Constraining the Structure of GRB Jets Through the log(N)-log(S) Distribution

Abstract

A general formalism is developed for calculating the luminosity function and the expected number $N$ of observed GRBs above a peak photon flux $S$ for any GRB jet structure. This new formalism directly provides the true GRB rate without the need for a `correction factor'. We apply it to the uniform jet (UJ) and universal structured jet (USJ) models for the structure of GRB jets and perform fits to the observed log(N)-log(S) distribution from the GUSBAD catalog which contains 2204 BATSE bursts. A core angle ${\theta }_c$ and an outer edge at ${\theta }_{max}$ are introduced for the structured jet, and a finite range of half-opening angles ${\theta }_{min}\le {\theta }_j\le {\theta }_{max}$ is assumed for the uniform jets. The efficiency ${\epsilon }_{\gamma }$ for producing gamma-rays, and the energy per solid angle $\epsilon$ in the jet are allowed to vary with ${\theta }_j$ (the viewing angle ${\theta }_{obs}$) in the UJ (USJ) model, ${\epsilon }_{\gamma }\propto {\theta }^{-b}$ and $\epsilon \propto {\theta }^{-a}$. We find that a single power-law luminosity function provides a good fit to the data. Such a luminosity function arises naturally in the USJ model, while in the UJ model it implies a power-law probability distribution for ${\theta }_j$, $P({\theta }_j)\propto {\theta }_j^{-q}$. The value of $q$ cannot be directly determined from the fit to the observed log(N)-log(S) distribution, and an additional assumption on the value of $a$ or $b$ is required. Alternatively, an independent estimate of the true GRB rate would enable one to determine $a$, $b$ and $q$. The implied values of ${\theta }_c$ (or ${\theta }_{min}$) and ${\theta }_{max}$ are close to the current observational limits. The true GRB rate for the USJ model is found to be $R_{GRB}(z=0)=0.86_{-0.05}^{+0.14}Gp{c}^{-3}y{r}^{-1}$.

arXiv comment9 pages, 5 figures, 1 table; accepted for publication in ApJ

References in corpus (31)

• Beaming in Gamma-Ray Bursts: Evidence for a Standard Energy Reservoir
• Jets in GRBs
• The Dynamics and Light Curves of Beamed Gamma Ray Burst Afterglows
• The Fourth BATSE Gamma-Ray Burst Catalog (Revised)
• How to Tell a Jet from a Balloon: A Proposed Test for Beaming in Gamma Ray Bursts
• Afterglow lightcurves, viewing angle and the jet structure of gamma-ray bursts
• GRB Energetics and the GRB Hubble Diagram: Promises and Limitations
• Off-Axis Afterglow Emission from Jetted Gamma-Ray Bursts
• On the Association of Gamma-ray Bursts with Massive Stars: Implications for Number Counts and Lensing Statistics
• Gamma-ray burst beaming: a universal configuration with a standard energy reservoir?
• The Evolution of a Structured Relativistic Jet and GRB Afterglow Light-Curves
• Redshift-Space Distortions and the Real-Space Clustering of Different Galaxy Types
• Physical Parameters for the Afterglows of GRB 980703, 990123, 990510, and 991216 Determined from Modeling of Multi-Frequency Data
• Linear Polarization in Gamma-Ray Bursts: The Case for an Ordered Magnetic Field
• The polarisation of afterglow emission reveals GRB jet structure
• Steepening of Afterglow Decay for Jets Interacting with Stratified Media
• Luminosity Function of Gamma-Ray Bursts Derived Without Benefit of Redshifts
• Constraining the Structure of GRB Jets Through the Afterglow Light Curves
• On Beaming Effects in Afterglow Light Curves
• Efficiency and spectrum of internal gamma-ray burst shocks
• Dynamical Evolution, Light-Curves and Spectra of Spherical and Collimated Gamma-Ray Burst Remnants
• On the Kinetic Energy and Radiative Efficiency of Gamma-Ray Bursts
• GRB as explosions with standard power
• Formation Rate, Evolving Luminosity Function, Jet Structure, and Progenitors for Long Gamma-Ray Bursts
• Jets in GRBs: Tests and Predictions for the Structured Jet Model
• Light Curves from an Expanding Relativistic Jet
• Inferring the Spatial and Energy Distribution of Gamma Ray Burst Sources. II. Isotropic Models
• Testing the Predictions of the Universal Structured GRB Jet Model
• Observational evidence for a correlation between peak-luminosities and beaming in GRBs
• The Decline of the Source Population of Gamma-Ray Bursts and Their Luminosity Function
• Luminosity Function of GRBs

Cited by in corpus (14)

• Gamma-Ray Bursts
• Gamma-Ray Bursts in the Swift Era
• The 3rd Fermi GBM Gamma-Ray Burst Catalog: The First Six Years
• Gamma-ray burst jet breaks revisited
• Afterglow Light Curves from Impulsive Relativistic Jets with an Unconventional Structure
• Luminosity function and jet structure of Gamma Ray Bursts
• Afterglow Observations Shed New Light on the Nature of X-ray Flashes
• Spectral analysis of Swift long GRBs with known redshift
• New constraints on GRB jet geometry and relativistic shock physics
• Intensity Distribution and Luminosity Function of the Swift Gamma-Ray Bursts
• Modelling the luminosity function of long Gamma Ray Bursts using SWIFT and FERMI
• Swift/UVOT: 18 Years of Long GRB Discoveries and Advances
• Astrophysics in 2005
• Automatic estimation of flux distributions of astrophysical source populations