The isotropic energy function and formation rate of short gamma-ray bursts
Zhiying Liu, Fu-Wen Zhang, Si-Yuan Zhu
Abstract
Abstract Gamma-ray bursts (GRBs) are brief, intense, gamma-ray flashes in the universe, lasting from a few milliseconds to a few thousand seconds. For short gamma-ray bursts (sGRBs) with duration less than 2 seconds, the isotropic energy ( E iso ) function may be more scientifically meaningful and accurately measured than the luminosity ( L p ) function. In this work we construct, for the first time, the isotropic energy function of sGRBs and estimate their formation rate. First, we derive the L p – E p correlation using 22 sGRBs with known redshifts and well-measured spectra and estimate the pseduo redshifts of 334 Fermi sGRBs. Then, we adopt the Lynden-Bell c − method to study isotropic energy functions and formation rate of sGRBs without any assumption. A strong evolution of isotropic energy E iso ∝ (1+ z ) 5.79 is found, which is comparable to that between L p and z . After removing effect of the cosmic evolution, the isotropic energy function can be reasonably fitted by a broken power law, which is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">iso</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∝</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">iso</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.45</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> for dim sGRBs and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">iso</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∝</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">iso</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1.11</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> for bright sGRBs, with the break energy 4.92 × 10 49 erg. We obtain the local formation rate of sGRBs is about 17.43 events Gpc −3 yr −1 . If assuming a beaming angle is 6° to 26°, the local formation rate including off-axis sGRBs is estimated as ρ 0,all = 155.79 – 3202.35 events Gpc −3 yr −1 .