An Amati-Like Correlation for Short Gamma-Ray Bursts

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1. Introduction

Gamma-ray bursts (GRBs) are immensely powerful stellar explosions with an equivalent isotropic energy, *E _{iso}* that can exceed 10

Over the past two decades, several GRB energy and luminosity correlations were discovered. Some of these correlations were obtained from the light curves, like the time-lag and variability relations [3] [4], while others were obtained from the spectra and include the Amati relation [5] [6] [7] [8], the Ghirlanda relation [9], the Yonetoku relation [10] [11], and the Liang-Zhang relation [12]. These correlations are important for two reasons. First, if calibrated properly they can be used as cosmological probes to constrain cosmological parameters [12] - [18]. Second, they are effective tools that might shed light on the physics of GRBs [19] [20].

This paper focuses on one of these correlations, namely the Amati relation, and examines whether it applies to both short (observed duration < 2 s) and long bursts (observed duration > 2 s) or only to long bursts, as currently believed. Section 2 provides essential background regarding peak-energy correlations, in general, and the Amati relation in particular. Our data sample, results, and physical interpretation are provided in Section 3, and our conclusions are provided in Section 4.

2. Peak Energy Correlations and the Amati Relation

Gamma-ray burst correlations involving the peak energy were first discovered in 1995 by [21], who studied 399 bursts observed by the BATSE instrument and discovered a correlation between *E _{p}*

A study by [22] later found a strong correlation between *E _{p}*

$\mathrm{log}\left({E}_{p,obs}\right)\approx 0.29\mathrm{log}\left({S}_{bol}\right)$, (1)

with a Kendall correlation coefficient τ = 0.80 and a chance probability *P* = 10^{−13}. However, it is important to remember that their selection criteria, *F _{p}* > 3 photons×cm

It is important to keep in mind that the peak energy correlations found by [21] and [22] were in the observer frame due to the paucity of data points with known redshift. The first rest-frame correlation involving the intrinsic peak energy, *E _{p}*

${E}_{p,i}=\left(\text{1}+z\right)\times {E}_{p,obs}$. (2)

On the other hand, *E _{iso}* can be calculated from the bolometric flux using:

${E}_{iso}=4\pi {d}^{2}{S}_{bol}/\left(1+z\right)$, (3)

where *d* is the luminosity distance, which can be calculated from *z* after assuming a certain cosmological model. In Amati’s original paper [5], a flat universe was assumed with Ω_{M} = 0.3, Ω_{Λ} = 0.7, and *H*_{0} = 65 km×s^{−1}×Mpc^{−1}. The Amati relation can also be expressed logarithmically as:

$\mathrm{log}\left({E}_{iso}\right)=A+B\times \mathrm{log}\left({E}_{p,i}/\langle {E}_{p,i}\rangle \right)$, (4)

where the normalization, *A*, and the slope, *B*, are constants, and where <*E _{p}*

${E}_{p,i}=K\times {\left({E}_{iso}/{10}^{52}\text{erg}\right)}^{m}$, (5)

where *E _{p}*

Another important peak-energy correlation is the Ghirlanda relation, which is a correlation between the peak energy and the total energy corrected for beaming, *E*_{g}, which is given by:

${E}_{\gamma}=\left[1-\mathrm{cos}\left({\theta}_{j}\right)\right]\cdot {E}_{iso}$, (6)

where θ_{j} is the jet’s half-opening angle. This correlation was discovered in 2004 by [9] who used 40 GRBs with known *E _{iso}* and

${\theta}_{j}=0.161{\left[{T}_{b}/\left(1+z\right)\right]}^{3/8}{\left[n\cdot {n}_{\gamma}\cdot {E}_{iso}\right]}^{1/8}$, (7)

where *T _{b}* (measured in days) is the time for the power-law break in the afterglow light curve,

· The jet break should be detected in the optical window

· The optical light curve should not end at *T _{b}*, but should continue beyond it

· The flux from the host galaxy and from any probable supernova should be subtracted out

After considering the above points, the Ghirlanda relation can be expressed as [9]:

$\mathrm{log}\left({E}_{peak}/100\text{\hspace{0.17em}}\text{keV}\right)=\left(0.48\pm 0.02\right)+\left(0.70\pm 0.04\right)\times \mathrm{log}\left[{E}_{\gamma}/4.4\times {10}^{50}\text{erg}\right]$. (8)

In what follows, we will focus on the Amati relation since in most data samples the jet’s half-opening angle in not always available.

3. Data Sample, Results, and Physical Interpretation

The data sample that we used in this study was taken from Table 1 and Table 2 of [28] which consists of 49 long bursts and 18 short bursts. First, we calculated the intrinsic peak energies for all the bursts using Equation (2), then a maximum-likelihood fit of the form expressed in Equation (4) was applied. For the long bursts, the best-fit parameters that we obtained were *A* = 53.41 and *B* = 0.85, with a mean intrinsic peak energy of 2151.8 keV and a linear regression coefficient *r* = 0.67, as shown in Figure 1.

For the short bursts, the best-fit parameters that we obtained were *A* = 51.69 and *B* = 2.03, with a mean intrinsic peak energy of 1929.3 keV and a linear regression coefficient *r* = 0.86, as shown in Figure 2.

The values obtained for the linear regression coefficient indicate that the fits obtained are statistically significant and that the correlations are strong. Although this is not surprising for long GRBs, which are known to follow the Amati relation, they are surprising for short GRBs, which were thought not to follow the Amati relation. However, it is important to keep in mind that since the fitting parameters, *A* and *B*, are appreciably different for the short bursts compared to the long bursts, it is more accurate to state that the short bursts seem to follow an Amati-like correlation rather than the Amati correlation in the strict sense because the Amati correlation is traditionally obtained from the fitting of long rather than short bursts.

The first attempt to provide a physical interpretation of correlations involving

Figure 1. The best-fit Amati relation applied to the sample of 49 long bursts.

Figure 2. The best-fit Amati relation applied to the sample of 18 short bursts.

the peak energy was carried out by [22] who investigated the *E _{peak}* -

The above results were confirmed by [5] who showed that the *E _{peak}* -

A recent study [31] investigated whether the *E _{peak}* -

4. Conclusion

The peak energy correlations of GRBs are important relations that can be utilized to probe the physics of GRBs. One of the most important peak energy correlations is the Amati relation, which correlates the peak energy and *E _{iso}*. Previous studies had found evidence that the Amati relation applied to long GRBs only. Our current study indicates that although the short bursts do not follow the Amati relation in the strict sense, they do follow an Amati-like relation albeit with a different slope and normalization. As more data on short bursts become available, it is important to confirm these results because they have important implications regarding the understanding of the physics behind short GRBs and their potential use, along with long GRBs, as tools to probe different cosmological models.

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