Solvepol: A Reduction Pipeline for Imaging Polarimetry Data

The data reduction procedure that the pipeline applies is based on the standard procedure explained by Massey (1997). Roughly the procedure is as follows:

• 1.  
  First, the bias frames are median combined. Then, using the idl sigma_filter procedure, sigma clipping is applied to remove pixels that deviate by more than 2.5 sigma from the median value of the neighboring 300 by 300 pixels. Finally, the overscan region is fit with a polynomial of order 2 and subtracted to create the master-bias frame.
• 2.  
  For flat field frames, the overscan region is fit as previously described and is subtracted to each flat field frame. Next, the flat fields are median combined. Then, sigma clipping is applied as before, and the master bias is subtracted to create the master flat frame.
• 3.  
  Each source image is reduced by first fitting a polynomial of second order to the overscan region, and subtracting this fit from the image. The master-bias is then subtracted and the result is then divided by the master-flat frame normalized to the median of the sky. Finally, the source images are trimmed to remove the overscan region.

If multiple images were observed per position angle of the waveplate, they are median combined after performing the reduction steps outlined above. The pipeline estimates and corrects for any possible shifts in X and Y between images before combining.

The pipeline finds all point sources, i.e., stars, and performs photometric measurements which are used to calculate their polarization. Solvepol is not going to find extended sources, and they will be treated as background. Aperture photometry is performed on both ordinary and extraordinary images of each star (F_o and F_e, respectively) with 10 concentric apertures ranging in size from one to 10 pixels in radius. For each aperture an annulus, with fixed inner and outer radii of 10 and 20 pixels respectively, is used to estimate the sky (the default annulus in Solvepol).

These apertures cover up to $\sim 5$×the full width at half maximum (FWHM) that the stelar disk may have when observing with the IAGPOL at the OPD. Figure 2 shows the seeing measured in standard stars observed with the 60 cm telescope. The seeing that we get ranges from 2 to 8 pixels (13 to 52), but we know that the seeing of stars distorted due to cirrus is 7 to 8 pixels (45 to 52). Under good observational conditions, we measured a seeing of ∼3.5 pixels (23). Thus, the 10 concentric apertures will cover all seeing conditions.

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The photometry is measured using the same aperture for all waveplate positions, and the aperture size that gives the lowest error in the polarimetry is selected to calculate the final polarimetric properties of a star.

The modulation of the intensity, z_i, where i is one of 16 orientations of the half-wave plate (i.e., i = 1 to $4,6,8$, or 16 positions), is

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{e,i}-{F}_{o,i}({F}_{e}^{T}/{F}_{o}^{T})}{{F}_{e,i}+{F}_{o,i}({F}_{e}^{T}/{F}_{o}^{T})}=Q\,\cos (4\,{\psi }_{i})+U\,\sin (4\,{\psi }_{i}),\end{eqnarray} \tag{ 1 }$$

where ${F}_{o,i}$ and ${F}_{e,i}$ are the fluxes of the ordinary and extraordinary images at the position i of the waveplate, and F^T_o and F^T_e are the ordinary and extraordinary total flux of a star summed on all the waveplate's positions, respectively (see Magalhães et al. 1984). Q and U refer to the instrumental Stokes parameters of the incoming beam's linear polarization. Equation (1) shows that a set of four waveplate positions (e.g., i = 1 through 4) measures Q, U, $-Q,$ and $-U$. A linear polarization measurement can then be obtained by doing exposures through 4, 8, 12, or 16 positions of the waveplate, although this cycle can be repeated as needed. The polarimetric properties are estimated following the formulation of Magalhães et al. (1984), which consists in solving for U and Q the expression applying the method of the least squares.

The solution for the stokes parameters Q and U for each source are

$$\begin{eqnarray}&&Q=\displaystyle \frac{2}{\mu }\displaystyle \sum _{i}^{\mu }{z}_{i}\cos (4{\psi }_{i})\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&U=\displaystyle \frac{2}{\mu }\displaystyle \sum _{i}^{\mu }{z}_{i}\sin (4{\psi }_{i}),\end{eqnarray} \tag{ 3 }$$

where μ is the number of positions of the half-wave plate and ${\psi }_{i}=[i-1]\ast 22\buildrel{\circ}\over{.} 5$ ($=0^\circ ,22\buildrel{\circ}\over{.} 5,45^\circ ,67\buildrel{\circ}\over{.} 5$, etc.). This yields the percent linear polarization, P, and the polarization angle, θ, measured from north to east after the zero-angle is calibrated using polarized standard stars:

$$\begin{eqnarray}&&P=\sqrt{{Q}^{2}+{U}^{2}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\theta =\displaystyle \frac{1}{2}{\tan }^{-1}\displaystyle \frac{U}{Q}.\end{eqnarray} \tag{ 5 }$$

The uncertainties of the polarimetric properties are in essence the residual of the actual measurements at each waveplate position angle with respect to the theoretical nonlinear curve of the modulation. Assuming that ${\sigma }_{Q}={\sigma }_{U}={\sigma }_{P}$, the error may be calculated as (see Magalhães et al. 1984; Naghizadeh-Khouei & Clarke 1993)

$$\begin{eqnarray}&&{\sigma }_{P}=\displaystyle \frac{1}{\sqrt{\mu -2}}\sqrt{\displaystyle \frac{2}{\mu }\displaystyle \sum _{i}^{\mu }{z}_{i}^{2}-{Q}^{2}-{U}^{2}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\sigma }_{\theta }=28\buildrel{\circ}\over{.} 65\displaystyle \frac{{\sigma }_{P}}{P}.\end{eqnarray} \tag{ 7 }$$

The reduced Chi square (${\chi }^{2}$) of the fitted modulation is calculated, and for this work, those stars with ${\chi }^{2}\gt 6.0$ are rejected. This limit was set by empirical grounds. Figure 3 shows the ${\chi }^{2}$ of the stars of the Musca field analyzed in this work (in Section 4). As can be seen, a ${\chi }^{2}=6.0$ separates reasonably good those stars belonging to the Musca cloud and those showing unexpected high polarization. Therefore, a ${\chi }^{2}\leqslant 6.0$ would include a reasonable sample with real polarization. This test was applied in other fields with same result. However, a different cutoff for ${\chi }^{2}$ can be selected by the user of Solvepol either interactively or a priori before running the code.

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Solvepol doesn't correct for Ricean bias affecting values with $P/{\sigma }_{P}\leqslant 3.0$ (Clarke & Stewart 1986). The users of Solvepol must be aware of this and correct their results. The results presented in this work were obtained with $P/{\sigma }_{P}\geqslant 5.0$ and 10.0, when indicated; therefore, they are not affected by Ricean bias—only values with $P/{\sigma }_{P}\leqslant 3.0$ are affected.

Solvepol uses the Astrometry.net software (Lang et al. 2010) to calibrate the astrometry of the image being analyzed to get the accurate celestial coordinates of the stars. The pipeline masks out one of the components of the doublet (always the most right component) of each star emerging from the calcite Savart prism on the first waveplate position's image, to get a "normal" image of the sky. Figure 1 shows the original image from the first waveplate position (left), and after applying the mask (right). For this case the right-up component of each star have been masked. The masked regions are replaced with the mode sky value of the image. The size of the mask is equal to the aperture that gives the lowest uncertainty in P. This "normal" image is processed by the Astrometry.net software.⁸

The Astrometry.net software finds, in the masked image, the relative positions of nearby groupings of three or four stars on the field, and finds their corresponding triples or quadruplets in the 2MASS and Tycho-2 catalogs. Each grouping generates a candidate matching sky, and Astrometry.net finds the true matching view. After an alignment of the image and the catalog, Astrometry.net calculates the astrometric solution that the pipeline includes in the masked image header. This image/header information is then used to translate from pixels to R.A. and decl. of the detected sources.

The mean accuracy in the position that we calculated by comparing the position of standard stars (see Section 4.1) with that of the literature is ±044. This accuracy allows to search the stars in the Guide Star Catalog, version 2.3 (GSC v2.3. Lasker et al. 2008) to perform the magnitude calibration.

New photometric measurements are applied to both the ordinary and extraordinary images within an aperture with diameter two times the mean FWHM and the default annulus in Solvepol to estimate the sky background of 10 pixel radius and 10 pixels width. The mean FWHM is estimated of at most 30 of the brightest stars in the field, in the first waveplate position's image. If the FWHM is greater than 10 pixels, an aperture of 10 pixels in radius is used for the photometry. We define the measured flux, F_m, as the sum of the ordinary and extraordinary fluxes. For our observations taken in the V-band, the pipeline obtains the visual magnitude

$$\begin{eqnarray}&&V={m}_{z}-2.5\mathrm{log}({F}_{m}),\end{eqnarray} \tag{ 8 }$$

where m_z is the zero point magnitude calibrated with the GSC v2.3 (Lasker et al. 2008). This catalog goes down to 18th magnitude.⁹ We define m_z as the mean of the absolute difference (to avoid that the sum is zero) between our measured magnitude, ${m}_{{\rm{m}},i}$, and the optical magnitude reported in the GSC v2.3, ${m}_{{\rm{c}},i}$, for at most the 25 brightest stars in the field, i.e., ${m}_{z}=\tfrac{1}{k}{\sum }_{i=1}^{k}| {m}_{{\rm{m}},i}-{m}_{{\rm{c}},i}|$, where k is at most 25. If there are no stars with measured magnitude in the catalog GSC v2.3, m_z is set to 0 (no calibration).

The analysis of polarized standard stars is essential to check the veracity of our results. For the 10 standard stars shown in Table 1, we preset polarization and magnitude results using Solvepol and compare these with reported values from the literature. The standard stars were observed during different observational campaigns, most of them in the 1997 Musca's campaign (PM04).

Table 1.  Standard Stars Analyzed

Star	R.A. J2000	decl. J2000	V	P	Obs.	References
			(mag)	(%)	Date
HD80558	$09\ 18\ 42.44$	$-51\ 33\ 37.69$	5.29 ± 0.20	3.25 ± 0.06	1997-04-09	...
	$09\ 18\ 42.36$	$-51\ 33\ 38.34$	5.94 ± 0.01	3.11 ± 0.01	...	a
HD84810	—	—	—	1.67 ± 0.07	1997-04-09	...
	$09\ 45\ 14.81$	$-62\ 30\ 28.44$	3.69 ± 0.03	1.58 ± 0.01	...	a
HD100623	—	—	—	0.04 ± 0.06	1997-04-10	...
	$11\ 34\ 29.49$	$-32\ 49\ 52.81$	6.05 ± 0.01	0.02	...	b
HD110984	$12\ 46\ 44.84$	$-61\ 11\ 11.91$	9.19 ± 0.18	5.64 ± 0.05	2001-03-02	...
	$12\ 46\ 44.84$	$-61\ 11\ 11.47$	8.49 ± 0.16	5.79 ± 0.04	1997-04-12	...
	$12\ 46\ 44.82$	$-61\ 11\ 11.47$	8.55 ± 0.16	5.86 ± 0.03	1997-04-13	...
	$12\ 46\ 44.84$	$-61\ 11\ 11.58$	9.04 ± 0.02	5.70 ± 0.01	...	c
HD111613	$12\ 51\ 18.05$	$-60\ 19\ 47.79$	5.47 ± 0.17	3.20 ± 0.04	1997-04-09	...
	$12\ 51\ 17.98$	$-60\ 19\ 47.24$	5.77 ± 0.01	3.06 ± 0.01	...	a
HD126593	$14\ 28\ 50.87$	$-60\ 32\ 24.94$	8.60 ± 0.29	4.94 ± 0.06	2001-03-02	...
	$14\ 28\ 50.87$	$-60\ 32\ 25.12$	8.70 ± 0.01	5.02 ± 0.01	...	c
HD147084	—	—	—	3.79 ± 0.35	1997-04-10	...
	$16\ 20\ 38.18$	$-24\ 10\ 09.55$	4.66 ± 0.01	4.17 ± 0.01	...	c
HD154445	—	—	—	3.79 ± 0.05	1997-04-10	...
	$17\ 05\ 32.26$	$-00\ 53\ 31.44$	5.65 ± 0.01	3.80 ± 0.08	...	c
HD155197	$17\ 10\ 15.74$	$-04\ 50\ 03.62$	9.60 ± 0.08	4.54 ± 0.09	1997-04-12	...
	$17\ 10\ 15.75$	$-04\ 50\ 03.67$	9.57 ± 0.03	4.63 ± 0.02	...	c
HD298383	$09\ 22\ 29.77$	$-52\ 28\ 57.65$	10.78 ± 0.72	5.17 ± 0.06	2001-03-01	...
	$09\ 22\ 29.77$	$-52\ 28\ 57.36$	10.74 ± 0.37	5.43 ± 0.04	1997-04-13	...
	$09\ 22\ 29.76$	$-52\ 28\ 57.25$	10.78 ± 0.34	5.38 ± 0.06	1997-04-12	...
	$09\ 22\ 29.77$	$-52\ 28\ 57.25$	9.75 ± 0.03	5.23 ± 0.01	...	c

Note.  Our measurements and those from the literature (last line for each star) are shown. R.A., decl. and V are from the GSC v2.3 catalog (Lasker et al. 2008); P is from: (a) Bastien et al. (1988), (b) Serkowski et al. (1975) (the error is not provided in this reference) and (c) Turnshek et al. (1990). "—" indicates that the position could not be obtained by the pipeline because of the lack of reference stars on the filed (see Section 4.1 for details), and then the magnitude could not be calibrated by Solvepol.

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Although the standard stars are used to get the zero-angle calibration of the observations, the disposition of the telescope relative to the sky or to different positions of the waveplate inside the polarimeter is not the same between observing campaigns. Hereby, the angle θ is meaningless to compare between stars observed in different campaigns, and it is not tabulated in Table 1. The position of HD100623, HD147084, and HD154445 could not be solved because less than three stars fall in their field—at least three stars are needed to solve the astrometry—therefore, its R.A. and decl. are not tabulated for its comparison with the literature.

Figure 4, shows plots comparing P and V of the standard stars calculated with Solvepol and from the literature. These results are consistent to well within $2\sigma$. The P that deviate by more corresponds to HD147084: our P is lower than the reported in Turnshek et al. (1990) by a factor of 0.9. Because we do not see in Figure 4 a systematic displacement from the line representing the equality, the scatter can be attributed to climatological differences along the observation (clouds, cirrus, etc.).

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We analyzed the field around HD110984. Figure 5 shows the polarization maps of the field centered on HD110984 solved by Solvepol and by PCCDPACK. The polarization (length of the lines) and position angle (measured from north to east) are indicated. Both maps will lead to the same conclusion about the distribution of the polarization on the field of HD110984. However, there are some discrepancies. Around R.A. 191.85 and decl. −61.22, PCCDPACK solves three stars, while Solvepol solves one star only, which is the brightest star of the group of three stars. Moreover, in the center of the image, corresponding to the standard star HD110984, Solvepol solves precisely for HD110984, while PCCDPACK solves three sources where clearly there is no stars but the spikes of HD110984. This examples indicate that Solvepol is producing reliable results.

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Figure 6 shows the distributions of P, θ and V for the HD110984 field measured with Solvepol and the PCCDPACK routines for comparison. We fixed $F/{\sigma }_{s}\geqslant 5.0$ and $P/{\sigma }_{P}\geqslant 5.0$ for this analysis. The distributions of P, θ and V produced by both routines are quite similar. In particular, the distributions of P and θ indicate that both routines lead to the same conclusion about the polarimetric properties on the field of HD110984. Therefore, both routines give polarimetric properties that will lead to the same conclusions when analyzing the same field.

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In Figure 7, we compare P, uncertainty in the polarization (${\sigma }_{P}$), θ and V using Solvepol versus those using the PCCDPACK routines. For the same stars found by Solvepol and the PCCDPACK routines, it can be seen that both routines produce the same results with each other to within $3\sigma$. Clearly the V-error bar given by PCCDPACK is underestimated because it is unexpectedly lower than the mean error reported in the catalog GSC v2.3 for this field, which is ±0.43. Solvepol, on the other hand, takes the mean error of the magnitude reported in the GSC v2.3 catalog as a lower limit of its V-error, i.e., if Solvepol measures an V-error lower than the mean V-error of the magnitudes reported in the GSC v2.3, Solvepol takes the mean V-error of the magnitude of the catalog as the error. About the mean error in θ in Figure 7, the mean error in theta is ±473 and ±306 for Solvepol and PCCDPACK, respectively. A difference as low as 167 indicates consistency in θ.

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In Figure 8 we compare the number of sources detected by Solvepol and PCCDPACK. The number of sources detected by Solvepol tends to be higher than the number of sources detected by the PCCDPACK routines for low values of $P/{\sigma }_{P}$ (with $F/{\sigma }_{s}\geqslant 5.0$). However, the number of sources detected by both routines converge towards greater values of $P/{\sigma }_{P}$. For instance, for $P/{\sigma }_{P}=3.0$, the number of stars detected by Solvepol is 154, and the number of sources detect by the PCCDPACK routines is 134—a difference of 20 detected stars—, while for $P/{\sigma }_{P}=5.0$ (usual ratio for a confident measurement) the difference is just 6 stars (101 and 95 detection by Solvepol and the PCCDPACK routines, respectively). Therefore, our new pipeline seems to be producing more complete catalogs.

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A comparison of our results with those reported in PM04 of the well studied field of the Musca Dark Cloud is presented in this section. We analyze all the fields observed in the 1997 campaign. The observational details are described in PM04. The source detection limits used in Solvepol were set to $F/{\sigma }_{F}\geqslant 5.0$ and $P/{\sigma }_{P}\geqslant 5.0$, same limits adopted by PM04, but using the PCCDPACK routines.

In Figure 9 the P, ${\sigma }_{P}$, θ, and V calculated by Solvepol, and those from the catalog of PM04 are presented. As can be seen, P, ${\sigma }_{P}$, θ and V are all consistent with each other, with some degree of scattering within $3\sigma$. The catalogs produced by Solvepol were processed with merge, therefore, there are no repeated stars from the overlapping field regions and the weighted mean of P, θ and V of those overlapped stars were calculated (see Section 3). On the other hand, the catalog (reported in VizieR) of PM04 includes two entries for stars in overlapping regions, increasing the total number of results. The uncertainty in V is not reported by PM04, and thus we cannot compare our uncertainties against theirs.

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While the error bars in P calculated by Solvepol and PCCDPACK indicated in Figure 9 are consistent, the error bar of θ calculated by Solvepol is slightly bigger than the error bar in θ calculated by PCCDPACK. The mean error in θ is ±4.23 and ±2.66 for Solvepol and PCCDPACK, respectively. This gives a difference of only 157, which indicates that Solvepol and PCCDPACK are in close agreement.

The distributions of P, θ, and V for Musca are shown in Figure 10. The sample reduced with Solvepol seems to be complete in magnitude up to 17 mag. This is 1.5 magnitudes deeper than the 15.5 mag reported in PM04 using PCCDPACK. However, a systematic shift of the estimated magnitudes is clearly seen. To know which magnitudes are offset, we compare both magnitudes with the magnitudes of the GSC v2.3, show in Figure 11. As it can be seen, the magnitudes estimated by Solvepol are closer to the equality line with the magnitude of the GSC v2.3 than those estimated by PCCDPACK: the Solvepol magnitudes fall very close to the equality line, while the magnitudes from PCCDPACK is offset bellow the equality line. Therefore, the magnitude calibration performed by Solvepol is more accurate with the GSC v2.3.

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Table 2 shows the field numbers (we have adopted the same notation used in Pereyra & Magalhães 2004), its central position calculated by our pipeline, the number of detected stars, and the weighted mean (as derived in PM04) of P, θ and V with their respective uncertainties calculated by Solvepol, and those from PM04 for comparison. We adopted the same limit of $P/{\sigma }_{P}\geqslant 10.0$ adopted by PM04, their table 5, for a direct comparison.

Table 2.  Fields of Musca Reduced by Solvepol Using the Values $P/{\sigma }_{P}\geqslant 10.0$ and $F/\sigma \geqslant 5.0$

Field	R.A. J2000	decl. J2000	N	mean P	mean θ	mean V
	Central Region			(%)	(${}^{\circ }$)	(mag)
03	$12\ 35\ 05.73$	$-71\ 00\ 30.80$	60	3.16 ± 0.02	101.455 ± 1.84	15.46 ± 1.22
05	$12\ 30\ 06.91$	$-71\ 00\ 35.20$	48	3.82 ± 0.02	105.507 ± 1.58	15.50 ± 1.06
08	$12\ 27\ 36.99$	$-71\ 08\ 40.13$	46	3.45 ± 0.03	107.602 ± 2.09	14.94 ± 0.92
11	$12\ 30\ 10.15$	$-71\ 24\ 38.34$	35	3.83 ± 0.03	104.835 ± 1.63	15.41 ± 1.24
13	$12\ 25\ 13.27$	$-71\ 24\ 40.64$	64	5.09 ± 0.02	111.129 ± 1.62	15.47 ± 1.08
14	$12\ 27\ 43.58$	$-71\ 32\ 35.64$	33	2.56 ± 0.03	105.917 ± 1.74	15.63 ± 1.15
16	$12\ 27\ 50.15$	$-71\ 41\ 00.17$	61	4.68 ± 0.03	100.026 ± 1.63	15.73 ± 1.00
20	$12\ 25\ 12.13$	$-72\ 00\ 56.48$	67	4.39 ± 0.02	107.791 ± 1.71	15.29 ± 1.11
22	$12\ 25\ 18.97$	$-72\ 09\ 19.19$	63	4.05 ± 0.02	109.961 ± 1.73	14.92 ± 1.16
24	$12\ 19\ 58.77$	$-72\ 09\ 33.49$	87	4.73 ± 0.02	105.628 ± 1.63	14.91 ± 1.18
26	$12\ 20\ 06.32$	$-72\ 17\ 26.06$	75	5.08 ± 0.02	104.501 ± 1.60	14.86 ± 1.08
27	$12\ 22\ 33.51$	$-72\ 25\ 16.68$	40	4.85 ± 0.03	116.587 ± 1.62	14.87 ± 0.98
29	$12\ 19\ 57.42$	$-72\ 33\ 32.76$	53	4.44 ± 0.02	102.888 ± 1.64	15.10 ± 1.39
30	$12\ 18\ 02.47$	$-72\ 33\ 39.58$	71	3.24 ± 0.02	109.563 ± 1.67	14.41 ± 1.12
34	$12\ 37\ 56.59$	$-70\ 41\ 10.27$	54	2.74 ± 0.02	106.526 ± 1.69	14.42 ± 1.20
03	$12\ 35\ 27.35$	$-71\ 00\ 31.29$	55	3.15 ± 0.02	102.31 ± 1.79	15.45 ± 1.00
05	$12\ 30\ 28.02$	$-71\ 00\ 34.25$	54	3.80 ± 0.02	106.17 ± 1.62	15.47 ± 1.01
08	$12\ 27\ 58.53$	$-71\ 08\ 35.56$	48	3.52 ± 0.03	107.38 ± 1.97	15.50 ± 1.06
11	$12\ 30\ 28.59$	$-71\ 24\ 34.25$	34	3.88 ± 0.03	105.95 ± 1.55	15.34 ± 1.11
13	$12\ 25\ 29.15$	$-71\ 24\ 36.75$	65	4.93 ± 0.02	111.53 ± 1.53	15.43 ± 0.97
14	$12\ 27\ 59.05$	$-71\ 32\ 35.56$	34	3.15 ± 0.03	106.96 ± 1.67	15.37 ± 1.10
16	$12\ 27\ 59.23$	$-71\ 40\ 35.56$	62	4.67 ± 0.03	102.71 ± 1.70	15.47 ± 0.97
20	$12\ 05\ 29.89$	$-72\ 00\ 36.75$	64	4.56 ± 0.03	105.13 ± 1.71	15.42 ± 0.98
22	$12\ 25\ 30.06$	$-72\ 08\ 36.75$	65	3.92 ± 0.02	107.74 ± 1.76	15.43 ± 0.97
24	$12\ 20\ 15.82$	$-72\ 08\ 38.87$	78	4.70 ± 0.02	102.45 ± 1.57	15.48 ± 1.00
26	$12\ 20\ 15.95$	$-72\ 16\ 38.87$	68	5.03 ± 0.02	101.81 ± 1.60	15.44 ± 1.00
27	$12\ 22\ 53.25$	$-72\ 24\ 37.87$	49	5.17 ± 0.03	113.36 ± 1.65	15.48 ± 1.05
29	$12\ 20\ 16.23$	$-72\ 32\ 38.87$	52	4.51 ± 0.02	99.72 ± 1.70	15.50 ± 1.03
30	$12\ 17\ 39.05$	$-72\ 32\ 39.74$	69	3.25 ± 0.02	107.20 ± 1.64	15.45 ± 1.00
34	$12\ 38\ 03.74$	$-70\ 40\ 29.55$	49	2.67 ± 0.02	104.24 ± 1.74	15.48 ± 1.05

Note.  R.A. and decl. are the central position of the field. N is he number of detected sources in each field. Weighted mean of P, θ and V and its standard deviation following the same procedure detailed in PM04. Below, the same fields of musca reported by PM04 and using the same $P/{\sigma }_{P}$ and $F/\sigma$ limits.

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The total number of stars with $P/{\sigma }_{P}\geqslant 10.0$ is 857, slightly higher than the 846 sources reported by PM04. About individual fields, the greatest difference in the number of detected stars is only 9 (Solvepol found more) which correspond to field number 24 (see Table 2). However, as it can be seen in Figure 12 (top left plot), that compares the number of sources versus $P/{\sigma }_{P}$ ($F/{\sigma }_{F}$ fix to 5.0) detected by Solvepol and the PCCDPACK routines, the number of sources detected by both routines tend to converge for high values of $P/{\sigma }_{P}$.

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Overall, mean values of P, θ, and V, tabulated in Table 2, and plotted in Figure 12 for an easier inspection, are consistent with the mean values of P, θ, and V reported by PM04 their Table 5, reproduced here in Table 2.

Figure 13 shows the polarization maps on the Musca region of the fields analyzed in this work and tabulated in Table 2 calculated by Solvepol. Each vector represents one object. Again, our polarization maps and those reported by PM04 are in agreement.

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It is important to highlight that an important difference between Solvepol and PCCDPACK is that Solvepol is completely automatic, a pipeline without human intervention. The results of PCCDPACK, on the other hand, have been cleaned of stars with unexpected polarimetric values by the user. The results of Solvepol compared in this work have not been post-processed in such a way. Therefore, the raw output of Solvepol is in reasonable agreement with the post-cleaning output of PCCDPACK.