An Amati-Like Correlation for Short Gamma-Ray Bursts ()

Walid J. Azzam^{}, Fatima S. Jaber^{}, Ambareena Naeem^{}

Department of Physics, College of Science, University of Bahrain, Sakhir, Bahrain.

**DOI: **10.4236/jamp.2020.811175
PDF HTML XML
133
Downloads
460
Views
Citations

Department of Physics, College of Science, University of Bahrain, Sakhir, Bahrain.

Gamma-ray bursts (GRBs) are by far the most powerful explosions in the universe. Over the past two decades, several GRB energy and luminosity correlations were discovered for long gamma-ray bursts, which are bursts whose observed duration exceeds 2 seconds. One important correlation, the Amati relation, involves the observed peak energy, *E*_{p,obs}, in the *v*F* _{v}* spectrum and the equivalent isotropic energy,

Share and Cite:

Azzam, W. , Jaber, F. and Naeem, A. (2020) An Amati-Like Correlation for Short Gamma-Ray Bursts. *Journal of Applied Mathematics and Physics*, **8**, 2371-2378. doi: 10.4236/jamp.2020.811175.

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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

[1] |
Atteia, J.-L., et al. (2017) The Maximum Isotropic Energy of Gamma-Ray Bursts. The Astrophysical Journal, 837, 119. https://doi.org/10.3847/1538-4357/aa5ffa |

[2] |
Le., T. and Mehta, V. (2017) Revisiting the Redshift Distribution of Gamma-Ray Bursts in the Swift Era. The Astrophysical Journal, 837, 17. https://doi.org/10.3847/1538-4357/aa5fa7 |

[3] |
Norris, J.P., et al. (2000) Connection between Energy Dependent Lags and Peak Luminosity in Gamma-Ray Bursts. The Astrophysical Journal, 534, 248-257. https://doi.org/10.1086/308725 |

[4] | Fenimore, E.E. and Ramirez-Ruiz, E. (2000) Redshifts for 220 BATSE Gamma-Ray Bursts Determined by Variability and the Cosmological Consequences. |

[5] |
Amati, L., et al. (2002) Intrinsic Spectra and Energetics of BeppoSAX Gamma-Ray Bursts with Known Redshifts. Astronomy and Astrophysics, 390, 81-89. https://doi.org/10.1051/0004-6361:20020722 |

[6] |
Amati, L. (2006) The E_{p,i}-E_{iso} Correlation in Gamma-Ray Bursts: Updated Observational Status, Re-Analysis and Main Implications. Monthly Notices of the Royal Astronomical Society, 372, 233-245. https://doi.org/10.1111/j.1365-2966.2006.10840.x |

[7] |
Amati, L., et al. (2008) Measuring the Cosmological Parameters with the E_{p,i}-E_{iso} Correlation of Gamma-Ray Bursts. Monthly Notices of the Royal Astronomical Society, 391, 577-584. https://doi.org/10.1111/j.1365-2966.2008.13943.x |

[8] |
Amati, L., et al. (2009) Extremely Energetic Fermi Gamma-Ray Bursts Obey Spectral Energy Correlations. Astronomy and Astrophysics, 508, 173-180. https://doi.org/10.1051/0004-6361/200912788 |

[9] |
Ghirlanda, G., et al. (2004) The Collimation-Corrected Gamma-Ray Burst Energies Correlate with the Peak Energy of their vF_{v} Spectrum. The Astrophysical Journal, 616, 331-338. https://doi.org/10.1086/424913 |

[10] |
Yonetoku, D., et al. (2004) Gamma-Ray Burst Formation Rate Inferred from the Spectral Peak Energy—Peak Luminosity Relation. The Astrophysical Journal, 609, 935-951. https://doi.org/10.1086/421285 |

[11] |
Ghirlanda, G., et al. (2010) Spectral-Luminosity Relation within Individual Fermi Gamma-Ray Bursts. Astronomy and Astrophysics, 511, A43-A53. https://doi.org/10.1051/0004-6361/200913134 |

[12] |
Liang, E. and Zhang, B. (2005) Model-Independent Multivariable Gamma-Ray Burst Luminosity Indicator and Its Possible Cosmological Implications. The Astrophysical Journal, 633, L611-L623. https://doi.org/10.1086/491594 |

[13] |
Ghirlanda, G., et al. (2006) Cosmological Constraints with GRBs: Homogeneous vs. Wind Density Profile. Astronomy and Astrophysics, 452, 839-844. https://doi.org/10.1051/0004-6361:20054544 |

[14] |
Capozziello, S. and Izzo, L. (2008) Cosmography by Gamma-Ray Bursts. Astronomy and Astrophysics, 490, 31-36. https://doi.org/10.1051/0004-6361:200810337 |

[15] |
Demianski, M. and Piedipalumbo, E. (2011) Standardizing the GRBs with the Amati E_{p,i}-E_{iso} Relation: the Updated Hubble Diagram and Implications for Cosmography. Monthly Notices of the Royal Astronomical Society, 415, 3580-3590. https://doi.org/10.1111/j.1365-2966.2011.18975.x |

[16] |
Azzam, W.J. and Alothman, M.J. (2006) Constraining Cosmological Parameters through Gamma-Ray Bursts. Advances in Space Research, 38, 1303-1306. https://doi.org/10.1016/j.asr.2004.12.019 |

[17] | Azzam, W.J. and Alothman, M.J. (2006) Gamma-Ray Burst Spectral-Energy Correlations as Cosmological Probes. Il Nuovo Cimento B, 121, 1431-1432. |

[18] |
Dai, Z.G., Liang, E.W. and Xu, D. (2004) Constraining ΩM and Dark Energy with Gamma-Ray Bursts. The Astrophysical Journal, 612, L101-L104. https://doi.org/10.1086/424694 |

[19] |
Zhang, B. and Mészáros, P. (2002) An Analysis of Gamma-Ray Burst Spectral Break Models. The Astrophysical Journal, 581, 1236-1247. https://doi.org/10.1086/344338 |

[20] |
Thompson, C., Mészáros, P. and Rees, M.J. (2007) Thermalization in Relativistic Outflows and the Correlation between Spectral Hardness and Apparent Luminosity in Gamma-Ray Bursts. The Astrophysical Journal, 666, 1012-1023. https://doi.org/10.1086/518551 |

[21] | Mallozi, R.S., et al. (1995) Gamma-Ray Bursts Spectra and the Hardness-Intensity Correlation. Proceedings Gamma-Ray Bursts 4th Huntsville Symposium, AIP Conference Series, Vol. 428, 273-277. |

[22] |
Lloyd, N.M., Petrosian, V. and Mallozi, R.S. (2000) Cosmological versus Intrinsic: The Correlation between Intensity and the Peak of the vF_{v} Spectrum of Gamma-Ray Bursts. The Astrophysical Journal, 534, 227-238. https://doi.org/10.1086/308742 |

[23] |
Azzam, W.J. and Alothman, M.J. (2013) Redshift Independence of the Amati and Yonetoku Relations for Gamma-Ray Bursts. International Journal of Astronomy and Astrophysics, 3, 372-375. https://doi.org/10.4236/ijaa.2013.34042 |

[24] |
Zitouni, H., Guessoum, N. and Azzam, W.J. (2014) Revisiting the Amati and Yonetoku Relations with Swift GRBs. Astrophysics and Space Science, 351, 267-279. https://doi.org/10.1007/s10509-014-1839-5 |

[25] |
Azzam, W.J. (2016) A Brief Review of the Amati Relation for GRBs. International Journal of Astronomy and Astrophysics, 6, 378-383. https://doi.org/10.4236/ijaa.2016.64030 |

[26] |
Zitouni, H., Guessoum, N. and Azzam, W.J. (2016) Determination of Cosmological Parameters from Gamma-Ray Burst Characteristics and Afterglow Correlations. Astrophysics and Space Science, 361, 383. https://doi.org/10.1007/s10509-016-2969-8 |

[27] | Azzam, W.J. and Al Dallal, S. (2015) Gamma-Ray Bursts: Origin, Types, and Prospects. Journal of Magnetohydrodynamics and Plasma Research, 20, 367. |

[28] |
Guo, Q., et al. (2020) Discovery of a Universal Correlation for Long and Short GRBs, and Its Application for the Study of Luminosity Function and Formation Rate. The Astrophysical Journal, 896, 83. https://doi.org/10.3847/1538-4357/ab8f9d |

[29] |
Dainotti, M.G., Del Vecchio, R. and Tarnopolski, M. (2016) Gamma-Ray Burst Prompt Correlations. Advances in Astronomy, 2018, Article ID: 4969503. https://doi.org/10.1155/2018/4969503 |

[30] |
Azzam, W.J. (2017) Peak Energy Correlations for Gamma-Ray Bursts. Journal of Applied Mathematics and Physics, 5, 1515-1520. https://doi.org/10.4236/jamp.2017.58124 |

[31] |
Mochkovitch, R. and Nava, L. (2015) The Ep-Eiso Relation and the Internal Shock Model. Astronomy and Astrophysics, 577, A31. https://doi.org/10.1051/0004-6361/201424490 |

Journals Menu

Contact us

customer@scirp.org | |

+86 18163351462(WhatsApp) | |

1655362766 | |

Paper Publishing WeChat |

Copyright © 2022 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.