Stellar longitudinal magnetic field determination through multi-Zeeman signatures



<span>A lot of effort has been put into the detection and determination of stellar magnetic fields using the spectral signal obtained from the combination of hundreds or thousands of individual lines, an approach known as "multi-line techniques". However, so far most of multi-line techniques developed that retrieve stellar mean longitudinal magnetic fields recourse to sometimes heavy simplifications concerning line shapes and Zeeman splittings.</span> <span> To determine stellar longitudinal magnetic fields by means of the Principal Components Analysis and Zeeman Doppler Imaging (PCA-ZDI) multi-line technique, based on accurate polarised spectral line synthesis.</span> <span>In this paper we present the methodology to perform inversions of profiles obtained with PCA-ZDI.</span> <span> Inversions with various magnetic geometries, field strengths and rotational velocities show that we can correctly determine the effective longitudinal magnetic field in stars using the PCA-ZDI method.</span>

Introduction
============

It is well known that the magnetic solar type stars host weak longitudinal fields, typically of the order of a few tens of Gauss (e.g. @Marsdenetal2014). For such field strengths, Stokes $V$ circular polarisation signatures of individual spectral lines are generally well below the noise level. The use of so-called multi-line techniques makes it possible to overcome this problem through the "addition" of multiple individual lines in Doppler space, resulting in a "mean" circular polarisation profile, the so-called Multi-Zeeman-Signature (MZS) which greatly increases the signal to noise ratio. Starting with the pioneering work of @SemelLi1996, different techniques have been developed over the following two decades for the establishment of Multi-Zeeman-Signatures. The most popular appears to be the Least Squares Deconvolution (LSD) technique first described in @Donatietal1997. Two assumptions underlie the LSD technique. The first one states that the local circular polarisation profiles to be added are all of similar shape; the second postulates that the Zeeman broadening is very small compared to the thermal Doppler broadening, such that the weak field approximation can safely be applied to the Stokes profiles. As a consequence, the coupled system of equations of polarised radiative transfer do not have to be formally solved. Instead, under a perturbative scheme the system of equations can be partially decoupled, permitting to find an analytical solution: to first order the circular polarisation profile is proportional to the first derivative of the intensity, and to second order, the linear polarised profiles are proportional to the second derivative of the intensity (e.g. @LandiLandolfi2004). In recent times, LSD and the weak field approximation (WFA) have been widely employed at different levels of sophistication with the ultimate goal of precisely measuring weak stellar magnetic fields on the basis of MZSs (@KochukhovMaPi2010, @Kochukhov2015, @MartinezAsensio2012, @Martinezetal2012, @CarrollStr2014, @AsensioPet2015).

An alternative technique based on the Principal Components Analysis (PCA) was proposed by @Martinezetal2008. In this work, the MZS is derived by means of the addition of many lines in Doppler space (as in LSD) but a denoising procedure, the filtering of uncorrelated noise, is applied to individual lines to increase the signal to noise ratio in the final MZS. It is an important feature of this technique that similarity between the individual polarised circular line profiles is not required; neither is it necessary to invoke the WFA. The validity of this robust technique for the analysis of magnetic fields has been proved with numerical simulations but the method has not yet been applied to the quantitative <span>*measurement*</span> of field strengths.

In fact, most current techniques devoted to the analysis of stellar magnetic fields by means of MZS are avoiding spectral line synthesis on account of the computing resources required in view of the wide spectral ranges involved (thousands of Angstroms). To contribute to remedy this situation, we will present the basis of a novel technique for the determination of stellar longitudinal magnetic fields, a method that is based on spectral line synthesis incorporating detailed polarised radiative transfer.

"Solar" and stellar Stokes profiles
===================================

In this section we shall use the Stokes profiles obtained from a given single point on the surface of the star – we call them "solar" profiles – to derive the Stokes profiles integrated over the entire visible hemisphere of the star; we shall in what follows denote the latter as "stellar" profiles. To start with, we first consider a star with zero rotational velocity, but later on, this constraint will be removed.

A number of parameters have to be specified for the calculation of Stokes profiles of magnetic stars. In the present case, that we employ the eccentric dipole oblique rotator model [@StiftSt1975], these are the effective temperature $T_{\rm eff}$, the surface gravity $\log g$, the metallicity \[$M/H$\], the projected rotational velocity of the star $v\,\sin i$, the macro-turbulent ($v_{\rm turb}$) and the micro-turbulent ($\xi$) velocities, the position of the magnetic dipole inside the star – given by the two coordinates $x_2$ and $x_3$ – and the dipole moment $\mathbf{m}$. In addition, it is necessary to specify the orientation of the magnetic axis with respect to the rotation axis with the help of the 3 Eulerian angles $\alpha, \beta, \gamma$, and the orientation of the rotational axis with respect to the line of sight (LOS), i.e. the inclination angle $i$. See @StiftSt1975 for details. For the synthesis of the Stokes spectra we have employed <span>Cossam</span> (@Stift2000, @StiftStLeCo2012), an LTE code that calculates the four Stokes parameters by detailed opacity sampling and by accurately solving the coupled equations of polarised radiative transfer.

Spatial grids
-------------

The so-called effective magnetic field $H_{\rm eff}$ – which represents a weighted average of the longitudinal component of the magnetic field vector over the visible hemisphere of the star – depends on the wavelength range covered by the spectral lines, on their atomic properties, on their strength, and on the limb darkening [@Stift1986b]. The spatial distribution over the stellar surface of the quadrature points may also play a role if the spatial resolution is too low in the presence of rotation, pulsation and/or strong magnetic fields. Therefore one has to ensure the establishment of a spatial grid figuring a decent number of well distributed points. We may distinguish two main types of grids, viz. fixed grids and adaptive grids. The <span>Cossam</span> code provides both grid types. Fixed grids can be co-rotating as found in Doppler mapping (see e.g. @VogtVoPeHa1987) or observer-centred. For the former one tries to create surface elements of almost identical size in bands equidistant in latitude, whereas for the latter the points are arranged in concentric rings about the centre of the visible stellar disk. The concentric rings are very popular because even a quite small number of such rings usually proves sufficient for the spatial integration required to obtain quantities like the mean longitudinal magnetic field $H_{\rm eff}$ or the mean field modulus $H_{\rm s}$.

Polarised line synthesis in fast rotating stars can constitute a challenge for these simple fixed grids which may lead to a uncomfortably large number of points. Adaptive grids are better in general, representing non-uniform distributions with no pre-determined distances between the individual points. The central idea behind adaptive grids in line synthesis is to ensure that the difference in monochromatic line opacity between two adjacent points should not exceed a certain percentage – refer to @Stift1985 for details. Let us for example consider the sampling of the central opacity of a line at a given point. If the grid resolution is not high enough, it may happen that at a neighbouring point the line opacity is already negligible as a result from differences in the magnetic field strength and angle, or from the Doppler effect due to rotation or pulsation. The integrated line intensity would thus reach its minimum at one point, whereas at the neighbouring point there would be nothing but the continuum. However sophisticated the quadrature scheme, the integrated line profile would always be in error for such a sparse grid.

In <span>Cossam</span> the density of points can be adjusted through five parameters. As a proxy for the limit in the relative percentage change in opacity between neighbouring quadrature points one takes a fraction of the Doppler width of the opacity profile of a typical metal line ($R_{\rm dop}$). In the polarised radiative radiation equations (see @AlecianAlSt2004) two angles enter: the angle between magnetic field vector and the LOS on the one hand, and the azimuth angle defining Stokes $Q$ and $U$ on the other hand; they are taken care of by $R_{\rm cos}$ and $R_{\rm azi}$ respectively. There remains the continuum intensity $R_{\rm flux}$ which may not be neglected in the establishment of the spatial grid. Finally one has to ensure the monotonicity of the resolution function in those cases where the Doppler shift between adjacent points is zero and where the field angles are identical. For this purpose a small empirical constant $\Delta$ is added to the resolution function. Adaptive grids are very useful thanks to the optimal placement of the quadrature points and their reduced number compared to fixed grids. This can significantly reduce the time needed to calculate Stokes profiles in strongly magnetic and/or non-radially pulsating stars [@Fensl1995].

![image](landstreet.ps){width="7.5cm"}

![image](stift.ps){width="7.5cm"}

![image](lands_beff_small.eps){width="10cm"}

![image](stift_beff_small.eps){width="10cm"}

"Stellar" (integrated) versus "solar" (point source) profiles
-------------------------------------------------------------

We adopted two particular magnetic field configurations and spatial grids for the demonstration of how well stellar Stokes profiles are approximated by "solar" profiles. In the first case, we assumed an oblique rotator seen equator-on, with the magnetic dipole located at the centre of the star and its axis perpendicular to the rotational axis (Model1 in Table \[tab:magmod\]). At phase zero, the magnetic pole is at the centre of the visible disk, the field is thus symmetric around the line of sight. In this case, we employed a fixed grid with 10 rings where the innermost one features 8 surface elements, for a total of 440 points, shown in the left panel of Fig.\[fig:grids\]. For the second magnetic configuration, labelled Model2 in Table\[tab:magmod\], we chose a decentred tilted dipole, resulting in a highly inhomogeneous distribution of field strengths and angles over the visible hemisphere. With the adaptive grid parameters $\Delta = 0.5$, $R_{\rm dop} = 0.5$, $R_{\rm cos} = 0.3$, $R_{\rm azi} = 0.3$, and $R_{\rm flux} = 0.3$ we arrived at a total of 406 points distributed as shown in the right panel of Fig.\[fig:grids\].

             i ($^{\circ}$)   $\alpha$ ($^{\circ}$)   $\beta$ ($^{\circ}$)   $\gamma$ ($^{\circ}$)   $x_2$ ($R_{\star}$)   $x_3$ ($R_{\star}$)
  --------- ---------------- ----------------------- ---------------------- ----------------------- --------------------- ---------------------
   Model 1         90                  -90                     0                       0                     0.0                   0.0
   Model 2         15                  10                      23                     -34                    0.2                  -0.3

  : Magnetic models used for the calculations of the Stokes profiles. For the definition of the model parameters see the text.<span data-label="tab:magmod"></span>

For the synthesis of the Stokes profiles with <span>Cossam</span> we selected from the Atlas9 grid [@CastelliKurucz2004] an atmospheric model with $T_{\rm eff} = 5\,500\,K$, $\log g = 4.0$, solar metallicity, zero macro-turbulence ($v_{\rm turb} = 0.0$) and a micro-turbulent velocity of $\xi = 2.0$kms$^{-1}$. We then adopted the same dipole moment ($\boldmath{m} = 10$G) for both models, giving an effective longitudinal field $H_{\rm eff}$ of 7.65 and 5.78G respectively (for the wavelength range from 4000 to 4050<span>Å</span>; please be aware that $H_{\rm eff}$ depends on the wavelength range considered). We also calculated Stokes profiles for the "solar" case, having the magnetic field vector at the centre of the visible disk point towards the observer and setting the value of the field strength equal to the value of $H_{\rm eff}$. In order to compare the "solar" to the "stellar" (integrated) Stokes profiles, we finally normalised them to the continuum.

In what follows we shall discuss only the Stokes profiles in intensity $I$ and in circular polarisation $V$ given that in this study we are solely interested in the determination of the longitudinal magnetic field. In Fig.\[fig:comp\] we show the stellar (integrated) Stokes profiles (dashed black line) over a narrow interval of 5<span>Å</span> and the respective "solar" Stokes profiles (solid line in light colour). The difference between the stellar and the "solar" profiles is shown at the bottom of each panel. The Stokes profiles in the left (right) side panels were calculated with the grid shown in the left (right) side panel of Fig.\[fig:grids\].

From straightforward visual inspection it is clear that the "solar" profiles very closely reproduce the Stokes profiles obtained by integration of the local profiles over the visible hemisphere (there are more than 400 points for the two spatial grids). We quantify this result by evaluating the mean absolute error (MAE) of the difference between the "solar" and the "stellar" $V$ profiles (lower panels of Fig.\[fig:comp\]). Additionally, in order to test how well the effective longitudinal magnetic field can be approximated, we repeat the same procedure for the "solar" profiles but with increments of $\pm$ 0.5G in field strength. The results are listed in Table \[grids\_0km\].

  ---------- ------------------- ----------- ----------------------
    Fixed      "Stellar" field   "Solar"     MAE for
     grid     $H_{\rm eff}$ (G)  field (G)   $V$ profiles
                    7.65         8.65        $4.0 \times 10^{-5}$
   Magnetic         7.65         8.15        $2.0 \times 10^{-5}$
                    7.65         7.65        $1.0 \times 10^{-5}$
   Model 1          7.65         7.15        $2.1 \times 10^{-5}$
                    7.65         6.65        $4.0 \times 10^{-5}$
  ---------- ------------------- ----------- ----------------------

  : Comparison of $V$ profiles for the two grids in Fig. \[fig:grids\].<span data-label="grids_0km"></span>

  ---------- ------------------- ----------- ----------------------
   Adaptive    "Stellar" field   "Solar"     MAE for
     grid     $H_{\rm eff}$ (G)  field (G)   $V$ profiles
                    5.78         6.78        $4.1 \times 10^{-5}$
   Magnetic         5.78         6.28        $2.1 \times 10^{-5}$
                    5.78         5.78        $1.6 \times 10^{-5}$
   Model 2          5.78         5.28        $2.1 \times 10^{-5}$
                    5.78         4.78        $4.1 \times 10^{-5}$
  ---------- ------------------- ----------- ----------------------

  : Comparison of $V$ profiles for the two grids in Fig. \[fig:grids\].<span data-label="grids_0km"></span>

We note that for both grids the "solar" fit to the stellar $V$ Stokes profiles is highly accurate. Using the value of $H_{\rm eff}$ for input in a "solar" type polarised line synthesis, one obtains a near perfect fit; even a difference of a mere $\pm 0.5$G already leads to a less satisfactory result.

Including stellar rotation {#sec2.3}
--------------------------

In the preceding section we showed that "solar" profiles can be used instead of stellar (integrated) profiles for the analysis of stellar magnetic fields in the case of negligible rotational broadening. In this section we shall discuss what happens if we include the effects of rotational broadening.

In order to establish a proper broadening function (BF) that can be applied to the "solar" profiles $S_{\rm p}$, producing a correct fit to the stellar rotationally broadened profiles $S_{\rm rot}$, one has to solve the equation $$S_{\rm rot}(\lambda) = BF(\lambda) * S_{\rm p}(\lambda),
 \label{ecBF}$$ where the symbol "$*$" denotes the convolution of the two functions and $S$ represents any of the Stokes $IQUV$ parameters. Two approaches are possible to obtain the BF, viz. the cross-correlation (CC) or the Fourier deconvolution methods respectively. The former produces a BF which acts more like a proxy, while the latter provides a more accurate broadening function. @Rucinski1999 showed the advantages of using the Fourier deconvolution rather than CC function, but, more importantly, also showed how the deconvolution approach can be transformed into a linear system of equations which in turn can be solved through Singular Value Decomposition (SVD).

Let $m$ be the number of wavelength points of both "solar" and stellar profiles, and let $n$ be the number of points (window length) of the broadening function. The central idea is to use $S_{\rm p}$ to construct a matrix $\hat{M}$ of dimensions $(m-n) \times n$. Each column in matrix $\hat{M}$ is composed of a segment (of length $m-n$) of the profiles $S_{\rm p}$ but shifted vertically by one wavelength element. Let $S_{\rm p}^*$ denote the segment of interest containing the first $m-n$ elements of $S_{\rm p}$. The column arrangement is as follows: the last column of $\hat{M}$ is represented by $S_{\rm p}^*$, the penultimate by $S_{\rm p}^*$ shifted by one wavelength element, and so on. Finally, for the proper handling of the edges of the profiles $S_{\rm rot}$, a reduction from $m$ to $ m-n $ elements is applied. This allows to remove $n/2$ points from each side of $S_{\rm rot}$ – more details can be found in @Rucinski1992 and in @Rucinski1999. By applying this rearrangement, Eq.(\[ecBF\]) is transformed to a linear system of equations:

$$\hat{M}(\lambda) \ BF(\lambda) = S_{\rm rot}(\lambda).
 \label{ecM}$$

This system of equations is solved applying SVD to $\hat{M}$, from which we obtain the orthonormal matrices $\hat{U}$ and $\hat{V}$ and the vector $w$, with respective dimensions of $(m-n) \times n$, $n \times n$, and $n$ (see e.g. @GolubvanLoan1996). With the elements of the vector $w$, it is possible to define the square matrix $\hat{W_1}$ (with dimensions $n \times n$) which is different from zero only in the diagonal elements as $W_1(i,i)=w(i)$. It is then possible to write the matrix $\hat{M}$ as the product of these 3 matrices $$\hat{M}(\lambda) = \hat{U}(\lambda)\hat{W_1}(\lambda)\hat{V^{\rm T}}(\lambda).$$

![Upper left panel: a wide broadening function using a spectral window of 301 points. Upper right panel: optimal BF$(x)$ with 91 spectral points. The independent variable $x$ has been substituted for $\lambda$ in order to emphasise that the sampling rate is at constant velocity steps (1.0kms$^{-1}$ in our case). Lower panel: comparison of $V$ Stokes parameters as a function of the number of eigenvectors included in the BF. The optimum fit achieved with 91 eigenvectors is indicated by an arrow.<span data-label="fig:BF"></span>](broad_funct_v10.eps "fig:"){width="10cm"} ![Upper left panel: a wide broadening function using a spectral window of 301 points. Upper right panel: optimal BF$(x)$ with 91 spectral points. The independent variable $x$ has been substituted for $\lambda$ in order to emphasise that the sampling rate is at constant velocity steps (1.0kms$^{-1}$ in our case). Lower panel: comparison of $V$ Stokes parameters as a function of the number of eigenvectors included in the BF. The optimum fit achieved with 91 eigenvectors is indicated by an arrow.<span data-label="fig:BF"></span>](eigen_vectors_v10.eps "fig:"){width="9cm"}

Let us now define $\hat{W}$ as the diagonal matrix whose elements are the inverse of the diagonal elements of $\hat{W_1}$, viz. $W(i,i)=1/w(i)$. Finally, taking the inverse of the orthonormal matrices $U$ and $V$ (their transposes), it becomes possible to recover the broadening function as (omitting the dependence on $\lambda$): $$BF = \hat{V} \hat{W} \hat{U^T}S_{\rm rot}.
 \label{ec_solBF}$$

SVD has many interesting properties, one of them being that the eigenvectors in $\hat{U}$ and $\hat{V}$ are arranged in order of importance. This fact allows a reduction of dimensionality without loss of useful information. In practice, we implement the dimensionality reduction by storing in the diagonal of the matrix $\hat{W}$ a number $l < n$ of elements. To illustrate how to find the optimal number $l$, we use the circular Stokes profiles shown in the right panels of Fig.\[fig:comp\]. Let $S_{0}$ denote the "solar" Stokes profiles and $S_{10}$ the stellar (integrated) Stokes profiles calculated with the adaptive grid and $v\,\sin i = 10\,{\rm km}\,{\rm s}^{-1}$. We first calculate the Stokes profiles with only one eigenvector ($l = 1$) using Eq.(\[ecBF\]), i.e. $S_{10} = {\rm BF}_{l=1} * S_{0}$. Subsequently, we increase $l$ gradually until the full length of the BF is reached ($l = n = 301$).

![image](bf_lands_5kms.eps){width="10cm"} ![image](bf_stift_5kms.eps){width="10cm"}

![image](bf_lands_10kms.eps){width="10cm"} ![image](bf_stift_10kms.eps){width="10cm"}

![image](bf_lands_20kms.eps){width="10cm"} ![image](bf_stift_20kms.eps){width="10cm"}

In Fig.\[fig:BF\] we show the BF using the full length of the spectral window (upper left panel), and the one that gives the best fit (upper right panel). The accuracy of the fits as a function of the number of eigenvectors included while constructing the BF is displayed in the lower panel. Note that the fit of the profiles improves quickly as the number of eigenvectors increases, it then enters a plateau phase, subsequently improving once more until reaching the optimum fit ($l = 91$), and finally entering a second plateau phase. The explanation for particular behaviour is twofold. From one side, the broadening regions do not contain information about the broadening process (e.g. @Rucinski1992), and from the other side, the numerical value in these continuum regions is very close to zero but is not exactly zero. These two properties make that the curve shows this two plateau behaviour. We have corroborated this fact by adding noise to the profiles, and in this case the fits improve monotonically until the optimum fit is reached. Of course, the fit is better when there is no noise. In the noise-free case, the difference between "solar" and stellar $V$ profiles for the optimum fit is gratifyingly small (MAE of 8.6$\times$ 10$^{-6}$), and in fact even better than the fits found for the case of $v\,\sin i$ = 0 listed in Table2. It has to be kept in mind that in real observations, the latest eigenvectors are associated with noise. Hence, it is important to carefully determine the optimum number of eigenvectors to use in the construction of the BF.

We applied the broadening procedure to the Stokes profiles shown in Fig.\[fig:comp\] with values of $v\,\sin i$ of 5, 10, and 20kms$^{-1}$. It turns out that the adaptive grid is very sensitive to the value of $v\,\sin i$. We have therefore modified the parameters of the adaptive grid in such a way as to obtain a number of points comparable to the one used for the fixed grid (close to 400). The resulting profiles are shown in Fig.\[fig:rota\_V\] and demonstrate that the broadening procedure applied to the "solar" profiles is working properly; please note that the BF properly reproduces both the broadening due to rotation and the broadening due to the distribution of magnetic field strength across the stellar surface. This will become even clearer in the next section where we will consider higher magnetic field strengths.

Multi-line technique
====================

We now want to investigate whether it is possible to generalise our results to spectral ranges of thousands of Angstroms, to strong magnetic fields and to non-zero rotational velocities. For this purpose, in what follows we shall employ the PCA-ZDI multi-line technique developed by @Semeletal2006. Details concerning the addition properties of individual lines in the context of this technique have been presented in @Semeletal2009, and a detailed description of the procedure used to obtain the MZSs and a discussion of some of their properties are given in @Ramirezetal2010. Please note that "Zeeman Doppler Imaging" is used here in the original sense introduced by @SemelSe1989, not to be confounded with the <span>*mapping*</span> of magnetic and/or abundance structure as e.g. in @Hussainetal2000. The PCA-ZDI technique is based on a representative database of synthetic Stokes spectra, calculated in the present case with the help of <span>Cossam</span>; PCA is used to obtain the eigenvectors of that database. Subsequently, these eigenvectors serve as detectors (in analogy to the line mask used in LSD), in order to obtain the MZSs through a cross correlation between the observed spectra and the eigenvectors. In order to prevent confusion, let us emphasise that the eigenvectors used in the PCA-ZDI technique have nothing to do with the eigenvectors used for the construction of the BFs described previously. In fact, the eigenvectors used in PCA-ZDI form the basis of the Stokes parameters in the synthetic database, such that any Stokes profile in that database can be represented by a linear combination of the PCA-ZDI eigenvectors. For more details see the three references given above.

With the PCA-ZDI technique it is possible to obtain one MZS per eigenvector, i.e. as many as the model spectra contained in the database (a total of 700 for this work, see below). However, the first eigenvector is the most significant one, exhibiting the most representative line shape pattern of all models included in the database. Thus, in order to simplify the interpretation of the results, for our tests and analyses we shall employ only the MZS obtained with the first eigenvector.

In addition, given that nowhere in the whole procedure has similarity of the individual lines been assumed nor any special regime of the magnetic field strength and direction, our technique does not suffer any of the constraints typical for LSD. Our only restriction pertains to a minimum ratio of line depth to continuum, which for this work we have fixed at $> 0.1$, resulting in a total number of individual lines used for the construction of MZSs of $\sim$ 38000.

The first magnetic detections with the help PCA-ZDI in linear and circular polarisation were presented a decade ago (@Semeletal2006, @Ramirezetal2006), but magnetic field quantities like $H_{\rm eff}$ have not yet been determined with this technique. In the following we show how stellar effective longitudinal magnetic fields can be retrieved by application of PCA-ZDI.

![Left panel: optimal BF, derived with 41 eigenvectors. Right panel: MZS profiles. The red line represents the "solar" MZS ($v\,\sin i = 0$kms$^{-1}$), the dashed black line the stellar MZS ($v\,\sin i = 10$kms$^{-1}$), and the solid line in cyan the result of the application of the BF to the "solar" MZS. The green line at the bottom of the panel, shifted for clarity's sake, represents the difference between the "solar" broadened MZS and the stellar MZS. The stellar Stokes profiles were calculated using Model1, a dipolar magnetic moment of $m=400$G and a fixed grid of 440 quadrature points.<span data-label="fig:fit_mzs1"></span>](bf_mzsv_lands10.eps){width="3.3cm"}

![Left panel: optimal BF, derived with 41 eigenvectors. Right panel: MZS profiles. The red line represents the "solar" MZS ($v\,\sin i = 0$kms$^{-1}$), the dashed black line the stellar MZS ($v\,\sin i = 10$kms$^{-1}$), and the solid line in cyan the result of the application of the BF to the "solar" MZS. The green line at the bottom of the panel, shifted for clarity's sake, represents the difference between the "solar" broadened MZS and the stellar MZS. The stellar Stokes profiles were calculated using Model1, a dipolar magnetic moment of $m=400$G and a fixed grid of 440 quadrature points.<span data-label="fig:fit_mzs1"></span>](fit_mzsv_lands10.eps){width="5.4cm"}

\[h\]

![Same as Fig. \[fig:fit\_mzs1\] but referring to Model2 (see Table \[tab:magmod\]).<span data-label="fig:fit_mzs2"></span>](bf_mzsv_lands10_misaligned.eps){width="3.3cm"}

![Same as Fig. \[fig:fit\_mzs1\] but referring to Model2 (see Table \[tab:magmod\]).<span data-label="fig:fit_mzs2"></span>](fit_mzsv_lands10_misaligned.eps){width="5.4cm"}

MZS using PCA-ZDI
-----------------

We have considered a wide spectral range from 350 to 1000nm in steps of 1kms$^{-1}$ (yielding a total of $\sim$ 315000 wavelength points). Adopting the same atmospheric model as before, we built a database of Stokes profiles for the "solar" case, with the magnetic field varying between 0 and 350G in steps of 0.5G. From this database we obtained the eigenvectors, but we then use only the first eigenvector to produce a total of 700 "solar" MZSs. Our goal is to retrieve $H_{\rm eff}$ – a "stellar" (integrated) quantity – by converting "solar" MZSs to stellar MZSs, similarly to what we did in the preceding section.

For this purpose, let us consider the stellar MZS obtained from Model1 of Table\[tab:magmod\], $v\,\sin i = 10$kms$^{-1}$ and a magnetic moment of $m = 400$G. The effective field for this configuration at phase zero is 256G. We then search in the "solar" space of MZSs for a model with field strength equal to the value of $H_{\rm eff}$. Now we are confronted with the same situation as before, i.e. we have to apply the proper broadening to the "solar" MZS to reproduce the stellar MZS but with the difference that instead of working with spectral lines, viz. Eq.\[ecBF\], we work with the MZSs: $$MZS_{\rm rot}(X) = BF(X) * MZS_{\rm p}(X),
\label{BFmzs}$$ where $MZS_{\rm rot}$ stands for the stellar MZS, $MZS_{\rm p}$ for the "solar" MZS and $X$ indicates that we are working in Doppler space.

The broadening function is found as explained in the preceding section (\[sec2.3\]). In the left panel of Fig.\[fig:fit\_mzs1\] we show the optimum BF and in the right panel the "solar" MZS (red line), the stellar MZS (dashed black line), and the "solar" broadened MZS (full line in cyan). The difference between the last two MZSs is shown at the bottom of the panel (full line in green).

![Upper panel: Inversion of MZSs using the fixed grid with a centred dipole pointing towards the observer (Model1 of Table \[tab:magmod\]). The various values of $v\,\sin i$ are shown with different symbols and colours. Lower panel: Inversion errors.<span data-label="fig:inv_l"></span>](mzs_inv_lands.eps){width="8.5cm"}

We now want to demonstrate that a BF established this way can be applied irrespective of the strength of the effective longitudinal field $H_{\rm eff}$. To this purpose, we simply apply the BF to all "solar" MZSs and we look whether the $H_{\rm eff}$ values for a set of stellar MZSs with different magnetic field strengths can correctly be recovered. Please note that the inversions concern only circular Stokes profiles, they do not include the MZSs in intensity. Varying the magnetic moment (m = \[5, 10, 20, 30 ... 490G\]), we produce 50 stellar MZSs, carrying out this procedure for the three values of $v\,\sin i$ previously used: 5, 10 and 20kms$^{-1}$. It has to be emphasised that for each value of $v\,\sin i$ it is mandatory to find the appropriate BF. The results displayed in Fig.\[fig:inv\_l\] clearly show that our approach leads to the correct retrieval of the stellar $H_{\rm eff}$.

Nevertheless, it is important to realise that the BF also depends on the orientations of the principal axes of the system – combinations of the 3 Eulerian angles $\alpha$, $\beta$, $\gamma$ and of the inclination angle $i$ between rotation axis and the LOS. To illustrate this dependence, we take the example shown in Fig.\[fig:fit\_mzs1\] and we recalculate it with the only difference that the angles of the system change from those of Model1 to those of Model2. For this new configuration, the longitudinal field value drops to $H_{\rm eff} = 212$G, but much more importantly, the shape of the MZSs and the BF change drastically (see Fig.\[fig:fit\_mzs2\]).

We have established that for the recovery of the effective longitudinal field $H_{\rm eff}$, one cannot use some arbitrary BFs. In other words, not only for each rotational velocity, but also for each orientation of the principal axes of the system there exists a particular associated BF that allows the proper recovery of $H_{\rm eff}$. To illustrate this fact, we repeat the field inversions of the 50 stellar MZSs calculated with Model1, but this time to broaden the "solar" MZSs we used the BF of Model2 (left panel of Fig.\[fig:fit\_mzs2\]). From the results (not shown) it becomes abundantly clear that the field inversions fail completely. In practice, when dealing with observational data, it is not possible to know a priori the orientation of the principal axes. We thus have to devise a method that generalises the field inversion procedure to any arbitrary orientation.

Magnetic field inversions for arbitrary orientations
----------------------------------------------------

\[ht\]

![image](plot_bf_inv_mzs_centred_dipole.eps){width="15cm"}

We now want to demonstrate that it is possible to correctly recover the effective longitudinal field from MZSs for arbitrary combinations of the angles $\alpha$, $\beta$, $\gamma$ and of the inclination $i$. Considering $v\,\sin i = 10$kms$^{-1}$, adopting a magnetic moment of $m = 400$G (not implying any loss of generality) and randomly varying the four angles, we calculated 600 different stellar MZSs at phase zero. For each combination of the angles, we obtained a particular stellar MZS with an attached value of $H_{\rm eff}$ and an associated BF. Each of these 600 BFs were obtained following the same process that we employed in Figs.\[fig:fit\_mzs1\] and \[fig:fit\_mzs2\], i.e. by solving Eq.(\[BFmzs\]) based on the method described in section\[sec2.3\]. All the BFs were constructed consistently with the same number (41) of eigenvectors. Now let BF$_{n}$ denote the broadening function associated with the n-th combination of angles. This BF$_{n}$ was directly applied to the 700 "solar" MZSs – each one having a different value of field strength from 0 to 350G in steps of 0.5G. Repeating the process for all the BFs results in a total of $600 \times 700$ "solar" broadened MZSs, all of which are characterised by a particular combination of angles and magnetic field strength.

Considering the same rotation value of $v\,\sin i = 10$kms$^{-1}$, we then calculated a set of 100 stellar MZSs with random variations of the four angles $\alpha$, $\beta$, $\gamma$, $i$ and of the dipolar magnetic moment – between 0 and 490G. We inverted these stellar MZSs in the database made up of the broadened "solar" MZSs. Fig.\[fig:fit\_mzs\_mvariable\] shows detailed fits of 3 among the 100 stellar MZSs, suggesting that our proposed technique is able to fairly accurately recover the effective longitudinal field of a star. The results for the whole bulk of 100 stellar MZSs is displayed in Fig.\[fig:mzs\_mvariable\] and definitely validates the excellent performance of the method, reflected in gratifyingly small errors.

These encouraging results were obtained, implicitly, only for the special case when the dipole position is known a priori (in our latest example at the centre of the star). More generally, one has however to consider all the parameters of the magnetic model to be unknown, a problem which we shall address in the following.

![Recovery of $H_{\rm eff}$ from stellar MZSs (at left) and uncertainties (at right), considering $\alpha$, $\beta$, $\gamma$, $i$ and the magnetic moment $m$ as variable parameters. The MZSs were calculated using an adaptive grid ($\sim$80 points).<span data-label="fig:mzs_mvariable"></span>](bf_inv_mzs_centred_dipole2.eps){width="9cm"}

Retrieving $H_{\rm eff}$ in the general case
--------------------------------------------

We performed a similar test to the one presented in the preceding section but now including as unknowns the position and direction of the dipole. We calculated 7500 stellar MZSs, varying randomly all the parameters in the magnetic model ($\alpha$, $\beta$, $\gamma$, $i$, $x_2$, $x_3$). The position of the dipole was constrained to less than 0.3$R_{\star}$. The angles ($\alpha$, $\beta$, $\gamma$) were allowed to vary between $-180\deg$ and $+180\deg$, and the inclination $i$ between $0\deg$ and $+180\deg$. Please note that with these limits to the range of the Eulerian angles, there is a certain redundancy in the resulting models, so that we can fix the rotational phase kristenherring@example.org. Finally, we arbitrarily adopted a magnetic moment of 400G as before and we also assumed a value of $v\,\sin i = 10$kms$^{-1}$.

For each of these MZS we repeated the procedure described in the last section, i,e. for any given combination of magnetic parameters we obtained an associated MZS with a particular value of $H_{\rm eff}$ and an attached BF. Considering all the calculated magnetic models we obtained a total of 7500 BFs which we applied to broaden the "solar" MZSs, reaching a total of $7500 \times 700$ "solar" broadened MZSs. Again, each of these "solar" MZSs is distinguished by a particular combination of magnetic geometry and field strength. Then, to test the inversions, we calculated a set of 100 stellar MZSs adopting $v\,\sin i = 10$kms$^{-1}$ and random values of the four angles, of the dipole moment (between 0 and 690G) and of the dipole position ($< 0.3 \, R_{\star}$). We show the results of the inversions in Fig.\[fig:mzs\_mvariable2\].

These plots reveal that none of the geometric model parameters ($\alpha$, $\beta$, $\gamma$, $i$, $x_2$, $x_3$) can be correctly retrieved. This is not surprising since for a given value of $H_{\rm eff}$ a multiplicity of combinations of angles, magnetic moments and dipole positions can produce the same stellar longitudinal field. Nevertheless the value of $H_{\rm eff}$ is in general well retrieved. The inversion errors for $H_{\rm eff}$ are larger than in the case of a fixed dipole position. Besides, in our tests we have considered as fixed the atmospheric model ($T_{\rm eff}$, $log\,g$, $[M/H]$, $v_{\rm turb}$, $\xi$) and for this reason we have not included Stokes $I$ in the inversions; when inverting simultaneously the Stokes $I$ and $V$ parameters, the case of real data, the inversion incertitudes will increase. However, please note that a way to reduce the inversions errors is to increase the number of BFs considered (7500 in this test). We thus may conclude that it is possible with our method to correctly measure stellar longitudinal magnetic fields.

![Inversion – in the general case – of stellar MZSs, using a fixed grid (80 points) and considering all the parameters in the magnetic model ($\alpha$, $\beta$, $\gamma$, $i$, $x_2$, $x_3$, $m$) as unknowns. In each panel the x-axis gives the values of the input parameters to the stellar MZS. The y-axis of the upper right panel gives the inversion errors for the $H_{\rm eff}$, for the rest of the panels it gives the parameter values retrieved by the inversions.<span data-label="fig:mzs_mvariable2"></span>](inv_mzs_dipole_descentrado.eps){width="8.2cm"}

Spatial grid resolution with PCA-ZDI profiles
---------------------------------------------

Throughout this work we have used both adaptive and fixed grids at different spatial resolutions. Now we want to show that the database used in the last test, based on 80 spatial quadrature points, can be used to invert MZSs calculated at much higher spatial grid resolution (up to 1000 points and more). For this purpose we have considered two arbitrary magnetic models whose parameters are listed in Table \[tab:grids\]. Taking an adaptive grid with 80 quadrature points and a magnetic moment of 39.26G, for Model1 one obtains $H_{\rm eff} = -24.32$G at phase zero. Similarly, adopting a value of $m = 395.78$G, we find $H_{\rm eff} = 228.90$G at phase 0.526 for Model2.

As mentioned right at the beginning, $H_{\rm eff}$ is a quantity that depends on the limb darkening, being thus potentially affected by the number and the spatial distribution of the quadrature points. We have inspected this problem with the help of the two magnetic models of Table \[tab:grids\], varying the grid resolution. In Fig. \[heff\_grid\] we plot $H_{\rm eff}$ as a function of the number of grid points; the values of $H_{\rm eff}$ are given relative to the value of $H_{\rm eff}$ calculated with an 80 point grid. We see a modest but significant change in $H_{\rm eff}$ when the number of grid points is increased, reaching a difference close to 4% and 3% respectively for Models1 and 2 at $\sim$500 grid points. Further increasing the number of points gives only very small variations in $H_{\rm eff}$. For Model1 $H_{\rm eff}$ remains essentially constant when the number of grid points is in excess of 600, whereas for Model2 the normalised longitudinal field passes from 97.2% to 96.9% between 502 and 1096 grid points. We therefore consider a grid resolution of about 1000 points sufficient for highly accurate $H_{\rm eff}$ determinations in both models.

             i ($^{\circ}$)   $\alpha$ ($^{\circ}$)   $\beta$ ($^{\circ}$)   $\gamma$ ($^{\circ}$)   $x_2$ ($R_{\star}$)   $x_3$ ($R_{\star}$)
  --------- ---------------- ----------------------- ---------------------- ----------------------- --------------------- ---------------------
   Model 1       124.7                107.4                   16.5                   -20.9                  0.22                  -0.11
   Model 2        26.1                -56.3                   25.0                   63.6                   0.18                  0.20

  : Magnetic models used to test the spatial grid resolution.<span data-label="tab:grids"></span>

![Percentual variation of $H_{\rm eff}$ as a function of the number of quadrature points considered in the adaptive grid. The $H_{\rm eff}$ values are normalised to the $H_{\rm eff}$ value obtained with a grid of 80 points.<span data-label="heff_grid"></span>](plot_heff_grids.eps){width="9cm"}

![Examples of 2 stellar MZSs constructed at very high grid resolution ($>$1000 points) using adaptive grids. The dashed black lines represent the stellar MZSs while the solid lines in cyan colour are their respective inversions. The retrieved values of the $H_{\rm eff}$ (shown at the top of the panels) clearly indicate that the PCA-ZDI profiles are quite insensitive to the grid resolution.<span data-label="hr_stift"></span>](hr_mzs_stift.eps){width="9cm"}

![Same as Fig. \[hr\_stift\] but considering a fixed grid of 1100 points in the construction of the stellar MZSs.<span data-label="hr_lands"></span>](hr_mzs_lands.eps){width="9cm"}

We thus established the stellar MZSs for Models1 and 2 at high resolutions (1096 and 1146 points respectively), employing an adaptive grid. We then inverted these two MZSs in the "solar" broadened database used for the test presented in the preceding section. Please note that the BFs applied to broaden the "solar" MZSs of that database were obtained from stellar MZSs established with a fixed grid of 80 points. We then performed the same inversions again, but with only difference that now we used a fixed grid of 1100 points to establish the two stellar MZSs. The results shown in Figs. \[hr\_stift\] and \[hr\_lands\] clearly reveal that we can properly retrieve $H_{\rm eff}$ from the stellar MZSs constructed at high grid resolution.

Let us explain the reasons for this gratifying result. After inspection of the $V$ profiles at different grid resolutions, we found that in general differences in the integrated stellar Stokes profiles are small; only for very few lines the spatial grid resolution can have some influence. For example, comparing the adaptive grids at high and low resolution, 1046 and 80 points respectively, for Model1 we found a MAE of the $V$ profiles of $4.5\, \times 10^{-5}$, whereas the maximum difference in the line profiles is 2.3%. Accordingly, when constructing the MZSs through the addition of thousands of lines, the differences in the Stokes profiles will cancel out statistically, permitting the correct retrieval of $H_{\rm eff}$. One would however expect the resolution of the spatial grid to have more important repercussions on the synthesis of Stokes profiles in the presence of very strong magnetic fields, fast stellar rotation or strong stellar pulsations (e.g. @Fensl1995), but this is a case outside the scope of the present study.

To conclude, the results of Figs. \[hr\_stift\] and \[hr\_lands\] are very encouraging since the most time consuming part of our method lies in the computation of thousands of stellar MZSs that serve to obtain the BFs that we use to broaden the "solar" MZSs. The fact that for the calculation of the stellar MZSs we can employ a relatively low spatial grid resolution without compromising the results makes our approach less expensive than might have been feared.

PCA-ZDI and (magnetic) broadening
=================================

\[ht\]

![image](h_sup3.eps){width="17cm"}

Given that we obtained the stellar spectra through the integration of the local Stokes profiles is that the surface distribution of the local magnetic fields is known.

Let us for example have a look at the 100 MZSs employed for the inversion test shown in Fig.\[fig:mzs\_mvariable2\]. Recall that for the establishment of this set of MZSs all the parameters of the magnetic model were randomly varied. In the upper panels of Fig.\[fig:hsup\] we plot the mean, minimum and maximum values of the field modulus as a function of $|H_{\rm eff}|$ for the full set of MZSs. The lower panels display their respective cumulative distributions. Not surprisingly, small absolute values of $H_{\rm eff}$ can be accompanied by large field moduli which means that $H_{\rm eff}$ cannot be taken as an indicator for the validity of the weak-field approximation. In our sample for example we encounter a value of $H_{\rm eff} = 24.6$G, associated with mean, minimum and maximum values of 1443, 797, and 2120G respectively. Unlike other multi-line techniques, the PCA-ZDI technique allows us to correctly recover $H_{\rm eff}$ irrespective of the distribution and intensities of the local magnetic fields (see upper panels of Fig.\[fig:mzs\_mvariable2\]).

To summarise, our approach does not require any of the following assumptions:

1.  That the local residual Stokes $I$ intensities are small (see e.g. discussion by @Sennhauseretal2009 and by @KochukhovMaPi2010),

2.  That the magnetic broadening is largely inferior to the non-magnetic broadening (thermal, rotational, micro/macro turbulence),

3.  That the Zeeman splitting results in a normal triplet for all lines.

In other words, with the PCA-ZDI technique we can properly deal with blended lines, take into account the anomalous Zeeman effect or negative Land<span>é</span> factors for individual lines, and work in the presence of strong magnetic fields such that the spectral lines are in the regime of Zeeman saturation, i.e. the amplitudes of the $V$ profiles do not grow anymore linearly with the intensity of the magnetic field. This considerably widens the application range of our method compared to other multi-line techniques like for example LSD.

In addition, we have verified that the weak field approximation can not be applied to retrieve $H_{\rm eff}$ from the MZSs. We find that the shape of the derivative of the Stokes $I$ MZS does not correspond to the shape of the Stokes $V$ MZS, indicating that PCA-ZDI is incompatible with the WFA. The reason for this, is simply that our technique does not take into account the requiered asummptions of the WFA to be valid.

Finally, we would like to emphasise that the fits to the MZSs (Figs.\[fig:fit\_mzs\_mvariable\],\[hr\_stift\],\[hr\_lands\]) clearly indicate that magnetic and rotational broadening are simultaneously well been taken into account by the BF method.

Conclusions
===========

Spectropolarimetry constitutes the optimum observational technique for the study of solar and stellar magnetism. These data are best analysed with the help of codes that implement spectral line synthesis in the presence of magnetic fields, solving the coupled equations of polarised radiative transfer. It is highly desirable that any multi-line approach to the determination of global magnetic quantities such as the effective longitudinal field $H_{\rm eff}$ or the mean magnetic field modulus $H_{\rm s}$ should incorporate full opacity sampling and a correct treatment of polarised radiative transfer when calculating local Stokes $IQUV$ profiles. This is exactly what we have done with the help of a database of Stokes profiles established with the help of the polarised line synthesis code <span>Cossam</span> and MZSs obtained by means of the PCA-ZDI technique.

Efforts to derive $H_{\rm eff}$ from MZSs obtained with PCA-ZDI date back to @Semeletal2009 who showed that the centre of gravity method can be applied to the MZSs for an estimate of magnetic field strengths. Additionally, using all four Stokes parameters, @Ramirezetal2010 demonstrated that MZSs correctly encode the information on the magnetic field, meaning that both strength and orientation of the magnetic field can successfully be recovered for field strengths of up to 10kG. In these two papers, the results were obtained using "solar" MZSs, i.e. any effect of rotation was neglected and no inhomogeneous spatial distribution of magnetic strengths over the surface was assumed.

The present study introduces a novel approach that can be applied to the analysis of spectropolarimetric data. The principal idea behind this method consists in the use of broadened "solar" MZSs to infer the effective longitudinal field $H_{\rm eff}$. We have shown that the BFs are an effective tool to properly reproduce the rotational and magnetic broadening effects on the Stokes profiles in wavelength space, much as they are for the MZSs in Doppler space. We have tested our approach for different moderate values of $v\,\sin i$, obtaining good results in all cases. Gratifyingly enough it has turned out that the results do not depend on whether one uses fixed grids or adaptive grids.

<span>Cossam</span>, the polarised spectral line synthesis code employed in this work, is considered a reference for the calculation of stellar Stokes parameters (@Wadeetal2001, @CarrollKopStr2008). For the inversion – in the general case – of the stellar MZSs, we adopted fixed stellar atmospheric model parameters ($T_{\rm eff}$, $log\,g$, $[M/H]$, $v_{\rm turb}$, $\xi$), but we freely varied the parameters of the magnetic model incorporated in <span>Cossam</span>, viz. the magnetic moment $m$, the Eulerian angles $\alpha$, $\beta$, $\gamma$, the vector of the dipole offset \[0, $x_2$, $x_3$\] and the inclination $i$. We were able to demonstrate that "solar" broadened MZSs are able to reproduce the shape of stellar MZSs, providing us with the possibility to determine $H_{\rm eff}$. Currently, <span>Cossam</span> assumes a tilted eccentric dipole but our technique should also work for quadrupolar or higher order configurations, even for magnetic fields concentrated in starspots.

Computing times for the (point-source) "solar" MZSs are of course largely inferior to those for (integrated) stellar MZSs which increase proportionally to the number of spatial quadrature points. Still, for the approach presented here extensive calculations are required to arrive at a significant number of stellar MZSs needed to establish the BFs that serve to broaden the "solar" MZSs. About 3 weeks had to be spent on a 56 core workstation to obtain 7500 stellar MZSs covering the interval from 350 to 1000nm in steps of 1kms$^{-1}$, and adopting an 80 point spatial grid. It might at this point legitimately be asked why one should spend such an effort on the determination of $H_{\rm eff}$ instead of trying to directly model the stellar MZSs. Given the huge number of combinations in the parameter space of tilted eccentric dipole model, an obvious reason for wanting to know the effective longitudinal field at various phases is to reduce this number, possibly by a very large amount. The availability of (even modestly sized) supercomputers can greatly increase the attractiveness of our method. In a forthcoming paper we shall present the application of this technique to observational data.

Acknowledgements {#acknowledgements .unnumbered}
================

JRV, SGN and LS acknowledge support from the <span>CONACyT</span> grants 240441, 168078 and 180817 respectively. Part of the results presented here have been obtained using the <span>*"Supercomputo - DGTIC"*</span> facilities, grant SC16-1-IR-40, of the UNAM, the computers from the <span>CONACyT</span> project 153985 and the UNAM-PAPIIT project 107215, and the computers "Tychos" (Posgrado en Astrofisica-UNAM, Instituto de Astronomia-UNAM and <span>PNPC-CONACyT</span>). LS also acknowledges support from the grant UNAM-DGAPA-PAPIIT IA101316. Thanks go to AdaCore for providing the GNAT GPL Edition of its Ada compiler.

{'source': 'arXiv_out/1609'}