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4 Evidence for an injection bend

The injection bend induces the gradual transition in the spectral energy distribution described by Eq. (4), occurring at a ``bend'' frequency:

 \begin{displaymath}\nu_{\rm b} \simeq {1.87\times 10^{15} \over 1+z}\,
{[\gamma...
...\,
\left[{n_p\over 10^{-3}\;{\rm cm}^3}\right]^{1/2}
{\rm Hz},
\end{displaymath} (14)

where we have used the characteristic synchrotron frequency of Eq. (3) for the magnetic field of Eq. (2), and transposed the result to the observer's frame.

For the reference CB parameters and z=1, $\rm\nu_b(t=0)\simeq 0.93\times 10^{15}$ Hz, above the optical band. Since the product $\gamma^3\, \delta$ typically declines by more than an order of magnitude within a couple of days, the bend frequency in many GRBs crosses the optical band into the NIR during the early afterglow. In Fig. 2 we present the time dependence of $\nu_{\rm b}(t)$ for $\gamma _0=1250$ and 750, characterizing the range of the observations, for various angles $\theta$, z=1, and the rest of the parameters at their reference values of Table 1. The figures show that, depending on the parameters, the bend frequency in the early AG may be above or below the optical band, and, if it is above, it will cross it later.

  \begin{figure}
\includegraphics[width=8.8cm,clip]{MS2654f2a.eps}\vspace*{3mm}
\includegraphics[width=8.8cm,clip]{MS2654f2b.eps}
\end{figure} Figure 2: Typical predictions for the bend frequency in the AG spectrum as a function of time, for $\theta =0,\,3/\gamma _0$and $10/\gamma _0$. The ``optical'' U to I band is shown as a horizontal band. Upper panel: for $\gamma _0=1250$. Lower panel: for $\gamma _0=750$.

The bend frequency of the CB model is not the break frequency of the traditional fireball model. The time evolution of the former is given by Eq. (14), and is different from that of the latter, which, prior to the ``break'' in the AG light-curve, can be shown to be t-1/2 (Granot & Sari 2002).

The evolution predicted by Eq. (4) from a  $\nu^{-0.5\pm 0.1}$ to a  $\nu^{-1.1\pm 0.1}$ spectral behaviour is affected by extinction. The early behaviour corresponds to times when CBs are not yet very far from their progenitors: extinction in the host galaxy may steepen the spectrum. After a day or more, when the CBs are further away, we do not expect strong extinction in the host. So the prediction (after extinction in the Galaxy is corrected for) is an evolution from a behaviour close to - or steeper than -  $\nu^{-0.5\pm 0.1}$, to a more universal $\nu^{-1.1\pm 0.1}$ at later times.

The predicted spectral behaviour has been observed, with varying degrees of significance, in the AG of several GRBs, listed in Table 2. The first column is the bend frequency $\rm\nu_b^0$ at t=0, computed with Eq. (14) and the optical AG parameters of Table 3 (the density np is extracted from the measured $x_\infty$ with use of Eq. (38) and our reference $R_{\rm max}$ and $N_{_{\rm CB}}$). For the listed GRBs the bend frequency is above the visible band at t=0 and the early AG measurements result in effective spectral slopes, $\beta(t_1)$, not far from the expectation $-0.5\pm 0.1$, or somewhat steeper. A few days later, the measured values, $\beta(t_2)$, are compatible with the expectation ${-1.1\pm 0.1}$. The second entry on GRB 990510 in Table 2 (Beuermann et al. 1999) requires an explanation. These authors argue that $\beta(t_2)=0.55\pm 0.10$, a result that assumes a strong extinction correction in the host galaxy. But, after a day or so, we do not expect such an extinction. For the latest points measured by Beuermann et al. (1999), at day 3.85 (well after the bend), $B-R=0.98\pm 0.07$and $R-I=0.49 \pm 0.06$. Converting these results - without extinction - to a spectral slope yields $\beta(t_1)=1.11\pm 0.12$, in agreement with expectation.

 
Table 2: The crossing of the bend frequency through the U to I bands [(10 to 3) $\times 10^{15}$ Hz]b.
\begin{table}\begin{displaymath}
\begin{array}{llcclc}
\hline\hline
\noalig...
...l. (\cite{Stanek01}); Masetti et~al.~(\cite{Masetti01}).
\end{list} \end{table}


 
Table 3: The afterglow parametersc.
\begin{table}\begin{displaymath}
\begin{array}{lccccc}
\hline\hline
\noali...
...ference parameters that we had chosen in previous works.
\end{list} \end{table}


 
Table 4: Frequencies, in GHz, at which the radio AGs of GRBs of known redshift were measuredd.
\begin{table}\begin{displaymath}
\begin{array}{ p{0.22\linewidth}l p{0.24\li...
...\;\;1.38& $\;\;\;$1.38& & \\
\hline
\end{array} \end{displaymath} \end{table}
d
References:
GRB 970508: Frail et al. (2000a).
GRB 980425: Kulkarni et al. (1998).
GRB 990123: Kulkarni et al. (1999b); Galama et al. (1999).
GRB 990510: Harrison et al. (1999).
GRB 991208: Galama et al. (2000).
GRB 991216: Frail et al. (2000b).
GRB 000301c: Berger et al. (2000).
GRB 000418: Berger et al. (2001a).
GRB 000926: Harrison et al. (2001).

The evolution from a softer to a harder spectrum should be a gradual change in time, rather than a sharp break, so that an AG's optical spectrum, if ``caught'' as the injection bend is ``passing'' should have an index evolving from $-0.5\pm 0.1$ to ${-1.1\pm 0.1}$ with the time dependence described by Eqs. (4) and (14). In Fig. 3 we test this prediction in the case of GRB 970508, for the time-dependent value of the ``effective'' slope $ \alpha \simeq \, \Delta\, [\log\, F_\nu]/\Delta\,[\log\,\nu]$, constructed from the theoretical expectation in the same frequency intervals used by the observers. The actual predicted $\nu_{\rm b}(t)$ in Eq. (14) is obtained by use of the optical-AG fitted parameters ($\theta$, $\rm\gamma_0$ and $x_\infty$) that determine $\gamma(t)$ and $\delta(t)$, and the density np deduced[*] from $x_\infty$ and the reference $N_{_{\rm CB}}$ and $R_{\rm max}$. The data are gathered by Galama et al. (1998a) from observations in the U, B, V, $R_{\rm c}$ and $I_{\rm c}$ bands (Castro-Tirado et al. 1998; Galama et al. 1998b; Metzger et al. 1997; Sokolov et al. 1998; Zharikov et al. 1998), by Chary et al. (1998) for K band results, and by Pian et al. (1998) for the H band.

  \begin{figure}
\includegraphics[width=8.8cm,clip]{MS2654f3.eps}
\end{figure} Figure 3: A comparison between the predicted evolution in time of the effective spectral slope through the optical/NIR band and the data collected by Galama et al. (1998a) for the U, B, V, $R_{\rm c}$ $I_{\rm c}$ band of the AG of GRB 970508 (upper panel), for the K and $R_{\rm c}$ band (full squares, lower panel, Chary et al. 1998) and for the H and $R_{\rm c}$ band (triangle, lower panel, Pian et al. 1998) The three coloured lines, in the same order, are the (parameter-less) predictions.

In spite of considerable uncertainties in the spectral slopes deduced from observations (Galama et al. 1998a), the results shown in Fig. 3 are satisfactory: the observed crossing of the injection bend is in agreement with the theoretical prediction, based on the fit in DDD 2001 to the overall R-band light curve from which the GRB 970508 AG parameters have been fixed; no extra parameters have been fit. A couple of points in the lower panel do not agree with the prediction, but they do not agree with the observations at very nearby frequencies reported in the upper panel, either.

A complementary analysis to that in the previous paragraph is the study of an AG's optical spectrum at a fixed time at which the injection bend is crossing the observed frequency range, or is nearby. A spectral ``snapshot'' at such time should have the intermediate slope and curvature described by Eq. (4) for $\rm\nu\sim\nu_b$. To test this prognosis, we compare in Fig. 4 the predicted spectral shape of the optical/NIR AG of GRB 000301c around March 4.45 UT ($\sim $3 days after burst) to its measured shape (Jensen et al. 2001). We have selected this GRB because its extinction correction in the galactic ISM is rather small: E(B - V)=0.05 (Schlegel et al. 1998), and there is no evidence for significant extinction in the host galaxy (Jensen et al. 2001). The theoretical line in Fig. 4 is given by Eq. (4) with the observer's $\rm\nu_b$of Eq. (14) ( $\nu_{\rm b}\,(1+z)=1.75\times 10^{14}$ Hz at t=3 days, for the density deduced from the value of $x_\infty$of this GRB, and the reference values of $N_{_{\rm CB}}$ and $R_{\rm max}$). In the figure the theory's normalization is arbitrary but the (slightly evolving) slope of the theoretical curve is an absolute prediction: it is based on the fit in DDD 2001 to the overall R-band light curve and, once more, no extra parameters have been fit. The result is astonishingly good, even for the curvature which - given the figure's aspect ratio as chosen by the observers - is not easily visualized (a look at a slant angle helps). The late-time spectral slope deduced from the HST observations (Smette et al. 2001) around day 33.5 after burst indicated a slope of $\sim $-1.1, again in agreement with our expectation.

  \begin{figure}
\includegraphics[width=8.8cm,clip]{MS2654f4.eps}
\end{figure} Figure 4: Comparison between the observations and the (parameter-less) prediction for the spectral shape of the optical AG of GRB 000301c, at $\sim $3 days after burst. Data from Jensen et al. (2001).

We conclude that the evidence is very strong for a spectral injection bend at the time-dependent frequency, Eq. (14), predicted in the CB model. As illustrated in Fig. 1 and contrasted with data in Sect. 8, further evidence for the injection bend is provided by the fact that it is essential to the description of the observed broad-band spectra of GRB afterglows.


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