3 Data reduction

Considering first the two frames obtained with the EFOSC1 rotated at 270 $\hbox{$^\circ$ }$ and 225 $\hbox{$^\circ$ }$, the normalized Stokes parameters q and u are given by

$$q = \frac{I^{\scriptscriptstyle \rm u}_{\scriptscriptstyle \rm 27... ...criptstyle \rm 225}+I^{\scriptscriptstyle \rm l}_{\scriptscriptstyle \rm 225}},$$ (1)

where $I^{\scriptscriptstyle \rm u}$ and $I^{\scriptscriptstyle \rm l}$ respectively refer to the intensities integrated over the upper and lower orthogonally polarized images of the object.

When the four frames with the HWP oriented at 0 $\hbox{$^\circ$ }$, 22.5 $\hbox{$^\circ$ }$, 45 $\hbox{$^\circ$ }$ and 67.5 $\hbox{$^\circ$ }$ are considered, the normalized Stokes parameters are derived using the following formulae:

$$\frac{R_q - 1}{R_q + 1} \hspace{0.5cm} \mbox{where} \hspace{0.5cm... ...ptscriptstyle \rm u}/I_{\scriptscriptstyle \rm 45}^{\scriptscriptstyle \rm l}},$$ (2)

$$\frac{R_u - 1}{R_u + 1} \hspace{0.5cm} \mbox{where} \hspace{0.5cm... ...scriptstyle \rm u}/I_{\scriptscriptstyle \rm 67.5}^{\scriptscriptstyle \rm l}},$$

$I^{\scriptscriptstyle \rm u}_{\beta}$ and $I^{\scriptscriptstyle \rm l}_{\beta}$ respectively denoting the intensities integrated over the upper and the lower images of the object, $\beta$ representing the position angle of the HWP. This combination of four frames obtained with different HWP orientations not only removes most of the instrumental polarization (di Serego Alighieri 1998), but is essential for correcting the effects of image distortions introduced by the HWP (Lamy & Hutsemékers 1999). Note that q and u are measured with respect to the instrumental reference frame.

 

 

Table 1: Polarized calibration stars

Date	Object	Ref
11-09-96	HD 161291	1
27-04-98	HD 111579, HD 155197, HD 298383	2,3
28-04-98	HD 155197	2,3
18-10-98	HD 298383	2
13-04-99	HD 164740, HD 126593, HD 161291, HD 298383	1,2
14-04-99	HD 111579, HD 126593, HD 161291, HD 298383	1,2
07-09-99	HD 283812	2,4

References: (1): Schwarz 1987; (2) Turnshek et al. 1990; (3) Schmidt et al. 1992; (4) Whittet et al. 1992.

It is clear from these relations that intensities must be determined with the highest accuracy. For this, the data were first corrected for bias and dark emission, and flat-fielded. A plane was locally fitted to the sky around each object image, and subtracted from each image individually. Since it appeared that standard aperture photometry was not accurate enough, we have measured the object center at subpixel precision by fitting a 2D Gaussian profile and integrated the flux in a circle of same center and arbitrary radius by taking into account only those fractions of pixels inside the circle. With this method, the Stokes parameters may be computed for any reasonable radius of the aperture circle. They were found to be stable against radius variation, giving confidence in the method even when the object images are distorted (Lamy & Hutsemékers 1999). In order to take as much flux as possible with not too much sky background, we fixed the aperture radius at $\alpha \times [(2\,\ln 2)^{-1/2}$ HWHM], where $\alpha=2.5$ with EFOSC1 and $\alpha=3.0$ with EFOSC2 to account for the image elongation introduced by the HWP. HWHM represents the mean half-width at half-maximum of the Gaussian profile. Note that in the few cases where the objects are resolved into multiple components, we use the smallest rectangular aperture encompassing all the components. The whole procedure has been implemented within the ESO MIDAS reduction package.

Figure 1: The QSO polarization degree p0 (in %) [ $\ifmmode\hbox{\rlap{$\sqcap$ }$\sqcup$ }\else{\unskip\nobreak\hfil \penalty50\h... ...x{\rlap{$\sqcap$ }$\sqcup$ } \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi$] is represented here as a function of the Galactic latitude of the objects ( $\vert b_{\scriptscriptstyle \rm II}\vert$, in degree), together with the de-biased polarization degree of field stars [$\times$] (also corrected for the small systematic trend reported in Table 2), and the maximum interstellar polarization degree $p_{\scriptscriptstyle \rm ISM}$ derived from the Burstein & Heiles (1982) reddening maps [+]. Only B1333+2840 (p0 = 5.9%) is not represented here

 

 

Table 2: Residual instrumental polarization

Date	$\overline{q}_{\star}$	$\overline{u}_{\star}$	$\overline{\sigma}_{\star}$	$n_{\star}$
	(%)	(%)	(%)
04/98	-0.07	+0.01	0.17	15
04/99	+0.00	+0.19	0.17	16

 

 

Table 3: Polarimetric measurements

Object	z		Date	Filter	q	u	$\sigma$		$q_{\star}$	$u_{\star}$	$\sigma_{\star}$
					(%)	(%)	(%)		(%)	(%)	(%)
B0059-2735$^{\star}$	1.59		11-09-96	V	1.38	-0.45	0.15		-	-	-
B0059-2735$^{\star}$	1.59		11-09-96	i	2.06	-1.14	0.17		-	-	-
B0846+1540$^{\star}$	2.91		14-04-99	V	0.43	-0.48	0.12		-	-	-
B0856+1714$^{\star}$	2.32		14-04-99	V	0.70	0.20	0.17		0.10	0.13	0.03
B1009+0222$^{\star}$	1.35		13-04-99	V	0.06	-0.58	0.08		0.07	0.26	0.07
J 1053-0058$^{\star}$	1.55		13-04-99	V	-1.90	0.16	0.08		-	-	-
J 1104-0004$^{\star}$	1.35		13-04-99	V	0.49	0.14	0.13		0.16	-0.02	0.05
J 1141-0141$^{\star}$	1.27		13-04-99	V	-0.10	0.56	0.16		0.14	0.01	0.08
B1151+1145	0.18		28-04-98	V	-0.74	-0.24	0.06		-0.14	-0.24	0.04
B1157-2354$^{\star}$	2.10		27-04-98	V	-1.38	-0.24	0.04		0.00	0.04	0.04
B1157-2354$^{\star}$	2.10		14-04-99	V	-1.33	-0.38	0.05		-0.03	0.18	0.05
B1157+0128	1.99		13-04-99	V	0.16	0.93	0.07		0.21	-0.09	0.10
B1158+0045	1.38		13-04-99	V	-0.37	0.42	0.10		-0.16	0.10	0.04
B1203+1530$^{\star}$	1.63		13-04-99	V	0.75	1.54	0.10		0.18	-0.02	0.17
B1205+1436$^{\star}$	1.64		27-04-98	V	0.58	-0.51	0.07		-	-	-
B1210+1942	1.24		13-04-99	V	-0.29	0.34	0.08		-0.19	0.35	0.11
B1215+1244$^{\star}$	2.08		28-04-98	V	0.45	0.35	0.17		-	-	-
B1216+1103$^{\star}$	1.62		27-04-98	V	-0.41	0.48	0.09		-0.85	0.09	0.24
B1219+1244$^{\star}$	1.31		27-04-98	V	0.29	-0.57	0.10		-0.32	-0.24	0.11
B1222+1437	1.55		27-04-98	V	-0.22	0.19	0.06		-0.08	-0.04	0.05
J 1225-0150$^{\star}$	2.04		14-04-99	V	-0.43	-0.48	0.14		-0.12	0.27	0.05
B1228+1216$^{\star}$	1.41		27-04-98	V	-0.04	-0.11	0.06		0.21	0.48	0.22
B1230+1705$^{\star}$	1.42		27-04-98	V	-0.35	-0.10	0.09		-0.09	0.24	0.06
B1230-2347	1.84		28-04-98	V	-0.11	0.04	0.08		-	-	-
B1234-0209	1.62		27-04-98	V	-0.51	0.32	0.07		-0.30	0.33	0.10
B1235-1813	2.19		13-04-99	V	0.97	-0.14	0.05		-	-	-
B1235+1807$^{\star}$	0.45		14-04-99	V	0.05	0.29	0.17		0.17	0.42	0.03
B1238-0944	2.09		28-04-98	V	-0.24	0.07	0.06		-	-	-
B1239+0955$^{\star}$	2.01		27-04-98	V	0.58	-0.49	0.06		0.03	-0.12	0.03
B1239+1435	1.95		28-04-98	V	0.06	0.13	0.10		0.21	0.58	0.20
B1242+0006	2.08		28-04-98	V	-0.15	0.21	0.08		-0.10	0.05	0.07
J 1252+0053$^{\star}$	1.69		14-04-99	V	-0.02	-0.02	0.06		-	-	-
B1256-1734	2.06		27-04-98	V	-0.78	0.58	0.08		0.28	0.23	0.08
B1258-1627	1.71		28-04-98	V	-0.13	-0.52	0.07		-0.13	-0.05	0.04
B1305+0011	2.11		27-04-98	V	0.30	-0.58	0.14		0.05	0.33	0.06
B1333+2840$^{\star}$	1.91		13-04-99	V	4.66	-3.39	0.11		-0.15	0.31	0.24
B1334+2614$^{\star}$	1.88		13-04-99	V	-0.14	0.01	0.08		-	-	-
B1416-1256	0.13		28-04-98	V	-0.21	0.35	0.08		-0.49	0.54	0.07
B1429-0053	2.08		13-04-99	V	0.29	0.37	0.09		-0.36	0.16	0.06
B1429-0036$^{\star}$	1.18		14-04-99	V	-0.06	0.15	0.10		-	-	-
B1443+0141$^{\star}$	2.45		13-04-99	V	1.00	-0.68	0.15		-0.60	-0.13	0.10
B1451-3735	0.31		14-04-99	V	0.11	-0.05	0.05		-0.23	0.16	0.06
B1500+0824	3.94		14-04-99	V	-1.09	-0.19	0.28		-0.26	0.16	0.12
B2118-4303$^{\star}$	2.20		28-04-98	V	-0.11	-0.65	0.11		-	-	-
B2149-2745$^{\star}$	2.03		18-10-98	V	-0.13	0.07	0.10		0.00	0.25	0.15
B2226-3905	1.13		07-09-99	V	-0.13	0.16	0.09		-	-	-
B2240-3702$^{\star}$	1.83		11-09-96	V	1.16	1.75	0.08		-	-	-
B2240-3702$^{\star}$	1.83		11-09-96	i	1.33	0.73	0.10		-	-	-
B2329-3828	1.19		07-09-99	V	-0.12	-0.42	0.08		-	-	-
B2341-2333$^{\star}$	2.82		18-10-98	V	-0.28	-0.58	0.11		-0.03	0.39	0.13

 

 

Table 4: Final polarimetric data

Object		q	u		p	$\sigma_{p}$		p0	$p_{\scriptscriptstyle \rm ISM}$		$\theta$	$\sigma_{\theta}$
		(%)	(%)		(%)	(%)		(%)	(%)		( $\hbox{$^\circ$ }$)	( $\hbox{$^\circ$ }$)
B0059-2735$^{\star}$		1.38	-0.45		1.45	0.23		1.43	0.16		171	5
B0059-2735$^{\star}$		2.06	-1.14		2.35	0.24		2.34	0.16		166	3
B0846+1540$^{\star}$		0.43	-0.67		0.80	0.21		0.77	0.17		151	8
B0856+1714$^{\star}$		0.70	0.01		0.70	0.24		0.66	0.09		0	10
B1009+0222$^{\star}$		0.06	-0.77		0.77	0.19		0.75	0.06		137	7
J 1053-0058$^{\star}$		-1.90	-0.03		1.90	0.19		1.89	0.17		90	3
J 1104-0004$^{\star}$		0.49	-0.05		0.49	0.21		0.45	0.12		177	13
J 1141-0141$^{\star}$		-0.10	0.37		0.38	0.23		0.32	0.06		53	21
B1151+1145		-0.67	-0.25		0.72	0.18		0.70	0.01		100	7
B1157-2354$^{\star}$		-1.31	-0.25		1.33	0.17		1.32	0.44		95	4
B1157-2354$^{\star}$		-1.33	-0.57		1.45	0.18		1.44	0.44		102	4
B1157+0128		0.16	0.74		0.76	0.18		0.74	0.01		39	7
B1158+0045		-0.37	0.23		0.44	0.20		0.40	0.01		74	14
B1203+1530$^{\star}$		0.75	1.35		1.54	0.20		1.53	0.22		30	4
B1205+1436$^{\star}$		0.65	-0.52		0.83	0.18		0.81	0.03		161	6
B1210+1942		-0.29	0.15		0.33	0.19		0.28	0.10		76	19
B1215+1244$^{\star}$		0.52	0.34		0.62	0.24		0.58	0.06		17	12
B1216+1103$^{\star}$		-0.34	0.47		0.58	0.19		0.55	0.02		63	10
B1219+1244$^{\star}$		0.36	-0.58		0.68	0.20		0.65	0.02		151	9
B1222+1437		-0.15	0.18		0.23	0.18		0.17	0.19		65	30
J 1225-0150$^{\star}$		-0.43	-0.67		0.80	0.22		0.77	0.10		119	8
B1228+1216$^{\star}$		0.03	-0.12		0.12	0.18		0.00	0.17		142	-
B1230+1705$^{\star}$		-0.28	-0.11		0.30	0.19		0.25	0.05		101	22
B1230-2347		-0.04	0.03		0.05	0.19		0.00	0.57		72	-
B1234-0209		-0.44	0.31		0.54	0.18		0.51	0.08		72	10
B1235-1813		0.97	-0.33		1.02	0.18		1.00	0.09		171	5
B1235+1807$^{\star}$		0.05	0.10		0.11	0.24		0.00	0.07		32	-
B1238-0944		-0.17	0.06		0.18	0.18		0.00	0.18		80	-
B1239+0955$^{\star}$		0.65	-0.50		0.82	0.18		0.80	0.00		161	6
B1239+1435		0.13	0.12		0.18	0.20		0.00	0.05		21	-
B1242+0006		-0.08	0.20		0.22	0.19		0.14	0.00		56	39
J 1252+0053$^{\star}$		-0.02	-0.21		0.21	0.18		0.14	0.00		132	37
B1256-1734		-0.71	0.57		0.91	0.19		0.89	0.27		71	6
B1258-1627		-0.06	-0.53		0.53	0.18		0.50	0.12		132	10
B1305+0011		0.37	-0.59		0.70	0.22		0.67	0.02		151	9
B1333+2840$^{\star}$		4.66	-3.58		5.88	0.20		5.88	0.03		161	1
B1334+2614$^{\star}$		-0.14	-0.18		0.23	0.19		0.16	0.03		116	34
B1416-1256		-0.14	0.34		0.37	0.19		0.33	0.56		56	16
B1429-0053		0.29	0.18		0.34	0.19		0.30	0.17		16	18
B1429-0036$^{\star}$		-0.06	-0.04		0.07	0.20		0.00	0.16		107	-
B1443+0141$^{\star}$		1.00	-0.87		1.33	0.23		1.31	0.27		159	5
B1451-3735		0.11	-0.24		0.26	0.18		0.21	0.56		147	25
B1500+0824		-1.09	-0.38		1.15	0.33		1.10	0.08		100	9
B2118-4303$^{\star}$		-0.04	-0.66		0.66	0.20		0.63	0.16		133	9
B2149-2745$^{\star}$		-0.13	0.07		0.15	0.20		0.00	0.14		76	-
B2226-3905		-0.13	0.16		0.21	0.19		0.12	0.00		65	45
B2240-3702$^{\star}$		1.16	1.75		2.10	0.19		2.09	0.00		28	3
B2240-3702$^{\star}$		1.33	0.73		1.52	0.20		1.51	0.00		14	4
B2329-3828		-0.12	-0.42		0.44	0.19		0.40	0.00		127	14
B2341-2333$^{\star}$		-0.28	-0.58		0.64	0.20		0.61	0.02		122	9

First applied to the calibration stars, the method provides polarization degrees in excellent agreement with the published values. During some nights more than one star has been observed (Table 1) in order to check the stability and the reliability of the zero-point offset of the polarization position angle. For all stars observed during a given night, the values of the angle offset do agree within 1 $\hbox{$^\circ$ }$, which is quite small given the fact that the EFOSC2 HWP is not achromatic.

The normalized Stokes parameters q and u were then computed for the QSO sample, and modified according to the zero-point offset determined for each night independently. The uncertainties $\sigma_q$and $\sigma_u$ are evaluated by computing the errors on the intensities $I^{\scriptscriptstyle \rm u}$ and $I^{\scriptscriptstyle \rm l}$, from the read-out noise and from the photon noise in the object and the sky background (after converting the counts in electrons), and then by propagating these errors in Eqs. (1) or (2). Uncertainties are typically around 0.1% for both q and u.

Since on most CCD frames field stars are simultaneously recorded, one can in principle use them to estimate the residual instrumental polarization, and to correct frame-by-frame the QSO Stokes parameters. However, the field stars (even when combined in a single "big'' one per frame) are often fainter than the QSO, and a frame-by-frame correction introduces uncertainties on the QSO polarization larger than the instrumental polarization itself. We then computed the weighted average ( $\overline{q}_{\star}$ and $\overline{u}_{\star}$) and dispersion ( $\overline{\sigma}_{\star}$) of the normalized Stokes parameters of field stars considering the ($n_{\star}$) frames obtained during a given run. These values are given in Table 2 for the two runs with enough data. Note that possible contamination by interstellar polarization is included in the uncertainties. These values indicate that the residual instrumental polarization is small, as expected since most of the instrumental polarization is removed by the observing procedure. We nevertheless take it into account in a rather conservative way by subtracting the systematic $\overline{q}_{\star}$ and $\overline{u}_{\star}$ from the QSO q and u, and by adding quadratically the errors. For those objects observed in other runs, no systematic correction was applied; only the errors were similarly combined assuming, quite reasonably, that they are typical of the instrument.