Gamma-ray Burst Emission Mechanisms - Afterglows and External Shocks

Afterglows and External Shocks

The GRB itself is very rapid, lasting from less than a second up to a few minutes at most. Once it disappears, it leaves behind a counterpart at longer wavelengths (X-ray, UV, optical, infrared, and radio) known as the afterglow that generally remains detectable for days or longer.

In contrast to the GRB emission, the afterglow emission is not believed to be dominated by internal shocks. In general, all the ejected matter has by this time coalesced into a single shell traveling outward into the interstellar medium (or possibly the stellar wind) around the star. At the front of this shell of matter is a shock wave referred to as the "external shock" as the still relativistically moving matter ploughs into the tenuous interstellar gas or the gas surrounding the star.

As the interstellar matter moves across the shock, it is immediately heated to extreme temperatures. (How this happens is still poorly understood as of 2007, since the particle density across the shock wave is too low to create a shock wave comparable to those familiar in dense terrestrial environments – the topic of "collisionless shocks" is still largely hypothesis but seems to accurately describe a number of astrophysical situations. Magnetic fields are probably critically involved.) These particles, now relativistically moving, encounter a strong local magnetic field and are accelerated perpendicular to the magnetic field, causing them to radiate their energy via synchrotron radiation.

Synchrotron radiation is well-understood and the afterglow spectrum has been modeled fairly successfully using this template. It is generally dominated by electrons (which move and therefore radiate much faster than protons and other particles) so radiation from other particles is generally ignored.

In general, the GRB assumes the form of a power-law with three break points (and therefore four different power-law segments.) The lowest break point, corresponds to the frequency below which the GRB is opaque to radiation and so the spectrum attains the form Raleigh-Jeans tail of blackbody radiation. The two other break points, and, are related to the minimum energy acquired by an electron after it crosses the shock wave and the time it takes an electron to radiate most of its energy, respectively. Depending on which of these two frequencies is higher, two different regimes are possible:

Fast cooling - Shortly after the GRB, the shock wave imparts immense energy to the electrons and the minimum electron Lorentz factor is very high. In this case, the spectrum looks like:

$F_\nu \propto \begin{cases} {\nu^{2}}, & \nu<\nu_a \\ {\nu^{1/3}}, & \nu_a<\nu<\nu_c \\ {\nu^{-1/2}}, & \nu_c<\nu<\nu_m \\ {\nu^{-p/2}}, & \nu_m<\nu \end{cases}$

Slow cooling – Later after the GRB, the shock wave has slowed down and the minimum electron Lorentz factor is much lower.:

$F_\nu \propto \begin{cases} {\nu^{2}}, & \nu<\nu_a \\ {\nu^{1/3}}, & \nu_a<\nu<\nu_m \\ {\nu^{-(p-1)/2}}, & \nu_m<\nu<\nu_c \\ {\nu^{-p/2}}, & \nu_c<\nu \end{cases}$

The afterglow changes with time. It must fade, obviously, but the spectrum changes as well. For the simplest case of adiabatic expansion into a uniform-density medium, the critical parameters evolve as:

Here is the flux at the current peak frequency of the GRB spectrum. (During fast-cooling this is at ; during slow-cooling it is at .) Note that because drops faster than, the system eventually switches from fast-cooling to slow-cooling.

Different scalings are derived for radiative evolution and for a non-constant-density environment (such as a stellar wind), but share the general power-law behavior observed in this case.

Several other known effects can modify the evolution of the afterglow:

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