffer EOS. ThEOS also results in maximum neutron star masses oforder of s2.1 2 2.2dMØ, comparable to those from ourelativistic EOS [13] (the relativistic effects can also binterpreted in terms of certain effective repulsive threbody forces [16]). These masses are too large compato the recent estimates of Finn [19]. Also, if we taka neutron star with total mass1.4MØ with our DBHFEOS, we would get central densities of the order0.4 fm23, with no possibility for direct URCA reactionsOur nonrelativistic calculations give a maximum massMmax 1.6MØ, in good agreement with present expermental constraints. The corresponding central density10 times the saturation density of nuclear matter, meanalso that the direct URCA process is a possible coolmechanism. However, our nonrelativistic BHF calcultion does not meet the empirical data for symmetric nclear matter, while these are met in the DBHF calculatioThus, there are strong model dependencies regardingEOS, and although relativistic degrees of freedom are ctainly present at densities larger thanr0, we will use ournonrelativistic scheme in the computation of the modifiURCA processes, since the expressions we use are gin a nonrelativistic scheme.

The next step is then to evaluate the gap equatfollowing the scheme of Baldoet al. [20]. These authors

1995


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introduced an effective interactionVk,k0 . This effectiveinteraction sums up all two-particle excitations abovecutoff momentumkM . It is defined according to

Vk,k0 Vk,k0 2X

k00.kM

Vk,k00

12Ek00

Vk00,k0 , (5)

where the quasiparticle energyEk is given by Ek qs´k 2 ´Fd2 1 D

2k, ´F being the single-particle energy

at the Fermi surface,Vk,k0 is the free nucleon-nucleonpotential in momentum space, andDk is the pairing gap

Dk 2X

k0#kM

Vk,k0

Dk0

2Ek0

. (6)

For notational economy, we have dropped the subscripion the single-particle energies.

In summary, first we obtain the self-consistent BHsingle-particle spectrum k for protons and neutrons inb-stable matter. This procedure gives the relevant protand neutron fractions at a given total density. Hereaftwe solve self-consistently Eqs. (5) and (6) in order to otain the pairing gapD for protons or neutrons. For stateabovekM , the quasiparticle energyEk is approximated byEk ´k 2 ´F , an approximation found to yield satisfactory results in neutron matter [21]. Our prescription thegives the pairing gap atT 0. The results for the pro-ton pairing gap are shown in Table I as functions of thtotal baryonic density. In this table we also show the efective masses for neutrons and protons, together withchemical potential for electrons. These results have beobtained in a nonrelativistic calculation with both muonand electrons present, which means that the chemicaltential for muons is equal to that of electrons for densitilarger than0.15 fm23.

The reader should notice that our calculation of thpairing gap does not include complicated many-bodcontributions like polarization terms. These may furth

TABLE I. Neutron and proton effective massesmpn and mp

p ,the proton pairing gapDp, and the electron (and muon)chemical potentialme as functions of total baryonic densityr. Densities are in units of fm23, andDp and me in units ofMeV. All results are for the nonrelativistic calculation.

r mpnymn mp

pymp Dp me

0.005 0.994 0.988 0.023 8.2910.01 0.988 0.977 0.054 25.620.05 0.947 0.917 0.331 57.640.10 0.899 0.861 0.688 86.320.15 0.855 0.823 0.862 108.680.20 0.814 0.788 0.906 126.390.25 0.776 0.757 0.815 141.530.30 0.743 0.734 0.617 155.300.35 0.714 0.716 0.390 168.040.40 0.686 0.694 0.233 180.120.50 0.625 0.630 0.0 203.090.60 0.460 0.511 0.0 231.11

1996

a

t

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n

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reduce the pairing gap and bring in an uncertainregarding the size, which strongly influences the modifiURCA processes. Although we have employed the BoA potential in Table A.2 of Ref. [16], the pairing gapobtained with other potentials such as the Paris or Bonnand Bonn C is rather similar. This is expected since t1S0 channel is determined by the central force compononly of the nucleon–nucleon interaction, and all modepotentials reproduce the phase shifts for this channel.

With the pairing gap, effective masses, and the Fermomenta for the various particles given in Table I, wfirst look at the neutrino production rate for Eq. (2) witelectrons only. The expressions for these processes wderived by Friman and Maxwell [22], and read [8]

Qn ø 8.5 3 1021

√mp

n

mn

!3 √mp

p

mp

! √re

r0

!1y3

T 89 anbn , (7)

in units of ergs cm23 s21, whereT9 is the temperature inunits of 109 K, and, according to Friman and Maxwellan describes the momentum transfer dependence ofsquared matrix element in the Born approximation fthe production rate in the neutron branch. Similarly,bn

includes the non-Born corrections and corrections duethe nucleon-nucleon interaction not described by one-pexchange. Friman and Maxwell [22] usedan ø 1.13 atnuclear matter saturation density andbn 0.68. In theresults presented below, we will not includean and bn.For the reaction of Eq. (2) with muons, one has to replare with rm and add a factor1 1 X with X k

mF yke

F[23,24]. For the proton branch of Eq. (3) with electronwe have the approximate equation [8]

Qp ø 8.5 3 1021

√mp

p

mp

!3 √mp

n

mn

! √re

r0

!1y3

T89 apbpF ,

(8)

with F °1 2 ke

Fy4kpF

¢Q, whereQ 1 if kn

F , 3kpF 1

keF and zero elsewhere. Yakovlev and Levenfish p

an ap andbp bn. We will, due to the uncertaintyin the determination of these coefficients, omit themour calculations of the reaction rates. With muons, tsame changes as in Eq. (7) are made.

The reaction rates for the URCA processes are redudue to the superconducting protons. Here we adoptresults from Yakovlev and Levenfish [8], their Eqs. (31and (32) for the neutron branch of Eq. (7) and Eqs. (3and (37) for the proton branch of Eq. (8). In thiwork we study proton singlet superconductivity onlyemploying the approximationkBTC Ds0dy1.76, whereTC is the critical temperature andDs0d is the pairinggap at zero temperature derived in our calculations. Tcritical temperature is then used to obtain the temperatdependence of the corrections to the neutrino reactrates due to superconducting protons. To achieve thwe employ Eq. (23) of [8].

In this Letter, we present results for the neutrino enerrates at a densityr 0.3 fm23. The critical temperature


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FIG. 2. Temperature dependence of neutrino energy loss rain a neutron star core at a total baryonic density of0.3 fm23

with muons and electrons. Solid line represents Eq. (2) welectrons, dotted line is Eq. (2) with muons, dashed linis Eq. (3) with electrons, while the dash-dotted line is thprocesses of Eq. (3) with muons. The corresponding resuwith no pairing are also shown.

is TC 3.993 3 109 K, whereas without muons we haveTC 4.375 3 109. The implications for the final neu-trino rates are shown in Fig. 2, where we show the resufor the full case with both muons and electrons for thprocesses of Eqs. (2) and (3). In addition, we also dispthe results when there is no reduction due to superconduing protons. At the density considered,r 0.3 fm23, wesee that the processes of Eqs. (2) and (3) with muonscomparable in size to those of Eqs. (2) and (3) with eletrons. The direct URCA process of Eq. (1) is not allowein our b-stable matter for densitiesr , 0.62 fm23, dueto momentum conservation. Similarly, the direct URCprocess with muons is also not allowed at densitiesr ,

0.7 fm23. Thus, Eqs. (2) and (3) with muons give an additional and important contribution to the neutrino production in the core of a neutron star. However, the protopairing gap is still sizable at densities up to0.4 fm23, andyields a significant suppression of the modified URCA prcesses discussed here, as seen in Fig. 2. We have omany discussion on neutron pairing in the3P2 state. Forthis channel we find the pairing gap to be rather sm[21], less than0.1 MeV and close to that obtained inRef. [25]. We expect therefore that the major reductionsthe neutrino rates in the core come from superconductprotons in the1S0 state. Moreover, the recent reinvestigation of neutrino-pair bremsstrahlung in the crust bPethick and Thorsson [26] indicates that this processmuch less important for the thermal evolution of neutronstars than suggested by earlier calculations. Thus, cobined with the present reduction due to superconductiprotons, our results may indicate that there is need for otprocesses than those studied here to explain the rapid cing of neutron stars. The contribution from neutrino-pabremsstrahlung in nucleon-nucleon collisions in the corealso, for most temperature ranges relevant for neutron st

tes

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smaller than the contribution from modified URCA processes [8]. Possible candidates are then the direct URprocesses due to hyperons and isobars, as suggesteRef. [4], or neutrino production through exotic states omatter, such as kaon or pion condensation [9,10] or quamatter [11,12]. However, the analysis of Page [3] indcates that it is hard to discriminate between fast and slcooling scenarios, although in both cases agreement wthe observed temperature of Geminga is obtained if barypairing is present in most, if not all, of the core of the sta

This work has been supported by the Istituto Trentindi Cultura, Italy, the Research Council of Norway, and thNordic Academy for Advanced Research.

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1997

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Emissivities of Neutrinos in Neutron Star Cores