The initial Lorentz factors of fireballs inferred from the early X-ray data of SWIFT GRBs

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Volume 498 / No 3 (May II 2009)

A&A, 498 3 (2009) 671-676

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Table 2:

Temporal behavior of the X-ray afterglow light curves in the case of 1<p<2.
	Emission regime	ISM	Wind
Thin shell case (FS)
$t<t_{\rm p}$	$\nu_{_{\rm X}}>\max\{\nu_{\rm c}^{\rm f},~\nu_{\rm m}^{\rm f}\}$	t²	t⁰
$t<t_{\rm p}$	$\nu_{\rm c}^{\rm f}<\nu_{_{\rm X}}<\nu_{\rm m}^{\rm f}$	t²	t^(p-1)/2
$t<t_{\rm p}$	$\nu_{\rm m}^{\rm f}<\nu_{_{\rm X}}<\nu_{\rm c}^{\rm f}$	t³	t^-1/2
Thin shell case (RS)
$t<t_{\rm p}$	$\nu_{\rm m}^{\rm r}<\nu_{\rm c}^{\rm r}<\nu_{_{\rm X}}$	t³	t⁰
$t<t_{\rm p}$	$\nu_{\rm m}^{\rm r}<\nu_{_{\rm X}}<\nu_{\rm c}^{\rm r}$	t⁴	t^-1/2
Thick shell case (RS)
$t<t_{\rm p}$	$\nu_{_{\rm X}}>\max\{\nu_{\rm c}^{\rm r},~\nu_{\rm m}^{\rm r}\}$	t^-3(2-p)/8	t⁰
$t<t_{\rm p}$	$\nu_{\rm c}^{\rm r}<\nu_{_{\rm X}}<\nu_{\rm m}^{\rm r}$	t^-3(2-p)/8	t^(p-1)/2
$t<t_{\rm p}$	$\nu_{\rm m}^{\rm r}<\nu_{_{\rm X}}<\nu_{\rm c}^{\rm r}$	t^(3p-2)/8	t^-1/2
Late FS emission
$t>t_{\rm p}$	$\nu_{_{\rm X}}>\max\{\nu_{\rm c}^{\rm f},~\nu_{\rm m}^{\rm f}\}$	t^-(10+3p)/16	t^-(6+p)/8
$t>t_{\rm p}$	$\nu_{\rm c}^{\rm f}<\nu_{_{\rm X}}<\nu_{\rm m}^{\rm f}$	t^-(22-9p)/16	t^-(12-5p)/8
$t>t_{\rm p}$	$\nu_{\rm m}^{\rm f}<\nu_{_{\rm X}}<\nu_{\rm c}^{\rm f}$	t^-(6+3p)/16	t^-(8+p)/8
Late FS emission (jet)
$t>t_{\rm j}$	$\nu_{_{\rm X}}>\max\{\nu_{\rm c}^{\rm f},~\nu_{\rm m}^{\rm f}\}$	t^-(p+6)/4	t^-(p+6)/4
$t>t_{\rm j}$	$\nu_{\rm c}^{\rm f}<\nu_{_{\rm X}}<\nu_{\rm m}^{\rm f}$	t^(3p-10)/4	t^(3p-10)/4
$t>t_{\rm j}$	$\nu_{\rm m}^{\rm f}<\nu_{_{\rm X}}<\nu_{\rm c}^{\rm f}$	t^-(p+6)/4	t^-(p+6)/4

Note. Observationally the crossing time $t_\times$ marks the beginning of the late sharp decline,
denoting such a timescale by $t_{\rm p}$ . $t_{\rm j}$ is the jet break time.

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