2 Gravitational lensing

2.1 The model of lens

As a model of mass distribution in the lens galaxy, we will use the so-called Isothermal Sphere with Core (ISC), which gives a quite good description of the matter distribution in galaxies of different types (Hinshaw & Krauss 1987). We use a standard set of parameters: $D_{\rm d}$, $D_{\rm s}$, $D_{\rm ds}$ (linear distances to the lens galaxy, to the GRB source and between the lens galaxy and the GRB source, respectively), $\sigma_{\rm v}$ (velocity dispersion in the lens galaxy), $r_{\rm c}$, $\theta_{\rm c}=\frac{r_{\rm c}}{D_{\rm d}}$ (linear and angular radius of galaxy core), $\beta_0=\frac{\beta}{\theta_{\rm c}}$ and $\theta_0=\frac{\theta}{\theta_{\rm c}}$ (real and visual angular position of the source from the "observer-lens" axis in units of $\theta_{\rm c}$).

The lens equation is the following (see e.g. Wu 1989):

$$\theta_0= \beta_0+D \frac{\sqrt{1+\theta_0^2}-1}{\theta_0},$$ (1)

where

$$D = \frac{4 \pi \sigma_{\rm v}^2}{c^2} \frac{D_{\rm d} D_{\rm ds}}{r_{\rm c} D_{\rm s}}$$ (2)

is a dimensionless parameter. This equation can have 1 or 3 real solutions, which correspond to 1 or 3 source images or one or three bursts (i.e. two repetitions). It can be shown that this equation has three real roots when D > 2. To obtain these roots, an additional restriction for $\beta_0$is necessary (Beskin et al. 1998a,b):

$$\begin{eqnarray} \beta_0 < \beta_{\rm 0max} = \\ \nonumber\end{eqnarray}$$ (3)

This means that we will have 3 GRBs from one source if its angular position $\beta_0$ is less than $\beta_{\rm 0max}$.

2.2 The application to real galaxies

Let us give quantitative estimates assuming $\Omega =1$ and H₀= 50 km s^-1 Mpc^-1. For the distances we have

$$D_{\rm d,s} = 12 \left [1-(1+z_{\rm d,s})^{-1/2} \right ] (1+z_{\rm d,s})^{-1}~{\rm Gpc}.$$ (4)

It can also be shown that

$$D_{\rm ds} = 12 \left [ \left( \frac{1+z_{\rm s}}{1+z_{\rm d}} \right)^{1/2} -1 \right] (1+z_{\rm s})^{-3/2}~{\rm Gpc}.$$ (5)

To estimate D we take $\sigma_{\rm v} \sim 300$ km s^-1 and $r_{\rm c} \sim 0.2$kpc for elliptical galaxies, $\sigma_{\rm v}\sim 150$ km s^-1 and $r_{\rm c}\sim 1$ kpc for spirals (Fukugita & Turner 1991; Hinshaw & Krauss 1987). As it is shown in Table 1, elliptical galaxies can give three images of GRB for any combination of $z_{\rm d}$ and $z_{\rm s}$. In the case of a spiral galaxy, only for $z_{\rm d} \sim 0.5-1$and $z_{\rm s} \sim 2-4$.We estimated the probability P of GRB repetitions for elliptical galaxies of size $\sim 15$ kpc, if the GRB is seen through it (see Table 1). We used for P the expression $P \sim (\beta_{\rm 0max} r_{\rm c}/R)^2$, where R is the size of the galaxy.

  

Table 1: Probability P of repetitions for GRB lensed by elliptical galaxies of size $\sim 15$ kpc for different $z_{\rm d}$ and $z_{\rm s}$

From Table 1 we see that about $40\%$ of the most distant GRBs will be repeated if their emissions pass close to elliptical galaxies placed not far from us.

2.3 Relative intensities and delay times of repeated bursts

As an example, we now compute the parameters of repeated transients -- intensities and delay times between two subsequent repetitions -- for $D \sim 60$. The estimate of the intensity ratios is obtained according to the expression of lens magnification (Wu 1996):

$$\mu = \left \vert\frac {\beta_0} {\theta_0} \left [1+ \left ... ...theta^2_0 \right )^{-\frac{1}{2}} \right ] \right \vert ^{-1}.$$ (6)

The time delay is instead defined with the expression (Beskin et al. 1998a):

$$\begin{eqnarray} \Delta t_{ij} = \frac{4 \pi \sigma_{\rm v}^2}{c^3} r_{\rm c}(1... ...ac{1+\sqrt{\theta_{0i}^2+1}}{1+\sqrt{\theta_{0j}^2+1} } \Biggr\}, \end{eqnarray}$$ (7)

which is obtained with the help of the Fermat's potential (see e.g. Schneider et al. 1992).

  

Table 2: Parameters of the repeated GRBs with respect to the first one

In Table 2 are reported the parameters of the repeated GRBs for $D\sim$ 60 (the source is assumed at $z\sim$ 4, while the elliptical galaxy has $z\sim$ 0.5). Thus, repeating transients can be perfectly observed, although they would be 4-7 magnitudes fainter than the first event. Delay times cover the range from days to years, while characteristic times of physical intensity variations span from days to months. For this reason, light curves can have complicated structures. To distinguish repeating transients we need to analyze all the available data, such as variations of intensity, astrometry and spectroscopy. It is necessary to note that observable angular distances between the first GRB and the repeated transients are $0\hbox{$.\!\!^{\prime\prime}$}2-3^{\prime\prime}$ i.e. they can be separated.