Modeling the late-time merger ejecta emission in short gamma ray bursts

Abstract

The short gamma ray bursts (GRBs) are the aftermath of the merger of binary compact objects (neutron star–neutron star or neutron star–black hole systems). With the simultaneous detection of gravitational wave (GW) signal from GW 170817 and GRB 170817A, the much-hypothesized connection between GWs and short GRBs has been proved beyond doubt. The resultant product of the merger could be a millisecond magnetar or a black hole depending on the binary masses and their equation of state. In the case of a magnetar central engine, fraction of the rotational energy deposited to the emerging ejecta produces late-time synchrotron radio emission from the interaction with the ambient medium. In this paper, we present an analysis of a sample of short GRBs located at a redshift of $z \le 0.16$, which were observed at the late-time to search for the emission from merger ejecta. Our sample consists of seven short GRBs, which have radio upper limits available from very large array and Australian telescope compact array observations. We generate the model light curves using the standard magnetar model incorporating the relativistic correction. Using the model light curves and upper limits we constrain the number density of the ambient medium to be $10^{-5}$–$10^{-3}$ cm$^{-3}$ for rotational energy of the magnetar $E_\mathrm{rot} \sim 5\times 10^{51}$ erg. Variation in ejecta mass does not play a significant role in constraining the number density.

1 Introduction

The most accepted progenitor model for the origin of short GRBs is the merger of binary compact objects (binary neutron stars (BNS) and neutron star–black hole (NS–BH)). These systems are also the prime candidates for producing gravitational waves and kilonovae. The joint detection of the GW 170817, GRB 170817A and AT2017gfo confirmed the connection among the three events (Abbott et al. 2017; Valenti et al. 2017; Savchenko et al. 2017; Andreoni et al. 2017; Drout et al. 2017; Tanvir et al. 2017). The most debated open question regarding the central engine of GRB is whether black hole, production is necessary for the emergence of short GRB jet or the central engine could be a highly magnetized and rapidly spinning magnetar (Zhang & Mészáros 2001; Metzger et al. 2008). In case of a BNS merger, the nature of the remnant depends on the initial masses of the BNS and the equation of state of the NSs. The massive binaries ($\ge$3 $M_\odot$) will directly collapse to a black hole whereas the less massive BNS merger creates a transient state in between the merger and the production of black hole which is a millisecond magnetar. In the Swift era, the X-ray light curves from X-ray telescope (XRT, Gehrels et al. 2004) show a complex light curve morphology with an early-time X-ray excess ($\le$10 s), mid time flattening or plateau (10–1000 s) and late time X-ray excess followed by a sharp decay ($\ge$1000 s). The plateau phase in the X-ray light curves is thought to be powered by the magnetar (Rowlinson et al. 2013).

The energy extraction from the central engine occurs via two channels. The first is through the emerging jets from the central engine which carries an enormous amount of energy and gets decelerated by the interaction with the nearby ambient medium. This emission is responsible for the prompt emission and afterglow of the GRBs. Another mode of energy injection is governed by the isotropic ejecta emerging after the merger. As the ejecta is neutron-rich, the matter released during the merger undergoes rapid neutron capture (r-process nucleosynthesis) producing heavy elements. The radioactive decay of these heavy and unstable elements power the isotropic and thermal emission known as kilonova (Yu et al. 2013; Metzger & Fernández 2014; Metzger 2017). The kilonova ejecta is comparatively much slower than the jet and this mildly relativistic ejecta interacts with the ambient medium on the timescale of nearly a few years since the explosion depending on the energy injection from the central engine as well as the density of the medium (Nakar & Piran 2011). If the resultant product of the merger is a millisecond magnetar, the ejecta would be energized because of the continuous energy injection from the magnetar through the spin-down process. This energized ejecta inevitably crashes into the ambient medium resulting in a long-lived synchrotron emission which peaks in the radio (cm) bands. Late-time observations are therefore, important to detect this emission which will not only help to get a complete picture of the central engine, but will also allow to put constraints on the nature of the ambient medium (Nakar & Piran 2011; Metzger & Bower 2014).

In the past, a few studies have been carried out by Metzger & Bower (2014), Horesh et al. (2016), Fong et al. (2016), Schroeder et al. (2020) and Ricci et al. (2021) using the late time observations acquired with the very large array (VLA) and Australian telescope compact array. All these observations were performed at a frequency of 1.4 GHz or more. The upper limits of flux density quoted in the study by Metzger & Bower (2014) were quite high and they concluded that the number density value of $n_0 \le 10^{-1}$ cm$^{-3}$ can be constrained for a stable magnetar remnant with the rotational energy of $E_\mathrm{rot} \sim 10^{53}$ erg. The study by Horesh et al. (2016) and Fong et al. (2016) constrained the number density to be $n_0 \le 10^{-3}$ cm$^{-3}$ for a large magnetar rotational energy of $E_\mathrm{rot} \sim 10^{53}$ erg and ejecta mass of $M_{ej} \sim 10^{-2} \ M$. In Schroeder et al. (2020) and Ricci et al. (2021), the ambient medium density was constrained between $2\times 10^{-3}$ and $2\times 10^{-1}$ cm$^{-3}$ for the maximum rotational energy of magnetar ${\le }10^{52}$ erg and ejecta mass of ${\le } 0.12 \ M_{\odot }$.

We have carried out a comprehensive study with a sample of short GRBs up to the year 2017 which lie at a redshift of $z \le 0.16$. Owing to the small distance where these bursts lie, they are ideal candidates for detecting any late-time emission from the merger ejecta. We assume that these events are the results of BNS merger and with the availability of detection upper limits, our model will allow to put stringent constraints on the remnant and ambient medium properties. In our model, we use the basic magnetar model taking into account the energy injection from the central engine, evolution of the synchrotron frequencies and generic hydrodynamics as well as propose modifications by incorporating the relativistic correction due to the evolution of the bulk Lorentz factor. A detailed discussion of our model is presented in Section 3.

The paper is organized as follows. In Section 2, we discuss the sample selection of short GRBs. Section 3 describes our model and the modifications we have incorporated as compared to the other previous studies. In Section 4, we provide model parameters for the selected bursts and conclude our findings in Section 5. Throughout the paper, we assume a $\Lambda$CDM cosmology with $H_0= 70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega _m = 0.27$ and $\Omega _{\Lambda } = 0.73$ (Komatsu et al. 2011).

2 Sample selection

We choose short GRBs in the local universe with spectroscopic redshift up to 0.16 and those showing a plateau phase in their X-ray light curves. The previous studies by Fong et al. (2016), Schroeder et al. (2020) and Ricci et al. (2021) present late-time data of short GRBs which exibhit a plateau phase in their X-ray light curves. We select GRBs from these studies, which have observations at 2.1 and 6.0 GHz. Seventeen GRBs in the literature show a plateau in their X-ray light curves. Among these, only seven GRBs lie within the redshift up to 0.16. GRB 170817A located at a very low redshift of 0.0097 is also included in our sample. The observational details of seven GRBs in our sample are given in Table 1.

Table 1 Observational details of seven GRBs in our sample.

3 Magnetar modeling

The late-time radio emission in short GRBs powered by magnetars is a result of the interaction between the tidally disrupted ejecta and the ambient medium (Nakar & Piran 2011). The first consideration is that the ejecta is emerging spherically from the central engine with an initial velocity $\beta _0$. The emission from the ejecta is characterized by several parameters like ejecta mass $M_{ej}$, rotational energy of the magnetar $E_\mathrm{rot}$, ambient medium density $n_0$, the fraction of post-shock energy into the accelerating electrons and magnetic field $\epsilon _e$ and $\epsilon _B$, respectively, and the power-law distribution index of electrons p with $N(\gamma ) \propto \gamma ^{-p}$. The simplistic magnetar modeling has earlier been used in Nakar & Piran (2011), Metzger & Bower (2014), Horesh et al. (2016), Fong et al. (2016), where they included the energy injection from the central engine and the synchrotron self-absorption effect. Some modifications like generic hydrodynamics and deep Newtonian phase were proposed in Liu et al. (2020), Schroeder et al. (2020) and Ricci et al. (2021). However, none of these models consider the evolution of the bulk Lorentz factor. We discuss the magnetar model in detail below, incorporating the relativistic correction.

3.1 Ejecta dynamics

The rotational energy of the stable NS remnant formed in a BNS merger can be calculated using

$$\begin{aligned} E_\mathrm{rot}=\frac{1}{2}I\Omega ^2\simeq 3\times 10^{52} \ \mathrm{erg} \ \bigg (\frac{P}{1 \ \mathrm{ms}}\bigg )^{-2}, \end{aligned}$$

(1)

where $I \simeq 1.5 \times 10^{15}$ g cm$^{-3}$ is the moment of inertia of a NS. In the presence of a magnetar central engine, a significant fraction of its rotational energy is transferred to the ejecta, which acts as a driving force behind the acceleration of the ejecta (Nakar 2007).

To calculate the deceleration radius and timescale, it is mandatory to consider the evolution of the bulk Lorentz factor ($\Gamma$) of the ejecta with the swept up mass ($M_s$) as described in Pe’er (2012) and is shown below:

$$\begin{aligned} \frac{\mathrm{d}\Gamma }{\mathrm{d}M_{s}} = -\frac{\hat{\gamma }(\Gamma ^2 - 1) -(\hat{\gamma } - 1) \Gamma \beta ^2}{M_{ej} + M_s[2 \hat{\gamma }\Gamma - (\hat{\gamma } - 1) (1 + \Gamma ^{-2})]}, \end{aligned}$$

(2)

where $\hat{\gamma }$ is the adiabatic index, $M_{ej}$ is the ejecta mass and $\beta$ is the velocity of the ejecta.

The evolution of swept-up mass from ambient medium ($M_s$) and ejecta radius (r) are given in Pe’er (2012) as follows:

$$\begin{aligned} \frac{\mathrm{d}M_s}{\mathrm{d}r} = 4\pi r^2n_0m_p \end{aligned}$$

(3)

and

$$\begin{aligned} \frac{\mathrm{d}r}{\mathrm{d}t} = \Gamma ^2 \beta (\Gamma ) c (1 + \beta (\Gamma )), \end{aligned}$$

(4)

where $m_p$ is the mass of proton and $n_0$ is the density of the ambient medium.

On solving Equations (2)–(4), we obtain expressions for $\Gamma$, swept-up mass $M_s$ and radius.

Initially, there is an enhancement in the kinetic energy of the ejecta because of continuous energy injection from the magnetar. When the ejecta collects a mass comparable to its own from the ambient medium, the ejecta starts decelerating at the characteristic timescale ($t_\mathrm{dec}$) and radius ($R_\mathrm{dec}$). To calculate $t_\mathrm{dec}$ and $R_\mathrm{dec}$, we have considered a spherical outflow from the central engine with the rotational energy ($E_\mathrm{rot}$) of the magnetar and initial Lorentz factor ($\gamma _0$) with the associated velocity $c\beta _0$, which propagates through the CSM of density $n_0$:

$$\begin{aligned} R_\mathrm{dec}\simeq \bigg (\frac{3E_\mathrm{rot}}{4\pi n_0m_pc^2\gamma _0(\gamma _0 - 1)}\bigg )^{1/3} \end{aligned}$$

(5)

and

$$\begin{aligned} t_\mathrm{dec}\sim \frac{R_\mathrm{dec}(1 - \beta _0)}{c\beta _0}. \end{aligned}$$

(6)

The minimum Lorentz factor ($\gamma _m$) and the magnetic field strength (B) in the shock can be calculated using the fraction of energy transferred to the electron energy and magnetic field ($\epsilon _e$ and $\epsilon _B$), respectively.

$$\begin{aligned} \gamma _{m} = 1 + \bigg (\frac{p-2}{p-1}\bigg ) \frac{m_p}{m_e}\epsilon _e(\Gamma - 1) \end{aligned}$$

(7)

and

$$\begin{aligned} B = c (32 \pi n_0 m_p \epsilon _B \Gamma (\Gamma - 1))^{1/2}. \end{aligned}$$

(8)

3.2 Calculation of synchrotron frequencies

The late-time radio spectrum is completely dominated by two frequencies. One of them is the typical synchrotron frequency $\nu _m$ of the electrons having the minimum Lorentz factor $\gamma _m$:

$$\begin{aligned} \nu _m = \bigg (\frac{3q}{4\pi m_e c} \bigg ) B \gamma _m^2 \Gamma (1+z)^{0.5}, \end{aligned}$$

(9)

where q and $m_e$ are charge and mass, respectively, of the electron, c is the velocity of light, B is the magnetic field and z is the redshift of the GRB.

The other is the synchrotron self-absorption frequency $\nu _a$. Radio emission becomes observable when the observed frequency is above $\nu _a$. $\nu _a$ plays a very significant role in low frequency radio regime. The expression for $\nu _a$ from Nakar & Piran (2011) is shown below:

$$\begin{aligned} \nu _a \approx 1 \ \mathrm{GHz} \bigg (\frac{R}{10^{17}}\bigg )^{\frac{2}{(p+4)}} \bigg (\frac{\epsilon _B}{0.1}\bigg )^{\frac{2+p}{2(p+4)}} \bigg (\frac{\epsilon _e}{0.1}\bigg )^{\frac{2(p-1)}{p+4}} n^{\frac{6+p}{2(p+4)}} \beta _0^{\frac{5p-2}{p+4}}. \end{aligned}$$

(10)

Another important frequency in the synchrotron spectrum is the cooling frequency $\nu _c$. For the late-time emission scenario, $\nu _c$ lies much above the observable frequency, and $\nu _m$ and $\nu _a$. The contribution of $\nu _c$ in the late-time radio light curves is therefore not significant.

3.3 Synchrotron flux calculation

If the observing frequency is greater than $\nu _m$ and $\nu _a$, then, the corresponding observed peak flux density can be calculated from Wijers & Galama (1999). The specific flux $f_{\nu }$ of a single electron can be written as

$$\begin{aligned} F_{{\nu },e}= \frac{P_{{\nu },e}}{4\pi d_L^2}, \end{aligned}$$

(11)

where $P_{\nu ,e}$ is the power emitted by a single electron per unit frequency in rest frame and $d_L$ is the distance to the source. $P_{{\nu }_m,e}$ can be written as (Wijers & Galama 1999):

$$\begin{aligned} P_{\nu _m,e}= \phi _p\frac{\sqrt{3}e^3 B}{m_e c^2}, \end{aligned}$$

(12)

where $\phi _p$ is the flux of the dimensionless maximum point of the spectrum ($f(x_p) = \phi _p$). e and $m_e$ are representing the electron charge and mass, respectively.

After accounting for the redshift and relativistic transformation, the total specific flux, $F_{\nu _m,\mathrm{obs}}$, can be written as:

$$\begin{aligned} F_{\nu _m,\mathrm{obs}} = \frac{N_e P_{{\nu _m,e}} \Gamma (1+z)}{4 \pi d_L^2}. \end{aligned}$$

(13)

Here, we consider that the ejecta is ploughing through the constant density inter-stellar medium (ISM), then, we can write $N_e = {4}/{3} \pi R^3 n$. Substituting $N_e$ and Equation (12) in Equation (13), we evaluate the total specific flux as

$$\begin{aligned} F_{\nu _m,\mathrm{obs}} = \frac{\sqrt{3} B e^3}{m_e c^2}\frac{R^3 n_0 \Gamma (1+z)}{3 d_L^2}. \end{aligned}$$

(14)

Substituting Equation (8) in Equation (14), we obtain the final flux equation in terms of n and $\epsilon _B$.

$$\begin{aligned} F_{\nu _m,\mathrm{obs}} = \frac{\sqrt{96 \pi n_0^3m_p\epsilon _B\Gamma ^3(\Gamma - 1)} }{m_e c}\frac{R^3 e^3 (1+z)}{3 d_L^2}. \end{aligned}$$

(15)

Figure 1

Model light curves and the evolution of $\nu _a$ and $\nu _m$. The ejecta mass is fixed at 0.04 $M_{\odot }$ and number density of the ambient medium is fixed at 10$^{-2}$ cm$^{-3}$. The solid horizontal lines in the bottom panels indicate the observed frequency.

4 Light curve analysis

Using the above formulation, we generate the model light curves and compare them with the 3$\sigma$ upper limits. While constructing the model light curves, we take into account the assumptions and fiducial values of certain parameters described below:

• Numerical simulations showed that long-lived merger remnants eject a large fraction of the remnant accretion disk mass (Metzger & Fernández 2014). We consider ejecta mass values of 0.04 $M_{\odot }$ and 0.1 $M_{\odot }$, which corresponds to velocities $\beta \sim 0.5$ and 0.3, respectively. Velocities below $\beta < 0.3$ are discarded because they will produce very weak and delayed radio emission.

• We varied the magnetar rotational energy from $10^{51}$ to $10^{53}$ erg. The upper limit of rotational energy corresponds to a stable neutron star of mass 2.2 $M_{\odot }$ (Metzger et al. 2015).

• The fraction of post-shock energy into the accelerating electron $\epsilon _e$ is fixed at 0.1.

• The fraction of post-shock energy into the magnetic field $\epsilon _B$ is at 0.1 and 0.01.

• The powerlaw index p is fixed at 2.4.

• The density of the ambient medium is not fixed. It generally varies in a vast range for different GRBs. We take the number density range between $10^{-5}$ and 1 cm$^{-3}$ in six equal intervals.

Figure 2

Model light curves along with the VLA 6 GHz 3$\sigma$ upper limits. The upper limits are shown as inverted triangles in the figure. The value of $\epsilon _B$ is fixed to 0.1. For two different values of ejecta mass 0.04 and 0.1 $M_{\odot }$, we generate the model light curves with different magenetar rotational energy and number density as indicated in the plot.

Figure 3

Same as Figure 2, but for a fixed value of ejecta mass and two different values of $\epsilon _B$ (0.01 and 0.1).

The top panel of Figure 1 show the model light curves for different values of rotational energy ($5\times 10^{51}$–$10^{53}$ erg) for a fixed ejecta mass (0.04 $M_{\odot }$) and number density ($10^{-2}$ cm$^{-3}$) at a frequency of 6 GHz. The evolution of $\nu _a$ and $\nu _m$ is shown in the bottom panel. $\nu _m$, which represents the average kinetic energy of the radiating electrons remains below $\nu _a$ for lower values of rotational energy. But for higher rotational energy, the electron population achieves higher $\gamma _m$, leading to $\nu _m$ to exceed $\nu _a$. The peak of the model light curves is consistent with the peak of $\nu _m$. At late times, the role of $\nu _c$ is insignificant as it lies much beyond the observable frequencies.

Figures 2–4 show the model light curves along with the 3$\sigma$ upper limits from VLA and ATCA of the GRBs in our sample (Table 1). The models are generated for different values of ejecta mass, $\epsilon _B$, magnetar rotational energy and number density as indicated in the Figures. The model light curves in these Figures depict that the peak time of the light curves are decreasing with increasing values of rotational energy of the magnetar and the ambient medium density. In addition to that, the model flux decreases for increasing values of ejecta mass and decreasing values of magnetar rotational energy.

From Figures 2 and 3 it is evident that the most energetic scenario ($E_\mathrm{rot} \sim 10^{53}$ erg) is disfavoured as the upper limits lie beyond the model light curves and would require much smaller ambient medium density (${<}10^{-5}$ cm$^{-3}$) and $\epsilon _B$ ($<0.01$) to satisfy the observational constraints. In a study by Ricci et al. (2021), it was found that the number densities are better constrained for GRBs at lower redshifts as compared to those lying at higher redshifts. In our sample of low redshift short GRBs, the models with ambient medium densities of $n_0 \ge 10^{-4}$ cm$^{-3}$ are ruled out for magnetar rotational energy of $10^{52}$ erg. Models with rotational energy ${\sim }5 \times 10^{51}$ erg constrains the number density values within $10^{-5}$–$10^{-3}$ cm$^{-3}$. The number density obtained in this work is consistent with the previous studies by Ricci et al. (2021).

Figure 4

Model light curves along with the ATCA 2.1 GHz 3$\sigma$ upper limits. The model light curves are generated for a fixed ejecta mass and two different values of $\epsilon _B$ (0.01 and 0.1).

The model light curves with high rotational energy ($10^{53}$ erg) and lower value of $\epsilon _B = 0.01$ are satisfied only for very low number density values ($n_0 \le 10^{-5}$ cm$^{-3}$). However, for lower magnetar rotational energy ($10^{51}$ and $10^{52}$ erg), stringent constraints for number density values (between $10^{-4}$ and $10^{-2}$ cm$^{-3}$) have been found.

In Figure 5, we show the model light curves, generated for fixed values of ejecta mass and $\epsilon _B$ considering different values of rotational energy and number density along with the 3$\sigma$ upper limit of GRB 170817A. The upper limit available is at $\sim$714 days since the burst. The available upper limit does not satisfy any of the models indicating that different values of the parameters may be required to generate the models. Further late-time monitoring of GRB 170817A would be useful to search for the emission at late-time due to merger ejecta.

Figure 5

Model light curves along with the VLA 6 GHz 3$\sigma$ upper limit of GRB 170817A. The ejecta mass and $\epsilon _B$ values are fixed at 0.04 $M_{\odot }$ and 0.1, respectively.

5 Conclusions

A faint radio emission is expected when the merger ejecta collides with the ambient medium in the presence of a magnetar central engine (Nakar & Piran 2011). Several studies in the past have reported upper limits in a quest to search for merger ejecta emission in radio bands from short GRBs (Metzger & Bower 2014; Fong et al. 2016; Horesh et al. 2016; Schroeder et al. 2020; Ricci et al. 2021). The upper limits are very useful to constrain the ambient medium density and the rotational energy of the magnetar. We select a sample of seven short GRBs located at a redshift of $z \le 0.16$ for which upper limits were reported in the literature. We used the standard magnetar model and modified it to incorporate the relativistic corrections. The model light curves thus generated are used to constrain the basic parameters of the central engine and ambient medium.

We find that models with high rotational energy are disfavoured, whereas models with lower values of rotational energy provide tight constraints on the ambient medium density ($n_0 \sim 10^{-5}$–$10^{-3}$ cm$^{-3}$) except the ATCA 2.1 GHz light curves for $\epsilon _B = 0.01$. The number density values obtained are very low. It is also seen that changing the ejecta mass does not have much effect on the late-time light curves. The radio emission from merger ejecta is expected to peak at the deceleration time (typically after a few years since the burst). Therefore, observations not very far out in time, will not be able to detect any radio emission, if any. It is important to acquire observations at late-time. In the case of GRB 170817A, the model light curves are unable to explain the radio upper limits. Late-time observations of GRB 170817A could substantially refine these constraints.

In the near future, it may be possible to detect merger ejecta emission from near by local short GRBs with the next generation radio telescopes and prove the existence of magnetar central engine. The upcoming radio telescopes like the square kilometer array (SKA) and new generation VLA (ng-VLA) with increased sensitivity of $\mu$Jy level will push the detection limits of merger ejecta emission at late time.