2 Data and analysis

Figure 1: The optical afterglow of GRB 020405 superimposed to the host galaxy during run 1 (left) and run 2 (right). Pictures were obtained composing the polarization images in the Bessel V-band filter. Box size is about $18\hbox {$^{\prime \prime }$ }\times 18\hbox {$^{\prime \prime }$ }$; North is up and East is left. The OT is marked by an arrow.

Our observations of GRB 020405 were obtained with the ESO's VLT-UT3 (Melipal), equipped with the Focal Reducer/low dispersion Spectrometer (FORS 1) and a Bessel filter V in the imaging polarimetry mode. Our first observation (hereafter run 1) started on April 7, 03:33 UT (2.1 days after the burst) and lasted $\sim$3 hours. At the beginning of this observation the V magnitude was $21.82 \pm 0.02$, with respect to the USNO-A2.0 stars 0525_16813005 and 0525_16815468 (Simoncelli et al. 2002). Our second observation (run 2) was performed during the following night, starting April 8, 4:01 UT (3.2 days after the burst), and lasted $\sim$3.5 hours. The V magnitude of the OA was $22.45 \pm 0.05$ again with respect to the two above reported stars. Observations were performed under good seeing conditions ( $0.5\hbox{$^{\prime\prime}$ }{-}0.9\hbox{$^{\prime\prime}$ }$) in standard resolution mode with a scale of  $0.2\hbox{$^{\prime\prime}$ }$/pixel (Fig. 1).

Standard stars were also observed. One polarized, Hiltner 652, in order to fix the offset between the polarization and the instrumental angles, and one non-polarized, WD 1615-154, to estimate the degree of artificial polarization possibly introduced by the instrument.

The data reduction was carried out with the Eclipse package (version 4.2.1, Devillard 1997). After bias subtraction, non-uniformities were corrected using flat-fields obtained with the Wollaston prism. The flux of each point source in the field of view was derived by means of both aperture and profile fitting photometry by the DAOPHOT II package (Stetson 1987), as implemented in ESO-MIDAS (version 01SEP) and the Graphical Astronomy and Image Analysis (GAIA) tools. For relatively isolated stars the various applied photometric techniques differ only by a few parts in a thousand. The general procedure followed for FORS 1 polarization observation analysis is extensively discussed in Covino et al. (1999, 2002d).

The average polarization of the field stars is low as shown by the normalized Stokes parameters Q and U: $\langle Q \rangle = -0.0021 \pm 0.0009$ and $\langle U \rangle = 0.0012 \pm 0.0009$, corresponding to $P = (0.24 \pm 0.09)$%.

The degree P and angle $\vartheta$ of polarization are obtained from the measurements of Q and U for the OA $\left[P = \sqrt{U^2 + Q^2}, \vartheta = \frac{1}{2}\arctan(U/Q)\right]$ after correcting for the polarization induced by the instrument or by the local interstellar matter. Moreover, for any low level of polarization ( $P/\sigma\le 4$), a correction which takes into account the bias due to the fact that P is a definite positive quantity (Wardle & Kronberg 1974) is required. At low polarization level, the distribution function of P (and of $\vartheta$, the polarization angle) is no longer normal and that of P becomes skewed (Clarke et al. 1983; Simmons & Stewart 1985; Fosbury et al. 1993).

 

 

Table 1: Polarization degree P and positional angle $\vartheta$ for the optical counterpart to GRB 020405. Observations were performed with the VLT-UT3 (Melipal) in the Bessel V-band filter.

Run	UT	V mag	P ($\%$)	$\vartheta$ ( $\hbox{$^\circ$ }$)
1	Apr 7.212	$21.82 \pm 0.02$	$1.96 \pm 0.33$	$154 \pm 5$
2	Apr 8.297	$22.45 \pm 0.05$	$1.47 \pm 0.43$	$168 \pm 9$

We then corrected our measurements for this bias (Simmons & Stewart 1985) and derived the normalized polarization Stokes parameters for the OA: $Q = 0.0126 \pm 0.0033$ and $U = -0.0150 \pm 0.0033$ for run 1 and $Q = 0.0137 \pm 0.0043$ and $U = -0.0054 \pm 0.0044$ for run 2. From these values of Q and U we have derived the polarization degree P and positional angle $\vartheta$ for both run 1 and 2, as reported in Table 1. Monte Carlo simulations confirmed the reported values and uncertainties.

2.1 Host galaxy contamination to photometry

Figure 1 clearly shows that the OA is superimposed to a rather bright and extended galaxy ($\sim$ $4\hbox{$^{\prime\prime}$ }\times 7\hbox{$^{\prime\prime}$ }$ in our VLT images, with some bright knots). Since the light of the galaxy is unavoidably mixed with that of the OA, it is important to estimate the effect of this contamination on the polarization angle and degree. If the emission of the galaxy is not polarized, the net effect is to effectively reduce the degree of polarization of the OA. It is easy to show that the observed polarization degree  $P_{\rm obs}$ can be corrected to yield the intrinsic value  $P_{\rm true}$, if we know the contributions to the total flux of the galaxy, $F_{\rm gal}$, and of the OA,  $F_{\rm OA}$:

$$P_{\rm true} = \left(1 + \frac{F_{\rm gal}} {F_{\rm OA}}\right) P_{\rm obs} = \frac{F_{\rm tot}}{F_{\rm OA}} P_{\rm obs},$$ (1)

where $F_{\rm tot} = F_{\rm OA} + F_{\rm gal}$. The polarization angle is of course not affected, even if the lower value of P eventually leads to a larger uncertainty.

To estimate the contribution of the galaxy within the point spread function (PSF), it is necessary to analyze late-time images, when the flux of the afterglow gives only a negligible contribution. For GRB 020405, only a rough R magnitude is reported to date (Bersier et al. 2002; see also Price et al. 2002c), suggesting that in the PSF area $V \ge 24$ depending on the color of the galaxy (e.g. Fukugita et al. 1995).

Although an accurate analysis of the late-time image would be required, the good seeing conditions in our images make these corrections, estimated by Eq. (1), essentially negligible.