The Evolution of Proto-Strange Stars
aa r X i v : . [ a s t r o - ph . H E ] S e p Compact Stars in the QCD Phase Diagram III (CSQCD III)December 12-15, 2012, Guaruj´a, SP, Brazil
The Evolution of Proto-Strange Stars
Omar G. BenvenutoFacultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata,Instituto de Astrof´ısica de La Plata (IALP-CONICET)Paseo del Bosque S/N, 1900 La Plata, Argentina
Email: [email protected] andJorge E. HorvathInstituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas, Universidade de S˜aoPaulo, 05570-010 Cidade Universit´aria, S˜ao Paulo, SP, Brazil
Email: [email protected]
Since the idea that strange quark matter (SQM) may be the ground state of hadronicmatter [1] it has been a topic of great interest to find ways to discern between neutronstars (NSs) and strange stars (SSs). There have been several proposals related to thisidea. For example, if the transition from nuclear matter to SQM occurs during thecooling of a spin-down pulsar it may be detected as a giant glitch [2] because the starshould undergo a sudden change in it moment of inertia. Another possibility is tolook for differences in the cooling of young presumed NSs [3]. Also, if it were possibleto measure the radius of low mass objects, we could distinguish SSs from NSs becauseof their smaller radius [4].If SQM actually were the ground state of matter, several interesting astrophys-ical consequences should be expected. For example, there may exist high densitycores inside otherwise standard white dwarf stars. If these stars undergo non-radialpulsation, the modes would be splitted on several close, detectable periods [5]. SQMformation may be the process that releases enough energy to produce the core collapsesupernova explosions [6].In the conditions present in the first seconds after core bounce, the hadronic mat-ter that made up a proto -NS (PNS) contains a gas of degenerate electron neutrinos.At the very beginning the evolution of the PNS is dominated by the release of its neu-trino content. It has been found that a gas of degenerate neutrinos pushes away thecritical density for the transition to quark matter. Thus, PNS deleptonization favors the occurrence of the transition [7]. During the first minutes of the evolution of PNSsthe thermodynamic conditions at their interiors change so strongly that we think this1o be the best place to detect the transition to SQM. It should be remarked thatsupernovae light curves are absolutely insensitive to the details of the explosion. So,the only way to observe the transition to SQM (if it really occurs) is by observing theneutrino emission of a forthcoming nearby core collapse supernova. If close enough,such an event will allow the detection of a large number of neutrinos allowing for de-tailed statistical studies. Unfortunately, the historical detections related to SN1987A[8] were not enough for this purpose because of the low number of neutrinos.In order to interpret future detailed neutrino observations it is relevant to haveavailable models of the behavior of NSs during its first seconds. This has been thesubject of several papers [9, 10, 11, 12]. Here we shall present the first results wehave found in the evolution of bare SSs. This represents the starting point of aneffort devoted to predict the details of the neutrino signal due to the transition fromnuclear matter to SQM. Here we do not consider the occurrence of the transition toSQM during evolution, but center our attention on the process of deleptonization ofthe SSs leaving the inclusion of the physics of the quoted transition to future works.The reminder of this paper is organized as follows: In Section 2 we describe ourGeneral Relativistic, hydrostatic stellar evolution code. In Section 3.1 we brieflydescribe the physical ingredients we employed and in Section 4 we present our firstresults. Finally, in Section 5 we make some concluding remarks.
We have adapted our Newtonian stellar evolution code [13] to solve the equations ofGeneral Relativistic, hydrostatic stellar evolution [11] in the diffusion approximation.The structure evolves with a timescale far larger than transport, thus we solve it withtwo coupled Henyey (finite differences, fully implicit) schemes: one for the structureand the other for the transport [11]. Initially, at the stellar interior, neutrino meanfree paths are far shorter than the size of the star. However this is not the case at theouter layers or at later times. So, we adopted a flux limiter to assure that causalityis fulfilled.The fluxes of lepton number H ν and energy F ν are H ν = − T e − Λ − φ π (cid:20) D ∂ ( T e − φ ) ∂r + ( T e − φ ) D ∂∂r (cid:18) µ e T (cid:19)(cid:21) , (1) F ν = − T e − Λ − φ π (cid:20) D ∂ ( T e − φ ) ∂r + ( T e − φ ) D ∂∂r (cid:18) µ e T (cid:19)(cid:21) (2) T is the temperature, Λ and φ are factors appearing in the spherical Schwarszchildmetrics, r is the radius and µ e is the electron neutrino chemical potential. We considerelectron, muon and tau neutrinos. In general µ e = 0 but muon and tau neutrinos2re due to pair creation, thus µ ν = 0, µ τ = 0. Then, diffusion coefficients are D = D ν e + D ν e , D = D ν e − D ν e , D = D ν e + D ν e + 4 D ν µ where D jn = Z ∞ dxx n f ( E ) (1 − f ( E )) P i ( σ i /V ) , j = ν e , ν e , ν µ . (3) σ i represent the cross section of the reactions that provide the neutrino opacity (seebelow) and f ( E ) = (1 + exp [( E − µ ) /T ] is the occupation number correspondingto the considered neutrino. The equation of lepton number per baryon Y L and energyconservation are ∂Y L ∂t + e − φ ∂∂a (4 πr e φ F ν ) = 0 , (4) e φ T ∂s∂t + e φ µ e ∂Y L ∂t + e − φ ∂∂a (4 πr e φ H ν ) = 0 . (5) s is the entropy per baryon and t is the time. The equations of hydrostaticequilibrium, gravitational mass, radius, and metrics are ∂P∂a = − e Λ πr n B ( ρ + P )( m + 4 πr P ) ,∂m∂a = ρn B e Λ ,∂r∂a = 14 πr e Λ n B ,∂φ∂a = e Λ πr n B ( m + 4 πr P ) .a represents the baryon number enclosed by a sphere of radius r which is anadequate Lagrangian coordinate for our purposes, P is the pressure and n B is thebaryon number density. At the center we have r (0) = 0; m (0) = 0; H ν (0) = 0; F ν (0) = 0 whereas, at the surface φ ( a s ) = log [2 m ( a s ) /r ( a s )] and P ( a s ) = P s . Theneutrino luminosity is L ν = e φ πr H ν . Neutrino opacity has been computed following the formalism presented in Ref. [14].As stated above, we considered electron, muon and tau neutrinos assuming that µ e = 0, µ ν = 0, µ τ = 0. We considered the cross section per unit volume given by3 V = g Z d p (2 π ) Z d p (2 π ) Z d p (2 π ) W fi f (1 − f )(1 − f )(2 π ) δ ( p + p − p − p ) , where the matrix element W fi is W fi = G F E E E E (cid:20) ( V + A ) ( p · p )( p · p )+ ( V − A ) ( p · p )( p · p ) − ( V − A )( p · p )( p · p ) (cid:21) .E i and p i are the energy and momentum of each particle participating in the re-actions (e.g., ν e + d → e − + u , i = 1 , , , V and A are coefficientscorresponding to each reaction, given in [14], and the other symbols have their stan-dard meaning. We have developed a Montecarlo integration scheme that providesthe quark matter neutrino opacity for the thermodynamic conditions relevant at SSinteriors. For the equation of state of the SQM we have adopted the standard descriptionprovided by the MIT bag model. In the zero strange quark mass limit, it is wellknown that the form of the equation is P = ( ρ − B ), with B a constant thatparametrizes the non-perturbative interactions giving rise to confinement. In thesimulations presented below we adopted the “standard” value of B = 60 M eV f m − .When a finite value for the quark mass is employed there are deviations from thesimple form given above, although the linearity still holds to a high degree. We have applied our new relativistic stellar code to the case of a 1.4 M ⊙ homogeneousproto SS. We considered an initial energy content compatible with that expected fora gravitational collapse of a massive star. The evolution of the temperature and theneutrino abundance per baryon of this object is depicted in Fig. 1.The proto SS is initially hot and plenty of neutrinos that tend to diffuse outwardson a timescale of tens of seconds. The net leptonic number carried by neutrinos is lostin the process termed “deleptonization” in the literature. Heat (mainly in neutrinopairs) takes a much longer timescale, as indicated by the comparison of both panels4igure 1: Left panel: The temperature profile during the first seconds of evolution ofa 1.4 M ⊙ homogeneous SS. Each curve is labeled with its age, given in seconds on theright. Notice that, due to the outgoing neutrino flux at the fist stages of evolutionthe inner layers of the star get hotter (Joule effect). Right panel: The neutrinoper baryon profile of the same SS. Notice that the timescale of deleptonization isappreciably shorter than that of cooling.of Fig. 1. Because the SS interior is strongly degenerate, the star also undergoes atiny contraction. In fact the gravitational mass and radius evolve less than in thecase of NSs, in which the outer layers are partially degenerate and more sensitive tothermal effects.In is interesting to compare these results with the Fig. 9 of Ref. [11]. Our cal-culations indicate that NSs undergo a much faster evolution as compared to SSs.Certainly, this is important in predicting the neutrino signal of the phase transitionto SQM and also in interpreting the observed signal in SN1987A within these models. In this conference we have presented the first results of the evolution of a 1D, bare,proto SSs. This represents a starting point in our effort devoted to predict the neutrinosignal to be expected to arrive from the next nearby core collapse supernova. Recent5ork has explored 2D simulations in which the acceleration of the conversion front onits way outwards could be addressed [15], although several open questions remain inboth type of modeling and their eventual consistency. We have performed a detailedcomputation of the neutrino opacity and coupled it to a flux limited, hydrostatic,General Relativistic stellar evolution code. Then, we applied the code to the evolutionof a 1.4 M ⊙ homogeneous SS finding a slower evolution as compared to standard PNSevolution. The net lepton number is lost faster than the thermal energy, and it maybe expected that the signal in a neutrino detector should last longer. However, acloser inspection to the Fig. 1 also shows that the temperature of the neutrinosphere(the imaginary surface at which most of the neutrinos freely escape) decreases a factor ∼ ∼ sec , causing the mean energy of the emitted neutrinos to fallbelow ∼ M eV or so. Thus, the emission of the leaking neutrinos stands, but theyare less energetic and easily missed unless the threshold of the detector is very low.This and other important features should be studied in depth and a more detaileddescription of the results is in preparation and will be published elsewhere.O.G.B. deeply acknowledges the financial support he received from the LOC thatallowed him to attend the meeting.
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