Magnetar giant flare origin for GRB 200415A inferred from a new scaling relation and its implications

GRB 790305 is a GF detected from SGR 0526-66 (Mazets et al., 1979). Its spectrum can be fitted by an optically thin thermal bremsstrahlung (OTTB) model $dN(E)/dE\propto E^{-1}\exp(-E/kT)$, with $kT=520\pm 100\rm\ keV$ (Mazets & Golenetskii, 1981; Mazets et al., 1982; Fenimore et al., 1996)). For this spectrum, the peak of the $E^{2}dN(E)/dE$ spectrum is at $E_{\rm p}=kT$. GRB 980827 is a GF detected from SGR 1900+14 (Hurley et al., 1999a). Its spectrum is also well represented by an OTTB model, so we take the peak energy $E_{\rm p}$ (averaged over a 1-s interval) as $kT=240\pm 20\rm\ keV$ (the error bar is obtained from Figure 1 (panel b) of Hurley et al. (1999a)). GRB 041227 is a GF detected from SGR 1806-20 (Hurley et al., 2005; Palmer et al., 2005; Frederiks et al., 2007a). The spike’s intensity drove all X- and $\gamma$-ray detectors into saturation, but particle detectors aboard RHESSI and Wind made reliable measurements (Hurley et al., 2005). The SOPA and ESP instruments located on geosynchronous satellites also measured the flux and spectrum(Palmer et al., 2005). The SOPA data of this GF were fitted with a CPL model, and we take the spectral peak energy and flux from Palmer et al. (2005). GRB 980618 is an intermediate flare detected from SGR 1627-41 and its spectrum can be modelled by an OTTB spectrum (Mazets et al., 1999a). From Figure 3 of Mazets et al. (1999a), we can infer that the spectral peak of the $E^{2}dN(E)/dE$ spectrum is equal to $kT=100-150\rm\ keV$ (here we set $E_{\rm p}$ to $125\pm 25\rm\ keV$). The hard spectrum of this flare, together with a total energy of $10^{43}{\rm\ erg}$, makes it quite similar to giant flares. Indeed, Mazets et al. (2008) consider this flare as a giant flare. For these reasons, we include the flare from SGR 1627-41 in the correlation analysis. Two GF candidates, GRB 051103 (Frederiks et al., 2007b) and 070201 (Mazets et al., 2008) are both well fitted by the CPL model. The spectral peak energies of these two GFs are taken from Mazets et al. (2008).

The spectral peak energy and the total isotropic energy of GRB 200415A, together with those of four GFs and two GF candidates, are summarized in Table 1. Note that, in Table 1, we also give the uncertainty in the total isotropic energy of GFs due to the uncertainty in the source distance.

With the observed properties of these magnetar GFs and candidates, we study relation between the spectral peak energy $E_{\rm p,z}$ and the isotropic energy $E_{\rm iso}$ of them. Since the samples in this work is small, the scaling relations are estimated by ordinary least squares for an ensemble of datasets where the data point are selected by bootstrap resampling and have positions jiggled by values consistent with their uncertainties (considering both the uncertainties of $E_{\rm p}$ and $E_{\rm iso}$). The best-fit results of the slopes and intercepts are set to be the median of the distribution of the ensemble, and the uncertainties of them are calculated by $1\sigma$ confidence intervals, which is obtained from the distribution of the ensemble.

We first use the three identified GFs and the intermediate flare for the study. The best-fit relation between the peak energy $E_{\rm p}$ and the total isotropic energy $E_{\rm iso}$ is $\log(E_{\rm{p}}/{\rm keV})=(2.78_{-0.05}^{+0.06})+(0.22_{-0.05}^{+0.04})\log(E_{\mathrm{iso}}/10^{46}\rm erg)$, with a Pearson correlation coefficient of $\kappa=0.91$ and a chance probability of $p=0.09$. If we consider only the three identified GFs, the correlation becomes weaker, but the slope is consistent with the above analysis, i.e., $\log(E_{\rm{p}}/{\rm keV})=(2.73_{-0.05}^{+0.06})+(0.15_{-0.05}^{+0.06})\log(E_{\mathrm{iso}}/10^{46}\rm erg)$, with a Pearson correlation coefficient of $\kappa=0.64$. As shown in Figure 2, plotting GRB 200415A onto the $E_{\rm p}$ vs. $E_{\rm iso}$ plane, we find that GRB 200415A is in good agreement with this relation and far away from the short GRB population. This indicates that GRB 200415A is more likely to be a magnetar GF from the Sculptor Galaxy.

Including the three GFs, the intermediate flare, and GRB 200415A to do the re-fitting, we find that the best-fit relation gives a tighter relation: $\log(E_{\rm{p}}/{\rm keV})=(2.85_{-0.05}^{+0.04})+(0.24_{-0.03}^{+0.04})\log(E_{\mathrm{iso}}/10^{46}\rm erg)$, with a Pearson correlation coefficient of $\kappa=0.93$ and a chance probability of $p=0.02$. Considering also the other two candidate GFs (GRB 051103 and GRB 070201) lying within the $3\sigma$ of the track of the GF population, we include these sources to do the re-fitting. We find that the best-fit relation gives a tighter relation: $\log(E_{\rm{p}}/{\rm keV})=(2.80_{-0.03}^{+0.02})+(0.23_{-0.02}^{+0.03})\log(E_{\mathrm{iso}}/10^{46}\rm erg)$, with a Pearson correlation coefficient of $\kappa=0.93$ and a chance probability of $p=0.003$. This correlation suggests that these sources and the identified GFs are the same class of sources. This relation is quite different from the correlation for short GRBs¹¹1If GRB 200415A is not physically associated with the Sculptor galaxy, but is instead a chance coincidence and originates from an unrelated galaxy at a cosmological distance, then it could satisfy the $E_{p,z}-E_{iso}$ relation for a redshift of $z>0.025$., i.e., the ”Amati relation” (Amati et al., 2002) with $\log(E_{\rm{p,z}}/{\rm keV})=(3.55\pm 0.07)+(0.54\pm 0.04)\log(E_{\mathrm{iso}}/10^{52}\rm erg)$ (Zhang et al., 2018), where $E_{\rm{p,z}}=E_{\rm p}\rm(1+z)$. As shown in Figure 2, GFs do not fall on the the Amati relation of short GRBs. We also find that the two correlations do not agree at $>5\sigma$ level, indicating that GFs are distinct from short GRBs.

Next we show that the correlation between $E_{\rm p}$ and $E_{\rm iso}$ for magnetar GFs can be understood in the framework of the fireball photosphere emission model. During a GF, strong shearing of a neutron star’s magnetic field, combined with growing thermal pressure, appears to have forced an opening of the field outward (Thompson & Duncan, 2001). A huge amount of magnetic energy $E=10^{46}{\rm\ erg}$ is subsequently released at the surface of the neutron star with a radius of $R_{0}$ over a short period of $t=0.1{\rm\ s}$, leading to the formation of a hot fireball (Thompson & Duncan, 1995, 2001; Nakar et al., 2005), which is quite similar to the case of classic GRBs (Mészáros & Rees, 2000). The optical depth for pair production is extremely high, regardless of the suppression of the effective cross-section due to the large magnetic field of the magnetar (Herold, 1979). With such a large optical depth, a radiation-pairs plasma is formed at a thermal equilibrium with an initial temperature

where $\sigma$ is the Stephan-Boltzmann constant. This plasma expands under its own pressure, launching a relativistic outflow. As the optically thick (adiabatic) outflow expands, the comoving internal energy drops as $e^{\prime}\propto V^{\prime-4/3}\propto n^{\prime 4/3}$, where $V^{\prime}$ is the comoving volume and $n^{\prime}$ is the comoving baryon number density. The baryon bulk Lorentz factor increases as $\Gamma\propto R$ and the comoving temperature drops $T^{\prime}\propto R^{-1}$. The $e^{\pm}$ pairs drop out of equilibrium (Paczynski, 1986) at $T^{\prime}_{p}=17{\rm\ KeV}$ at a radius of $R_{p}=R_{0}(T_{0}/T^{\prime}_{p})$. This is the radius of an $e^{\pm}$ pair photosphere. If the outflow carries enough baryons, the baryonic electrons lead to a photosphere larger than $R_{p}$. As long as the outflow remains optically thick, it is radiation dominated and continues to expand with $\Gamma\propto R$. For large baryon loads, $\Gamma$ reaches the saturation value $\eta$ at a radius $R_{s}=\eta R_{0}$, where the baryon load is parameterized by $\eta=E/Mc^{2}$. Above the saturation radius, the flow continues to coast with $\Gamma=\eta$.

On the other hand, for low baryon loads, a baryonic electron photosphere appears in the accelerating portion $\Gamma\propto R$. An electron scattering photosphere is defined by $\tau=\sigma_{\rm T}n^{\prime}R_{ph}/\Gamma=1$, where $\tau$ is the Thomson optical depth, $\sigma_{\rm T}$ is the Thomson cross-section, $n^{\prime}=(L/4\pi m_{p}c^{3}\Gamma\eta)$ is the comoving baryon density, $r/\Gamma$ is a typical comoving length, $m_{p}$ is the proton mass, and $L$ is the total power of the burst. There is a critical value of $\eta$ at which $R_{ph}=R_{s}$, i.e.,

Thus, for low baryon loads where $\eta\geq\eta_{*}$, the outflow becomes optically thin in the accelerating portion. The outflow emits a quasi-blackbody spectrum as it became optically thin, with a spectral temperature comparable to the temperature at its base, because declining temperature in the outflow is compensated by the relativistic blueshift, i.e., $T_{ph}=\Gamma T^{\prime}_{ph}=T_{0}$. The observed photospheric thermal luminosity is $L_{ph}=4\pi R^{2}\Gamma^{2}\sigma T^{\prime 4}_{ph}=4\pi R_{0}^{2}\sigma T_{ph}^{4}$. Thus, the observer-frame photospheric temperature is

where $E_{\rm iso}=L_{\rm ph}t$ and we have assumed a duration of $t=0.1{\rm\ s}$ for the all the GFs. The peak energy of the spectrum is at $E_{\rm p}=2.8\ kT_{\rm ph}$. It is remarkable to see that Eq.(3) agrees well with the $E_{\rm p}-E_{\rm iso}$ relation that we found for GFs.

The spectra of GRB 200415A and some other GFs are not purely blackbody and better modelled by CPL spectrum. This can be interpreted as being due to the extra contribution by some other radiation components to the low-energy spectrum, such as multi-temperature blackbody emission or synchrotron emission arising from some shocks in the relativistic outflow (Mészáros & Rees, 2000). There is evidence for multi sub-pulses and fast variability in the initial spike in this and other GFs. Different sub-pulses have different temperatures, so multi-temperature blackbody is naturally expected. The fast variability implies unsteady outflow, so internal shocks could occur, which may produce a flat synchrotron spectrum. If this synchrotron spectrum is subdominant compared with the photosphere emission, the low-energy spectrum will be flatter while the peak energy remains unchanged (Mészáros & Rees, 2000). A detailed study of the spectrum considering these effects is beyond the scope of the present paper.

Note that Eq.(3) holds only when $R_{ph}<R_{s}$, corresponding to $\eta>\eta_{*}$. If $R_{ph}>R_{s}$ ($\eta<\eta_{*}$), the observer-frame photospheric temperature and photospheric thermal luminosity evolve as $T_{ph}\propto R^{-2/3}$ and $L_{ph}\propto R^{-2/3}$, respectively. In this case, the relation is different from Eq.(3). That is, the temperature (or the peak energy of the spectrum) would be smaller than observed if the flow is baryon-rich, and therefore, in the present case, the outflow must be baryon-poor. This implies that baryon load in the outflow of GRB 200415A should satisfy the condition $\eta>\eta_{*}$.

The outflow reaches a Lorentz factor of $\Gamma=R_{ph}/R_{0}=\eta_{*}(\eta/\eta_{*})^{-1/3}$ at the photosphere. Beyond the photosphere, most of the electrons above the photosphere can still scatter with a decreasing fraction of free-streaming photons as long as the comoving Compton drag time is less than the comoving expansion time (Mészáros & Rees, 2000). As a result, for $\eta>\eta_{*}$, the terminal Lorentz factor of the outflow is $\eta_{*}$ (instead of $\eta$). The energy that remains in the baryons of the outflow is $E_{b}=E(\eta_{*}/\eta)$. The energy in the afterglow emission of GFs could place a limit on the energy in the final baryonic ejecta²²2The kinetic energy of the outflow in the giant flare of SGR 1806-20 is constrained by the radio afterglow (Gaensler et al., 2005; Gelfand et al., 2005; Granot et al., 2006).. The fluence of the GeV emission of GRB 200415A is about $(2.9\pm 0.2)\times 10^{-6}{\rm\ erg\ cm^{2}}$ in the time interval $0-300$ s, which is only a factor of 3 smaller than the fluence of the GF. This implies that $\eta$ should not be much larger than $\eta_{*}$. As a result, we estimate that the baryon load in the GF of GRB 200415A is about $M\sim E/(\eta_{*}c^{2})\sim 10^{23}{\rm g}$.