The injection bend induces the gradual transition
in the spectral energy distribution described by Eq. (4), occurring
at a ``bend'' frequency:
For the reference CB parameters and z=1,
Hz,
above the optical band. Since the product
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
for
and 750, characterizing the range
of the observations, for various angles
,
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.
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
to a
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 -
,
to a more universal
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
at t=0,
computed with Eq. (14) and the optical AG parameters of
Table 3
(the density np is extracted from the measured
with use of Eq. (38)
and our reference
and
). 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,
,
not far from the expectation
,
or somewhat steeper.
A few days later, the measured
values,
,
are compatible with the expectation
.
The second entry on GRB 990510 in Table 2
(Beuermann et al. 1999)
requires an explanation. These authors argue that
,
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),
and
.
Converting these results - without
extinction - to a spectral slope yields
,
in agreement with expectation.
|
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
to
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
,
constructed from the theoretical expectation in the
same frequency intervals used by the observers.
The actual predicted
in Eq. (14) is obtained by use of
the optical-AG fitted parameters (
,
and
)
that determine
and
,
and
the density np deduced
from
and the reference
and
.
The data are gathered by Galama et al. (1998a) from
observations in the U, B, V,
and
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.
![]() |
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, ![]() ![]() ![]() ![]() |
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
.
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
(
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
of Eq. (14) (
Hz at
t=3 days, for the density deduced from the value of
of this GRB, and the reference values of
and
).
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
-1.1, again in agreement
with our expectation.
![]() |
Figure 4:
Comparison between the observations and the
(parameter-less) prediction
for the spectral shape of the optical AG of GRB 000301c, at
![]() |
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.
Copyright ESO 2003