Images, Light Curves and Spectra of GRB Afterglow

We calculate the light curve and spectra near the peak and the self absorption break, for an adiabatic blast wave described by the Blandford-McKee solution, considering the emission from the whole region behind the shock front. The expected light curve and spectra are flat near the peak. This rules out the interpretation of the sharp peak observed in the optical afterglow of GRB970508 as the expected peak of the light curve. The observed image of an afterglow is calculated for a broad range of frequencies. We show that for frequencies below the self absorption frequency the image is rather homogeneous, as opposed to the bright ring at the outer edge and dim center, which appear at higher frequencies. We fit the observed spectra of GRB970508 to the detailed theory and obtain estimates of the physical parameters of this burst. 1. The Physical Model We consider emission from the whole volume behind an adiabatic highly relativistic spherical blast wave expanding into a cold and uniform medium. The hydrodynamics is described by the Blandford-McKee (1976 denoted BM hereafter) self similar solution. For typical parameters, the evolution becomes adiabatic fairly early, about an hour after the initial burst (Sari, Piran & Narayan 1998, Granot, Piran & Sari 1998a and 1998b, hereafter GPSa and GPSb, respectively). The BM solution is valid from this time, and as long as γ∼2 (Kobayashi et. al. 1998), typically a few months after the burst. We assume that νa ≪ νm, where νm is the peak frequency and νa is the self absorption frequency, which is reasonable for the first few months. The dominant radiation emission mechanism is assumed to be synchrotron radiation, while Compton scattering and electron cooling are ignored. We denote quantities measured in the local rest frame of the matter with a prime, while quantities without a prime are measured in the observer frame. We assume that the energy of the electrons is everywhere a constant fraction of the internal energy: e′el = ǫee , and consider a power law electron distribution: N(γe) ∝ ν << νa νa << ν << νm νm << ν << νc Fig. 1. The observed image of a GRB afterglow at a given observed time, for different frequencies. γ e for γe ≥ γmin. The magnetic field is also assumed to hold a constant fraction of the internal energy: e′B = ǫBe , where eB = B /8π is the energy density of the magnetic field. Alternative magnetic field models were considered in GPSa and GPSb, and we obtained that our results are not sensitive to the assumptions on the magnetic field. 2. The Observed Image The observed images, at various frequencies, are shown in Figure 1 (GPSa, GPSb). For ν ≫ νm a thin bright ring appears on the outer edge of the image, while the center is much dimmer (only a few percent of the maximal surface brightness). For νa ≪ ν ≪ νm the surface brightness at the center is 34% of the maximal surface brightness, and 58% of the average surface brightness. For ν ≪ νa the surface brightness at the center of the image is 77% of its average value, resulting in an almost uniform disk. 3. Light Curve, Spectra & The Burst Parameters The light curve and spectra of an afterglow are flat near the peak. The exact shape, and the value of the peak frequency and peak flux depend on the values of the physical parameters of the burst (see GPSa). In Figure 2 we see the peak in the optical light curve of GRB970508 (Sokolov et al. 1997, Metzger et al. 1997), with three theoretical light curves. These light curves are for p = 2.57, which corresponds to the power law decay that follows the peak, and differ by the values of the remaining parameters. 2 Jonathan Granot, Tsvi Piran & Re’em Sari: Images, Light Curves and Spectra of GRB Afterglow 10 −1 10 0 10 1 10 2 18