Discovery of a New Electron Spectrum Produced by Fast Cooling of Relativistic Electrons
Abstract
In energetic astrophysical sources like the gamma-ray bursts (GRBs) and blazars, the relativistic electrons lose their energy efficiently through synchrotron radiation and inverse Compton (IC) scattering and can rapidly cool to energies below the minimum acceleration energy γm. It is widely known that such electrons develop a power-law spectrum γe−2 below γm. However, this power-law spectrum is valid only if a steady-state population of electrons is assumed. In general, the cooled electrons below γm evolve in a non-steady state and their spectrum deviates significantly from the power-law γe−2. We present here a new electron spectrum fU that describes this fast-cooled, generally-evolving, and non-steady-state population of electrons below γm. We consider two cooling cases: (1) when the electrons undergo radiative (i.e., synchrotron + IC) cooling, and (2) when the electrons experience both radiative and adiabatic cooling. We first derive fU analytically and find an explicit expression of fU for each cooling case. Secondly, we show that fU is a solution to a Fokker-Planck equation in the entire energy range below γm and derive a general version of fU for each cooling case. We also compare fU to a detailed numerical calculation of a blazar electron spectrum and find an excellent agreement for the whole energy range below γm. Thus, fU gives us a general electron spectrum to describe the entire cooled electrons below γm. Our discovery of a new fast-cooling solution fU below γm may find it applicable in a wide range of energetic phenomena in high-energy astrophysics.
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BibTeX
@misc{uhm2026discovery,
title = {{Discovery of a New Electron Spectrum Produced by Fast Cooling of Relativistic Electrons}},
author = {Z. Lucas Uhm},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37044},
eprint = {2610.37044},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37044},
}
RIS
TY - UNPB
TI - Discovery of a New Electron Spectrum Produced by Fast Cooling of Relativistic Electrons
AU - Z. Lucas Uhm
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37044
UR - https://arxiv.org/abs/2610.37044
AB - In energetic astrophysical sources like the gamma-ray bursts (GRBs) and blazars, the relativistic electrons lose their energy efficiently through synchrotron radiation and inverse Compton (IC) scattering and can rapidly cool to energies below the minimum acceleration energy γ_m. It is widely known that such electrons develop a power-law spectrum γ_e^−2 below γ_m. However, this power-law spectrum is valid only if a steady-state population of electrons is assumed. In general, the cooled electrons below γ_m evolve in a non-steady state and their spectrum deviates significantly from the power-law γ_e^−2. We present here a new electron spectrum f_U that describes this fast-cooled, generally-evolving, and non-steady-state population of electrons below γ_m. We consider two cooling cases: (1) when the electrons undergo radiative (i.e., synchrotron + IC) cooling, and (2) when the electrons experience both radiative and adiabatic cooling. We first derive f_U analytically and find an explicit expression of f_U for each cooling case. Secondly, we show that f_U is a solution to a Fokker-Planck equation in the entire energy range below γ_m and derive a general version of f_U for each cooling case. We also compare f_U to a detailed numerical calculation of a blazar electron spectrum and find an excellent agreement for the whole energy range below γ_m. Thus, f_U gives us a general electron spectrum to describe the entire cooled electrons below γ_m. Our discovery of a new fast-cooling solution f_U below γ_m may find it applicable in a wide range of energetic phenomena in high-energy astrophysics.
ER -