SEARCH FOR A MAGNETIC FIELD VIA CIRCULAR POLARIZATION IN THE WOLF–RAYET STAR EZ CMa^*

The W-R star EZ CMa (WR6 = HD 50896) is one of the brightest and best studied stars of its kind. It has now been observed at almost all wavelengths available with ground-based and space observatories in an attempt to understand its puzzling nature and ultimately the extreme stellar wind properties of W-R stars and the consequences on their circumstellar environment. EZ CMa is a WN4 star of magnitude m_V = 6.94 (van der Hucht 2001), the sixth brightest W-R star in the Galaxy and in the night sky. EZ CMa exhibits unusual properties that still need to be explained and confirmed observationally. It shows periodic optical and UV line and continuum variability with a period of P = 3.766 days (Lamontagne et al. 1986). The multiple signs of variability also point toward long-term dynamic changes in the large electron-density distribution and geometry. These wind variations are seen in photometric, polarimetric, and spectroscopic observations over the same timescale, suggesting that a common mechanism is responsible for the slow and continuous evolution of the observed changes (Robert et al. 1992). Matthews et al. (1992) and Duijsens et al. (1996) also reported flare-type variability in EZ CMa. The former authors suggested that the intriguing brightness increase is consistent with a field strength of B_core ∼ 10³ G, if due to the reconnection of magnetic field lines at the stellar surface. Robert et al. (1992) argued that the lack of long-term coherency in the shape of the phase variations could be accounted for by a slow random growth and fading of disturbances or spots in the visible parts of the rotating star. Similar modulations are observed in T Tauri and RS CVn stars. Constrained but variable magnetic loops are implied, and a field strength of B_wind > 90 G in the accelerating part of the wind is required. Furthermore, Duijsens et al. (1996) proposed a 400 day timescale for the maximum light amplitude variations that would be caused by a potential magnetic cycle. To explain the variable properties of EZ CMa, a likely scenario is the modulation of a rotating single star with CIRs, as seen in the Sun (Mullan 1986). Additionally, a corotating inhomogeneous density distribution in the stellar wind causing variable transparency could create a variable mass loss. CIRs could emanate from hot, magnetically active regions near the surface (due to a global magnetic field) and would fade away when the star is in the low regime of its magnetic cycle (St-Louis et al. 1995; Duijsens et al. 1996).

In spite of the multiple studies conducted on EZ CMa, a number of questions remain unresolved. Is the variability related to a dynamic, structured, and rotating envelope? If so, what is its shape? Can magnetic fields account for some of the observed characteristics? If so, does EZ CMa harbor a global, large-scale field or does it only have small-scale localized magnetic loops at its surface? Can modern spectropolarimetry be exploited to directly detect the magnetic activity of the star? In an attempt to address these questions, we have collected circular and linear spectropolarimetry of EZ CMa throughout three epochs of observations.

In this section, we present the different methods used to increase the detectability of a potential magnetic field. A typical observing run consisted of the following sequence of Stokes parameter exposures: Q, U, V₁,..., V_n, U, Q, where n corresponds to the total number of Stokes V exposures recorded during the night. After passing through the reduction pipeline, five quantities are computed at each wavelength; e.g., for a Stokes V exposure, we get I_V, V, N_{1, V}, N_{2, V}, and σ_V. The latter represents the error on all quantities since these are generated using similar manipulations of light beams (Donati et al. 1997).

Stokes Q and U spectra were taken at the beginning and at the end of a Stokes V observing sequence to check if significant variations had occurred in linear polarization while the V exposures were taken. In the event that linear line polarization has changed significantly, Q and U spectra could be, for example, linearly interpolated to correct the individual Stokes V exposures. In actual fact, no significant variations were observed in linear polarization during any sequence, although Q and U spectra do vary over longer timescales.

To reach a sufficiently high S/N, we needed about two hours of exposure time (estimated by the ESPaDOnS exposure-time calculator) on EZ CMa each night. However, we had to split that time into several shorter observations, as the detector would otherwise saturate. Therefore, multi-exposures of circular polarization were taken during each observing sequence. Furthermore, strong emission lines forced us to split the exposures even more than if based on the continuum intensity, depending on the peak line intensity. Once processed by the Upena pipeline, the spectra were then combined to generate one high-S/N Stokes V profile per night. The same reduction was applied to the Stokes Q and U and check spectra (N₁ and N₂). Processing one order at a time, a weighted mean was calculated at each wavelength with the rms errors added accordingly (see Bevington & Robinson 2003). In the end, three polarization profiles were generated each night. Stokes Q and U measurements were later used to correct Stokes V observations for crosstalk effects (see below).

Figure 1 shows the four Stokes parameter spectra collected during the first night of observation of the 2010 run. The top panel presents the relative intensity spectrum in arbitrary units. When being processed by Upena, the actual number of counts recorded at each wavelength is lost during the optimal extraction of data.⁴ The bottom three panels show the combined (flux) Stokes parameters V, Q, and U. Another feature of ESPaDOnS échelle data is the order-to-order overlaps. In Figure 1, these typically correspond to regions where the noise level in Stokes V, Q, and U significantly increases. Figure 2 offers a better view of this characteristic property of échelle spectra as it shows the same spectrum of EZ CMa but now focuses only on three orders. The dashed vertical lines indicate where the orders overlap and especially at the extremities, where the noise level is higher. This is a property of spectrographs using échelle gratings in which a spectrum will be underexposed at the ends of the orders since the échelle efficiency varies across each order (Schroeder 1970). To reduce the impact of noisy order extremities, we decided (following the work of Semel & Li 1996) to eliminate the overlap region of each order while at the same time joining together the 40 orders into one continuous spectrum. For each Stokes parameter, the junctions between orders were determined based on their uncertainty profiles. Figure 2 shows an example of this operation done on Stokes V. The top and bottom panels show the Stokes I_V profile and its associated error σ_V as a function of wavelength, with order overlaps, clearly visible (bottom profiles in black and blue). This shows that the error (noise) increases at the beginning and end of each order. Consequently, the order junctions were set at wavelengths where the uncertainty profiles from two successive orders cross each other. The result is a continuous spectrum with no overlaps. The dashed vertical lines in the figure illustrate where the order junctions were cut and joined together. The top curves in these panels show in red the final spectrum and associated error once the orders were trimmed and joined together. The middle panels of Figure 2 show the three Stokes parameters V, Q, and U.

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While it is known and reported that crosstalk affects measurements made with ESPaDOnS, the two null profiles are not expected to be influenced by such an effect. We discuss here a curious characteristic observed in N_{V, 1} and N_{V, 2}. As mentioned previously, two null spectra for each Stokes parameter are generated during an observing sequence. Theoretically, these should show a flat profile centered at the zero polarization level. However, as shown in Figure 5, N_{V, 1} and N_{V, 2} profiles exhibit structure inside the spectral line. Moreover, those patterns are sometimes of amplitude comparable to or larger than the Stokes V signatures. Furthermore and perhaps more obvious is the fact that N_{V, 1} and N_{V, 2} do not seem to be centered at zero as they are supposed to be but appear shifted in polarization level. This suggests that, at the very least, a constant amount of spurious polarization is added to the null spectra when these are generated. Since signatures in N_{V, 1} and N_{V, 2} also seem to follow the shape of the Stokes Q and U profiles (much like what was observed in Stokes V before the crosstalk removal; see Figure 4), perhaps crosstalk effects are also present in the null spectra. Mueller matrix calculations allow us to verify that there is a potential for crosstalk effects to appear in the null spectra (e.g., if the Fresnel rhombs and their respective positions are not without imperfections; see the Appendix). Since the four sub-exposures needed to compute Stokes V are independent, i.e., different combinations of their azimuthal axis are used, an error on any of the prism positions can yield the presence of Q and/or U in V, N_{V, 1}, and N_{V, 2}. Moreover, rotational variability and stress birefringence from optical elements located before the polarization module could also be an additional source of crosstalk, even though these elements have been studied and improved throughout the years. According to the ESPaDOnS support team, some of the crosstalk left, along with its variation, is due to the atmospheric dispersion corrector, which probably still exhibits stress birefringence, and since its configuration changes as a function of time (as a function of the telescope's position), this small stress birefringence introduces a small level of variable crosstalk. With that in mind, we have decided to be cautious and eliminate crosstalk from N_{V, 1} and N_{V, 2} using the same method described in Section 4. The crosstalk parameters are shown in Table 4. When parameters a_Q, b_Q, a_U, and b_U are considered, there is no systematic trend present in the results obtained from Stokes V that appear in N_{V, 1} or in N_{V, 2}. They are, however, of similar magnitude. Since Stokes V is fundamentally different in nature (what it represents and how it is generated) than the null spectra—it potentially contains a polarization signal intrinsic to the star—this is expected as the crosstalk removal method is aimed at eliminating all signal present in a given parameter. On the other hand, the δ parameter value obtained from Stokes V, N_{V, 1}, and N_{V, 2} should be consistent because it is independent of Q and U and constant within the line profile. Comparing δ values presented in Tables 2 and 4 confirms this.

Within most of the emission line, the crosstalk model behaves like a polarization constant that centers the null profiles at the zero polarization level once it is subtracted from the observed data. Equations (A24)–(A26) along with Equations (A27)–(A34) (see the Appendix) indicate that the net crosstalk effect on N_{V, 1} and N_{V, 2} due to positional errors of the Fresnel rhombs is likely more complex than a simple linear combination of Q and U. Thus, the model used to remove the crosstalk in the two check spectra is likely oversimplified. Nonetheless, since terms in Q and U represent the most important contributions to crosstalk, the model still is a good first-order approximation.

As mentioned in Section 3, we have used a binning procedure aimed at reducing noise and thus favoring the detection of any potential signal due to the presence of a magnetic field. The crosstalk removal could, at worst, eliminate a fraction of the Stokes V profile intrinsic to the star. However, using the same crosstalk parameters for all the nights in one run helps to minimize this because there is no reason to believe that any intrinsic, variable polarization would exhibit the same pattern as the crosstalk, which depends entirely on the behavior of Q and U. Nonetheless, we checked (by visual inspection) that the crosstalk model was a smooth, continuous function within the line profile and that it did not resemble, in shape and amplitude, the Stokes V signature predicted by Gayley & Ignace (2010). Moreover, except for the 2005 data (see Figure 4), the crosstalk model is approximatively constant throughout the line profiles. The presence of a magnetic field in the wind of EZ CMa observed via Zeeman splitting was looked for in two ways: (1) calculation of the reduced-χ² within the Stokes V line profile and (2) measurement of the longitudinal field B_l directly from the first-order moment of the Stokes V profile. In the first method, the reduced-χ² within the Stokes V line profile is computed to verify if the signal diverges significantly from a flat, null profile characteristic of a non-magnetic star. To diagnose the presence of a magnetic field in EZ CMa, we followed the detection diagnostic of Donati et al. (1992) and Donati et al. (1997). A circularly polarized signature is definitely detected if the associated detection probability is larger than 99.999%. A detection probability between 99.9% and 99.999% corresponds to a marginal detection, and below 99.9%, no formal detection is reported. Moreover, any positive detection across the spectral line is considered real if the detection probabilities calculated in both null profiles and in the continuum are below 99.9%.

For reasons discussed in Section 3, the level of binning that is applied to the data is important as it could influence the detectability of any potential Stokes V signature present inside the line profile. Because the expected signatures are extremely small in amplitude (Donati et al. 1997), especially for spectral lines formed in a stellar wind (Gayley & Ignace 2010), they need to be measured with high accuracy, i.e., with relative noise levels lower than 10⁻⁴. Since choosing a different S/N_threshold for each spectral line is somewhat arbitrary, variable binning was thus conducted on the data so that individual error bars reflected a 0.01% uncertainty. With such a relative noise level, a strong magnetic field (∼10^2–3 G) present in the wind of EZ CMa would yield Stokes V signatures detectable at the 3σ level. On the other hand, the presence of a weaker field at ≲ 100 G level requires an even higher S/N level to be clearly detected. Ignace & Gayley (2003) and Gayley & Ignace (2010) predict that such a magnetic field intensity, in a split monopole configuration, would generate Stokes V line profiles reaching ∼0.01%–0.04% in amplitude for emission lines formed in the wind and viewed under ideal conditions. Based on these considerations, we computed the detection probability within the Stokes V line profiles after binning the data so that the relative error bars reflected σ_V ≃ 10⁻⁴. Table 5 reports those detection probabilities calculated for the Stokes V profiles of He ii λ4686, as well as for the two null spectra, N_{V, 1} and N_{V, 2}. We present only the results obtained from the He ii λ4686 emission line because it is the strongest emission line in the spectrum with the smallest error bars. Although this might be compensated for by a lower signal, since it is formed farther from the star, no other lines nor a combination thereof show any better signal.

Table 5. Detection Probabilities Computed for the He ii λ4686 Emission Line

Year	Phase	PV	$P_{N_{1}}$	$P_{N_{2}}$
	(ϕ)	(%)	(%)	(%)
2005	0.155	65.4	83.4	98.8
	0.442	59.1	78.4	99.1
	0.682	62.3	37.7	38.7
2009	0.242	72.7	99.779	78.2
	0.479	94.4	99.843	77.5
	0.749	33.9	61.5	3.0
	0.012	87.5	90.3	59.3
2010	0.155	83.7	85.8	68.1
	0.418	28.7	81.1	67.8
	0.674	10.8	95.2	64.4
	0.954	90.0	35.8	50.0

Notes. Values for P_V, $P_{N_{1}}$, and $P_{N_{2}}$ represent the detection probabilities derived from within the Stokes V, N₁, and N₂ line profile, respectively, and binned in phase according to the ephemeris of Lamontagne et al. (1986). The results are presented at the three-decimal precision level when the probability reaches 98% so the criteria of Donati et al. (1997) can be adequately evaluated. The phase ϕ corresponds to the first observation taken each night, and detection probabilities are obtained from profiles binned to a precision of σ_V ≃ 10⁻⁴.

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In a few instances, the criteria are met to report, at best, a marginal detection in Stokes V (99.915% for He ii λ5411 on the third night of the 2005 observing run; 99.904% and 99.964% for N iv λ7111 on the third and fourth nights of the 2009 observing run, respectively). However, no systematic detections can be reported for all spectral lines or for a given emission line at all phases. Therefore, this test does not allow us to infer the presence of a magnetic field, as most of the Stokes V profiles are not significantly different (according to the χ² statistics) from a null, non-magnetic signature. Strangely, marginal detections are also achieved on a few occasions within check profiles and continuum regions. This would suggest that excess noise in the data can sometimes randomly affect the shape of the signal, leading to an erroneous marginal detection on the basis of a χ² test when compared to a Stokes V = 0 profile. As a whole, these results probably indicate that if there is a large-scale magnetic field in the wind of EZ CMa, it is not strong enough to consistently generate detectable signatures. By binning the data further to increase the S/N, any potential signature could consequently be smoothed out, thus lowering the upper detection limit. Also, the approach taken to eliminate crosstalk effects might remove a fraction of a real magnetic signature and hence decrease its detectability. In the end, the fact that no Zeeman-splitting signature is, in general, unambiguously detected points toward the following conclusion: any magnetic field present in the wind of EZ CMa (in the line formation zones) is too weak for the detection capabilities of both the instrument and the methods used in this study.

Of course, at this stage no information about the strength and structure of a potential magnetic field is obtained. Ionization stratification in the wind of EZ CMa (Hillier 1987; Schulte-Ladbeck et al. 1995; Herald et al. 2000) implies that emission lines present in the spectrum are formed in different conditions (e.g., different distances from the stellar surface, different velocities), and hence line-emitting regions would be threaded by different field intensities, no matter what large-scale magnetic configuration could be present. Moreover, emission lines often suffer blends with other emitting ions, which can dilute the circular polarization component of the dominating emitting ion. For these reasons, averaging the circularly polarized profiles from different individual emission lines (as is done for stars showing a large number of photospheric absorption lines) is inadequate. Thus, all results presented in this paper must come from the analysis of single emission lines. Consequently, this could explain why no significant detections are made according to the strict criteria mentioned above, which are adapted for profiles analyzed by the least-squares deconvolution method. In the context of hot stars, this technique combines several (∼10^1–2) photospheric absorption lines in order to increase the detectability of a potential Stokes V signal (Donati et al. 1997) and has proven its efficiency of exposing magnetic signatures that are hidden by noise within single line profiles. Donati et al. (1992) used different criteria when analyzing individual lines, and so perhaps the analysis of W-R emission lines should be based on better adapted criteria. Nonetheless, we feel that analyzing single emission lines of EZ CMa with the highest standards represents the safest, most conservative way of detecting a magnetic field. Moreover, eye inspection of Stokes V data led us to conclude that profiles resembling an organized, s-wave type, Zeeman signature do not necessarily result in a marginal (or even significant) detection, whereas unorganized, relatively flat profiles with a few divergent data points sometimes do. We feel that the search for a Zeeman-splitting signature by computing the χ² statistics within individual emission lines would yield a positive detection only if a strong, organized magnetic signature is present in the data.

The other technique used to evaluate the presence of a magnetic field (and its amplitude) in the wind of EZ CMa is the determination of its mean longitudinal intensity by calculating the first-order moment of the Stokes V profile within the line according to

$$\begin{equation} B_{l}=-2.14\times 10^{11} \frac{\int vV(v) dv}{c \lambda g_{{\rm eff}} \int [I_c-I(v)] dv} \end{equation} \tag{ 5 }$$

(Rees & Semel 1979; Mathys 1989; Wade et al. 2000), where the wavelength λ of the center of gravity of the line is in nm, g_eff is the effective Landé factor, and B_l is expressed in gauss. The uncertainties associated with B_l were computed by propagating the errors of each spectral bin through Equation (5). The center of gravity (from which the velocity was determined) was calculated according to the following equation:

$$\begin{equation} \lambda = \frac{1}{W_{\lambda }}\int I(\lambda) \lambda \,{d}\lambda, \end{equation} \tag{ 6 }$$

where W_λ is the equivalent width of the line and I(λ) is the unpolarized line profile. The effective Landé factors were computed through the formula

$$\begin{equation} g_{{\rm eff}} = 0.5(g_1+g_2)+0.25(g_1-g_2)(J_1(J_1+1)-J_2(J_2+1)), \end{equation} \tag{ 7 }$$

where g₁, g₂ and J₁, J₂ are, respectively, the Landé factors and total angular momentum quantum numbers of the involved levels of the line transition. For the emission lines considered in this work, the Landé factor of a certain level was computed by the LS-coupling approximate formula:

$$\begin{equation} g_i = 1 + \frac{J_i(J_i+1)-L_i(L_i+1)+S_i(S_i+1)}{2J_i(J_i+1)}, \end{equation} \tag{ 8 }$$

where L and S are the orbital angular momentum and spin quantum numbers of the considered level, respectively, and subscript i refers to the transition level of interest. Using Equations (7) and (8), the effective Landé factors for He ii λλ4686, 5411, 6560, and N iv λ7111 emission lines are g_eff = 1.07, 1.06, 1.06, and 1.17, respectively. The integration ranges for both λ and B_l were chosen to be symmetric about the center of gravity of the line going from −2000 to 2000 km s⁻¹. These limits of integration were chosen to encompass all information about the magnetic field in the Stokes V profile while avoiding the inclusion of excess continuum and line blends outside of the profile. In the end, the longitudinal field measured from Stokes V is detected significantly (at the 3σ level and over) if |B_l|/σ ⩾ 3. For the sake of conciseness, only the results for the He ii λ4686 emission line are presented in Table 6 since a longitudinal magnetic field is significantly detected at the 3σ level (and higher) within the observed Stokes V profiles on only a few occasions (one such instance can be found in Table 6). The only other significant detection occurs for the He ii λ5411 line, on the third night of the 2009 observing run (B_l = −128 ± 35 G ⇒ z = 3.7). Also, the first moment technique, when applied to both N₁ and N₂, never results in a detection, as expected.

Table 6. Longitudinal Field Values Calculated for the He ii λ4686 Emission Line

Year	Phase	Stokes V		Null N1		Null N2
	ϕ	Bl ± σB	zV	Bl ± σB	$z_{N_{1}}$	Bl ± σB	$z_{N_{2}}$
2005	0.155	−20 ± 17	1.2	−22 ± 17	1.2	21 ± 17	1.2
	0.442	−13 ± 15	0.8	2 ± 15	0.1	37 ± 15	2.5
	0.682	29 ± 11	2.7	−17 ± 11	1.6	−19 ± 11	1.7
2009	0.242	−8 ± 16	0.5	5 ± 16	0.3	−12 ± 16	0.8
	0.479	52 ± 16	3.2	41 ± 16	2.5	−2 ± 16	0.1
	0.749	6 ± 14	0.4	38 ± 14	2.7	−6 ± 14	0.4
	0.012	−18 ± 15	1.3	1 ± 15	0.1	8 ± 15	0.6
2010	0.155	−24 ± 17	1.4	−7 ± 17	0.4	−8 ± 17	0.5
	0.418	−11 ± 17	0.7	16 ± 17	1.0	−9 ± 17	0.5
	0.674	−27 ± 18	1.6	14 ± 17	0.8	−1 ± 17	0.1
	0.954	8 ± 16	0.5	16 ± 16	1.0	25 ± 16	1.6

Notes. The detection significance of the longitudinal magnetic field is expressed by z = |B_l/σ_B|. The longitudinal magnetic field is given in gauss, and ϕ corresponds to the phase of the first observation taken each night according to the ephemeris of Lamontagne et al. (1986). The significant detection (z > 3σ) appears in bold font.

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Figure 6 illustrates the phase variation of the longitudinal field for all three epochs of observations and for all four emission lines analyzed. A least-squares fit of a cosine curve to the data was performed using the form B_l(ϕ) = B₀ + B₁cos (2π(ϕ − ϕ₀)). The resulting parameters along with the reduced χ² value are presented in Table 7. The least-squares fit was applied assuming that the variations are continuous and consistent from one epoch of observations to another. This continuity has not been confirmed, but it still represents a reasonable hypothesis. According to the least-squares statistic, the longitudinal field is not significantly different from zero and the variation of the field is detected at only a 2σ level. Based on this test, there is not enough evidence to infer the presence of a magnetic field in the wind of EZ CMa. Moreover, the reduced χ² statistic obtained through the curve fit is essentially the same as that computed for the magnetic null hypothesis, i.e., B = 0. These calculations were also applied to the N profiles, and the results suggest that they are also compatible with a null profile, as expected. For completeness, we employed a Kolmogorov–Smirnov test to investigate if the distribution of the detection significance from Stokes V (z_V) is significantly different from that derived from the null profiles, N₁ and N₂. For the cumulative distributions of z_V and $z_{N_{1}}$, we obtain D = 0.1363, and for z_V and $z_{N_{2}}$, D = 0.2045, indicating that the null hypothesis (that the distributions are the same) can be rejected at confidence levels of about 36% and 79%. In other words, the two-sided Kolmogorov–Smirnov test indicates that the distribution of z_V does not appear to be significantly different from the $z_{N_{1}}$ and $z_{N_{2}}$ distributions. This supports our claim that the longitudinal magnetic field test does not provide enough evidence to infer the presence of a detectable magnetic field.

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The longitudinal magnetic field considered here is only used as a statistical indicator of the field strength. It is important to note that the longitudinal magnetic field is not used as the primary diagnostic of the presence of a magnetic field. This is because of a number of considerations. First, a variety of magnetic configurations can produce a null mean longitudinal field component, one for which the first-order moment of the Stokes V profile is zero (i.e., a profile roughly symmetric about the center of gravity of the emission-line profile). Nonetheless, almost all of these configurations generating a null mean longitudinal field component will produce a detectable Stokes V signature in the line profile, given that the magnetic field is strong enough. In the next section, we will discuss how this might actually be the type of profile expected for the magnetized stellar wind of W-R stars. Second, Equation (5) was derived for magnetic dipoles in rotating stellar photospheres under some simplifying assumptions. It is thus important to verify if it can be applied to W-R star winds in which the field topology is not dipolar and where the dominant spectral line broadener is not in rotation. The derivation of Equation (5) assumes that the emergent Stokes V line profiles are antisymmetric about their center. As shown by Gayley & Ignace (2010), this is expected in the context of the present study even if the magnetic field topology in W-R stellar winds resembles a split monopole rather than a dipole. The derivation of Equation (5) shows that it is independent of the projected rotational velocity of the star under very general conditions, i.e., the effect of the rotation on the line profiles has been fully accounted for and the first-order moment technique takes advantage of the rotational Doppler effect to get an enhanced diagnosis (Mathys 1988, 1989). However, the Zeeman broadening produced by a line observed in the visible band and a modest magnetic field is much smaller than the velocity broadening in the winds of hot stars. This leads to severe polarimetric cancellation of the σ components of the standard Zeeman triplet, and hence the ability to detect the longitudinal magnetic field through the Zeeman effect depends generally on the amount of Zeeman shift relative to the relevant broadening processes (Ignace & Gayley 2003). As described by these authors, this cancellation is, however, mitigated by velocity gradients in the wind, which allows for some Zeeman signals to persist in the line profile and produce an antisymmetric Stokes V profile about its center. What the exact physical meaning of Equation (5) is, in the context of W-R stars, is not clearly established, and thus it cannot a priori be taken for granted that it is the average over the emitting volume of the line-of-sight component of the magnetic field. Furthermore, Equation (5) should be used for the analysis of single spectral lines that are effectively unblended (Mathys 1989). In the case of W-R stars, their strong stellar winds generate very broad lines that often consist of a blend of different ion species. Even if one ion usually dominates (in intensity) at the wavelength of interest, contributions from other species can affect the results of the first-order moment technique. To minimize the contributions due to other spectral lines on the edges of the line profile, the limits of integration were chosen to only include the information of the Stokes V profile within the emission line. Moreover, Mathys (1989) states that the mean longitudinal field obtained from the observation of different spectral lines is not a priori identical, especially for an element inhomogeneously distributed over the star. This is also relevant in the case of EZ CMa, and possibly for W-R stars in general, where ionization stratification occurring in its stellar wind (Hillier 1987; Schulte-Ladbeck et al. 1995; Herald et al. 2000) implies that different spectral lines are formed in regions where the local field strength varies due to the radial dependence of the magnetic field intensity. While the spectral lines considered in this work are, in first approximation, homogeneously distributed over the stellar surface (the radial extent of individual formation zones is larger than the stellar core radius), their measurements may still yield discrepant field values. If different radial distributions are found for the various ions considered, as appears to be the case for EZ CMa, then Equation (5) will depend in a different manner on the spectral line, leading to a discordant value of the longitudinal magnetic field. As mentioned in Section 1.1, the variations observed in EZ CMa are epoch dependent. If the magnetic field is also subject to such a dependency, the fit of a cosine curve through the phased-bin variations of its longitudinal value (recorded over a five-year span) should be interpreted with extra care, even more so when a straight line (B(ϕ) = 0) yields a similar fit according to the reduced-χ². Finally, another shortcoming of this technique may come from the process of line formation in the winds of W-R stars. The first-order moment method is based on several hypotheses (e.g., Milne–Eddington approximation, weak-line approximation, and LTE) that are not relevant for radiation-driven winds from hot stars. In all, the determination of the longitudinal magnetic field using Equation (5) appears to be an approximation of its true value. Certainly, a better understanding of spectral line formation in the atmospheres of W-R stars and how the moment technique should be modified for different ions formed in a hot star wind would help understand the exact physical meaning of Equation (5). Still, Figure 6 illustrates that when the longitudinal field values derived from all four emission lines (for all three epochs of observations) are combined together, the cosine fit finds a field value at a ∼2σ significance level in Stokes V, while in the null N₁ and N₂, it is significant at ∼1σ. Even if 2σ is not statistically significant enough to report a detection, it is worth nothing that, when all the above considerations are taken into account, the cosine-fitted longitudinal field reaches such a significance level while, for N₁ and N₂, it is around unity, as expected. In summary, the longitudinal magnetic field variation does not provide enough evidence to infer the presence of a magnetic field, but it should not be on its own interpreted as the absence of magnetism in EZ CMa. We look closer into this below.

Despite the negative results on the presence of a detectable global magnetic field in EZ CMa using techniques relying on the amplitude of the Stokes V signature and its (non)symmetric nature, it is useful to take yet another approach: fitting a simple model to the data. To this end, we assume that the magnetic field of EZ CMa can be described by the split monopole model of Gayley & Ignace (2010), allowing us to place an upper limit on its intensity and to investigate its potential geometry. The first step taken by these authors to establish a model describing the effect of a magnetic field on the stellar wind of hot stars came earlier when they incorporated the weak-field longitudinal Zeeman effect in the Sobolev approximation (Ignace & Gayley 2003). They derived expressions for the circularly polarized emission from a magnetized wind. Particularly interesting in the case of W-R stars, they investigated the split monopole configuration to schematically illustrate the expected polarized profile. They found that the signature resembles an s-wave pattern and that for a surface field of about 100 G (seen pole-on), the largest polarizations peaking at approximately 0.05% in Stokes V result for optically thin but strong recombination emission lines. More recently, Gayley & Ignace (2010) published the results of an improved model of Stokes V(λ) profiles for emission lines formed in hot-star winds threaded with a weak magnetic field. For the most favorable orientation (i.e., pole-on), they found that the circularly polarized profile has a characteristic heartbeat shape exhibiting multiple sign inversions across the spectral line (see Gayley & Ignace 2010, Figure 3). For a wind magnetic field intensity of B ≃ 100 G and at depths where the wind speed is v ≃ 100 km s⁻¹, the overall scale of V/I is of the order of 0.1%.

A split monopole is considered under the assumptions that the magnetic field has been radially combed by the wind and rotation does not play a central role. It is important to note that the overall competition between the wind's mechanical energy density and the magnetic energy density can be described by the magnetic wind confinement parameter $\eta _{\star } = ({B_{\star }^{2}R_{\star }^{2}}/{\dot{M} v_\infty })$, where B_⋆ and R_⋆ are, respectively, the equatorial magnetic field and the radius at the star's surface. Based on the work of Hamann et al. (2006), R_⋆ corresponds to a radial optical depth of τ_Ross = 20. In regions far from the stellar surface, ud-Doula & Owocki (2002) demonstrated that the magnetic field lines are stretched out to follow the approximately radial wind outflow since the wind dominates the interaction. However, closer to the surface where the wind velocity is smaller, a sufficiently strong magnetic field can channel the wind plasma along closed field lines and hence significantly influence the wind flow. Quantitatively, for η_⋆ < 1, the field is fully opened by the wind outflow. For stronger confinements, η_⋆ > 1, the magnetic field remains closed over a limited range of latitudes and heights above the equatorial surface but eventually is opened into a nearly radial configuration at large radii. ud-Doula & Owocki (2002) and Owocki & ud-Doula (2004) also demonstrated that, for confinements as small as η_⋆ = 0.1, the magnetic field can have a significant influence in enhancing the density and reducing the wind speed near the magnetic equator. In the case of EZ CMa, η_⋆ < 1 requires a magnetic field intensity of B_⋆ < 4 kG and η_⋆ = 0.1 would imply a field strength of B_⋆ ∼ 1.3 kG, for the following wind and stellar parameters: $\dot{M}=10^{-4.3}\, M_{\odot }\ {\rm yr}^{-1}$, v_∞ = 1700 km s⁻¹, and R_⋆ = 2.65 R_☉ (Hamann et al. 2006). Because of the enormous mass-loss rates of W-R stars, even relatively small values of η_⋆ require large field strengths. Hence, in the weak-field limit, a split monopole configuration, resulting from the interaction of a dipolar field with a strong stellar wind, is a relatively good representation of the magnetic field present in the wind of W-R stars.

When modulated with phase, the model of Gayley & Ignace (2010) can be described by four parameters linked to the magnetic field geometry: (1) the viewing inclination angle of the stellar rotation axis, i₀; (2) the obliquity angle β between the field axis and the rotation axis; (3) the phase of closest approach of the magnetic pole to the line of sight, ϕ₀; and (4) the magnetic field intensity at the line formation region B_line, contained in the quantity b₁ in their parameterization (see Gayley & Ignace 2010, Equations (30) and (32)). Additionally, in order to account for lines of different widths, one more parameter is used to determine the crossover wavelengths (|x| = 1 in their parameterization) on each wing of the line profile. Along with the reversal existing at line center, this is where the polarization reversals occur in the Stokes V model. It is typically the wavelength that resonates at the line formation region, which appears around the half-width at half-maximum for rounded profiles, expressed as λ_{x = 1} in what follows.

Our approach was to simultaneously fit all Stokes V profiles recorded during one epoch of observation (within a chosen spectral line). Thus, for the 2005 data, three nights were simultaneously modeled while four nights were used for both fitting processes of the 2009 and 2010 data. At first, a χ² minimization was used to compare the observed Stokes V signatures to a grid of computed, theoretical profiles so the best set of parameters can be determined. The minimization was done in two successive steps: find the value of λ_{x = 1} first, and then determine the magnetic geometry [B_line, i₀, β, ϕ₀]. We proceeded this way since the shape of Stokes V model profiles is not a variable; only its amplitude with time is. In this framework, [B_line, i₀, β, ϕ₀] are free parameters, but constants for a given model. Because the geometry of the magnetic field does not influence where |x| = 1 occurs on the emission-line profile, we chose to dissociate both while performing the χ² minimization. This means that a first grid (five parameters) is generated to determine λ_{x = 1} and a second grid (four parameters) to find the best set of [B, i₀, β, ϕ₀] with the previously found value of λ_{x = 1} fixed.

To begin with, wavelengths λ were transformed into scaled wavelengths x (as used by Gayley & Ignace 2010) according to Equation (9):

$$\begin{equation} x = \frac{\lambda - \lambda _{o}}{\Delta \lambda _{{\rm HWHM}}\kappa }, \end{equation} \tag{ 9 }$$

where λ_o is the central wavelength of the line, Δλ_HWHM = λ_HWHM − λ_o is the half-width at half-maximum of the line profile, and κ is a constant scale factor. When κ = 1, this definition of x assumes that λ_{x = 1} = λ_HWHM. To find the correct position of |x| = 1, we allow κ to vary by no more than ±20%, i.e., the scale factor was sampled from 0.8 to 1.2 at every 0.02. Ideally, eye inspection of the observed profiles would clearly indicate where |x| = 1, and finding its position on the line profile would be straightforward. Since the recorded data do not yield such a clear visual picture, and because the binning process smooths out the polarization reversal regions, the κ parameter sampling is sufficient to obtain reasonable precision. Furthermore, Gayley & Ignace (2010) mention that the sharp wing spike occurring at |x| = 1 is an artifact of their treatment, and a smoother transition (which would involve some cancellation near the discontinuity) is to be expected for emission lines with rounded top. The following specifies how the first grid samples the other parameters: 0 ⩽ B_line ⩽ 3000 G at every 20 G; 0° ⩽ i₀ ⩽ 180° at each 10° (idem for β); and 0 ⩽ ϕ₀ ⩽ 1 at each 0.1 phase. Once the best κ value is found (according to χ² statistics), the second grid of computed profiles is generated sampling each of the remaining four parameters within the same range of values but using a smaller step size: ΔB_line = 10 G, Δi₀ = Δβ = 10°, and Δϕ₀ = 0.025. In the end, the parameters that provided the lowest reduced-χ² were chosen to represent the best-fitting model. This χ² minimization procedure was applied to the four strongest emission lines present in the spectrum of EZ CMa: He ii λλ4686, 5411, 6560, and N iv λ7111.

Figure 7 shows an example (for the 2010 data set) of the χ² landscapes computed from the observed Stokes V profiles of the He ii λ4686 emission line. Each landscape is a two-dimensional slice of the grid (with the other three parameters being fixed at their best-fit values), showing the reduced-χ² values with the minimum indicated by a white star. Additionally, the 1σ, 2σ, and 3σ uncertainty contours are superimposed on the landscapes, and the former allows us to estimate the uncertainty of the best-fitting model parameters. These are summarized in Table 8.

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The i₀–β landscape shows the presence of two minimum values of χ²_ν. This is a direct consequence of modulating the heartbeat Stokes V profile according to the phase of observation. To achieve this, Equations (30) and (32) of Gayley & Ignace (2010) need to be corrected by a phase-dependent scale factor, F(ϕ) = 1 − sin (i_B), where i_B represents the effective viewing inclination of the field axis at any moment. It is defined as cos (i_B) = cos (i₀)cos (β) + sin (i₀)sin (β)cos (ϕ) (Ignace et al. 2011). According to this definition and because i₀ and β are allowed to vary from 0° to 180°, there exist two pairs of i₀ − β values that generate the same modulating factor F(ϕ), and this degeneracy is reflected by the presence of two minima in Figure 7 (i₀–β landscape). Furthermore, due to the trigonometric properties of the sine and cosine functions, four pairs of i₀ − β values can also yield the same minimum in a χ²_ν landscape. When reading Table 8, it is important to remember that values of i₀ and β can be swapped; the order in which they are presented does not reflect any preferences. This method of determining the best estimates for the split monopole model parameters gives poor constraints on the angles i₀ and β. It would be arbitrary to invoke any trends or preferred values for both parameters given the degeneracy of the phase-dependent modulation applied to the model. The magnetic field intensity at the line formation region appears to decrease from the first epoch of observation to the last. Assuming that the emission lines considered here are formed roughly at the same distance from the star and under the same conditions, the average of B_line is ∼163 G in 2005, ∼85 G in 2009, and ∼68 G in 2010. The same decreasing trend is noticeable, in general, when each spectral line is considered individually. However, B_line estimates suffer from large uncertainties, and this epoch-dependent variation cannot be confirmed. It is important to note that the quality of ESPaDOnS observations has increased over time, which is reflected by the generally decreasing error bars on B_line at each epoch of observation. This could also explain the observed decreasing trend as better data were recorded in 2010 than in 2005. At best, this analysis allows us to propose an upper-limit value for the magnetic field strength in the wind of EZ CMa of B_line ∼ 100. Conversely, the reduced-χ² values are generally close to unity while the field strength approaches zero, within its lower uncertainty limit. This could also mean that the non-magnetic hypothesis offers a reasonable fit to the data.

Still within the framework of the split monopole model, we performed a Bayesian analysis of the spectropolarimetric observations following the work of Petit & Wade (2012). This analysis describes the relative likelihood of the split monopole model by generating a multidimensional posterior probability density that covers the parameter space of the model. Since only ϕ is allowed to change between different observations, rotation-independent magnetic configurations [B_line, i₀, β] can be computed in order to determine which configuration yields good posterior probabilities for all observed Stokes V profiles. The set of parameters that provides the maximum likelihood model is then considered the most probable magnetic configuration fitting the data. In this Bayesian statistics approach, we do not constrain the phases (ϕ being treated as a nuisance parameter). The inclination angle or the obliquity angle can vary from 0° to 180°, while the magnetic intensity has an upper bound at 3 kG. The rotational phase and obliquity both have simple flat priors, and the inclination angle prior is described by a random orientation probability distribution of the form p(i₀) = 0.5sin (i₀). Since the magnetic field strength is a parameter that can vary over several decades, we need to set an equal probability per decade while avoiding the singularity at B = 0. Therefore, we used a modified Jeffreys prior (Gregory 2005; Petit & Wade 2012) with a cut at B = 20 G, which corresponds to twice the grid step. The computed probability distribution functions (PDFs) marginalized for i₀, β, and B_line are presented, from top to bottom, in Figure 8 for our 2010 observations. The PDFs have been normalized by their maximum values to facilitate the display. Also, credible regions corresponding to 68.3%, 95.4%, and 99.7% of the total posterior probability distribution (analogous to the 1σ, 2σ, and 3σ contours of a Gaussian-like distribution) appear (superposed) in shades of green, from dark to pale colors, respectively. To express the best estimates of model parameters, we present the maximum of the joint posterior probability density (MAP) and the modes of the marginalized PDF for each parameter. These are computed for the four strongest lines, for all epochs of observations, and are summarized in Table 9. According to Petit & Wade (2012), MAP and mode values are not always equal, especially if the probability distribution is complex. The MAP values generally produce the best fit to the data but do not necessarily reflect the bulk of each PDF. On the other hand, the mode of each parameter represents adequately the bulk of the probability but does not necessarily give a good fit to the observations. The Bayesian analysis was performed assuming κ values found previously when applying the χ² goodness-of-fit test. We nonetheless verified that the same κ values were obtained through the Bayesian framework by making no assumptions on the scaling parameter and running the analysis. Since this verification yielded the same results, we then focused on the parameters describing the large-scale magnetic split monopole.

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Another useful application of Bayesian statistics resides in testing the plausibility of one model versus another by computing the so-called odds ratio, $\mathcal {O}(M_{1}/M_{0})$. It provides a more sensitive diagnostic of the presence of a weak magnetic signature embedded in noise than the detection probability technique or the longitudinal field method (Petit & Wade 2012). In this case, we can quantitatively evaluate the plausibility of the magnetic split monopole hypothesis, M₁, by calculating its odds ratio with the null model, M₀, representing the non-magnetic hypothesis (Stokes V = 0). The evidence supporting model M₁ is considered very strong when $\mathcal {O}(M_{1}/M_{0})$ is >10² (100:1), strong when >10^1.5 (30:1), moderate when >10¹ (10:1), and weak when >10^0.5 (3:1) (Jeffreys 1998). The odds ratios computed from Stokes V profiles are also summarized in Table 10. As a sanity check of these values, the same analysis was applied to the check spectra N₁ and N₂.

Table 10. Odds Ratio Computed for Stokes V, N₁, and N₂ Observations

Epoch	Line	Log($\mathcal {O}(M_{1}/M_{0})$)
		Stokes V	Null N1	Null N2
(1)	(2)	(3)	(4)	(5)
2005	He ii λ4686	−0.30	−0.33	−0.46
	He ii λ5411	−0.31	−0.34	−0.51
	He ii λ6560	−0.16	−0.47	−0.49
	N iv λ7111	−0.18	−0.46	−0.48
	Combined H ii lines	−0.50	−0.47	−0.64
2009	He ii λ4686	−0.35	−0.50	−0.17
	He ii λ5411	−0.45	−0.40	−0.49
	He ii λ6560	0.62	−0.54	−0.43
	N iv λ7111	−0.26	−0.52	−0.52
	Combined H ii lines	−0.14	−0.57	−0.49
2010	He ii λ4686	0.66	−0.53	−0.50
	He ii λ5411	0.23	−0.38	−0.14
	He ii λ6560	−0.35	−0.50	−0.35
	N iv λ7111	1.95	−0.32	−0.13
	Combined H ii lines	0.00	−0.68	−0.22

Notes. For all epochs of observations, the odds ratios are computed for the four strongest emission lines (individually) and for all He ii lines of that subset (simultaneously). The values are expressed in a base-10 logarithm form, i.e., log($\mathcal {O}(M_{1}/M_{0})$).

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The Bayesian analysis of individual emission lines in the spectrum of EZ CMa results, in most applications, in a non-detection. In fact, Tables 9 and 10 illustrate that field intensities of B_line = 0 are usually preferred to describe the observations, which is reflected by the associated odds ratio favoring the non-magnetic hypothesis. The magnetic split monopole hypothesis is preferred on four occasions, three of them being part of the 2010 observing run. This could be explained by the upgrades done on ESPaDOnS through the years, increasing the spectropolarimeter efficiency and precision. On the other hand, it could also reflect an intrinsic change in EZ CMa's magnetic properties. Although less likely, this hypothesis is still a possibility given the variable nature of the W-R star with epoch (Robert et al. 1992). Taking into account all nights simultaneously, the odds ratio for the He ii λ6560 emission line Stokes V profiles observed in 2009 is ∼4: 1 in favor of the magnetic hypothesis, while the N₁ and N₂ profiles are both ∼3: 1 in favor of the null, non-magnetic hypothesis, as expected. In 2010, the odds ratio is ∼5: 1, ∼2: 1, and ∼89: 1 in favor of the magnetic split monopole for Stokes V profiles for the He ii λ4686, He ii λ5411, and N iv λ7111 emission lines, respectively. Meanwhile, all odds ratios computed from their respective N₁ and N₂ profiles favor the non-magnetic scenario, as expected. It is interesting to note that Bayesian model comparison, through the calculation of the odds ratio, automatically disadvantages more complicated models. It has a built-in Occam's razor implying that simpler explanations are to be preferred, unless there is sufficient evidence in favor of more complicated explanations (Gregory 2005). The theorem of Bayes both quantifies such evidence and determines how much additional evidence is sufficient by assigning a large probability to a more complicated model only if the complexity of the data justifies the additional complication of the model. Since the expected Stokes V amplitudes expected for a magnetized stellar wind are at the noise level of our ESPaDOnS observations, it is not surprising to obtain an odds ratio in favor of a non-magnetic scenario. Only observations showing larger (in amplitude) and more clearly organized Stokes V signatures are assessed additional evidence via the calculation of the split monopole model global likelihood. It is thus interesting to note the existence (in a few cases) of the odds ratio favoring the magnetic hypothesis even if the evidence is, in general, weak.

With this method, when a magnetic field is detected, only limited information can be obtained for angles i₀ and β. The magnetic geometry is recovered by the mode values of the marginalized PDFs (most probable inclination and obliquity), but the credible regions are too extended to put strong constraints on these parameters. On average, the field strength (when detected) is 〈B_line〉 = 78⁻³¹₊₈₈, which is significant at approximately 2.5σ. The latter assumes that the emission lines considered are formed roughly under the same conditions, in the wind of EZ CMa. Since it was reported (Hillier 1987; Schulte-Ladbeck et al. 1995; Herald et al. 2000) that EZ CMa exhibits ionization stratification in its stellar wind, averaging the field strength from different emission lines is a simplistic way of quantifying the magnetic intensity. Nonetheless, even if different spectral lines form at different distances for the star's surface, their formation zones are quite extended and often overlapping. Consequently, 〈B_line〉 should be considered as an order-of-magnitude indication of the field strength. Hoping that the combination of more observations (from the same epoch) would yield better constraints on the split monopole model parameters, we combined data from the He ii λ4686, He ii λ5411, and He ii λ6560 emission lines and performed the Bayesian analysis. These He ii spectral lines (same ionic species) are roughly formed at the same distance from the stellar core, 8–10 R_core according to Hillier (1987), and should therefore be generated in a similar magnetized plasma. In 2009 and 2010, the most probable values for B_line are 20⁻²⁰₊₄₅ and 30⁻³⁰₊₅₅, respectively, while for the 2005 observing run, B_line = 0₊₄₀ is the most probable estimate of the field strength (see Table 9). The corresponding odds ratios (see Table 10) are all in favor of the non-magnetic hypothesis, which is not surprising given the weak estimates obtained for the field strength.

To calculate the outcome of an optical path, Mueller calculus requires the application of several matrices that must be organized in the appropriate order (Kliger et al. 1990; Clarke 2010). Below are the matrices representing the optical elements of the ESPaDOnS polarimeter: λ/2-Fresnel rhomb,

$$\begin{equation} M_{\lambda /2}= \left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&0 \\ 0&0&-1&0\\ 0&0&0&-1\end{array} \right], \end{equation} \tag{ A1 }$$

and λ/4-Fresnel rhomb,

$$\begin{equation} M_{\lambda /4}= \left[\begin{array}{@{}cccc@{}} 1&0&0&0\\ 0&1&0&0 \\ 0&0&0&1\\ 0&0&-1&0\end{array} \right]. \end{equation} \tag{ A2 }$$

For a general linear polarizer, such as the Wollaston prism, with its optical axes rotated by an angle ω from the x-axis of a right-handed Cartesian frame, the Mueller matrix is given by

$$\begin{equation} P_{\omega }= \frac{1}{2}\, \left[\begin{array}{@{}cccc@{}}1&\cos (2\omega)&\sin (2\omega)&0\\ \ms \cos (2\omega)&\cos ^{2}(2\omega)&\cos (2\omega)\sin (2\omega)&0 \\ \ms \sin (2\omega)& \cos (2\omega)\sin (2\omega)&\sin ^{2}(2\omega) &0\\ \ms 0&0&0&0\end{array} \right]. \end{equation} \tag{ A3 }$$

In the reference frame, the light beam travels along the z-axis and the x–y plane is perpendicular to direction of propagation. Angle ω is positive and is measured anti-clockwise, as seen looking against the direction of propagation.

ESPaDOnS is designed to record two orthogonal spectra, side by side, on its CCD detector. The first one corresponds to the ∥ beam of light (ω = 0°), and the second one to the ⊥ beam of light (ω = 90°). From Equation (A3), two Mueller matrices can be calculated for the two components emerging from a Wollaston prism: Wollaston ordinary beam (∥),

$$\begin{equation} M_{\parallel }= \frac{1}{2}\, \left[\begin{array}{@{}cccc@{}}1&1&0&0\\ 1&1&0&0 \\ 0&0&0&0\\ 0&0&0&0\end{array} \right], \end{equation} \tag{ A4 }$$

and Wollaston extraordinary beam (⊥),

$$\begin{equation} M_{\perp } = \frac{1}{2}\, \left[\begin{array}{@{}cccc@{}}1&-1&0&0\\ -1&1&0&0 \\ 0&0&0&0\\ 0&0&0&0\end{array} \right]. \end{equation} \tag{ A5 }$$

Equations (A4) and (A5) assume that the Wollaston prism exhibits perfect transmission. If transmission happens to be different from the ideal case, the matrices should be multiplied by coefficients (<1) to allow for this.

The individual matrices above represent the different elements of the polarization analyzing device of ESPaDOnS, but they do not account for their orientation, when applicable. For instance, a λ/2-Fresnel rhomb with a horizontal transmission axis has a different Mueller matrix from the same rhomb that is turned by some degrees. The Stokes vector representing the beam of light is expressed in a particular reference frame and, to determine the effect of any polarizing element, it is first necessary to set the description of the polarization to the frame corresponding to the principal axes of the device (Clarke 2010). Thus, the operation of a rotatable λ/2-Fresnel rhomb, say, may be represented by

$$\begin{equation} \left[\begin{array}{@{}c@{}}\it I_{o}\\ {\it Q_{o}} \\ {\it U_{o}}\\ {\it V_{o}}\end{array} \right] = [R(-\theta)][M_{\lambda /2}][R(\theta)]\left[\begin{array}{@{}c@{}}\it I_{i}\\ {\it Q_{i}} \\ {\it U_{i}}\\ {\it V_{i}}\end{array} \right], \end{equation} \tag{ A6 }$$

where the indices "o" and "i" stand for output and input, respectively. In the above example, the third 4 × 4 matrix rotates the output Stokes vector back to the original reference frame. Rotation of the direction of vibration of the linear polarization or of the major axis of the polarization ellipse can be expressed as

$$\begin{equation} R_{\theta } = \left[\begin{array}{@{}cccc@{}}1&0&0&0\\ 0&\cos \left(2\theta \right) &\sin \left(2\theta \right) &0\\ 0 &-\sin \left(2\theta \right) &\cos \left(2\theta \right) &0 \\ 0&0&0&1\end{array} \right]. \end{equation} \tag{ A7 }$$

Positive θ is measured anti-clockwise from the x-axis toward the y-axis as seen looking against the z-direction (Clarke 2010).

In calculating the outcome of ESPaDOnS polarimeter optics, Mueller calculus requires the application of several matrices in combination that must be organized in the correct order. Following matrix algebra, we can calculate the four Stokes parameters outputted by the polarimeter with the equation below:

$$\begin{equation} \left[\begin{array}{@{}c@{}}\it I_{o}\\ {\it Q_{o}} \\ {\it U_{o}}\\ {\it V_{o}}\end{array} \right] = [P]\left[\begin{array}{@{}c@{}}\it I_{i}\\ {\it Q_{i}} \\ {\it U_{i}}\\ {\it V_{i}}\end{array} \right], \end{equation} \tag{ A8 }$$

where

$$\begin{eqnarray} [P] &=& [{\rm Wollaston}][R(-\phi)][M_{\lambda /2}] \nonumber\\ && \times [R(\phi)][M_{\lambda /4}][R(-\theta)][M_{\lambda /2}][R(\theta)]. \end{eqnarray} \tag{ A9 }$$

Here θ and ϕ are, respectively, the angle of the first and second λ/2-rhomb, and the [Wollaston] matrix stands for both the parallel and perpendicular components. So if one wants to calculate the ordinary component, [Wollaston] = [M_∥], and for the extraordinary component of the output beam, [Wollaston] = [M_⊥]. Equations (A10) and (A11) present the polarization matrix of ESPaDOnS for both components:

$$\begin{equation} P_{\parallel }= \frac{1}{2}\left[\begin{array}{@{}cccc@{}}1&a&2b&2c\\ 1&a&2b&2c \\ 0&0&0&0\\ 0&0&0&0\end{array} \right] \end{equation} \tag{ A10 }$$

$$\begin{equation} P_{\perp }= \frac{1}{2}\left[\begin{array}{@{}cccc@{}}1&-a&-2b&-2c\\ -1&a&2b&2c \\ 0&0&0&0\\ 0&0&0&0\end{array} \right], \end{equation} \tag{ A11 }$$

where

$$\begin{eqnarray} a &=&(\cos ^{2}(2\phi)-\sin ^{2}(2\phi))(\cos ^{2}(2\theta)-\sin ^{2}(2\theta)), \end{eqnarray} \tag{ A12 }$$

$$\begin{eqnarray} b &=&(\cos ^{2}(2\phi)-\sin ^{2}(2\phi))\cos (2\theta)\sin (2\theta), \end{eqnarray} \tag{ A13 }$$

$$\begin{eqnarray} c &=&{-}\cos (2\phi)\sin (2\phi). \end{eqnarray} \tag{ A14 }$$

Following the double-ratio method described by Donati et al. (1997), four exposures using each a different combination of θ and ϕ are used to calculate Stokes parameters Q, U, and V. The general expressions for the parallel and perpendicular components of the intensity beam are described in Equations (A15) and (A16):

$$\begin{eqnarray} I^{\parallel }_{\theta,\phi } = & \frac{1}{2}I + \frac{1}{2} ((\cos ^{2}(2\phi)-\sin ^{2}(2\phi))(\cos ^{2}(2\theta)-\sin ^{2}(2\theta)))Q \nonumber \\ & +... +((\cos ^{2}(2\phi)-\sin ^{2}(2\phi))\cos(2\theta)\sin(2\theta)) U +...\nonumber \\ & -(\cos(2\phi)\sin(2\phi)) V, \end{eqnarray} \tag{ A15 }$$

$$\begin{eqnarray} I^{\perp }_{\theta,\phi } = & \frac{1}{2}I - \frac{1}{2}((\cos ^{2}(2\phi)-\sin ^{2}(2\phi))(\cos ^{2}(2\theta)-\sin ^{2}(2\theta)))Q \nonumber \\ & +... {-}((\cos ^{2}(2\phi)-\sin ^{2}(2\phi))\cos(2\theta)\sin(2\theta)) U +... \nonumber \\ & +(\cos(2\phi)\sin(2\phi)) V. \end{eqnarray} \tag{ A16 }$$

To measure Stokes V, the four different combinations⁸ of θ and ϕ are

• 1.  
  θ = 0°, ϕ = 675,
• 2.  
  θ = 90°, ϕ = 225,
• 3.  
  θ = 45°, ϕ = 225,
• 4.  
  θ = 135°, ϕ = 675.

In the end, two spectra(I^∥_{θ, ϕ} and I^⊥_{θ, ϕ}) are recorded side by side on the detector and are described as follows, depending on θ and ϕ, by the equations below:

$$\begin{eqnarray} &&I^{\parallel }_{0,67.5}=\frac{1}{2}(I+V) \quad I^{\perp }_{0,67.5}=\frac{1}{2}(I-V), \end{eqnarray} \tag{ A17 }$$

$$\begin{eqnarray} &&I^{\parallel }_{90,22.5}=\frac{1}{2}(I-V) \quad I^{\perp }_{90,22.5}=\frac{1}{2}(I+V), \end{eqnarray} \tag{ A18 }$$

$$\begin{eqnarray} &&I^{\parallel }_{45,22.5}=\frac{1}{2}(I-V) \quad I^{\perp }_{45,22.5}=\frac{1}{2}(I+V), \end{eqnarray} \tag{ A19 }$$

$$\begin{eqnarray} &&I^{\parallel }_{135,67.5}=\frac{1}{2}(I+V) \quad I^{\perp }_{135,67.5}=\frac{1}{2}(I-V). \end{eqnarray} \tag{ A20 }$$

With these equations, we can calculate the normalized Stokes V parameter along with its two null spectra with

$$\begin{eqnarray} &&\frac{V}{I} = \frac{R_{V}-1}{R_{V}+1}, \end{eqnarray} \tag{ A21 }$$

$$\begin{eqnarray} &&\frac{N_{V,1}}{I} = \frac{R_{N_{1}}-1}{R_{N_{1}}+1}, \end{eqnarray} \tag{ A22 }$$

$$\begin{eqnarray} &&\frac{N_{V,2}}{I} = \frac{R_{N_{2}}-1}{R_{N_{2}}+1}, \end{eqnarray} \tag{ A23 }$$

where R_V, $R_{N_{1}}$, and $R_{N_{2}}$ are defined by

$$\begin{eqnarray} &&R_{V} = \sqrt[4] { \frac{ \frac{ I_{0,67.5 \parallel } }{I_{0,67.5 \perp }} \cdot \frac{I_{135,67.5 \parallel }}{I_{135,67.5 \perp }} }{ \frac{ I_{90,22.5 \parallel } }{I_{90,22.5 \perp }} \cdot \frac{I_{45,22.5 \parallel }}{I_{45,22.5 \perp }} }}, \end{eqnarray} \tag{ A24 }$$

$$\begin{eqnarray} &&R_{N_{1}} = \sqrt[4] { \frac{ \frac{ I_{0,67.5 \parallel } }{I_{0,67.5 \perp }} \cdot \frac{I_{45,22.5 \parallel }}{I_{45,22.5 \perp }} }{ \frac{ I_{90,22.5 \parallel } }{I_{90,22.5 \perp }} \cdot \frac{I_{135,67.5 \parallel }}{I_{135,67.5 \perp }} } }, \end{eqnarray} \tag{ A25 }$$

$$\begin{eqnarray} &&R_{N_{2}} = \sqrt[4] { \frac{ \frac{ I_{0,67.5 \parallel } }{I_{0,67.5 \perp }} \cdot \frac{ I_{90,22.5 \parallel } }{I_{90,22.5 \perp }} }{ \frac{I_{45,22.5 \parallel }}{I_{45,22.5 \perp }} \cdot \frac{I_{135,67.5 \parallel }}{I_{135,67.5 \perp }} } }. \end{eqnarray} \tag{ A26 }$$

One can then explore how an angular error on θ and ϕ can affect the values of I^∥_{θ, ϕ} and I^⊥_{θ, ϕ}. Applying a third-order multivariate Taylor expansion on, say, I^∥_{90, 22.5}, we get

$$\begin{eqnarray*} I^{\parallel }_{\Delta _{\theta =90},\Delta _{\phi =22.5}} &\simeq & \frac{1}{2}(I - V) - 1.5\times 10^{-10}Q \nonumber\\ && + \Delta _{\phi =22.5}(-2Q+ 1.4\times 10^{-9}V) +... \\ && -6\times 10^{-10}\Delta _{\theta =90}U + 1.2\times 10^{-9}\Delta ^{2}_{\theta =90} Q +...\\ && -8\Delta _{\theta =90}\Delta _{\phi =22.5}U + \Delta ^{2}_{\phi =22.5}\nonumber\\ && \times(2.7\times 10^{-9} Q + 4 V), \end{eqnarray*}$$

where Δ_{τ = t} = τ' − t are the angular errors on the Fresnel rhombs orientations, expressed in radians(τ' and t being the instrumental and theoretical orientations, respectively). For example, Δ_{ϕ = 67.5} = ϕ' − 675, expressed in radians. Neglecting terms with coefficients of order 10⁻⁹ or less, the full set of spectra recorded can be expressed as

$$\begin{eqnarray} I^{\parallel }_{\Delta _{\theta =0},\Delta _{\phi =67.5}} &=& \frac{1}{2}(I+V)+2\Delta _{\phi =67.5}Q \nonumber\\ &&+ 8\Delta _{\phi =67.5}\Delta _{\theta =0} U - 4 \Delta ^{2}_{\phi =67.5}V, \end{eqnarray} \tag{ A27 }$$

$$\begin{eqnarray} I^{\perp }_{\Delta _{\theta =0},\Delta _{\phi =67.5}}&=& \frac{1}{2}(I-V)- 2\Delta _{\phi =67.5}Q \nonumber\\ && - 8\Delta _{\phi =67.5}\Delta _{\theta =0} U +4 \Delta ^{2}_{\phi =67.5}V, \end{eqnarray} \tag{ A28 }$$

$$\begin{eqnarray} I^{\parallel }_{\Delta _{\theta =90},\Delta _{\phi =22.5}} &=& \frac{1}{2}(I-V)-2\Delta _{\phi =22.5}Q \nonumber\\ && - 8\Delta _{\phi =22.5}\Delta _{\theta =90}U+4\Delta ^{2}_{\phi =22.5} V, \end{eqnarray} \tag{ A29 }$$

$$\begin{eqnarray} I^{\perp }_{\Delta _{\theta =90},\Delta _{\phi =22.5}} &=& \frac{1}{2}(I+V) +2\Delta _{\phi =22.5}Q \nonumber\\ && + 8\Delta _{\phi =22.5}\Delta _{\theta =90}U -4\Delta ^{2}_{\phi =22.5} V, \end{eqnarray} \tag{ A30 }$$

$$\begin{eqnarray} I^{\parallel }_{\Delta _{\theta =45},\Delta _{\phi =22.5}} &=& \frac{1}{2}(I-V) +2\Delta _{\phi =22.5}Q \nonumber\\ && +8\Delta _{\phi =22.5}\Delta _{\theta =45} U + 4\Delta ^{2}_{\phi =22.5} V, \end{eqnarray} \tag{ A31 }$$

$$\begin{eqnarray} I^{\perp }_{\Delta _{\theta =45},\Delta _{\phi =22.5}} &=& \frac{1}{2}(I+V) -2\Delta _{\phi =22.5}Q \nonumber\\ && -8\Delta _{\phi =22.5}\Delta _{\theta =45} U - 4\Delta ^{2}_{\phi =22.5} V, \end{eqnarray} \tag{ A32 }$$

$$\begin{eqnarray} I^{\parallel }_{\Delta _{\theta =135},\Delta _{\phi =67.5}} &=& \frac{1}{2}(I+V) - 2\Delta _{\phi =67.5}Q \nonumber\\ && - 8\Delta _{\phi =67.5}\Delta _{\theta =135}U - 4\Delta ^{2}_{\phi =67.5} V, \end{eqnarray} \tag{ A33 }$$

$$\begin{eqnarray} I^{\perp }_{\Delta _{\theta =135},\Delta _{\phi =67.5}} &=& \frac{1}{2}(I-V) + 2\Delta _{\phi =67.5}Q \nonumber\\ && + 8\Delta _{\phi =67.5}\Delta _{\theta =135}U + 4\Delta ^{2}_{\phi =67.5} V. \end{eqnarray} \tag{ A34 }$$

From these, we can see that the calculation of R_V, $R_{N_{1}}$, and $R_{N_{2}}$ does not yield the same results as in Equations (A24)–(A26). In particular, $R_{N_{1}}$ and $R_{N_{2}}$ do not equal one as they are design to be. Hence, the presence of errors on the orientations of the Fresnel rhombs could potentially yield crosstalk effects N_{V, 1} and N_{V, 2} from Stokes Q and U. For example, an error of ±1° on each position of the two Fresnel rhombs, in the first sub-exposure only, yields ±0.035Q and ±0.002U to enter the calculation of $R_{N_{V,1}}$ and $R_{N_{V,2}}$. Even though the crosstalk model presented in Section 4 is oversimplified to adequately characterize the effects of crosstalk on observed N_{V, 1} and N_{V, 2} profiles, it still offers a good approximation. For example, using the values presented in Table 4, parameters α_λ and β_λ(on top of the He ii λ4686 emission line, λ ∼ 468.8 nm) are, for the 2010 data set, α = −0.013 and β = −0.012 within the N_{V, 1} line profile and α = −0.023 and β = −0.005 within the N_{V, 2} line profile. Similar values(order of magnitude) are obtained for the 2005 and 2009 observations.