An analytic derivation of the empirical correlations of gamma-ray bursts

Abstract

Empirical correlations between various key parameters have been extensively explored ever since the discovery of gamma-ray bursts (GRBs) and have been widely used as standard candles to probe the Universe. The Amati relation and the Yonetoku relation are two good examples that enjoyed special attention. The former reflects the connection between the peak photon energy (Ep) and the isotropic $\gamma$-ray energy release (Eiso), while the latter links Ep with the isotropic peak luminosity (Lp), both in the form of a power-law function. Most GRBs are found to follow these correlations well, but a theoretical interpretation is still lacking. Some obvious outliers may be off-axis GRBs and may follow correlations that are different from those at the on-axis. Here we present a simple analytical derivation for the Amati relation and the Yonetoku relation in the framework of the standard fireball model, the correctness of which is then confirmed by numerical simulations. The off-axis Amati relation and Yonetoku relation are also derived. They differ markedly from the corresponding on-axis relation. Our results reveal the intrinsic physics behind the radiation processes of GRBs, and they highlight the importance of the viewing angle in the empirical correlations of GRBs.

arXiv comment12 pages, 7 figures, 2 tables. A&A accepted

References in corpus (66)

• Spectra and Light Curves of Gamma-Ray Burst Afterglows
• Intrinsic spectra and energetics of BeppoSAX Gamma-Ray Bursts with known redshifts
• Unsteady Outflow Models for Cosmological Gamma-Ray Bursts
• Optical and Long Wavelength Afterglow from Gamma-Ray Bursts
• Beaming in Gamma-Ray Bursts: Evidence for a Standard Energy Reservoir
• The Internal-Collision-Induced Magnetic Reconnection and Turbulence (ICMART) Model of Gamma-Ray Bursts
• GRB Formation Rates inferred from the Spectral Peak Energy - Luminosity Relation
• The collimation--corrected GRB energies correlate with the peak energy of their $\nu F_{\nu }$ spectrum
• Can internal shocks produce the variability in GRBs?
• Dissipative Photosphere Models of Gamma-ray Bursts and X-ray Flashes
• The Ep,i - Eiso correlation in GRBs: updated observational status, re-analysis and main implications
• Discerning the physical origins of cosmological Gamma-ray bursts based on multiple observational criteria: the cases of z=6.7 GRB 080913, z=8.3 GRB 090423, and some short/hard GRBs
• A mildly relativistic wide-angle outflow in the neutron star merger GW170817
• Off-Axis Afterglow Emission from Jetted Gamma-Ray Bursts
• Late time afterglow observations reveal a collimated relativistic jet in the ejecta of the binary neutron star merger GW170817
• Low Luminosity Gamma-Ray Bursts as a Unique Population: Luminosity Function, Local Rate, and Beaming Factor
• The observable effects of a photospheric component on GRB's and XRF's prompt emission spectrum
• Detection of a Thermal Spectral Component in the Prompt Emission of GRB 100724B
• Constraining GRB Initial Lorentz Factor with the Afterglow Onset Feature and Discovery of a Tight Gamma_0-E_iso Correlation
• GRB110721A: An extreme peak energy and signatures of the photosphere
• Temporal Evolution Of Thermal Emission From Relativistically Expanding Plasma
• Fast cooling synchrotron radiation in a decaying magnetic field and $\gamma$-ray burst emission mechanism
• Cosmology with gamma-ray bursts: I. The Hubble diagram through the calibrated $E_{\mathrm{p},\mathrm{i}}$ - $E_{\mathrm{iso}}$ correlation
• The Cosmic Rate, Luminosity Function and Intrinsic Correlations of Long GRBs
• A complete sample of bright Swift Long Gamma Ray Bursts: testing the spectral-energy correlations
• Extra-galactic high-energy transients: event rate densities and luminosity functions
• Prompt high-energy emission from gamma-ray bursts in the internal shock model
• A theory of photospheric emission from relativistic, collimated outflows
• Failed gamma-ray bursts and orphan afterglows
• Radiative transfer in ultra-relativistic outflows
• The Ep,i - Eiso correlation: type I gamma-ray bursts and the new classification method
• Bulk Lorentz factors of Gamma-Ray Bursts
• A Theory of Mulicolor Black Body Emission from Relativistically Expanding Plasmas
• An Interpretation of the $h{\nu }_{peak}$ - $E_{iso}$ Correlation for GRB
• Gamma-ray bursts as cool synchrotron sources
• Synchrotron Origin of the Typical GRB Band Function - A Case Study of GRB 130606B
• Regulation of the spectral peak in gamma-ray bursts
• Lessons from the short GRB170817A - the First Gravitational Wave Detection of a Binary Neutron Star Merger
• The Afterglow of GRB 990123 and a Dense Medium
• Gamma-ray burst jet breaks revisited
• Gamma-Ray Burst Prompt Emission Light Curves and Power Density Spectra in the ICMART Model
• The X-ray fundamental plane of the Platinum Sample, the Kilonovae and the SNe Ib/c associated with GRBs
• On the Optical Light Curves of Afterglows from Jetted Gamma-ray Burst Ejecta: Effects of Parameters
• An Off-Axis Model for GRB 031203
• An Off-Axis Jet Model For GRB980425 and Low Energy Gamma-Ray Bursts
• Peak Energy-Isotropic Energy Relation in the Off-Axis Gamma-Ray Burst Model
• GRB 161219B / SN 2016jca: A low-redshift gamma-ray burst supernova powered by radioactive heating
• Low-energy Spectra of Gamma-ray Bursts from Cooling Electrons
• Cosmological X-ray Flashes in the Off-Axis Jet Model
• Measuring cosmological parameters with a luminosity-time correlation of gamma-ray bursts
• Modeling the Optical Afterglow of GRB 030329
• On The Origin Of High Energy Correlations in Gamma-ray Bursts
• The photospheric origin of the Yonetoku relation in gamma-ray bursts
• Single- and Two-Component GRB Spectra in the Fermi GBM-LAT Energy Range
• X-ray Plateaus in Gamma-Ray Burst Afterglows and Their Application in Cosmology
• Imprints of Electron-positron Winds on the Multi-wavelength Afterglows of Gamma-ray Bursts
• $E_{\mathrm{iso}}$-$E_{\mathrm{p}}$ correlation of gamma ray bursts: calibration and cosmological applications
• Cosmic Evolution of Long Gamma-Ray Burst Luminosity
• Pseudo Redshifts of Gamma-Ray Bursts Derived from the L-T-E Correlation
• Testing High-latitude Curvature Effect of Gamma-Ray Bursts with {\it Fermi} Data: Evidence of Bulk Acceleration in Prompt Emission
• Is the Birth of PSR J0538+2817 Accompanied by a Gamma-ray Burst?
• On Spectral Peak Energy of Swift Gamma-Ray Bursts
• Characteristics of Long Gamma-Ray bursts in the Comoving Frame
• The $E_{\mathrm{p}}$ - $E_{\mathrm{iso}}$ relation and the internal shock model
• Kinematic Origin OF Correlations Between Gamma Ray Burst Observables
• Inferring the emission regions for different kinds of gamma-ray bursts

Cited by in corpus (6)

• EP241021a: a months-duration X-ray transient with luminous optical and radio emission
• Possible origin of AT2021any: a failed GRB from a structured jet
• Prospects for Time-Domain and Multi-Messenger Science with eXTP
• Exploring the cosmic microwave background dipole direction using gamma-ray bursts
• Estimating the peak energy of Swift gamma-ray bursts using supervised machine learning
• Einstein Probe Fast X-ray Transients Extend the Physical Parameter Space of Relativistic Jets