Thermal evolution of neo-neutron stars. I: envelopes, Eddington luminosity phase and implications for GW170817
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THERMAL EVOLUTION OF NEO-NEUTRON STARS. I: ENVELOPES, EDDINGTONLUMINOSITY PHASE AND IMPLICATIONS FOR GW170817
Mikhail V. Beznogov, Dany Page, and Enrico Ramirez-Ruiz
2, 3 Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico, Ciudad de M´exico, 04510, Mexico Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark (Dated: December 10, 2019)
Submitted to ApJAbstractA neo-neutron star is a hot neutron star that has just become transparent to neutrinos. In a corecollapse supernova or accretion induced collapse of a white dwarf the neo-neutron star phase directlyfollows the proto-neutron star phase, about 30 to 60 seconds after the initial collapse. It will alsobe present in a binary neutron star merger in the case the “born-again” hot massive compact stardoes not immediately collapse into a black hole. Eddington or even super-Eddington luminositiesare present for some time. A neo-neutron star produced in a core collapse supernova is not directlyobservable but the one produced by a binary merger, likely associated with an off-axis short gamma-rayburst, may be observable for some time as well as when produced in the accretion induced collapseof a white dwarf. We present a first step in the study of this neo-neutron star phase in a sphericallysymmetric configuration, thus neglecting fast rotation, and also neglecting the effect of strong magneticfields. We put particular emphasis on determining how long the star can sustain a near-Eddingtonluminosity and also show the importance of positrons and contraction energy during neo-neutron starphase. We finally discuss the observational prospects for neutron star mergers triggered by LIGO andfor accretion-induced collapse transients.
Keywords:
Neutron stars — Type II supernovae — Gamma-ray bursts — Accretion induced collapse INTRODUCTIONNeutron stars are by far the most intriguing objectsin the Universe. They are superdense, can be superfastrotators, may have superstrong magnetic fields, and aresurrounded by the strongest gravitational fields (see, e.g.Haensel et al. 2007). They are born in core collapsesupernova events (Baade & Zwicky 1934) or in accretioninduced collapse of white dwarves (Canal & Schatzman1976) and start their life as proto-neutron stars (Burrows& Lattimer 1986). Moreover, a hot born-again massive [email protected]@[email protected] neutron star may also be produced in the merging of abinary neutron star system and survive as such (Klu´zniak& Ruderman 1998), or collapse into a black hole. Duringthe first hot phase, neutrinos are copiously produced butare trapped in the stellar interior and only escape byslowly diffusing outward. This early evolution, lastingless than a minute, has been extensively studied theo-retically, in large part because a Galactic core-collapsesupernova would allow us to follow it observationallythrough the detection of the emitted neutrinos. The sub-sequent phase, which we will call the neo-neutron star phase, from an age of a minute after the birth/re-birthto a few hours/days, has, however, never been carefullyconsidered. Later phases have been the object of numer-ous studies (see, e.g., Yakovlev & Pethick 2004; Page,Geppert, & Weber 2006). a r X i v : . [ a s t r o - ph . H E ] D ec Beznogov, Page, and Ramirez-Ruiz
After the supernova, it may take decades till the ejectabecome transparent to electromagnetic radiation fromthe central object (Bahcall et al. 1970). In the case of thesupernova SN 1987A it is only recently, after more thanthirty years, that credible evidence of the presence of acompact object has been found (Cigan et al. 2019). Theyoungest observed neutron star is the compact objectin the center of the Cassiopeia A supernova remnant(Tananbaum 1999), with an age of about 340 years (Fesenet al. 2006). It is thus doubtful we will have, in the nearfuture, valuable observational data on the very earlycooling history of a neutron star, and even less of a neo-neutron star. The neo-neutron phase is, however, thephase during which the neutron star crust is formed andit is, thus, establishing the basic structure for a largeamount of neutron star phenomenology.A complementary set-up is provided by binary neutronstar mergers (Lee & Ramirez-Ruiz 2007; Faber & Rasio2012). Although it is often considered that the outcomeof such event would be the formation of a low-mass blackhole (Eichler et al. 1989; Rezzolla et al. 2011; Murguia-Berthier et al. 2014), there is a possibility that the mergedobject survives as a massive neutron star (Usov 1992;Klu´zniak & Ruderman 1998; Metzger et al. 2008). In sucha scenario we would have a “born-again” neutron star,with trapped neutrinos because of its high temperature,followed by a massive neo-neutron star. This possibilityis real only if the high density equation of state (EOS)is stiff enough to have a high maximum mass, M max .The maximum mass of a cold slowly- or non-rotatingneutron star is at least 2 M (cid:12) , from the masses of thepulsars PSR J1614-2230 (Demorest et al. 2010) and PSRJ0348+0432 (Antoniadis et al. 2013) and possibly higherthan 2 . (cid:12) from the upper value of the mass of PSRJ0740+6620 (Cromartie et al. 2019). Analyses of theGW170817/GRB170817A gravitational wave/gamma-ray burst event have also provided new constraints on M max based on the delayed collapse of the merged objectinto a black hole. Margalit & Metzger (2017) obtain M max ≤ .
17 M (cid:12) (90%) and Rezzolla et al. (2018) find2 M (cid:12) ≤ M max ≤ . (cid:12) while the more detailed studyof Shibata et al. (2019) conclude that M max ≤ . (cid:12) .If such is the case only mergers of binaries containinglow mass neutron stars could produce a stable mergedobject so that our neo-neutron star description would beof interest.As a first step, in the present paper, we consider theevolution of the outer layer of the neo-neutron star, itsenvelope, just after the formation of nuclei when thesurface temperature is high enough to be of the order ofthe Eddington luminosity, i.e., of the order of 10 erg s − .An important question we tackle is the duration of a possible Eddington or super-Eddington phase, and thenconsider the subsequent evolution. Based on previousstudies of proto-neutron star we explore the impact ofdifferent possible initial temperature/luminosity profilesin the envelope on the Eddington phase.This paper is structured as follows. In Sect. 2 and 3we setup the problem and present our results on the low-density regime of an Eddington envelope. In Sect. 4 wedefine what a neo-netron star phase is and contrast themethod for study of long term cooling of isolated neutronstars versus the neo-neutron star case. Our resulted aredescribed in Sect. 5 and their observational relevanceis discussed in Sect. 6. A summary and conclusionsare presented in Sect. 7. Finally in Appendix A andB we describe the physical properties of hot neutronstar envelopes and our numerical scheme is detailed inAppendix C. THERMAL EVOLUTION EQUATIONSWe consider a spherically symmetric problem, neglect-ing the effects of rotation and magnetic fields. Since thestructure of the outer layers of our stars will expand orcontract we employ the enclosed baryon number a as a(Lagrange) radial variable instead of the circumferentialradius r . The full set of general relativistic structureand mechanical evolution equations can be found, e.g.,in Potekhin & Chabrier (2018). The thermal evolutionequations we will solve are: (cid:101) L = − K (cid:0) πr (cid:1) n e φ ∂ (cid:101) T∂a , (1)e φ ∂ ( (cid:101) T e − φ ) ∂t = − C V ( (cid:101) Q L + (cid:101) Q ν + (cid:101) Q V ) (2)where (cid:101) L = L e φ and (cid:101) T = T e φ are red-shifted luminosityand temperature, respectively, e φ being the time com-ponent of the metric. C V is the heat capacity and K thethermal conductivity. The energy sources/sinks are (cid:101) Q L ≡ n ∂ (cid:101) L∂a (3)which gives the heat loss/injection from the luminositygradient, (cid:101) Q ν = e φ Q ν which gives the neutrino energyloss and (cid:101) Q V ≡ − (cid:101) T (cid:18) ∂P∂T (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) n ∂ ln n∂t (4)that will be called the contraction energy . This last termcomes from the “ P d V ” work and the volume dependentpart of the internal energy and gives the gravitationaland internal energy release owing to the contraction ofthe star during its cooling. volution of Neo Neutron Stars. I. OUTER BOUNDARY AND ENVELOPEBoundary conditions at the center, where a = 0, areobvious: (cid:101) L (0) = 0 , r (0) = 0 , m (0) = 0 , (5)while P ( a = 0) ≡ P c is an arbitrary parameter that willdetermine the mass of the star. Boundary conditionsat the surface are more delicate and “surface” must beproperly defined. The simplest and naive condition is the“zero condition”: P s = ρ s = T s = 0, and one takes R ≡ r s and M ≡ m s . Subscript “s” refers to the quantitiesat the surface. However, this is too naive since P , ρ ,and T , likely never really reach zero and, more likely,there is a smooth transition from the stellar interiorto the surrounding magnetosphere (or the interstellarmedium in the case of a non magnetized star). It is moreappropriate, when studying the thermal evolution of thestar, to define its surface as located at the photosphere,i.e., the layer where the outflowing thermal radiation isproduced. We adopt the commonly used Eddington, orphotospheric, condition (see, e.g., Hansen, Kawaler, &Trimble 2004) in which detailed radiative transfer (wherethe energy dependence of the opacity is wholly takeninto account) is replaced by a diffusion approximation(where the energy dependent opacity is replaced by itsRosseland mean), i.e., the same equation (1), and thephotosphere is defined as the layer where the opticaldepth is 2 /
3. This lead to the conditions L s = 4 πσ SB R T (6)and P s = 23 g s κ s (cid:18) L s L Edd (cid:19) (7)where g s = e λ GM/R (8)is the free-fall acceleration at the surface, e λ beingthe radial component of the metric, σ SB the Stephan-Boltzmann constant, κ s the Rosseland mean opacity, atthe surface, and L Edd ( R ) = 4 πc GM e λ κ s = 4 πR c g s κ s (9)the Eddington luminosity at the stellar surface. These are then complemented by the obvious relationsthat define the mass M and radius R of the star M = m s and R = r s , (10)and the continuity of the metric coefficient with theexternal Schwarzschild solutione φ ( R ) = e − λ ( R ) = (cid:112) − GM/Rc . (11)It is numerically inconvenient to directly apply theouter boundary conditions of Eq. (6) and (7) and werather apply the standard scheme of separating out an envelope (Gudmundsson, Pethick, & Epstein 1982) asdescribed below.In these outer layers the equation of hydrostatic equi-librium is simply d P d l = − g s ρ (12)when written in terms of the proper radial length l whichis defined through d l ≡ e − λ ( r ) d r . From a given layerat ρ a and P a , hydrostatic equilibrium can be integratedoutward giving the well known classical result P a = y a g a with y a ≡ (cid:90) ∞ a ρ ( r )d l (13)being the proper column density of matter above point“a” and where it has been assumed that the upper layersare sufficiently concentrated that g can be consideredconstant. In practice we can replace g a by g s .3.1. High Luminosity Envelopes
The envelope is defined as the outer layers, from thesurface down to a bottom layer at some pressure P b or,equivalently, density ρ b , in which the EOS is temperaturedependent and thus require a special treatment comparedto the highly degenerate interior. For our present purposewe extended the previous models of Beznogov, Potekhin,& Yakovlev (2016) to higher temperatures by addingradiation pressure to the EOS of fully ionized plasma ofPotekhin & Chabrier (2010) and by adding the L s /L Edd term to the surface condition as in Eq. (7). As we justifybelow in Sect. 4, we restrict ourselves to envelopes madeof pure iron.Fig. 1 shows six envelope temperature profiles labelledby log T s and the locations of several critical loci: themelting curve (i.e., ions form a Coulomb crystal atdensities above this curve), the appearance of electron-positron pairs, the onset of electron degeneracy (labelled Beznogov, Page, and Ramirez-Ruiz log ⇢ [g / cm ] l og T [ K ] e l e c t r o n - p o s i t r o n p a i r s P m = P r T = T F m e lti ng c u r v e nu c l e i f o r m a ti on l og T [ K ]
Envelope temperature profiles for six surfacestemperature, as labelled by log T s . ( g s = 10 cm s − isassumed.) See details in the text. The grey shadowed partabove 10 g cm − is shown for illustration but not used inour evolutionary calculations. as “ T = T F ”), the transition from matter to radiationpressure dominated regimes (labelled as “ P m = P r ”),and the temperature below which nuclei are formed (Lat-timer & Swesty 1991). One can also see that the profileswith the highest temperatures (log T s [K] >
7) crossthe electron-positron pairs curve at the density about10 g cm − and also the nuclear formation/dissociationline when T > K. Neither pairs nor nuclei disso-ciation are included in our envelope models and theseparts of the profiles should not be trusted. However, asdiscussed in Sect. 4, we will locate our outer boundarydensity ρ b at 10 g cm − and this regime of high densityinaccurately modeled envelopes will not be actually usedand is plotted here only for illustrative purpose.Notice that in the matter dominated regime, and withopacity dominated by free-free-absorption, ρ s increaseswith T s while in the radiation dominated regime, andopacity dominated by electron scattering, the relation-ship is inverted. One sees from Fig. 1 that the transitionbetween these two regimes occurs just above T s ∼ K.These envelope models provide us with a relationshipbetween the temperature at the bottom of the envelope, T b , and at its surface, T s , the so-called “ T b − T s rela-tionship”: T b = T b ( T s ). Since energy sources and sinksare neglected within the envelope, the luminosity at itsbottom, L b , is equal to the surface luminosity and thuswe obtain a relationship between the two searched forsolutions of Eq. (1) and (2): L b = L b ( T b ). This allows to replace the outer boundary condition L s = L s ( T s ) ofEq. (6) at P s by a new one applied deeper at P b .It was shown by Gudmundsson et al. (1982) that inthe resulting relationship T s = T s ( T b ) the dependenceon M and R is only through g s in the form T s ( T b , g s , ) = g / , T s ( T b , g s , = 1) (14)where g s , ≡ g s / (10 cm s − ). We have explicitlychecked that this result is still valid for our hot envelopeswith a lower density inner boundary at ρ b = 10 g cm − .We notice that the approximations that lead to theEddington boundary condition of Eq. (7) are actuallyself-inconsistent (Hansen et al. 2004) and the identifi-cation of a “surface” layer at temperature T s has to berather seen as a convenient ansatz for a more realisticatmospheric boundary condition. It is however well-known (see, e.g., Kippenhahn, Weigert, & Weiss 2012)that envelope models will converge toward the “zerocondition” and the exact definition of the “surface” isnot important when studying the deeper layers. We willhenceforth adopt the common notation of writing theoutflowing luminosity in terms of an effective tempera-ture T eff as L = 4 πσ SB R T and use red-shifted quan-tities as L ∞ = 4 πσ SB R ∞ T ∞ with L ∞ ≡ e φ L = (cid:101) L s , T ∞ eff ≡ e φ T eff and R ∞ ≡ e − φ R , and in our case T eff = T s . NEO-NEUTRON STARSThe early evolution of a newly-born, or a born-again,neutron star can be divided in two separate phases: • proto-neutron star phase, 0 ≤ t (cid:46) −
60 s. Thestar is opaque to neutrinos, T (cid:29) K. Thechemical composition of the core slowly evolvestoward the zero-temperature one as neutrinos leakout and the star’s lepton number decreases. • neo-neutron star phase, 30 −
60 s (cid:46) t (cid:46) T (cid:28) K.The crust is being formed.The standard approach used in long term cooling stud-ies (Yakovlev & Pethick 2004; Page et al. 2006) needsadjustments to study neo-neutron stars since we are nowinterested in very short term evolutions and very hightemperatures. To be able to resolve short timescales itbecomes necessary to push the outer boundary to muchlower densities and we will typically use ρ b = 10 g cm − resulting in an envelope with a thermal time of the orderof a second. A direct consequence of this is that theouter layers of the interior have an EOS that becomestemperature dependent. We thus distinguish three re-gions • Outer (heat blanketing) envelope at densities ρ s ≤ ρ ≤ ρ b , treated separately in a time independentway (see Sect. 3.1). It has Eq. (6) and (7) as a volution of Neo Neutron Stars. I. ρ s , P s , and T s , for every given L s . • Inner envelope in the regime ρ b ≤ ρ ≤ ρ c in whichthe EOS is still temperature dependent and whereboth structure and thermal equations have to besolved simultaneously. The outer envelope providesthe outer boundary condition L b = L b ( T b ) for T and L while for P and ρ we use Eq. (13) to write P b ( t ) = g b ( t ) y b , the time dependence coming fromthe contraction of this inner envelope. With theEOS T b ( t ) and P b ( t ) give us ρ b ( t ) [and even if P b is constant ρ b will still change as long as T b does].In the absence of mass loss y b is constant, which iswhat we will assume in the present work. The factthat both P and ρ , and consequently the radius r , change with time in the inner envelope is thereason we prefer to use the baryon number a , aconserved quantity, as radial variable. • Stellar interior at ρ ≥ ρ c where the EOS is temper-ature independent and only the thermal equationshave to be solved at each time step.Models of both proto-neutron stars (see, e.g., Burrows& Lattimer 1986) and neutron star mergers (see, e.g.,Rosswog & Liebend¨orfer 2003) show that the star relaxesto a temperature of a few times 10 K in less thana minute and so we will take as initial temperature(2 − × K above ρ c . Taking ρ c = 10 g cm − asthe inner boundary of the inner envelope is sufficient toguarantee that the stellar interior EOS can be consideredas temperature independent. The microphysics we applyin the interior is the same as in long term cooling modelsand was described in Page et al. (2004, 2011) whilethe microphysics of the inner envelope is described inAppendix A.Numerically, solving the equations of mechanical struc-ture and thermal evolution at very high temperatureswhere the radiation and pair pressure are significant inthe inner envelope is much more challenging that in thelater evolution. We describe in Appendix C the detailsof our solver. RESULTS5.1.
Initial Configurations
We begin our modeling once the star is transparent toneutrino, i.e. after the ∼
30 s long proto-neutron starphase in the case of a core-collapse supernova or after asimilar duration after the fusion of the two stars in thecase of a neutron star-neutron star merger (in the casethe merged object survives instead of having collapsedinto a black hole). In both cases the interior tempera-tures are of the order of 2 − × K (see, e.g., Ponset al. 1999 and Rosswog & Liebend¨orfer 2003). We will thus take as an initial temperature T (cid:39) . × K atall densities above ρ c = 10 g cm − . At such densitiesthis T is just below the transition temperature wherenuclei are formed in the crust (Lattimer & Swesty 1991;see, e.g., Nakazato et al. 2018 for a proto-neutron starevolution study with formation of nuclei). We then intro-duce a temperature gradient in the inner envelope, from ρ c down to ρ b = 10 g cm − , our initial outer boundarypoint. Numerical simulations of neither proto-neutronstars nor mergers resolve the temperature profile at lowdensities (outside the neutrinosphere) and we have thusno information about this outer layer temperature gradi-ent. (Simulation of core-collapse supernovae do modelthe lower density layers but they typically follow theevolution of the system for less than a second, see, e.g.,Janka 2012.) We want to start with a star emitting atthe Eddington limit at its surface and this uniquely fixesthe initial temperature T b , at ρ b : the “Eddington effec-tive temperature” T eff , Edd is obtained, see Eq. (9), from L Edd = 4 πc GM e λ /κ s = 4 πR cg s /κ s ≡ πR σ B T , Edd or T eff , Edd = g / (cid:18) cσ B κ s (cid:19) / (15)while envelope models relate T eff to T b with the same g / scaling, see Eq. (14), implying that the T b resultingin an Eddington luminosity is a unique temperature, T b , Edd , determined by the boundary density ρ b and thechemical composition of the envelope, but independentof M and R . For a pure iron envelope we find that T b , Edd = 1 . × K at ρ b = 10 g cm − .We will consider three series of stellar models, withthree different surface gravities, and implement in themdifferent initial inner envelope luminosity or temperatureprofiles: • models A, A (cid:48) , B1, B2, B3, B4, and E, F: M =1 . (cid:12) and R (cid:39) . − . g s , (cid:39) . − . • model C: M = 2 M (cid:12) and R (cid:39)
11 km with g s , (cid:39) . • models D and D (cid:48) : M = 0 .
25 M (cid:12) and R (cid:39) − g s , (cid:39) . (cid:48) areaimed at mimicking the effect of fast rotation wherecentrifugal acceleration can be seen as resulting in asmall effective surface gravity: a complete treatment ofrotations would need a 2D code and our results are onlyintended to give a first approximation to the possibleeffects of fast rotation. In model C we do not includethe fast neutrino emission by the direct Urca process(Boguta 1981; Lattimer et al. 1991) acting deep in the Beznogov, Page, and Ramirez-Ruiz log ⇢ [g / cm ] L [ e r g / s ] (a) ACDE B1 B2B3B4 L [ e r g s ]
Panel (a): initial local luminosity profiles of all our models. Panel (b): corresponding initial local temperature profiles.See details in the text. log t [s] L [ e r g / s ] AA EF B1B2B3B4 (a) l og L [ e r g s ]
Figure 3.
Cooling curves of our 1.4 M (cid:12) models A, A (cid:48) , B1, B2, B3, B4, and E, F. Panel (a) shows the red-shifted luminosity L ∞ and panel (b) the red-shifted effective temperature T ∞ eff . See details in the text. inner core since it has no effect on the evolution of theouter parts of the star at early stages.As explained in Appendix C we find it more convenientnumerically to define the initial luminosity profile, L ( ρ ),in the envelope rather that defining directly T . Weshow in Fig. 2a our choices: models A, B1, B2, B3,C, and D, have L = L Edd at ρ b , with the value of L Edd for their corresponding M , and the variation of L with increasing density is constrained so that T reaches T (cid:39) . × K at ρ c = 10 g cm − . The models E is, in contradistinction, defined by the temperature profile,following Eq. (C5), and results in a super-Eddington L at the surface. Model B4 is obtained from the L profileof model E scaled down so that its resulting surfaceluminosity is again the Eddington one (but then it cannotreach T at ρ c ). In Fig. 2b we plot the correspondingtemperature profiles. For the reason discussed above,all models with L ( ρ b ) = L Edd start at the same T b , = T b , Edd = 1 . × K, while the super-Eddington modelE has a higher T b , . volution of Neo Neutron Stars. I. κ is much smallerin the inner envelope than at the surface, the local Ed-dington luminosity L Edd ( r ) = 4 πr c g s /κ is much largerthan L Edd ( R ) and, hence, in all models the luminosityin the inner envelope is always below L Edd ( r ).Finally, model F represents a cold start with an ini-tial T eff about twice lower and hence an initial surfaceluminosity about 15 times below L Edd ( R ). To avoidsaturating the figure the initial L and T profiles of thismodel are not displayed in Fig. 2.5.2. Evolution of a . (cid:12) Star
The cooling curves resulting from our initial L and T profiles are presented in Fig. 3 for our 1 . (cid:12) case.One sees that all models converge, i.e., forget their initialconditions, in about 10 s (except model A (cid:48) , see below)and this initial relaxation phase is denoted as phase “1”.After this, during phase “2”, the cooling is driven byneutrino emission from the pair annihilation process and,after the knee , at age ∼ × s, by neutrino from theplasmon decay process, the phase “3”. The model A (cid:48) hasthe same initial temperature and luminosity as model Abut the neutrino emission by the pair annihilation processhas been arbitrarily turned off: this model confirmsthat pair annihilation is responsible for the evolutionduring phase “2”, while during phase “3” (driven byplasmon decay) model A (cid:48) converges toward model A. Atan age of about one year the luminosity, and the surfacetemperature, reach a stagnation phase, “4”: this is the“early plateau” already well-known in neutron star coolingstudies, that will last for a few decades and corresponds tothermal relaxation of the whole neutron star crust whichwill eventually reach thermal equilibrium with the core(see Nomoto & Tsuruta 1987, Page 1989, Lattimer et al.1994, and Gnedin, Yakovlev, & Potekhin 2001). Theshift from phase “3” to “4” is due to the inner envelopetemperature dropping below the plasma temperatureand the consequent exponential suppression of plasmonformation and decay: the main neutrino process availableis then the very inefficient electron-ion bremsstrahlungresulting in a significant slow-down of the cooling.In the right panel of Fig. 3 we show a close up of theearly evolution of T ∞ eff . It is interesting to notice herethat these cooling curves map their initial temperatureprofiles that were displayed in Fig. 2b: it results froma mapping of T ( ρ ) into T eff ( t ). This mapping is goodup to time ∼ s with ρ up to 10 g cm − : it wasshown by Brown & Cumming (2009) that as long as T ∞ eff is controlled by heat transport from deeper layers This knee can already be seen in the results of Nomoto & Tsuruta(1987), but with no interpretation provided, and in Page (1989). | Q V |
Plot of dominant energy term in the energybalance Eq. (2) in model A: blue for Q ν , red for | Q V | andgreen for | Q L | ; other colors are where two contribution arewithin 20% of each other while in the black region all threeare within 20% of each other. (We use absolute values forquantities Q V and Q L that can be either positive or negative.) − log t [s] . . . . R b [ k m ] AB1B2B3B4E
Figure 5.
Time evolution of the radius of our 1 . (cid:12) models.See text for details Beznogov, Page, and Ramirez-Ruiz log t [s] ⇢ b [ g / c m ] AEB1B2B3B4 ⇢ b [ g c m ]
Time evolution of the boundary density ρ b of our1 . (cid:12) models. See text for details its value is determined mostly by the initial T ( ρ ) at adepth whose thermal diffusion time scale to the surface isequal to the time elapsed from when this initial T ( ρ ) wasset. In our case, the mapping ends when t − ρ reachesa density where the evolution is driven by neutrinosmore than by heat diffusion toward the surface and thishappens when approaching the phase “2” dominated bypair-annihilation neutrinos. In Fig. 4 we show the timeevolution of the dominant energy term in the energybalance Eq. (2) as a function of density for model A:the details of such a plot are dependent on the assumedinitial T profile but that Q ν eventually dominates at highdensities (which, as one can see, turn out to be above ∼ g cm − ) is a simple result of the high T dependenceof neutrino processes and the strongly raising T profileas ρ increases.In the Fig. 5 we show the evolution of the boundaryradius R b of our 1 . (cid:12) models. The different radii atearly times are a direct reflection of the inner envelopetemperature profiles: hotter envelopes are naturally moreexpanded. Excluding the model E we find contractions of R b of the order of 50 to 100 meters. Similarly, in Fig. 6,we show the evolution of the outer boundary density ρ b of the same models. Since ρ b evolves with T b in such away that P b remains almost constant , and since T b isinstantaneously correlated with T eff through the outer There is a small time evolution of P in the outer layers becauseof contraction and the resulting small change in g , as seen, e.g.,from Eq. (13). envelope, one sees that ρ b is directly anti-correlated with T eff shown in Fig. 3b. On the contrary, R b results formthe integral of the thickness of underlying layers andits evolution is not directly correlated with the detailedevolution of ρ b or T eff during phase “1”.Considering, again, our model A in more detail as anillustrative case, we present in Fig. 7 a series of envelopetemperature profiles. Since we have no mass-loss in ourmodels, the column density y a of any layer is constantduring the evolutions and hence each layer evolves at(almost) constant pressure . We display in the back-ground of Fig. 7 the pressure of the medium and a seriesof isobars: matter evolves along these isobars during thecooling. The profile at 600 s corresponds to the end ofthe early plateau during which T eff is locked to T eff , Edd :we can divide the inner envelope in two regions, layer “a”at densities above ∼ g cm − where neutrino losseshave had a significant effect (compare with Fig. 4) andlayer “b” below ∼ g cm − where the temperatureprofile has almost not evolved. During this phase R b has decreased by some 40 meters (Fig. 5) but this is dueto the contraction of layer “a” while layer “b” has notcontracted but has rather been slowly sinking, keepingits initial density and temperature profile. After thisfirst phase T b and the whole layer “b” are thermallyconnected to the temperature in layer “a”: T b , and T eff ,begin to drop following the cooling of “a”. As a resultthe layer “b” begins to contract and ρ b to increase asexhibited in Fig. 6. The evolution of the temperatureprofile from 600 up to 10 s shows a clear differencebetween the region dominated by pair neutrinos, layera , versus plasmon neutrinos, layer a , (separated in thefigure by the (white) dotted line). As time runs the layera encompasses less and less, while layer a encompassmore and more, mass. (At these phases layer “b”, whoseenergetics is dominated by either Q V or Q L , always startat densities around 10 as seen in Fig. 4.) This differentevolution of layers a versus a is easily understood byconsidering the difference in temperature dependence ofthese two neutrino processes (see Fig. 17) that resultin the strongly different cooling time scales displayed inFig. 18a. As long as part of the envelope is in the pairneutrino regime this layer a will drive the evolution ofthe outer layers and we are in phase “2” while after ∼ the layer a has disappeared , the cooling of the outerlayers is driven by layer a and we entered phase “3”. Itis interesting to see in Fig. 4 that at age ∼ . s, whenthe cooling curve passes through the “knee”, the decreasein the pair neutrino emission is so strong that the ener-getics of layers that were previously dominated by Q ν arenow dominated by Q L up to densities of 10 . g cm − . volution of Neo Neutron Stars. I. log ⇢ [g / cm ] l og T [ K ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . × × × b a a a a a a log ⇢ [g cm ]
Selected local temperature profiles of model A.Ages, in seconds, are indicated on the right margin. Back-ground color shows the pressure and contours are isobarslabelled with decimal logarithm of pressure [in dyn cm − ].The dashed (yellow) contour corresponds to the initial P b and the thick dotted (white) line reproduces the one fromFig. 17. log ⇢ [g / cm ] l og T [ K ] × × × log ⇢ [g cm ]
Selected local temperature profiles of model F(dotted lines) compared to model A (solid lines). Ages, inseconds, are indicated on the right margin.
Robustness of our . (cid:12) Star Results.
After this thorough study of our model A let us havea look at models B1 to B4. They are all based on thesame two starting points, an interior initially at a tem-perature T (cid:39) . × K as implied by studies of proto-neutron stars and binary mergers, and a surface lu-minosity initially at L Edd , but they have different L and T profiles in-between. These four scenarios have differentevolutions only during the early relaxation phase “1” asclearly seen in Figs. 3, 5, and 6, a phase where T eff isdriven by the heat diffusion in the low density part of theinner envelope. However, once the cooling is controlledby neutrino emission, phases “2” and later ones, theirevolutions are identical to scenario A: neutrinos are soefficient that they rapidly erase any remembrance of theinitial conditions. Nevertheless, during the first 10 sthese different scenarios only span a range of T ∞ eff between1 . . × K and a surface luminosity L ∞ between1 to 3 × erg s − . Hence, we have very similar lumi-nosity evolutions during the first half an hour and thena basically universal evolution for the first year . Noticethat the neutrino processes from either pair annihilationand plasmon decays depend only on the temperatureand the electron density and do not depend on the typeof nuclei present in the medium. It is only later, duringphase “4” controlled by neutrino emission from electron-ion bremsstrahlung that the actual chemical compositionof the medium becomes important.On the other side, it is well known that the chemicalcomposition of the outer envelope has a strong effecton the T b − T s relationship, lighter elements having alarger thermal conductivity and resulting in higher T s for a given T b . How large is this effect and how likely isthe presence of light elements in the high temperatureenvelope we employ is an open question. Notice thatat densities ∼ g cm − and temperatures ∼ Kthermonuclear rates are enormous and the survival oflight elements is doubtful. We intent to tackle theseissues in a forthcoming work.As a distinct initial configuration let us consider ourmodel F which started with the same T c , = 2 . × Kat density ρ c = 10 g cm − but a much lower outerboundary temperature T b , (cid:39) . × K at ρ b =10 g cm − . This model started with a clearly sub-Eddington L ∞ as seen in Fig. 3 but after a few hundredsseconds its surface layers heat up because of the highflux coming from the inner envelope. In Fig. 8 we showthe temperature profile evolution in the inner envelope:compared to model A the initial profile has no choicebut to have a stronger gradient in the inner part in orderto reach a lower T b , and this is the cause of the risein T eff at later times when this enhanced flux reachesthe surface. As in the other models, after ∼ secondsthe temperature profiles have converged to the univer-sal profiles and are indistinguishable from the ones ofmodel A.0 Beznogov, Page, and Ramirez-Ruiz log t [s] l og L [ e r g s / s ] ACDD l og L [ e r g s ]
Cooling curves of our models A (1 . (cid:12) ), C(2 M (cid:12) ), D, and D (cid:48) (0 .
25 M (cid:12) ). See details in the text.
As described in Appendix B there is some uncertaintyregarding the nuclei contribution to the specific heat, butit is only relevant at densities above 10 g cm − andtemperatures well above 10 K. This implies that thisuncertainty has no effect on the duration of the initialEddington luminosity phase: this phase terminates whenneutrino cooling in region a (see Sect. 5.2) drives theevolution of T eff and in this region the nuclei specificheat is negligible.5.4. Evolution of High and Low Gravity Stars
In the Fig. 9 we show the cooling curves for differentgravities. Model C with g s , (cid:39) . g s , (cid:39) .
6: we are plottingthe red-shifted luminosity and its intrinsically higherluminosity is in large part compensated by a higher red-shift. For the low gravity models, D and D (cid:48) , the lowerEddington luminosity clearly shows and moreover therelaxation time is much longer: the initial relaxationphase “1” last much longer and an Eddington luminositycan be sustained for more than 10 s versus less than10 s for models A and C. As a curiosity, in model D (cid:48) wehave arbitrarily switched-off the contraction energy ofEq. (4): as a result during the initial relaxation phase theluminosity slightly decreases instead of staying almostconstant as in model D. Nevertheless, at ages between10 up to 10 s (i.e., between three hours up to threemonths), luminosities of these three models with verydifferent surface gravities are still very similar and onlyactually differ in details (as, e.g., the time at which the“knee” occurs). 5.5. At the crossroads of different physical regimes
We finally describe our model E which has a surfaceluminosity twice higher that L Edd implying a significantmass-loss. However, in the inner envelope the luminosityin this model is still sub-Eddington due to the fact thatthe opacity K is much lower in this region than at thephotosphere. We can thus still model the inner envelopewithin our quasi-static formalism. As seen in Fig. 3this super-Eddington phase can last longer than theEddington phase of our other models: about 2,600 safter which time T eff suddenly drops.In Fig. 10 we illustrate the evolution of this modelthrough it T -profiles. Notice that at early times thelow-density part of the inner envelope is clearly in theradiation/pair dominated regime. This regime corre-sponds in this figure to the region where the isobarsare horizontal, i.e., ρ independent, with P ∝ T . Incontradistinction, in all our other Eddington luminos-ity models the inner envelopes were always in a regimewhere matter made a strong contribution fo the pressureas can be seen, e.g., in Fig. 7. This super-hot modelE results in a strongly puffed-up envelope, because ofradiation pressure, as seen from the larger radius R b in Fig. 5. The first four T -profiles displayed in Fig. 10show a rapid contraction at the lowest densities. Thiscontraction occurs at (nearly) constant pressure , henceat (nearly) constant temperature maintaining a (nearly)constant T eff , and the gravitational energy released bythis contraction is used to power the super-Eddingtonsurface luminosity. This contraction wave propagates in-ward until it reaches the cooling wave from the neutrinocooling propagating outward from the denser regions.After ∼ ,
600 s further evolution along the isobars im-plies a significant temperature drop, the inner envelopeentering a different pressure regime, and the end of thesuper-Eddington phase. After ∼ s this model hasforgotten its initial configuration and follows the sameevolution as all our other 1 . (cid:12) Eddington luminositymodels. 5.6.
Effects of a Strong Magnetic Field
Our neo-neutron star study assumes spherical symme-try and neglect the effect of both fast rotation and thepresence of a strong magnetic field and is, thus, onlya very first step in this direction. Since most realisticscenarios where a neo-neutron star may be observableare likely to produce a fast rotating and strongly mag-netized star and it is imperative to estimate expectabledeviations of realistic models from our idealized ones.A simple model intended to mimick fast rotation witha low gravity model was presented in subsection 5.4.The case of a strong magnetic field is more involved as volution of Neo Neutron Stars. I. log ⇢ [g / cm ] l og T [ K ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . × × × log ⇢ [g cm ]
Selected local temperature profiles of model E.Ages, in seconds, are indicated on the right margin. Back-ground color shows the pressure and contours are isobarslabeled with Log P [dyn cm − ]. The dashed (yellow) contourcorresponds to the initial P b . it not only breaks spherical symmetry, but introducesstrong anisotropies in almost every physical ingredientof our models at both the micro- and macroscopic levels,and modifies the equation of state, opacities, thermalconductivities and also possibly neutrino emission (forreviews about magnetic field effects we refer the readerto Yakovlev & Kaminker 1994 and Potekhin et al. 2015).The heat transport anisotropy in the presence of astrong magnetic field and its effects on the thermal emis-sion have been amply studied for a long time in neutronstar envelopes (starting from, e.g., Greenstein & Hartke1983 and Page 1995) and also in deeper layers of the crust(see, e.g., Geppert et al. 2004, 2006 and P´erez-Azor´ınet al. 2006). The overall effect is that regions of thesurface where the magnetic field is tangential to it willbe much colder that regions where the field is radial. Dueto quantum effects on the thermal conductivity the hotregions with a radial field are moreover slightly hotterthan they would be in the absence of a magnetic field.As a result, when integrating the outcoming flux overthe whole stellar surface, one generally obtains lumi-nosities similar to the non magnetic case with uniformsurface temperature (see, e.g., Page & Sarmiento 1996for examples with dipolar+quadrupolar magnetic fieldgeometries). We, thus, do no expect that the magnet-ically induced anisotropy in heat transport results insignificant deviations from our results. A second point of interest is the duration of the earlyEddington luminosity phase. It is controlled by neutrinoemission from the pair annihilation process which occursat densities above ∼ g cm − , the layer “a” in Fig. 7and clearly illustrated by the difference between themodels A and A (cid:48) in panel (a) of Fig. 3. This process isnot affected by magnetic field at least up to a strength ∼ G (Kaminker & Yakovlev 1994). It is only for muchstronger fields, which become strongly quantizing evenat temperature above 10 K in this density range, thatthe pair annihilation process may increase the neutrinolosses and reduce the duration of an Eddington phase, butunfortunately there is no complete calculation available inthis density-temperature range to confirm this statement.Synchrotron neutrino emission cannot compete at veryhigh temperatures with the pair annihilation process, atleast for fields below 10 G, and thus is not expected toaffect the Eddington luminosity phase for these magneticfield intensities. However, in later phases dominated inthe non magnetized cases by the plasmon decay process(phase “3” shown in panel (a) of Fig. 3) synchrotronneutrinos will increase losses when the field is above ∼ G and accelerate the cooling.The third interesting effect of the presence of a strongmagnetic field is the strong suppression of the opac-ity for extraordinary mode (XO) photons, κ XO ( ω ) ≈ ( ω/ω c ) κ Th when ω (cid:28) ω c , where ω c is the electron cy-clotron frequency and κ Th the electron-scattering (Thom-son) opacity. As a results, when mode switching betweenthe XO and the ordinary (O) mode is taken into ac-count, the critical luminosity L c (i.e., the Eddingtonluminosity in the presence of the magnetic field) is in-creased compared to the zero field L Edd of Eq. (9) by L c ≈ ( ω c /ω ) L Edd and can easily reach 10 − erg s − (Miller 1995). Even higher luminosities, ∼ erg s − ,have been observed during the magnetar Giant Flares(see, e.g., Kaspi & Beloborodov 2017) where a hotplasma is likely confined in the magnetosphere (Thomp-son & Duncan 1995). How long could a strongly mag-netized neo-neutron star sustain a high thermal lumi-nosity close to L c ∼ − erg s − is the crucial ques-tion. In terms of energetics, our model A sustained L (cid:39) × erg s − for about 1000 second and emitteda total of ∼ × ergs while our extreme model Ekept L (cid:39) × erg s − for about 2500 second emittinga total of ∼ × ergs: this second model had a muchhotter and bloated envelope that contained much moreenergy, both gravitational and thermal, allowing it tosustain a higher luminosity for a long time. Whether astrongly magnetized neo-neutron star envelope will behot enough and contains sufficient energy to sustain aluminosity close to its possible L c for a long time (a2 Beznogov, Page, and Ramirez-Ruiz l og L ∞ [ e r g s − ] ACDZ
Figure 11.
Plot of luminosity as a function of the age ofthe remnant for nearby neutron stars together with the pre-dictions from neutron star cooling models. The (green) dotsare measurements and the (blue) squares are upper limitsfrom detected neutron stars (data taken from Beznogov &Yakovlev 2015) while the (red) dotted error bars are up-per limits on the compact objects, black holes or neutronstars, expected to be presents in five core collapse super-nova remnants. These remnants are, in order of increasingage: G043.3–0.2 (a.k.a. W49B, marked by a diamond) fromLopez et al. (2013), G127.1+0.5, G084.2+0.8, G074.0–8.5,and G065.3+5.7 from Kaplan et al. (2004, 2006). Shadedareas show model predictions of Page et al. (2004, 2009) forthe minimal cooling of neutron stars that cover uncertaintieson the chemical composition of the envelope and nucleonpairing at high densities. In contradistinction the dashed-pentadotted curve, Z, exemplifies the effect of fast neutrinoemission from the direct Urca process expected to occur inmassive neutron stars (Boguta 1981; Lattimer et al. 1991)resulting in very cold stars (Page & Applegate 1992). Alsoplotted are the three different models (A, C, D) shown inFig. 9. significant fraction of an hour) is difficult to assess atpresent time and definitely requires a detailed calculationwith all the appropriate physics included. We hope toaddress this issue in a future paper. OBSERVATIONAL PROSPECTS6.1.
Core Collapse Supernova and Supernova Remnants
Neutron stars are great laboratories for studying theequation of state of nuclear-density matter. The study ofsupernova remnants, on the other hand, help us elucidatethe composition and structure of their stellar progenitors(e.g. Lopez et al. 2011). By associating neutron stars withsupernova remnants, we can obtain unique informationabout these systems that is unavailable when we study them separately. What is more, supernova associationsprovide a way to independently constrain the age ofthe neutron star as well as searching for former binarysurviving companions.There are, however, clear limitations that prevent usfrom uncovering young systems; the most noticeablebeing that the ejecta gas needs to be transparent to theneo-neutron star radiation. For optical (in the absenceof dust) and high X-ray energies ( (cid:38)
10 keV), electronscattering provides the main opacity (Bahcall et al. 1970).Let’s consider a cloud of gas with mass M ej ejectedfrom the explosion. The cloud radius expands freely as R = v ej t where v ej = (cid:112) E ej /M ej is the characteristicvelocity, t is the time since ejection and E ej is the totalkinetic energy. We thus expect an homogeneous envelopeto become transparent after a time t τ =1 ≈ . (cid:18) E ej erg (cid:19) − / (cid:18) M ej (cid:12) (cid:19) yrs , (16)which is frustratingly about the duration of the Edding-ton luminosity phase. Here we assume κ ≈ . g − ,which is a reasonable value for ordinary supernova mate-rial. At lower X-ray energies ( (cid:15) x (cid:46) ∼ (cid:15) x / − / .Despite this, the youngest neutron star we have uncov-ered has an age of about 340 years (Fesen et al. 2006). Ina few instances, a surviving binary companion has beendetected in post-explosion deep optical imaging of extra-galactic supernova (Maund & Smartt 2009; Folatelli et al.2014). In the case of 1993J, for example, the brightnessof the transient dimmed sufficiently after about a decadeso that its spectrum showed the features of a massivestar superimposed on the supernova (Maund et al. 2004).It is perhaps a stinging fact that despite the expectedmanifestations of neo-neutron stars, one of the mainissues in the field has been that most Galactic supernovaremnants have no detectable central source. Fig. 11shows the current detections and upper limits of thermalemission in nearby neutron stars with model predictions.Observational selection effects are clearly at play whenuncovering young objects yet there is the possibility thata sizable fraction of massive star collapses might produceblack holes rather than neutron stars, with the clearestexample being W49B (Lopez et al. 2013) and the otherfour examples, from Kaplan et al. (2004, 2006) all plottedin Fig. 11.6.2. Neutron Star Mergers, Short Gamma-Ray Burstsand LIGO Events
The merger of binary neutron stars and the subsequentproduction of a beamed, relativistic outflow is believed to volution of Neo Neutron Stars. I. -4 -3 -2 -1 Time (days) X - r ay Lu m i no s i t y ( e r g / s ) SGRBsNeo-HMNS -4 Time [days] 10 -3 -2 -1 Neo-HMNSSGRBs L X [ e r g s ]
Plot of the on-axis X-ray afterglow light curvesfrom a sample of 36 short GRBs with well sampled lightcurves and redshifts compiled by Fong et al. (2017). Alsoplotted is the X-ray luminosity of the cooling model C shownin Fig. 3, labeled as hyper-massive neutron star (HMNS). trigger short gamma-ray bursts (e.g. Abbott et al. 2017a)and expel metallic, radioactive debris referred to as akilonova (e.g. Kasen et al. 2017). The ultimate fate ofthe post-merger remnant remains unclear and is depen-dent on the mass limit for support of a hot, differentiallyrotating neutron star. The merged remnant can eithercollapse and form a low-mass black hole (Eichler et al.1989; Rezzolla et al. 2011; Murguia-Berthier et al. 2014)or survive as a hyper-massive neutron star (Usov 1992;Klu´zniak & Ruderman 1998; Metzger et al. 2008, 2018).In that case, the detectability of the hyper-massive rem-nant will depend primarily on the orientation of themerging binary.When the relativistic jet points in the direction of theobserver, the event will likely be detected as a classicalshort gamma-ray burst (Nakar 2007; Lee & Ramirez-Ruiz2007) and the X-ray emission emanating from the sur-viving remnant will be buried by the luminous afterglowemission. This can be seen in Fig. 12, where we comparethe X-ray luminosity of the cooling model C shown inFig. 9 to the on-axis X-ray afterglow luminosities of asample of short gamma-ray bursts.Our ability to directly uncover the emission of thenewly formed, hyper-massive neutron star drasticallyincreases when the event is off-axis, as was the case forGW170817 (Abbott et al. 2017a,b). In August 2017,Coulter et al. (2017) discovered the first optical coun-terpart to a gravitational wave source. In this case, thecataclysmic merger of two neutron stars. This landmarkdiscovery initiated the field of gravitational wave astron-omy and enabled an exhaustive observational campaign
Time (days)1 10 10010 G
10 G L ( e r g / s ) Neo-HMNSMagnetar
100 10 Neo-HMNSMagnetars 10 G10 GL X
10 Time [days] L [ e r g s ]
Plot of the X-ray light curves of the counterpartto GW170817 from
Chandra (0.3-10 keV) and cooling modelC shown in Fig. 3 (labeled as Neo-HMNS). Also plotted,for comparison, are the spin-down (bolometric) luminositiesexpected for a stable hyper-massive magnetar. Adapted fromMargutti et al. (2018). (Abbott et al. 2017b). In Fig. 13 we show the luminosityof the X-ray counterpart to GW170817 (Margutti et al.2018) that is seen to be significantly dimmer than theone expected from the spin-down of a highly-magnetized,rapidly rotating remnant but only slightly brighter thanthe X-ray luminosity predicted for the relevant cooling model C that is plotted in Fig. 9.The constraints imposed by the afterglow observationsfavor the idea that GW170817 was a typical GRB jetseen off axis (Abbott et al. 2017a,b). This interpretationassumes that our line of sight is tens of degrees fromthe core of the jet and thus suggests that the prospectsfor detecting the remnant directly might be doable forfuture events, in particular if they are seen further awayfrom the axis of the jet as can be seen in Fig. 14.As discussed in the case of core collapse supernova,one of the challenges for direct detection is that the neo-neutron star is likely to be surrounded by a thick andexpanding radioactive ejecta. In the case of GW170817,the optical depth is expected to be dominated by ther-process radioactively powered, kilonova ejecta (e.g. Met-zger et al. 2010; Roberts et al. 2011; Kasen et al. 2017).Given the quantities derived for the neutron star mergeroutflow of GW170817 (e.g. Villar et al. 2017; Kasenet al. 2017; Murguia-Berthier et al. 2017; Kilpatrick et al.2017; Ramirez-Ruiz et al. 2019), we expect the ejecta tobecome transparent after a time t τ =1 ≈ . (cid:18) E ej erg (cid:19) − / (cid:18) M ej − M (cid:12) (cid:19) days , (17)where we have used κ ≈
10 cm g − for the much moreopaque r-process rich ejecta (Barnes & Kasen 2013).Given the low mass ejecta, double neutron star mergers4 Beznogov, Page, and Ramirez-Ruiz -9 -8 -7 -6
1 10 100 X -r a y F l ux d e n s it y ( m J y ) Time [days] 36 o o o o Neo-HMNS
Figure 14.
Plot of the X-ray flux density at 1 keV for an offaxis model with 10 erg and θ obs = 36 o from Margutti et al.(2017) aimed at providing a reasonable description of theX-ray data of GW170817. Also shown are the correspondingmodels for observers seeing the same event but further awayfrom the axis of the jet and the cooling model C (labeled asNeo-HMNS). appear to be a viable system for uncovering a neo-neutronstar provided that the surviving remnant is stable. Al-ternatively, the lack of X-ray detection of the coolingsignal could be used to argue in support of a collapse toa black hole.6.3. Neo-Neutron Stars in Accretion Induced CollapseEvents
Another relevant progenitor avenue for our study isthe formation of a neutron star through the collapse ofoxygen-neon white dwarf stars in interacting binaries(Canal & Schatzman 1976; Miyaji et al. 1980; Canal et al.1990; Nomoto & Kondo 1991; Wang 2018a). A oxygen-neon white dwarf in a binary system might be able toaugment its mass near the Chandrasekhar mass leadingto accretion-induced collapse by accreting steadily ordynamically (e.g. Ruiter et al. 2019; Wang 2018b). Theformation of neutron stars from interacting oxygen-neonwhite dwarfs in binaries is likely to be accompanied bylow mass ejecta (Woosley & Baron 1992; Dessart et al.2006; Metzger et al. 2009; Darbha et al. 2010), whichmight help direct detection.The prospects for detection of the predicted transientsappear promising (Darbha et al. 2010), yet their charac-terization might be difficult as they might be confusedwith other thermal transients predicted to occur on sim-ilar timescales ( ≈ few days) such as failed deflagrations(Livne et al. 2005) and type .Ia supernovae (Bildstenet al. 2007). Such events are, however, not expected tobe accompanied by an X-ray transient. For an ejectamass of M ej = 10 − M (cid:12) , we expect these optical tran-sients to be uncovered by upcoming surveys to distancesof a few 100 Mpc (Darbha et al. 2010), which will make the X-ray characterization of the neo-neutron star doablewith current space-based facilities. SUMMARY OF RESULTS AND CONCLUSIONSWe have presented a detailed study of the evolution ofthe outer layers of a neo-neutron star. We started justafter the end of the proto-neutron star phase when the in-ternal temperature has dropped to ∼ . × K at den-sities ∼ g cm − and above. At these temperaturesthe nuclei in the crust have already been formed. We de-veloped model of the outer envelope, i.e. the region fromthe photosphere up to densities around ∼ − g cm − ,at luminosities close to the Eddington luminosity, in sta-tionary state, presented in Fig. 1. Using an extension ofthe neutron star cooling code NSCool (Page 1989, 2016)we then modeled the whole neutron star, but focused onthe description of the evolution of the inner envelope,at densities between 10 to 10 g cm − . The evolutionof the surface temperature, and hence the star’s sur-face thermal luminosity, during this early neo-neutronstar phase in controlled by the evolution of the innerenvelope and is thermally decoupled from the deeperlayers on such short time scales. The initial conditionof temperature ∼ . × K at high densities andsurface Eddington luminosity leaves some, but not much,space for variability of the temperature and luminosityin the inner envelope as shown in Fig. 2. As a result, thesurface luminosity remains close to the Eddington value,i.e., above 10 erg s − , for a few thousand seconds witheffective temperatures of the order of 1 . − × K,as presented in Fig. 3 for a 1.4 M (cid:12) star. After ∼ seconds the surface temperature evolution is controlledby neutrino emission, initially by pair annihilation in theinner envelope for some 10 seconds followed by plasmondecay until it has decreased to a few millions K andreached the “early plateau”, well known from isolatedneutron star cooling theory. Models with either largeror lower surface gravity have an initially different Ed-dington luminosity but later follow a very similar coolingtrajectory during their first year of evolution, as illus-trated in Fig. 9. At ages between 10 and 10 secondsthe luminosity drop is roughly, within a factor of a few,a power law L ( t ) (cid:39) × (cid:0) t/ s (cid:1) − / erg s − . (18)Neutron stars in the universe could have very differentorigins, including core-collapse supernovae, neutron starmergers, white dwarf collapses, and the so-called electron-capture supernovae that are somewhat similar to theaccretion induced collapse. In the case of birth in a corecollapse supernova it is very unlikely that the neo-neutronstar could be observed since it takes at least a few months volution of Neo Neutron Stars. I. Software:
NSCool (Page 1989, 2016) APPENDIX A. PHYSICAL STATE OF MATTER IN THE INNER ENVELOPEAs discussed in Sect. 4 in the inner envelope we are dealing with the matter at densities ρ b = 10 < ρ < ρ c =10 g cm − and temperatures (1 − × < T < (1 − × K. At this temperatures and densities one has totake into account presence of positrons and photons. Below we briefly describe the physical ingredients employed in ourmodel and present illustrative plots that allow to better understand our results.A.1.
Equation of State
We assume the presence of Ni at ρ c (following Haensel, Zdunik, & Dobaczewski 1989) and Fe at ρ b , while weinterpolate in both A and Z linearly in log ρ at intermediate densities. Pressure is obtained as the sum of radiation, freegases of electrons and positrons, a free gas of nuclei plus Coulomb interaction corrections following Potekhin & Chabrier(2010). Crystallization of ions takes place when the Coulomb coupling parameter Γ ≡ ( Ze ) (cid:14) ( a WS k B T ) reaches 175[ a WS = ( 4 πn i /3 ) − / is the Wigner-Seitz cell radius, n i being the number density of ions and Ze their electric charge].A.2. Opacity and Thermal Conductivity
The thermal conductivity K is taken as the sum of the electron, K e , and photon K ph conductivities. In the innerenvelope the plasma is fully ionized. Thus, there is no need to take into account the effects of partial ionization onopacity. The radiative opacity κ rad consists of two terms: free-free absorption and electron scattering. The former wascalculated based on the fits of Schatz et al. (1999). The latter is based on the modern fit of Poutanen (2017), whichtakes into account electron degeneracy and pair production (the fit handles both Thompson and Compton scattering). The version of the code used in the paper is not publicly avail-able at the present time, but the modifications are described inAppendix C Beznogov, Page, and Ramirez-Ruiz log ⇢ [g / cm ] l og T [ K ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . γ : ee e:l e:qse:cs γ : e (a) log ⇢ [g cm ]
Thermal conductivity K and total opacity κ for the conditions of the inner envelope. Panel (a) shows the contourplot of conductivity, panel (b) of opacity (from Eq. (A1)), panels (c) and (d) demonstrate the thermal conductivity as a functionof density, for a constant temperature, and temperature, for a constant density, respectively. Values on the contour lines on panel(a) are decimal logarithms of the conductivity [in erg s − cm − K − ], on panel (b) decimals logarithms of opacity [in cm g − ],on curves on panel (c) are decimal logarithms of temperature [in K] and values on curves on panel (d) are decimal logarithms ofdensity [in g cm − ]. The boxed labels in panels (a) and (b) indicate the dominant contribution to the thermal conductivity for agiven temperature and density region, as described in the text, and a few of them are reproduced in the other two panels. A correction factor of Potekhin & Yakovlev (2001) was used for adding free-free and electron-scattering opacities. Theelectron thermal conductivity is taken from Yakovlev & Urpin (1980) when ions are in a liquid phase and from Potekhinet al. (1999) in the solid phase.The resulting thermal conductivity K is illustrated in Fig. 15 as well as the corresponding total opacity κ defined by κ = 4 acT Kρ . (A1) volution of Neo Neutron Stars. I. log ⇢ [g / cm ] l og T [ K ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . γ e + e - e l e c t r o n s i o n s : l i o n s : c s i o n s : q s log ⇢ [g cm ]
Specific heat for the conditions of the innerenvelope. Values on the contour lines are decimal logarithmsof the specific heat capacity [in erg cm − K − ]. Boxed labelsindicate the dominant contributor in the various temperatureand density regions. See details in the text. log ⇢ [g / cm ] l og T [ K ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e l e c t r o n - i o n b r . p a i r a n n i h i l a t i o n p l a s m o n d e c a y log ⇢ [g cm ]
Neutrino emissivity for the conditions of theinner envelope. Values on the contour lines are decimal loga-rithms of the neutrino emissivity [in erg cm − s − ]. Boxedlabels indicate the dominant process in the various tempera-ture and density regions (where “electron-ion br.” stands forelectron-ion bremsstrahlung) and the thick dotted (white)line explicitly marks the transition form pair annihilation toplasmon decay for later reference (the line is not shown atlow densities because neutrino losses become negligible inthis regime). See details in the text. This κ , which include contributions from photons and electrons, should not be confused with the more restricted(Rosseland mean) radiative opacity.The different shape of the contour lines in panel (a) clearly exhibit different regimes that are indicated by the boxedlabels: • γ : ee – conductivity dominated by photons and controlled by Thomson/Compton scattering on electrons andpositrons; • γ : e – conductivity dominated by photons and controlled by Thomson scattering on electrons; • e : l – conductivity dominated by electrons and controlled by scattering on ions in the liquid phase; • e : cs – conductivity dominated by electrons and controlled by scattering on ions in a classical Coulomb solid; • e : qs – conductivity dominated by electrons and controlled by scattering on ions in a quantum Coulomb solid;In the density regime considered here for the inner envelope, photon opacity is dominated by free-free absorption onlyin a very narrow region at the transition between photon dominated to electron dominated transport. We have adiscontinuity in the electron conductivity along the melting curve which may be fictitious as argued by Baiko et al.(1998) but which is small enough and occurring at densities high enough that it has a negligible effect on our results.Notice the dramatic effect of pairs in limiting K at high temperatures simply due to the strong increase of the numberof scatterers, electrons and positions, with temperature in this regime.A.3. Specific Heat
The specific heat is computed as described in Potekhin & Chabrier (2010) to which we added the contribution ofradiation and pairs when present. As we are interested in temperatures up to about 1 − • γe + e − – photons and electron-positron pairs;8 Beznogov, Page, and Ramirez-Ruiz ! log ⇢ [g / cm ] l og T [ K ] - . . . . . . . . . . . . . . . . . . . . . . . (a) ⌧ ⌫
Time-scales in the inner envelope. Panel (a) shows contours of the neutrino cooling time scale τ ν ≡ C V /Q ν , the curvelabels giving the decimal logarithms of τ ν in units of second per GK (since 10 K is a typical temperature in our neo-neutronstar envelopes). The thick dotted (white) line reproduces the one from Fig. 17. Panel (b) shows contours of the heat diffusiontime scale τ h ≡ C V /K , the curve labels giving the decimal logarithms of τ h in units of second per (10 meters) (since 10 metersis a typical length-scale in our neo-neutron star envelopes). • electrons – electrons; • ions : l – ions in a Coulomb liquid; • ions : cs – ions in a classical Coulomb solid; • ions : qs – ions in a quantum Coulomb solid.Nuclear excitation never dominate but make a significant contribution at the highest densities ( ≥ g cm − ) andtemperatures ( (cid:29) K). A.4.
Neutrino Emission
In the density range of our inner envelope, 10 − g cm − , neutrino emission is dominated by three process, inorder of decreasing temperature importance: e + − e − -pair annihilation, plasmon decay and e-ion bremsstrahlung. Forthe first two we follow Itoh et al. (1996) and Kaminker et al. (1999) for the third one. We present in Fig. 17 contourplots of the total neutrino emissivity. Notice the dramatic change in temperature dependence when crossing the (dottedwhite) line from pair annihilation to plasmon decay dominance. The very strong temperature dependence of the pairannihilation process when approaching this line is due to the exponential suppression of pairs when electrons becomedegenerate. Similarly, when shifting from plasmon decay to electron-ion bremsstrahlung the temperature dependence ofthe plasmon process increase rapidly due to the exponential suppression of plasmons below the plasmon temperature.A.5. Time-Scales
Besides the micro-physics ingredients, κ , C V , and Q ν , the two evolutionary time-scales dictated by them are alsovery illustrative: the neutrino cooling time scale τ ν ≡ C V /Q ν and the heat diffusion time scale τ h ≡ C V /K . We displayboth of them in Fig. 18 as they are very helpfull to understand our results. B. NUCLEI SPECIFIC HEATIn principle, calculation of the specific heat is straightforward as it can be directly derived from the system’s partitionfunction Z = Z ( T ) (see, e.g., Landau & Lifshitz 1993) as C V = k B T (cid:32) Z (cid:48) Z − T (cid:18) Z (cid:48) Z (cid:19) + T Z (cid:48)(cid:48) Z (cid:33) , (B2) volution of Neo Neutron Stars. I. log T [K] l og C V [ e r g / ( c m K ) ] w nucl . C V w / o nucl . C V Few nucl . C V w / o nucl . C V Fe log T [K] l og C V [ e r g / ( c m K ) ] w nucl . C V w / o nucl . C V Fe Ni Ni 5 MeV Ni LD (a) (b) ⇢ = 10 g cm
Heat capacity, of both the nuclei and the total, as a function of temperature at two fixed densities ρ = 10 , panel(a), and 10 g cm − , panel (b). Solid black curve corresponds to Fe, dashed black curve – to Ni, dot-dashed black curve –to total heat capacity including contribution from Fe nuclei, dot-dashed gray curve – to total heat capacity excluding nucleicontribution. Solid gray curve demonstrates Ni nuclei heat capacity for the energy cutoff of 5 MeV. Dashed gray curve shows Ni nuclei heat capacity calculated employing the level density ρ LD instead of the density of states Ω DS . See details in the text. where primes denote derivative with respect to the temperature T and k B is the Boltzman constant. We describe belowhow do we proceed to calculate the partition function.The nucleus has a discrete excitation energy spectrum but only low lying energy levels (up to a few MeV) are knownreliably from experiments. Moreover, at higher energies the density of states grows so rapidly that it is more convenientto approximate with a continuous distribution. If ρ LD ( E, J ) is the density of energy levels of angular momentum J atenergy E , the “observable level density” is ρ LD ( E ) = (cid:80) J ρ LD ( E, J ) while the density of sates or “true level density”,that takes into account the spin degeneracy, is Ω DS ( E ) = (cid:80) J (2 J + 1) ρ LD ( E, J ) (Gilbert & Cameron 1965; Huizenga &Moretto 1972). We will follow the commonly used Back-Shifted Fermi Gas (BSFG) approximation (see, e.g., von Egidy& Bucurescu 2005), an extension of the non-interactive Fermi gas model of Bethe (1936), in whichΩ DS ( E ) = √ π exp (cid:16) (cid:112) a ( E − E ) (cid:17) a ( E − E ) = √ πσ ρ LD ( E ) (B3)where E is the energy back-shift, a the level density parameter, and σ the spin cutoff, whose values are obtained byfitting experimental data. Within this approximation we can calculate Z ( T ) as Z ( T ) = i = i cf (cid:88) i =0 g i exp (cid:18) − E i k B T (cid:19) + (cid:90) ∞ E cf Ω DS ( E ) exp (cid:18) − Ek B T (cid:19) d E , (B4)where E cf = E i cf is some arbitrary energy cutoff level to switch from discrete to continuous regime; g i is the spindegeneracy factor, which is obtained experimentally together with the energy levels E i . In the continuous spectrumrange the spin degeneracy is in principle taken into account in Ω DS ( E ). This spin degeneracy is not experimentallydetermined in the high excitation (continuous) regime and there are large uncertainties in its value (see, e.g., von Egidy& Bucurescu 2009) and for this reason many authors prefer to use ρ LD ( E ) instead of Ω DS ( E ) in the evaluation of Z ( T )in Eq. (B4).We used the procedure described above to calculate the contributions of Fe and Ni nuclei to the heat capacity.For Fe we used experimental data on the energy levels and spin degeneracy factors from Junde et al. (2011) and for0
Beznogov, Page, and Ramirez-Ruiz Ni – from Browne & Tuli (2013). The values of E and a were taken from the recent fits of Bucurescu & von Egidy(2015). The choice of the cutoff energy is important, so we tried two approaches: cutoff at 5 MeV and cutoff at the firstenergy level for which experimental value of g is not known. In the latter case the cutoff energy was ∼ . Fe and ∼ . Ni. The heat capacity calculated using Eqs. (B2), (B3) and (B4) is per nucleus. Thus, onehave to multiply it by the ion number density n i = ρ /( Am u ) , where A is the nucleus atomic mass number and m u isatomic mass unit. The results are presented on Fig. 19, which shows the heat capacity, of both the nuclei and the total,as a function of temperature at two fixed densities ρ = 10 and 10 g cm − . Solid black curve corresponds to Fe,dashed black curve – to Ni, dot-dashed black curve – to total heat capacity including contribution from Fe nuclei,dot-dashed gray curve – to total heat capacity excluding nuclei contribution. Solid gray curve demonstrates Ni nucleiheat capacity for the energy cutoff of 5 MeV instead of ∼ . Ni nuclei heat capacity calculated employing the level density ρ LD instead of the density of states Ω DS in Eq. (B4).From Fig. 19 one can make several conclusions. First, at low temperatures ( T ∼ K) Fe and Ni heat capacitiesare considerably different. This is not a surprise because at these temperatures the heat capacity is governed by a firstfew low lying energy levels, which can differ rather noticeably even for similar nuclei. On the other hand, this does notmatter much as at T ∼ K nuclei heat capacity is much less than the total heat capacity and, thus, can be neglected.Second, the maximum contribution of the nuclei heat capacity to the total heat capacity is achieved at T ∼ . K andcan be around 50% of the total heat capacity at ρ = 10 g cm − . Nuclei heat capacity is directly proportional to thedensity of matter (see paragraph before previous), thus, its contribution at lower densities is lower and at sufficientlylow densities ( ρ (cid:46) g cm − ) can be neglected at any temperature. Third, at temperatures T ∼ . − K nucleiheat capacity is sensitive to the particular nuclear species and to the choice of the energy cutoff. The difference can beup to ∼ C. NUMERICAL METHODWe base our calculations on the code
NSCool (Page 1989, 2016) with important adjustments to solve for hydrostaticequilibrium in the inner envelope in the conditions where radiation and pairs pressures are important.The structure equations are initially solved from the center of the star down to ρ c = 10 g cm − employing the zerotemperature EOS and this interior structure is not modified afterward. At densities between ρ c and ρ b the structureequations are solved at every time step. The thermal evolution equations (1) and (2) are solved in the whole star, i.e.,from the center down to ρ b , at every time step.So, in the inner envelope structure and thermal evolution equations have to be solved at each time step. Thereare several ways to do it and we had tried some of them until we have found a suitable one. The most considerabledifficulty lies in the fact that the outer parts of the inner envelope are dominated by photons and electron-positronpairs. Thus, the adiabatic index is close to 4/3 and the system is close to being unstable.In the standard long term cooling calculation scheme (see, e.g., Page 1989; Gnedin et al. 2001) thermal evolutionequations are usually solved fully implicitly employing Newton-Raphson method (Henyey scheme Henyey et al. 1959).The easiest way to modify this scheme to handle neo-neutron stars is to solve structure equations separately fromthermal equations at each Newton-Raphson iteration for the thermal equations. Unfortunately, this idea does not work.The thermal equation (2) [which is basically the energy conservation law] and the hydrostatic equilibrium equation[Eq. (4) of Potekhin & Chabrier (2018)] have a tendency to create oscillations in pressure, radius and temperature. Thisis easy to understand: if we solve them separately , some decrease in the radius will cause an increase in the temperaturedue to the injection of contraction energy [Eq. (2)], which will increase the pressure and cause an increase in the radiusdue to the hydrostatic equilibrium equation. This will, in turn, cause the temperature and pressure to drop and adecrease in the radius. Clearly, this method is prone to instability and should not be used. We implemented it andfound out that it indeed resulted in diverging iterations and in oscillations.So, to deal with this tendency to oscillate one has to solve structure and thermal equations together in a singleNewton-Raphson iteration scheme. In this case the changes in the pressure, radius and temperature are coordinatedwith each other at each iteration and consistent solution can be obtained. As it turns out there is now need to solveall six equations together in a single Newton-Raphson scheme. Actually, it is sufficient to solve only four equations volution of Neo Neutron Stars. I. L s (cid:54) = L b ) and the matching will occur automatically at thefirst time step. In our neo-neutron stars models the Henyey method would not converge at the first time step if suchinconsistent surface luminosity is employed. So, we have developed a special matching procedure for the luminosity tostart with the consistent initial and boundary conditions: T t =0 ( ρ ) = F (cid:0) ρ, { p , p , . . . } , p match (cid:1) , where p , p , . . . , p match are free parameters of the parametrization of an arbitrary initial temperature profile. The procedure is as follows: wefix the values of p , p , . . . and use Newton-Raphson method to search for the value of p match until the initial profilesatisfies the boundary condition to the desired precision [i.e, we stop when L s { T s ( T b ) } = L b ]. Typically, this takes 5-6Newton-Raphson iterations. Employment of such a procedure means that our initial temperature profile is no longercompletely arbitrary.In particular, as an initial temperature we take a uniform value T at densities above ρ c and in the inner envelope wechoose T ( ρ ) = T c , − ∆ T (cid:18) log[ ρ c /ρ ]log[ ρ c /ρ b ] (cid:19) γ (C5)where ∆ T = T c , − T b , and γ > γ is 1, then T ( ρ ) is just linear in log ρ . We usually fixedthe value of T c , to be 2 . × K and for various values of γ we solved for ∆ T to match the initial and boundaryconditions.Unfortunately, with the parametrization (C5) the matching occurs only at super-Eddington surface luminosities forany tested value of γ . As we do not consider mass loss and stellar winds in the current work we had two options:change the initial temperature parametrization or explicitly set the initial luminosity in the inner envelope. We decidedto do the latter. Setting the initial luminosity directly requires a separate step in the algorithm to solve for the initialtemperature given the initial luminosity. We incorporated matching of the initial and boundary conditions in this step.In such scheme we have lost direct control over the initial temperature, but, as we show in Sect. 5, direct control overluminosity might be more useful for studying neo-neutron stars. Besides, we can still control T b , via T b − T s relationsand the fact that L s ( T s ( T b )) = L b . We can also control T c , by adjusting the initial luminosity (see details in Sect. 5).We kept the parametrization (C5) to demonstrate how a relatively small change in the initial temperature profile canconsiderably affect the cooling during the first ∼ s.REFERENCES Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017a,ApJL, 848, L13—. 2017b, ApJL, 848, L12Akmal, A., Pandharipande, V. R., & Ravenhall, D. G. 1998,Phys. Rev. C, 58, 1804 Of course, with the surface boundary condition (7) we cannot havesuper-Eddington surface luminosity; thus, we had to extrapolate T b − T s relations of Sect. 3.1 to higher temperatures to obtainthe matching. Antoniadis, J., Freire, P. C. C., Wex, N., et al. 2013, Science,340, 448Baade, W., & Zwicky, F. 1934, Proceedings of the NationalAcademy of Science, 20, 254Bahcall, J. N., Rees, M. J., & Salpeter, E. E. 1970, ApJ, 162,737Baiko, D. A., Kaminker, A. D., Potekhin, A. Y., & Yakovlev,D. G. 1998, Physical Review Letters, 81, 5556Barnes, J., & Kasen, D. 2013, ApJ, 775, 18Bethe, H. A. 1936, Phys. Rev., 50, 332 Beznogov, Page, and Ramirez-Ruiz
Beznogov, M. V., Potekhin, A. Y., & Yakovlev, D. G. 2016,MNRAS, 459, 1569Beznogov, M. V., & Yakovlev, D. G. 2015, MNRAS, 447,1598Bildsten, L., Shen, K. J., Weinberg, N. N., & Nelemans, G.2007, ApJL, 662, L95Boguta, J. 1981, Physics Letters B, 106, 255Brown, E. F., & Cumming, A. 2009, ApJ, 698, 1020Browne, E., & Tuli, J. K. 2013, Nuclear Data Sheets, 114,1849Bucurescu, D., & von Egidy, T. 2015, in EPJ Web ofConferences, Vol. 93, EPJ Web of Conferences, 06003Burrows, A., & Lattimer, J. M. 1986, ApJ, 307, 178Canal, R., Garcia, D., Isern, J., & Labay, J. 1990, ApJL,356, L51Canal, R., & Schatzman, E. 1976, A&A, 46, 229Cigan, P., Matsuura, M., Gomez, H. L., et al. 2019, ApJ,886, 51Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017,Science, 358, 1556Cromartie, H. T., Fonseca, E., Ransom, S. M., et al. 2019,arXiv e-prints. https://arxiv.org/abs/1904.06759Darbha, S., Metzger, B. D., Quataert, E., et al. 2010,MNRAS, 409, 846Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts,M. S. E., & Hessels, J. W. T. 2010, Nature, 467, 1081Dessart, L., Burrows, A., Ott, C. D., et al. 2006, ApJ, 644,1063Eichler, D., Livio, M., Piran, T., & Schramm, D. N. 1989,Nature, 340, 126Faber, J. A., & Rasio, F. A. 2012, Living Reviews inRelativity, 15, 8Fesen, R. A., Hammell, M. C., Morse, J., et al. 2006, ApJ,645, 283Folatelli, G., Bersten, M. C., Benvenuto, O. G., et al. 2014,ApJL, 793, L22Fong, W., Berger, E., Blanchard, P. K., et al. 2017, ApJL,848, L23Geppert, U., K¨uker, M., & Page, D. 2004, A&A, 426, 267—. 2006, A&A, 457, 937Gilbert, A., & Cameron, A. G. W. 1965, Can. J. Phys., 43,1446Gnedin, O. Y., Yakovlev, D. G., & Potekhin, A. Y. 2001,MNRAS, 324, 725Greenstein, G., & Hartke, G. J. 1983, ApJ, 271, 283Gudmundsson, E. H., Pethick, C. J., & Epstein, R. I. 1982,ApJL, 259, L19 Haensel, P., Potekhin, A. Y., & Yakovlev, D. G. 2007,Astrophysics and Space Science Library, Vol. 326, NeutronStars. 1. Equation of State and Structure (Springer, NewYork)Haensel, P., Zdunik, J. L., & Dobaczewski, J. 1989, A&A,222, 353Hansen, C. J., Kawaler, S. D., & Trimble, V. 2004, Stellarinteriors : physical principles, structure, and evolution(Spinger-Verlag, New York)Henyey, L. G., Wilets, L., B¨ohm, K. H., Lelevier, R., &Levee, R. D. 1959, ApJ, 129, 628Huizenga, J. R., & Moretto, L. G. 1972, Annual Review ofNuclear and Particle Science, 22, 427Itoh, N., Hayashi, H., Nishikawa, A., & Kohyama, Y. 1996,ApJS, 102, 411Janka, H.-T. 2012, Annual Review of Nuclear and ParticleScience, 62, 407Junde, H., Su, H., & Dong, Y. 2011, Nuclear Data Sheets,112, 1513Kaminker, A. D., Pethick, C. J., Potekhin, A. Y., Thorsson,V., & Yakovlev, D. G. 1999, A&A, 343, 1009Kaminker, A. D., & Yakovlev, D. G. 1994, AstronomyReports, 38, 809Kaplan, D. L., Frail, D. A., Gaensler, B. M., et al. 2004,ApJS, 153, 269Kaplan, D. L., Gaensler, B. M., Kulkarni, S. R., & Slane,P. O. 2006, ApJS, 163, 344Kasen, D., Metzger, B., Barnes, J., Quataert, E., &Ramirez-Ruiz, E. 2017, Nature, 551, 80Kaspi, V. M., & Beloborodov, A. M. 2017, ARA&A, 55, 261Kilpatrick, C. D., Foley, R. J., Kasen, D., et al. 2017,Science, 358, 1583Kippenhahn, R., Weigert, A., & Weiss, A. 2012, StellarStructure and Evolution (Springer Verlag, Astronomy andAstrophysics Library)Klu´zniak, W., & Ruderman, M. 1998, ApJL, 505, L113Landau, L. D., & Lifshitz, E. M. 1993, Statistical Physics,Part 1 (Pergamon, Oxford)Lattimer, J. M., Pethick, C. J., Prakash, M., & Haensel, P.1991, Phys. Rev. Lett., 66, 2701Lattimer, J. M., & Swesty, D. F. 1991, Nuclear Physics A,535, 331Lattimer, J. M., van Riper, K. A., Prakash, M., & Prakash,M. 1994, ApJ, 425, 802Lee, W. H., & Ramirez-Ruiz, E. 2007, New Journal ofPhysics, 9, 17Livne, E., Asida, S. M., & H¨oflich, P. 2005, ApJ, 632, 443Lopez, L. A., Ramirez-Ruiz, E., Castro, D., & Pearson, S.2013, ApJ, 764, 50 volution of Neo Neutron Stars. I.23