Can newly born neutron star collapse to low-mass black hole?

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1 Can newly born neutron star collapse to low-mass black hole?. Miyazaki Abstract We investigate for the rst time the newly born neutron star (NS) containing both hyperons and kaons as strange hadrons within the relativistic mean- eld theory. It is found that the maximum baryonic mass of the newly born NS is lower than that of the cold deleptonized NS. Against the suggestion by Brown and Bethe, there is no possibility of the delayed collapse of NS to low-mass black hole. After Brown and Bethe [] suggested low-mass black holes (BHs) from the delayed collapses of neutron stars (NSs), there have been several investigations [2-6] for the metastability of newly born NS. he NS is in quasi-stationary -equilibrium state at di erent times during its evolution because the time scale of weak interaction is much shorter than the time scale of neutrino di usion. he static approach adopted in Refs. [2-6] is therefore useful to newly born NSs. hen, it is expected that the baryon number or the baryonic mass is conserved in NSs. A newly born NS, whose baryonic mass is larger than the maximum baryonic mass of cold deleptonized NS, is therefore metastable and so collapses to a BH after deleptonization. In Refs. [2-6] there appear low-mass BHs whose gravitational masses are lower than two times of the solar mass. o the contrary, the observed low-mass BH candidates [7,8] have the masses near 4M. Consequently, the possibility of low-mass BHs is still open to question. he suggestion by Brown and Bethe was based on the phase transition in NS matter due to antikaon condensation, while Refs. [2-6] did not take into account it. So far, only two works [9,] investigated the antikaon condensation in hot protoneutron star (PNS). heir results also supported the delayed collapses of NSs. However, both the works considered only nucleons as baryons but not hyperons. his is insu cient because the hyperons have strangeness as well as kaons. On the other hand, there are several works [-6] for the cold deleptonized NS consisting of antikaons and all the baryons. In their results the antikaon condensations lead to noticeable soft equations-of-state (EOSs) for NS matter. hose EOSs would be much sti er in hot PNS matter because of the thermal contribution to pressure, and so the maximum baryonic mass of newly born NS would be larger than that of the cold deleptonized NS. Consequently, the delayed collapses of NSs

2 Can newly born neutron star collapse to low-mass black hole? to low-mass BHs would be still possible even if the models of Refs. [-6] were applied to newly born NS. o the contrary, we have recently developed a new relativistic mean- eld (RMF) model [7] that predicts a sti er EOS of NS matter with the antikaon condensed phase than the EOS of normal NS matter without it. he model is based on the extended Zimanyi- Moszkowski (EZM) model [8-25] that has the eld-dependent meson-baryon coupling constants in dense nuclear medium but the free meson-kaon coupling constants. It has been found that the abundance of antikaons severely suppresses hyperons. Because the abundance of hyperons generally leads to a rather soft EOS [26], we have inversely a rather sti EOS due to antikaon condensations. Consequently, we can reproduce the recently observed massive NSs [27-3] being heavier than :6M, which are not reproduced in the EZM model [25] consisting of only baryons as hadrons. Because the result of Ref. [7] is remarkably di erent from the other models, it is worthwhile to apply the model to hot PNS. he purpose of the present work is to revisit the possibility of low-mass BHs. he formulation of the hot isentropic PNS matter consisting of baryons and leptons is the same as Ref. [6]. hen, we have only to add the kaon contributions. Here, we follow the formulation of Ref. [9] but take into account both the ( + ) and ( ) in the isospin doublet. he kaons are treated [3] on the same footing as the baryons. he thermodynamic potential per volume ~ =V at a given temperature is ~ = 2 m2 hi m2 h 3 i m2 h i 2 2 m2! h! i 2 2 m2 h 3 i 2 2 m2 h i 2 Z B EkB 2 (2) 3 ln + exp + + B=p;n;; + ; ; ; ; l=e;; e; l = ; = ; Z (2) 3 B EkB + ln + exp l ln + exp 2 (f ) V 2 Z (2) 3 ln exp! e kl + ln + exp + ln exp l e kl! : () As compared with Ref. [7], Ref. [5] did not take into account the strange mesons while Ref. [6] employed the eld-dependent meson-kaon coupling constants. Moreover, Refs. [5] and [6] employed the di erent formulation and calculus of antikaon condensation from Ref. [7], although they did not have signi cant e ects on the results. 2

3 . Miyazaki he Boltzmann constant is set as a unit. he rst to third lines in the right hand side are formally the same as Eq. () in Ref. [6]. he fourth line is the contribution of the s- wave antikaon condensations, in which f is the pion decay constant, is the condensate amplitude, V is the vector potential of kaon [7], is the chemical potential of kaon and is de ned by = M 2 V: 2 (2) M is the e ective mass of kaon [7] in strange baryonic matter. he fth line in Eq. () is the thermal kaon contribution, in which! = p k 2 + M 2 ; (3) = V : (4) he nonlinear equations, which determine the three independent e ective masses M p = m pm N, M n = m nm N and M = m M and the three independent vector potentials V p, V n and V, are C V N p + gnn gy Y! + gnn! gy Y I 3Y C V gnn gy Y Y Y 6= g nn g! g nn! g I 3 C V g nn g m 2! g nn h! i m 2 g nn! h 3 i + C V m 2 g nn h i = ; (5) C V N n + gpp gy Y! gpp! gy Y I 3Y C V gpp gy Y Y Y 6= gpp g! + gpp! g I 3 C V gpp g m 2! g pp h! i + m 2 g pp! h 3 i + C V m 2 g pp h i = ; (6) gy Y Y Y g m 2 h i = ; (7) M N Sp + Y 6= + m 2 p m 2! h! h! Y Y SY Y Y + Y 6= p + m 2 h 3 h p m 2 h 3 h p + m 2 h h p m 2 h h p p = ; (8)

4 Can newly born neutron star collapse to low-mass black hole? M N Sn + Y 6= + m 2 n m 2! h! h! Y Y SY Y + M N Y 6= S n + m 2 h 3 h n m 2 h 3 h n + m 2 h h n m 2 h h n = ; (9) Y Y SY Y + Y 6= + m 2 h h m 2 h @m = ; () where C V N = g pp! g nn + g nn! g pp; () C V = g!=g : (2) he terms for baryons and mean- elds in Eqs. (5)-() are formally the same as Eqs. (42)-(47) in Ref. [6]. 2 he isospin of kaon is de ned by I 3 = (; ) for = ;. he terms for kaons include the densities de ned by = (f ) 2 + S = (f ) 2 M + Z Z (2) 3 [ n k ( ) n k ( )]; (3) (2) 3 M! [ n k ( ) + n k ( )]; (4) where the Bose-Einstein distribution functions are n k ( ) = n k ( ) =! exp! + exp ; (5) : (6) 2 In Ref. [6] we mistyped Eqs. (45) and (46) so that their third terms include unnecessary factor M Y. 4

5 . Miyazaki he energy density is given by E = 2 m2 hi m2 h 3 i m2 h i 2 2 m2! h! i 2 2 m2 h 3 i 2 2 m2 h i < Z + = : (2) 3 E kb [ n kb ( ) + n kb ( )] A + V B B ; + + B=p;n;; + ; ; ; ; l=e;; e; l = ; Z Z (2) 3 e kl [ n kl ( ) + n kl ( )] + = ; 2 (f ) (2) 3 [(! + V ) n k ( ) + (! V ) n k ( )] ; (7) where the rst to third lines in the right hand side are formally the same as Eq. (55) in Ref. [6]. As mentioned rst, our investigation is based on a static approach, which assumes that a newly born NS is in chemical-equilibrium state: i = b i n q i e e ; (8) = e e ; (9) where i, b i and q i are the chemical potential, the baryon number and the charge of each particle. he properties of a newly born NS matter below the condensation threshold < M + V, is determined by solving Eqs. (5)-() with = = under the baryon number conservation = B=p;n;; + ; ; ; ; B ; (2) the charge neutral condition q i i = ; (2) the lepton number conservation [32] i=b;l; Y Le = e + e = :4; (22) Y L = + = ; (23) 5

6 Can newly born neutron star collapse to low-mass black hole? and the isentropic condition s = E + P n + Y Le e ; (24) by means of -dimensional Newton-Raphson method so that the e ective masses Mp, Mn and M, the vector potentials V p, V n and V, the chemical potentials n, e, e and and the temperature are determined selfconsistently. he pressure is given by P =. ~ Above the condensation threshold but below the condensation threshold < M + V, Eqs. (5)-() with = and Eqs. (2)-(24) are solved together with M + V = ; (25) by means of 2-dimensional Newton-Raphson method so that we have the e ective masses, the vector potentials, the chemical potentials, the temperature and the condensation amplitude f. Above the condensation threshold, Eqs. (5)-() and (2)-(25) are solved together with M + V = ; (26) by means of 3-dimensional Newton-Raphson method so that we have the e ective masses, the vector potentials, the chemical potentials, the temperature and the and condensation amplitudes. In the present calculation, we use the same meson-baryon coupling constants as the EZM-P in Ref. [25] and the same meson-kaon coupling constants as Ref. [7] for the antikaon optical potential U ( NM ) = 7MeV in a saturated nuclear matter. Within the EZM model their values are the most reasonable ones at present. We assume [3,6] that the newly born NS is composed of three layers. he rst is the neutrino-opaque inner core in the region :fm 3 with the entropy per baryon s =. he second is the neutrinotrapped hot shocked envelope in the region 6 4 fm 3 :2fm 3 with s = 5. he third is the neutrino-transparent cold outer crust in the region < 6 4 fm 3. he EOSs of the rst and second layers are calculated using our EZM model. Of course, the kaon-condensed phases appear only in the inner core. For the third layer we employ the EOSs of Feynman-Metropolis-eller, Baym-Pethick-Sutherland and Negele-Vautherin from Ref. [33]. he EOSs of the transition regions between the rst and second layers and between the second and third layers are obtained by linear interpolation. Figure shows the particle fractions in the inner core of newly born NS. Our calculation is stopped at = :37fm 3, above which the e ective mass of neutron becomes negative. his is not a problem because the central baryon density = :944fm 3 of the most massive PNS is lower than the limit. here appear no hyperons because 6

7 . Miyazaki their fractions are lower than 4. he threshold densities of and condensed phases are = :53fm 3 and = :74fm 3, respectively. hey are clearly seen as the rapid decreases of hyperons, which are therefore poor in newly born NS. As compared with the cold deleptonized NS, the thermal distribution in newly born NS shifts the threshold of condensation to higher density because of the charge neutral condition. (See Fig. 4 in Ref. [7].) Above the threshold, the proton fraction becomes larger, and so the increase of becomes slow because of the baryon number conservation. o the contrary, the thermal distribution in newly born NS shifts the threshold of condensation to lower density as compared with the cold NS. his is because the thermal distribution reduces the left hand side of Eq. (26). Above the threshold, the NS matter tends to recover the isospin symmetry. Consequently, the increase of is restrained, and the proton turns to decrease but the neutron turns to increase. he isospin symmetry is recovered at = :7fm 3, above which the isovector-scalar mean- eld changes its sign and so the symmetry is broken again. he red and blue curves in Fig. 2 show the EOSs of the inner core in newly born NS and cold NS, respectively. he latter is the result of Ref. [7]. It is remarkable that both the EOSs are almost the same. In our EZM model [7] the condensation leads to a sti EOS of the cold NS. (he condensation does not contribute directly to the EOS.) herefore, the delayed condensation threshold in Fig. inversely softens the EOS of newly born NS. Consequently, the thermal contributions to the EOS are canceled out. he kaon condensations in the EOSs are the second-order phase transitions. his is an attractive feature of our EZM model as was discussed in Ref. [7]. hen, we calculate the properties of NSs by integrating the olman-oppenheimer-volkov equation [34]. Figure 3 shows the gravitational masses of newly born NS and cold NS as functions of the central energy density. Because of the EOSs in Fig. 2 the red and blue curves coincide above E C = 6 4 g=cm 3. he maximum mass is M G = :95M on both the curves, while the minimum mass of newly born NS is M G = :4M. Figure 4 shows the relations between the gravitational mass and the radius. Because of the hot shocked envelope the radius R = 4:27km of the newly born NS with maximum mass is larger than the radius R = :64km of the most massive cold NS. he radius of the newly born NS with minimum mass is R = 4:22km. Figure 5 shows the relations between the gravitational and baryonic mass of newly born NS and cold NS. he maximum baryonic mass M B = 2:5M on the red curve is lower than M B = 2:25M on the blue curve. he result indicates that the delayed collapse of a newly born NS to a low-mass BH cannot be expected against the suggestion [] by Brown and Bethe. Is our prediction is reliable? Because our previous work without kaons [6] supports low-mass BHs, the present result is certainly due to the contributions of kaons. Moreover, the minimum mass of newly born NS plays a fundamental role in assessing the reliability, because it is sensitive [3,6,35-37] to our choice for the pro le of hot shocked envelope. As shown by the lower vertical dashed line in Fig. 5, the 7

8 Can newly born neutron star collapse to low-mass black hole? minimum baryonic mass M B = :54M of newly born NS determines the minimum gravitational mass of cold NS. he obtained value M G = :8M is consistent with the range M G = :35 :27M [38] of the well observed NS masses except for the recently observed massive NSs [27-3]. his indicates that our calculation is reasonable. For fairness we should mention a defect of our result. As shown by the upper vertical dashed line in Fig. 5, the maximum baryonic mass of newly born NS also determines the maximum gravitational mass of cold NS, which is lower than the maximum value M G = :95M on the blue curve. he obtained most massive NS with M G = :88M and R = 2:54km lies just below the mass-radius relation [3,39] of EO he defect is however solved by mass accretion of only :2M. Consequently, we conclude that the delayed collapse due to kaon condensation is questionable, although there still remain uncertainties of the kaon interactions in dense strange hadronic matter. 8

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11 . Miyazaki e p n Λ Particle Fraction 2 ν e Ξ Ξ otal Baryon Density (fm 3 ) Figure : he particle fractions in the inner core of newly born NS as functions of total baryon density.

12 Can newly born neutron star collapse to low-mass black hole? 6 Newly born NS Cold NS Pressure (MeV/fm 3 ) Energy Density (MeV/fm 3 ) Figure 2: he EOSs of the inner core in newly born NS and cold NS. 2

13 . Miyazaki M G (M sun ) Newly born NS Cold NS.6 3x 4 5 4x 5 ε c (g/cm 3 ) Figure 3: he gravitational masses of newly born NS and cold NS as functions of the central energy density. 3

14 Can newly born neutron star collapse to low-mass black hole? 2.5 Newly born NS Cold NS M G (M sun ) Radius (km) Figure 4: he relations between the gravitational mass and the radius of newly born NS and cold NS. 4

15 . Miyazaki 2.8 Newly born NS Cold NS M G (M sun ) M B (M sun ) Figure 5: he relations between the gravitational and baryonic masses of newly born NS and cold NS. he vertical dashed lines indicate the baryonic masses corresponding the minimum and maximum gravitational masses of newly born NSs. 5