2 Basic equations

The basic equations, describing this problem are:  

$$\frac{\partial \varrho }{\partial t} + \frac{1}{r^2} \frac{\partial}{\partial r} (r^2\varrho u) = 0 \ ,$$ (1)

$$\begin{eqnarray} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial r} = - \frac{1}{\varrho }\frac{\partial p}{\partial r} + F_{\gamma} \ ,\end{eqnarray}$$ (2)

$$\begin{eqnarray} \frac{\partial \epsilon}{\partial t} + u\frac{\partial \epsilon... ...2}\frac{\partial} {\partial r}(r^2u) + H_{\gamma} - C_{\gamma} \ .\end{eqnarray}$$ (3)

Where

$$\begin{eqnarray} \begin{array} {c} p = (n_{\rm i} + n_{\rm n} + n_{\rm e})kT,\qquad \epsilon = \frac{3}{2} \frac{kT}{\rm \mu m_u},\\ \end{array}\end{eqnarray}$$ (4)

$n_{\rm n}$, $n_{\rm i}$, $n_{\rm e}$ are concentrations of neutral atoms, ions and electrons correspondently.

  

Figure 1: The magnitudes of the counterparts (upper limit - solid line, lower limit - dashed line) as a function of time after burst for GRB with total flux near the Earth $F_{\rm GRB}=10^{-4}\mbox{ erg cm}^{-2}$:1a - for the case $E=10^{52}\mbox{ erg},\ n_0 = 10^5\mbox{ cm}^{-3}$; 1b - for the case $E=10^{51}\mbox{ erg},\ n_0 = 10^5\mbox{ cm}^{-3}$; 2a - for the case $E=10^{52}\mbox{ erg},\ n_0 = 10^4\mbox{ cm}^{-3}$; 2b - for the case $E=10^{51}\mbox{ erg},\ n_0 = 10^4\mbox{ cm}^{-3}$

For the $\gamma$-ray signal interacting with matter due to Thomson scattering of photons on electrons with cross-section $\sigma \rm _T$ we have:

$$F_{\gamma} = \frac{1}{c}\frac{L}{4\pi r^2}\frac{\mu_{\rm e} \sigma \rm _T}{m_{\rm u}}\cdot$$ (5)

For $\gamma$-rays with $h\nu\gg{B}_{\rm e}^{({a,i})}$($B_{\rm e}$ is the binding energy of electrons in atoms or ions) the cross-section is almost the same for free and bound electrons. Heating of the gas by the light signal with the spectrum $\frac{{\rm d}L}{{\rm d}E}=\frac{L}{E_{\rm max}}{\rm e}^{-E/E_{\rm max}}$is given by (Bisnovatyi-Kogan & Blinnikov [1980]):

$$\begin{eqnarray} H_{\gamma} = \frac{L}{4\pi r^2}\frac{\mu_{\rm e} \sigma\rm _T}{m_{\rm u}} \frac{E_{\rm max} -4\,kT}{m_{\rm e}c^2} ,\end{eqnarray}$$ (6)

where L is the energy flux of the signal; E is the energy of photons. This formula is valid for $E \ll m_{\rm e}c^2$.For photons with energies <10 keV, the main process of interaction with the gas is photoabsorption by ions of heavy elements. To take in to account gas heating by soft X-rays photons we enhance term (6) using $E_{\rm max} \sim 2$ MeV. Because of a lack of information about soft X-ray spectra of GRBs and variations in interstellar media density such assumption, seems to be quite good. On the other hand, completely ignoring gas heating by X-rays we get a counterpart by less then $2^{\rm m}$ fainter. Cooling of optically thin plasma by free-free and free-bound transitions is given by approximating function (Cowie et al. [1981]):

$$\begin{eqnarray} C_{\gamma} = \frac{\Lambda(T) n^2}{\varrho }\ , \qquad n = n_{\rm n} + n_{\rm i}\end{eqnarray}$$ (7)

Electron density ($n_{\rm e}$) is given by the Elvert formula.

This system was solved numerically using a full conservative difference scheme with flux corrected transport, because there is a strong density gradient in the solution (shock wave).