A Complete Reference of the Analytical Synchrotron External Shock Models of Gamma-Ray Bursts

Gamma-ray bursts are most luminous explosions in the universe. Their ejecta are believed to move towards Earth with a relativistic speed. The interaction between this "relativistic jet" and a circum burst medium drives a pair of (forward and reverse) shocks. The electrons accelerated in these shocks radiate synchrotron emission to power the broad-band afterglow of GRBs. The external shock theory is an elegant theory, since it invokes a limit number of model parameters, and has well predicted spectral and temporal properties. On the other hand, depending on many factors (e.g. the energy content, ambient density profile, collimation of the ejecta, forward vs. reverse shock dynamics, and synchrotron spectral regimes), there is a wide variety of the models. These models have distinct predictions on the afterglow decaying indices, the spectral indices, and the relations between them (the so-called "closure relations"), which have been widely used to interpret the rich multi-wavelength afterglow observations. This review article provides a complete reference of all the analytical synchrotron external shock afterglow models by deriving the temporal and spectral indices of all the models in all spectral regimes, including some regimes that have not been published before. The review article is designated to serve as a useful tool for afterglow observers to quickly identify relevant models to interpret their data. The limitations of the analytical models are reviewed, with a list of situations summarized when numerical treatments are needed.Comment: 119 pages, 45 figures, invited review accepted for publication in New Astronomy Review

Finite edge-transitive dihedrant graphs

AbstractIn this paper, we first prove that each biquasiprimitive permutation group containing a regular dihedral subgroup is biprimitive, and then give a classification of such groups. The classification is then used to classify vertex-quasiprimitive and vertex-biquasiprimitive edge-transitive dihedrants. Moreover, a characterization of valencies of normal edge-transitive dihedrants is obtained, and some classes of examples with certain valences are constructed

Methyl (R)-2-(2-chloro­phen­yl)-2-(3-nitro­phenyl­sulfon­yloxy)acetate

The reaction between methyl (R)-2-(2-chloro­phen­yl)-2-hy­droxy­acetate and 3-nitro­benzene­sulfonyl chloride gave the title compound, C15H12ClNO7S, which is a promising inter­mediate for the synthesis of Clopidrogel, an anti­platelet drug used in the prevention of strokes and heart attacks. In the crystal, mol­ecules are linked through C—H⋯O interactions, and there is also a short Cl⋯O contact present [Cl⋯O = 3.018 (2) Å]

Methyl pyrazine-2-carboxyl­ate

The title compound, C6H6N2O2, is approximately planar [r.m.s. deviation = 0.0488 (3) Å]. In the crystal, weak inter­molecular C—H⋯O and C—H⋯N inter­actions join the mol­ecules into an infinite three-dimensional network