Quark deconfinement in the proto-magnetar model of LGRB

Quark deconfinement in the proto-magnetar model of Long Gamma-Ray Bursts

Abstract

We investigate the possible implications of quark deconfinement on the phenomenology of Long Gamma-Ray Bursts focusing, in particular, on the possibility to describe multiple prompt emission phases in the context of the proto-magnetar model. Starting from numerical models of rotating Hadron Stars and Quark Stars in full general relativity we track the electromagnetic spin-down evolution in both the hadronic and quark phase, linking the two families through conservation of baryon number and angular momentum. We give estimates of the timescales and the energetics involved in the spin-down process deriving, in the relevant spin range, the relation between the initial and the final masses and rotational energies, whenever hadron-quark conversion is possible. We show how the results can be used in relevant astrophysical cases such as the double burst GRB 110709B.

keywords:
gamma-ray bursts - star: magnetar - star: neutron - dense matter - gravitation

1 Introduction

The nature of the inner engine of Long Gamma-Ray Bursts (LGRBs) is still one of the most interesting and unsolved problems in astrophysics. While there is a compelling evidence that these events are associated with the core collapse of massive stars, it is not yet established whether the burst is produced by a disk accreting onto a black hole (within the so-called Collapsar model, Woosley 1993) or by the outflow emerging from a fast rotating and highly magnetized proto-neutron star (within the so-called proto-magnetar model, Metzger et al. 2011). It is also possible that both these systems contribute in producing LGRBs, depending on the initial conditions of the progenitor (in particular its mass, spin frequency and magnetic field). The SWIFT discovery of a late time activity, lasting up to s and present in a sizable fraction of LGRBs, is more naturally interpreted within the proto-magnetar model as due to the pulsar-like energy injection powered by the residual rotational energy left after the prompt emission (Dall’Osso et al., 2011).

Within the proto-magnetar model, an important ingredient which regulates the temporal evolution of the jet and its gamma-ray luminosity is the neutrino signal released by the star due to deleptonization (the gradual neutronization of matter) and cooling. In particular, neutrinos ablate baryonic matter from the surface and provide a tiny amount of baryonic load which is crucial for an efficient internal dissipation of the kinetic energy of the jet into gamma emission. When the neutrino luminosity drops below erg/s (after a few tens of seconds), the magnetization (or the Lorentz factor) is too large and the prompt emission ends. It is not easy within the proto-magnetar model to reactivate the inner engine and therefore to describe multi-episodes of the prompt emission of LGRBs (e.g. Zhang et al. 2012) or X-ray flares occurring during the afterglow (Zhang et al., 2006).

The proto-magnetar model has been developed assuming that the newly born Compact Star (CS) is and remains a nucleonic star. Motivated by the measurements of very massive CSs (Demorest et al., 2010; Antoniadis et al., 2013) and the hints of the existence of very compact stellar objects (with radii close to 10km, Guillot et al. 2013; Özel et al. 2016), a two-families scenario has been recently proposed where both Hadronic Stars (HSs) and Quark Stars (QSs) exist in nature (Drago, Lavagno & Pagliara, 2014; Drago et al., 2016; Drago & Pagliara, 2016). Within this picture, a conversion of a HS into a QS can take place. In this paper we will discuss how this transition modifies the proto-magnetar model, revitalizing the inner engine. In particular, we will investigate the following scenario: i) a proto-magnetar is formed after a successful supernova explosion and its spinning down, which is responsible for the emission of a LGRB, leads to a gradual increase of the central density; ii) the increase of the density allows the formation of heavy hadrons such as Delta resonances and hyperons; iii) once a critical amount of strangeness is formed through hyperons at a density , the HS converts (on a time-scale  s) into a QS. This picture has been qualitatively discussed in Drago, Lavagno & Pagliara (2014) for the case of non-rotating stars. Here we will improve that work by including rotation and spin-down evolution along the line of Haensel et al. 2016, and Refs. therein. Since both the initial hadronic and the final quark configurations are fast rotating stars, we compute their structure as rigid rotating stars beyond the Hartle approximation (Hartle, 1967) through the XNS code, which adopts the conformally-flat approximation for the space-time metric (see Bucciantini & Del Zanna 2011; Pili, Bucciantini & Del Zanna 2014, 2015; Bucciantini, Pili & Del Zanna 2015).

A distinctive feature of the two-families model is that the formation of a QS is accompanied by a rather large amount of energy released in the conversion, of the order of erg. Moreover, we will show that since the final QS configuration has a larger radius than the initial HS configuration, the conversion is accompanied by a significant increase of the momentum of inertia with a corresponding decrease of the rotational frequency. We will discuss possible phenomenological implications of the two-families scenario for the light curves of LGRBs in connection with late time activity. For the case of QSs there are two possible ways to produce a jet with the appropriate Lorentz factor. The first mechanism is based on the ablation of baryonic material from the surface of the star as long as the conversion front has not yet reached the surface (Drago & Pagliara, 2016). A second mechanism is based on the rather large emission of electron-positron pairs from the surface of the bare quark star (Usov, 1998, 2001; Page & Usov, 2002). Both mechanism can be at work to produce late time activities.

In section 2 we describe our model and present our numerical results. In section 3 we conclude, discussing the phenomenological implications.

2 The quasi-stationary evolution of Compact Stars

In order to investigate the possible implications of the two-families scenario for the phenomenology of GRBs we have computed a large set of rigidly (rotation rate ) rotating equilibrium models for both HSs and QSs. Given the typical cooling evolution of newly formed proto-neutron stars (Pons et al., 1999), the assumption of a cold star is justified at times larger than  s after formation (the radius is relaxed to its final value). These numerical models are then linked together to describe the quasi-stationary evolution of CSs under the effect of the magnetic braking. With this simple approach we can give a preliminary estimate of both the timescales and the energetics involved in the spin-down before and after quark deconfinement, which we assume to happen when the central baryon density reaches a critical value of the order of (Drago, Lavagno & Pagliara, 2014). Here we set . We also derive the range of initial masses for which a delayed HS to QS conversion is possible.

Numerical models have been obtained using the equations of state (EoSs) discussed in Drago et al. (2016). The stars have been discretized with grid points in both the radial and angular direction and Einstein equations are solved with a semi-spectral method using 20 spherical harmonics. Since the typical surface magnetic field invoked by the magnetar model for GRBs are of the order of G our models are computed neglecting magnetic field. Indeed, as discussed in Pili, Bucciantini & Del Zanna (2015), only central magnetic field strengths higher than G, corresponding to surface magnetic fields stronger than a few G, are able to modify the stellar global properties to a level appreciable with respect to the overall accuracy of the numerical scheme ( ). Hence we can safely assume that the stellar structure and the associated global quantities such as the gravitational mass , the baryonic mass (a proxy for the total baryon number), the circumferential radius and the Komar angular momentum (for definitions see Pili, Bucciantini & Del Zanna 2014 and Refs. therein) do not depend on the magnetic field strength.

In the millisecond magnetar model for GRBs the typical rotation periods invoked in the literature (Metzger et al., 2011; Bucciantini et al., 2012) can be as high as ms. The maximum rotation rate of compact stars at the end of deleptonization has been investigated in several papers (Goussard, Haensel & Zdunik, 1998; Villain et al., 2004; Camelio et al., 2016). Its value depends on the initial physical conditions of the proto-neutron star (e.g. the entropy profile and a possible differential rotation) and on the compactness of the final cold and deleptonized configuration. If one adopts rather stiff EoS, the maximum rotational frequency ranges between and Hz (Goussard, Haensel & Zdunik, 1998; Villain et al., 2004; Camelio et al., 2016). In our analysis we use a rather soft hadronic EoS due to the formation of Delta resonances and hyperons and the maximum rotation frequency is expected to be larger. In the following we limit the rotation of our HS models to a maximum frequency Hz. On the other hand, for those QSs resulting from the conversion of a HS it is sufficient to limit the spin frequency to Hz.

Figure 1: Gravitational mass against baryon central density for sequences of HSs (left panel) and QSs (right panel) with constant spin frequency. In both cases the green solid lines represent the non-rotating sequences while the black solid lines are sequences rotating at the maximum spin frequency considered in this work: for HSs and for QSs. Dashed black lines are sequences with intermediate values of while blue lines are equilibrium sequences at constant baryon mass . The red solid lines locates HSs with while the red dashed line locates QSs originating from the conversion of HSs lying on the red solid line. Detailed results on the configurations labeled A and Bs are listed in table 1.

The obtained equilibrium models are shown in figure 1 in terms of central baryon density and gravitational mass for both HSs (left panel) and QSs (right panel). Here blue lines are sequences of equilibria for given baryonic mass and therefore they represent the evolutionary path of HSs and QSs undergoing spin-down. The same configurations are also shown in figure 2 where we plot the mass-radius relation for different sequences of CSs at constant spin frequency. There are two limiting configurations: the non-rotating configuration A with , and km; the 1 ms rotating configuration B with , and km. These define the mass range of interest for delayed quark deconfinement. The value of the critical mass of the configuration A cannot be determined with high precision not even in the scheme of the two families. The main uncertainties are due to the dynamics regulating the appearance of baryonic resonances and on the estimate of the nucleation time of the first droplet of quark matter (Bombaci et al., 2016).

Massive HSs, that after their initial cooling (lasting for at most  s after core collapse) end above the red line in figure 2, have and they will decay immediately to a QS. On the other hand those HSs having will never experience this transition1. Since for isolated CSs the baryonic mass is a conserved quantity, only those HSs having baryonic mass between and can migrate into the QS branch when their baryon central density rises to as a consequence of spin-down. The yellow shaded region of figure 2 shows these configurations. All the configurations having the same baryonic mass as A (for example the 1 ms rotator configuration C) take an infinite time to deconfine (deconfinement takes place at zero frequency). Notice that along the evolutionary paths the variation of the gravitational mass, and hence of the total energy, is at most of the order of %.

Figure 2: Gravitational mass as a function of the circumferential radius for both HSs and QSs. Thin dashed lines are sequences of stars at a fixed frequency from the non-rotating configurations (thick solid blue and green lines) to the configurations rotating at the maximum frequency (thin solid blue and green lines) and spaced by Hz. The yellow region shows hadronic configurations centrifugally supported against deconfinement. Red lines and labels are the same as in figure 1.

As soon as the central density reaches the critical value the HS decays into a QS in  s. Given the typical spin-down timescales we assume that this transition is instantaneous and both the baryon number (baryonic mass) and the angular momentum are conserved. The red dashed line in the right panel of figure 1 and in figure 2 maps upon the QS sequences the configurations which originate from quark deconfinement of HSs. With reference to our limiting configurations, the HS labeled with A migrates to the configuration with gravitational mass , while the HS B migrates to the configuration with . In both cases the total energy released in the transition is of the order of . Finally, after the formation, QSs can spin-down following the evolutionary path of constant shown in the right panel of figure 1. Detailed results are presented in table 1 where we list the variation of the spin frequency , of the gravitational mass and of the rotational kinetic energy (where ) during the phase transition for a set of selected models. Since QSs have larger radii and hence also a larger moment of inertia, conservation of the angular momentum implies a drop in spin frequency. Interestingly we found that the variation of the stellar radius is slightly larger (% between cases A and B) at higher . Hence transitions at higher are characterized by a bigger percent variation of the spin frequency and of the rotational energy. only accounts for the variation of the kinetic energy while the total change of energy is given by the variation of the gravitational mass and it will produce a reheating of the star. In the range of interest,  erg and  erg. This significant amount of energy released by quark deconfinement can be associated with a late time activity as we will discuss in the following.

Label
[] [Hz] [Hz] [] [] [ erg]
A 1.666 0.00 0.00 1.452 0.132 0.00
1.677 300 130 1.462 0.132 0.20
1.687 400 177 1.470 0.133 0.35
1.698 500 224 1.480 0.133 0.57
1.733 700 317 1.520 0.136 1.18
1.785 900 413 1.555 0.139 2.14
B 1.820 1000 462 1.585 0.142 2.80
Table 1: Global quantities at the phase transition for the configurations labeled in figure 1: gravitational mass and spin frequency of the hadronic configuration at deconfinement; difference between the spin frequency , the gravitational mass , and the rotational energy before and after deconfinement.

In the magnetar GRB model the spin-down evolution is governed by magnetic torques. Therefore in order to evaluate the evolutionary timescales of our models we solve, for simplicity, the spin-down formula for an aligned dipole rotator (Spitkovsky, 2006):

(1)

where is the magnetic field strength at the pole and we assume that the magnetic flux is constant during the evolution. This choice gives only an upper limit to the timescale for deconfinement: oblique rotators or mass loaded winds can have higher torques (Metzger et al., 2011) that can lead to faster spin-down (up to a factor 10) in the first tens of seconds. Furthermore we neglect the possibility of late time mass accretion (Bernardini et al., 2014), because it introduces extra degree of freedom that cannot be easily constrained. Moreover we also neglect gravitational waves emission which is relevant only if the star owns a very strong internal toroidal field larger than  G (Dall’Osso, Shore & Stella, 2009). In the mass range of interest we have evaluated the spin-down timescale up to deconfinement, which is shown in figure 3 and table 2, for each equilibrium sequence of constant , assuming an initial surface magnetic field  G. Note that the spin-down age scales with . Conservation of the magnetic flux is also assumed to hold during the phase transition. The resulting magnetic field strength is of the order of due to the change of the stellar radius.

Figure 3: Time evolution of the spin-frequency (left panel) and of the rotational energy (right panel) for dipole magnetic field losses along sequences of constant baryonic mass . Details about these sequences are listed in tables 1 and 2.
[ms] [ erg] [erg]
1.666 1.0 5.91 - -
1.677 1.0 3.3 2.7 hr 5.48 37 hr 0.19
2.0 3.3 1.8 hr 0.82
3.0 3.3 37 min 0.13
1.687 1.0 2.5 1.5 hr 5.13 21 hr 0.33
2.0 2.5 36 min 0.46
1.698 1.0 2.0 55 min 4.68 14 hr 0.53
1.733 1.0 1.4 23 min 3.37 8.2 hr 1.20
1.785 1.0 1.1 6 min 1.37 5.4 hr 1.95
1.820 1.0 1.0 0 0 4.6 hr 2.41
Table 2: Spin-down timescales to start quark deconfinement together with the associated variation of the rotational kinetic energy starting from an initial spin period for the equilibrium sequences shown in figure 3. We also report the spin-down timescales (defined as the time needed to half the rotational frequency of the QS) and the corresponding rotational energy loss after quark deconfinement. The initial magnetic field is of G.

The time evolution of the spin frequency and of the rotational energy is shown in figure 3 for the HS configurations labelled . It is evident that HSs with higher mass have a shorter lifetime before deconfinement, since they reach the critical density at higher spin frequencies. In table 2 we list the time it takes to start deconfinement and the associated rotational spin-down energy loss for different values for the initial spin period.

3 Discussion and conclusions

Let us consider now the phenomenological implications of our scenario for the evolution of proto-magnetars. The variation of the rotational energy displayed in table 2 gives an estimate of the energy reservoir available to the HS before deconfinement which occurs after a spin-down time scale that ranges from several minutes to hours. Comparing with the typical energetics of the millisecond magnetar model for GRBs (figure 19 of Metzger et al. 2011) one notices that the values shown in table 2 are compatible with the requirement for classical GRBs, being much larger than  erg, i.e. the typical energy emitted in X-rays and in -rays during GRB events. We also remark that since is much larger than the typical duration of the prompt phase of LGRBs, deconfinement does not spoil the nice description of the prompt emission of LGRBs within the proto-magnetar model.

However, as discussed before, also the HS to QS transition is characterized by a huge release of energy (table 1) that in principle could manifest itself as a second transient. Hence the time for deconfinement is indicative of the delay between the prompt GRB emission due to the HS and the possible flare or second prompt emission associated to quark deconfinement. Let us summarize how the deconfinement process proceeds and how is the energy released. As studied in Drago & Pagliara (2015), deconfinement can be described as a combustion process which can be separated in two phases. The first phase is very rapid due to turbulence and it converts the bulk of the star in a time scale of few ms (Drago, Lavagno & Parenti, 2007; Herzog & Ropke, 2011). The second phase is dominated by diffusion of strangeness and it is therefore much slower, typically lasting a few tens of seconds (Drago & Pagliara, 2015). The huge energy associated to deconfinement is released via thermal neutrinos whose luminosity can similarly be divided into an initial peak associated with the deconfinement of the bulk of the star and a lower quasi-plateau emission associated with the burning of the external layer of the star. The interaction of neutrinos with the material of the crust of the star causes the ablation of baryons which plays a crucial role in the proto-magnetar model of GRBs. A distinctive feature of the formation of a QS is the rapid suppression of the baryonic flux once the conversion front reaches the surface of the star. This poses an upper limit to the duration of an event of the prompt emission if described within the proto-magnetar model and if associated with the baryonic emission from the surface of a star undergoing quark deconfinement. However, also the lepton emission from the surface of the bare QS (Usov, 1998, 2001; Page & Usov, 2002) can be used to produce a jet.

As an example we discuss the case of GRB 110709B characterized by two sub-bursts of observed duration respectively  s and  s, separated by a delay  s (Zhang et al., 2012; Penacchioni et al., 2013). Neither the cosmological redshift nor the energetics of the event are well determined and in the following we adopt values compatible with the analysis by Penacchioni et al. (2013), being in agreement with the pulse-wise Amati relation (Basak & Rao, 2013). In particular we assume an isotropic energy  erg for the first burst and  erg for the second one at redshift . We further assume a beaming correction of in line with average jet opening angle of LGRB (Guidorzi et al., 2014). Requiring that during the bursting events (radiatively-efficient phases) the mass loading enhances the spin-down by a factor with respect to the force-free case (Metzger et al., 2011), we find that, assuming an initial magnetic field  G, HSs with initial spin period in the range  ms and baryon masses in have enough rotational energy to power the first event in a time , deconfine with a delay and originate a QS rotating fast enough to power also the second event on the required time . Notice that the requirements of short delays and fast rotating QS select for configurations with high baryon mass. On the contrary HSs with lower take longer to decay and form a slowly rotating QS that at most could power weak flaring events in the afterglows, on timescales of few hours (see table  2), similarly to what is seen in GRB 050916 (Chincarini et al., 2007).

Finally, let us clarify how deconfinement can be responsible of rather long gamma emissions, as in the case of GRB 110709B, and also of the prompt phase of short GRBs, as discussed in Drago & Pagliara (2016); Drago et al. (2015). The main differences are related to the neutrinos flux and energy which determine the baryonic mass ejection rate. In the case of short GRBs, the neutrino flux is significantly larger (of at least one order of magnitude) mainly because of the larger mass of the forming QS. Also the neutrino energy is larger roughly by a factor of three. This implies that the phase during which the QS forms cannot produce a large enough Lorentz factor in the case of short GRBs. At variance, in the case of LGRBs, the mass ejection rate is not too large and the prompt emission can start almost immediately at the beginning of deconfinement.

Although our model requires further investigation for a detailed assessment, our simple analysis of the spin-down evolution of CSs provides us with a new characterization, in terms of energetics and timescales, of the possible observational signatures associated with the two-families scenario. In particular we have shown, with reference to double GRBs, how, once the EoS of HSs and QSs are chosen, the associated phenomenology is constrained by the initial magnetic field strength, initial rotational period and the stellar mass. This implies that, in principle, new observations could validate the two-families hypothesis, gaining new insights in the physics of dense matter.

Footnotes

  1. For simplicity, we are not considering here the process of nucleation of quark matter (Bombaci et al., 2016). In a more realistic case the red lines connecting the configurations A and B would transform in a strip whose width is connected to the nucleation time.

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