Least-squares Deconvolution Method Research Articles

Applications of the non-negative least-squares deconvolution method to analyze energy-dispersive x-ray fluorescence spectra.

We used the Monte Carlo simulation method to establish a detector response matrix and the non-negative least-squares method to deconvolute x-ray spectra. The simulation and experimental data verified the effectiveness of this method, and the influence of full-width at the half of the maximum calibration accuracy on the deconvolution results was investigated. The non-negative least-squares method had high accuracy and efficiency compared with others. The results showed that, except for Zn, the relative errors between the inversion and the standard values were less than 0.1% for the simulated spectra. For the experimental data, the relative errors were within 0.2%. The peaks with similar characteristic energies can be better distinguished in the deconvolution spectra, reducing the errors caused by overlapping peaks in subsequent analysis.

Pushing least-squares deconvolution to the next level: Application to binary stars

Context. Eclipsing, spectroscopic double-lined (SB2) binaries remain the prime source of precise and accurate fundamental properties of stars. Furthermore, high-cadence spectroscopic observations of the eclipse phases allow us to resolve the Rossiter-McLaughlin (RM) effect, whose modelling offers the means to probe spin-orbit misalignment in binaries. Aims. We aim to develop a method that provides precise and accurate measurements of radial velocities (RVs) of both binary components, including the in-eclipse orbital phases where line profiles are subject to large distortions due to the RM effect. We also intend to separate spectral contributions of the primary and secondary components in the velocity space in order that a time series of the separated spectroscopic signals can be obtained throughout the binary orbit, preserving any line-profile variability (LPV) that might be present in either or both of those spectroscopic contributions. Methods. In this study, we provide a generalisation of the least-squares deconvolution (LSD) method to SB2 systems. Our LSD-Binary algorithm is capable of working with both in-eclipse and out-of-eclipse spectra as input, and delivers the LSD profiles, LSD-based model spectra, and precise RVs of both binary components as output. We offer an option to account for the RM effect in the calculation of the initial guess LSD profiles and components’ flux ratio, such that the effect can be modelled within the algorithm itself. In that case, the algorithm delivers both the LSD profiles and RVs, which are no longer distorted by the RM effect. Otherwise, when geometry of the RM effect is ignored in the calculation of the initial guess, the LSDBinary algorithm delivers an RV curve that contains contributions from both the orbital motion of the star and spectral line distortions due to the RM effect. Results. In this study, we provide an extensive test of the LSDBinary software package on simulated spectra of artificial binaries resembling Algol-type systems and detached binaries with similar components. We study the effects of signal-to-noise ratios (S/N) of input spectra, the resolving power of the instrument, uncertain atmospheric parameters of stars, and orbital properties of the binary system on the resulting LSD profiles and RVs measured from them. We find that atmospheric parameters have a negligible effect on the shape of the computed LSD profiles while affecting mostly their global scaling. High-resolution (R ≳ 60 000) spectroscopic observations are required in order to investigate the RM effect in detail, although a medium resolving power of R ≈ 25 000–30 000 might suffice when the amplitude of the effect is large. Our results are barely sensitive to the S/N of the input spectra provided they contain a sufficient number of spectral lines, such as in A-type and later stars. Finally, the orbital inclination angle and the components’ radii ratio are found to have the largest effect on the shapes of the LSD profiles and RV curves extracted from them. Conclusions. The LSDBinary algorithm is specifically developed to perform detailed spectroscopic studies of eclipsing SB2 systems whose orbital configuration and components’ atmospheric parameters are estimated by other means. The algorithm is well suited to study the RM effect, as well as to compute the separated LSD profiles of both binary components from the observed composite in-eclipse spectra of SB2 systems.

Analysis of high-resolution FEROS spectroscopy for a sample of variable B-type stars assembled from TESS photometry,

Context. Spectroscopic data are necessary to break degeneracies in the asteroseismic modelling of the interior structure in high- and intermediate-mass stars. With the TESS mission, the number of bright intermediate-mass B-type stars with long photometric light curves that are suitable for detailed asteroseismic studies has increased substantially compared to the pre-TESS era. Aims. We derive precise photospheric stellar parameters for a sample of 166 B-type stars with TESS light curves through a homogeneous spectroscopic analysis. The variability types of these sample stars are also classified based on all currently available TESS sectors, and they are ultimately prioritised according to their astrophysical potential. Methods. We obtained high-resolution spectra for all 166 targets with the FEROS spectrograph in the context of a large program. The spectra were reduced with the CERES pipeline, which we adapted to improve the quality of the reduced spectra. These spectra were subsequently analysed with ZETA-PAYNE, a machine-learning-based spectrum analysis algorithm, to infer precise stellar labels for all stars in the sample. Furthermore, the least-squares deconvolution (LSD) method was employed to investigate spectral line profile variability (LPV) and isolate binary systems from presumably single stars. Results. The LSD profile analysis identified 26 spectroscopic double-lined binaries; the remainder of the sample are 42 supergiants in the Large Magellanic Cloud galaxy and 98 Galactic stars, both with and without apparent LPV. For the Galactic single stars and single-lined spectroscopic binaries, we determine their five main surface parameters: effective temperature (Teff), surface gravity (log g), global metallicity ([M/H]), projected rotational velocity (v sin i), and microturbulent velocity (ξ) with average formal precisions of 70 K, 0.03 dex, 0.07 dex, 8 km s−1, and 0.7 km s−1, respectively. The average internal uncertainties we find for FEROS spectra with our spectrum analysis method are 430 K(Teff), 0.12 dex (log g), 0.13 dex ([M/H]), 12kms−1 (v sin i), and 2 kms−1 (ξ). Conclusions. We find spectroscopic evidence that 8 of the 98 galactic single or SB1 variables are fast-rotating gravity-mode pulsators occurring in between the slowly pulsating B (SPB) stars and δ Scuti instability strips. The g-mode frequencies of these pulsators are shifted to relatively high frequency values due to their rotation, and their apparently too low Teff relative to the SPB instability region can in most cases be explained by the gravity darkening effect. We also discover 13 new HgMn stars in the Galactic sample of which only one is found in a spectroscopic binary, resulting in a biased and therefore unreliable low binary rate of only 8%.

The GAPS Programme at TNG

Context. Giant planets in short-period orbits around bright stars represent optimal candidates for atmospheric and dynamical studies of exoplanetary systems. Aims. We aim to analyse four transits of WASP-33b observed with the optical high-resolution HARPS-N spectrograph to confirm its nodal precession, study its atmosphere, and investigate the presence of star-planet interactions. Methods. We extracted the mean line profiles of the spectra using the least-squares deconvolution method, and we analysed the Doppler shadow and the radial velocities. We also derived the transmission spectrum of the planet, correcting it for the stellar contamination due to rotation, centre-to-limb variations, and pulsations. Results. We confirm the previously discovered nodal precession of WASP-33b, almost doubling the time coverage of the inclination and projected spin-orbit angle variation. We find that the projected obliquity reached a minimum in 2011, and we used this constraint to derive the geometry of the system, and in particular its obliquity at that epoch (ϵ = 113.99° ± 0.22°) and the inclination of the stellar spin axis (is = 90.11° ± 0.12°). We also derived the gravitational quadrupole moment of the star J2 = (6.73 ± 0.22) × 10−5, which we find to be in close agreement with the theoretically predicted value. Small systematics errors are computed by shifting the date of the minimum projected obliquity. We present detections of Hα and Hβ absorption in the atmosphere of the planet, with a contrast almost twice as small as that previously detected in the literature. We also find evidence for the presence of a pre-transit signal, which repeats in all four analysed transits and should thus be related to the planet. The most likely explanation lies in a possible excitation of a stellar pulsation mode by the presence of the planetary companion. Conclusions. A future common analysis of all available datasets in the literature will help shed light on the possibility that the observed Balmer lines’ transit depth variations are related to stellar activity and pulsation, and to set constraints on the planetary temperature–pressure structure and thus on the energetics possibly driving atmospheric escape. A complete orbital phase coverage of WASP-33b with high-resolution spectroscopic (and spectro-polarimetric) observations could help us to understand the nature of the pre-transit signal.

On the use of the first-order moment approach for measurements of Heff from LSD profiles

ABSTRACT The big majority of the reported measurements of the stellar magnetic fields that have analysed spectropolarimetric data have employed the least-squares deconvolution method (LSD) and the first-order moment approach. We present a series of numerical tests in which we review some important aspects of this technique. First, we show that the selection of the profile widths, i.e. integration range in the first-order moment equation, is independent of the accuracy of the magnetic measurements, meaning that for any arbitrary profile width it is always possible to properly determine the longitudinal magnetic field. We also study the interplay between the line depth limit adopted in the line mask and the normalization values of the LSD profiles. We finally show that the rotation of the stars has to be considered to correctly infer the intensity of the magnetic field, something that has been neglected up to now. We show that the latter consideration is crucial, and our test shows that the magnetic intensities differ by a factor close to 3 for a moderate fast rotator star with vsini of 50 ${\rm km\, s^{-1}}$. Therefore, it is expected that in general the stellar magnetic fields reported for fast rotators are stronger than what was believed. All the previous results shows that the first-order moment can be a very robust tool for measurements of magnetic fields, provided that the weak magnetic field approximation is secured. We also show that when the magnetic field regime breaks down, the use of the first-order moment method becomes uncertain.

Spectroscopic time-series analysis of R Canis Majoris

R Canis Majoris is the prototype of a small group of Algol-type stars showing short orbital periods and low mass ratios. A previous detection of short-term oscillations in its light curve has not yet been confirmed. We investigate a new time series of high-resolution spectra with the aim to derive improved stellar and system parameters, to search for the possible impact of a third component in the observed spectra, to look for indications of activity in the Algol system, and to search for short-term variations in radial velocities. We disentangled the composite spectra into the spectra of the binary components. Then we analysed the resulting high signal-to-noise spectra of both stars. Using a newly developed program code based on an improved method of least-squares deconvolution, we were able to determine the radial velocities of both components also during primary eclipse. This allowed us to develop a better model of the system including the Rossiter–McLaughlin effect and to derive improved orbital parameters. Combining the results with those from spectrum analysis, we obtain accurate stellar and system parameters. We further deduce at least one oscillation frequency of 21.38 c d−1. It could be detected during primary eclipses only and confirms a previous photometric finding. Results point to an amplitude amplification of non-radial pulsation modes due to the eclipse mapping effect. The presence of a He I line in the spectra indicates mass transfer in the R CMa system. Calculations of its Roche geometry give evidence that the cool secondary component may fill its Roche lobe. No evidence of a third body in the system could be found in the observed spectra.

Evolution of the magnetic field of Betelgeuse from 2009–2017

Context. Betelgeuse is an M-type supergiant that presents a circularly polarized (Stokes V) signal in its line profiles, interpreted in terms of a surface magnetic field. Aims. The weak circular polarization signal has been monitored over 7.5 years in order to follow its evolution on different timescales, and eventually to determine its physical origin. Linear polarization measurements have also been obtained regularly in the last few years. Methods. We used both the ESPaDOnS and Narval spectropolarimeters to obtain high signal-to-noise ratio spectra, which were processed by means of the least-squares deconvolution method. In order to ensure the reality of the very weak circular polarization, special care has been taken to limit instrumental effects. In addition, several tests were performed on the Stokes V signal to establish its stellar and Zeeman origin. Results. We confirm the magnetic nature of the circular polarization, pointing to a surface magnetic field of the order of 1 G. The Stokes V profiles present variations over different timescales, the most prominent one being close to the long secondary period (LSP; around 2000 d for Betelgeuse) often invoked in red evolved stars. This long period is also dominant for all the other Stokes parameters. The circular polarization is tentatively modeled by means of magnetic field concentrations mimicking spots, showing in particular that the velocity associated with each “spot” also follows the long timescale, and that this signal is nearly always slightly redshifted. Conclusions. From the coupled variations of both linear and circular polarization signatures in amplitude, velocity and timescale, we favour giant convection cells as the main engine at the origin of polarization signatures and variations in all the Stokes parameters. This strengthens support for the hypothesis that large convective cells are at the origin of the LSP.

A differential least-squares deconvolution method for high precision spectroscopy of stars and exoplanets – I. Application to obliquity measurements of HARPS observations of HD189733b

High precision measurements of stellar spectroscopic line profiles and their changes over time contain very valuable information about the physics of the stellar photosphere (stellar activity) and can be used to characterize extrasolar planets via the Rossiter–McLaughlin effect or from reflected light from the planet. In this paper, we present a new method for measuring small changes in the mean line profile of a spectrum by performing what we call differential least-squares deconvolution (dLSD). The method consists in finding the convolution function (or kernel) required to transform a high-signal-to-noise ratio template of the star into each observed spectrum. Compared to similar techniques, the method presented here does not require any assumptions on the template spectrum (e.g. no line-list or cross-correlation mask required). We show that our implementation of dLSD is able to perform at least as good as other techniques by applying it to star–planet obliquity measurements of exoplanet HD183799 during its transit.1 Among other things, the method should enable model-independent detection of light reflected by an exoplanet.

First discovery of a magnetic field in a main-sequence δ Scuti star: the Kepler star HD 188774

Abstract The Kepler space mission provided a wealth of δ Sct–γ Dor hybrid candidates. While some may be genuine hybrids, others might be misclassified due to the presence of a binary companion or to rotational modulation caused by magnetism and related surface inhomogeneities. In particular, the Kepler δ Sct–γ Dor hybrid candidate HD 188774 shows a few low frequencies in its light and radial velocity curves, whose origin is unclear. In this work, we check for the presence of a magnetic field in HD 188774. We obtained two spectropolarimetric measurements with an Echelle SpectroPolarimetric Device for the Observation of Stars (ESPaDOnS) at Canada–France–Hawaii Telescope. The data were analysed with the least-squares deconvolution (LSD) method. We detected a clear magnetic signature in the Stokes V LSD profiles. The origin of the low frequencies detected in HD 188774 is therefore most probably the rotational modulation of surface spots possibly related to the presence of a magnetic field. Consequently, HD 188774 is not a genuine hybrid δ Sct–γ Dor star, but the first known magnetic main-sequence δ Sct star. This makes it a prime target for future asteroseismic and spot modelling. This result casts new light on the interpretation of the Kepler results for other δ Sct–γ Dor hybrid candidates.

The magnetic fields at the surface of active single G-K giants

We investigate the magnetic field at the surface of 48 red giants selected as promising for detection of Stokes V Zeeman signatures in their spectral lines. We use the spectropolarimeters Narval and ESPaDOnS to detect circular polarization within the photospheric absorption lines of our targets and use the least-squares deconvolution (LSD) method. We also measure the classical S-index activity indicator, and the stellar radial velocity. To infer the evolutionary status of our giants and to interpret our results, we use state-of-the-art stellar evolutionary models with predictions of convective turnover times. We unambiguously detect magnetic fields via Zeeman signatures in 29 of the 48 red giants in our sample. Zeeman signatures are found in all but one of the 24 red giants exhibiting signs of activity, as well as 6 out of 17 bright giant stars.The majority of the magnetically detected giants are either in the first dredge up phase or at the beginning of core He burning, i.e. phases when the convective turnover time is at a maximum: this corresponds to a 'magnetic strip' for red giants in the Hertzsprung-Russell diagram. A close study of the 16 giants with known rotational periods shows that the measured magnetic field strength is tightly correlated with the rotational properties, namely to the rotational period and to the Rossby number Ro. Our results show that the magnetic fields of these giants are produced by a dynamo. Four stars for which the magnetic field is measured to be outstandingly strong with respect to that expected from the rotational period/magnetic field relation or their evolutionary status are interpreted as being probable descendants of magnetic Ap stars. In addition to the weak-field giant Pollux, 4 bright giants (Aldebaran, Alphard, Arcturus, eta Psc) are detected with magnetic field strength at the sub-gauss level.

Doppler images and the underlying dynamo

The (Zeeman-)Doppler imaging studies of solar-type stars very often reveal large high-latitude spots. This also includes F stars that possess relatively shallow convection zones, indicating that the dynamo operating in these stars differs from the solar dynamo. We aim to determine whether mean-field dynamo models of late-F type dwarf stars can reproduce the surface features recovered in Doppler maps. In particular, we wish to test whether the models can reproduce the high-latitude spots observed on some F dwarfs. The photometric inversions and the surface temperature maps of AF Lep were obtained using the Occamian-approach inversion technique. Low signal-to-noise spectroscopic data were improved by applying the least-squares deconvolution method. The locations of strong magnetic flux in the stellar tachocline as well as the surface fields obtained from mean-field dynamo solutions were compared with the observed surface temperature maps. The photometric record of AF Lep reveals both long- and short-term variability. However, the current data set is too short for cycle-length estimates. From the photometry, we have determined the rotation period of the star to be 0.9660+-0.0023 days. The surface temperature maps show a dominant, but evolving, high-latitude (around +65 degrees) spot. Detailed study of the photometry reveals that sometimes the spot coverage varies only marginally over a long time, and at other times it varies rapidly. Of a suite of dynamo models, the model with a radiative interior rotating as fast as the convection zone at the equator delivered the highest compatibility with the obtained Doppler images.

High-precision measurements of global stellar magnetic fields

This paper presents a brief history of the development of devices and techniques for high-precision measurements of stellar magnetic fields. Two main approaches for the processing of spectral-polarimetric observations are described: the method of least-squares deconvolution (LSD), which is used to find a mean-weighted average of the normalized polarization profile using a set of spectral lines, and a method in which each individual spectral line is used to determine the magnetic field, viz., the single line method (SL). The advantages and disadvantages of the LSD and SL methods are discussed.

Denoising spectroscopic data by means of the improved least-squares deconvolution method

The MOST, CoRoT, and Kepler space missions led to the discovery of a large number of intriguing, and in some cases unique, objects among which are pulsating stars, stars hosting exoplanets, binaries, etc. Although the space missions deliver photometric data of unprecedented quality, these data are lacking any spectral information and we are still in need of ground-based spectroscopic and/or multicolour photometric follow-up observations for a solid interpretation. Both faintness of most of the observed stars and the required high S/N of spectroscopic data imply the need of using large telescopes, access to which is limited. In this paper, we look for an alternative, and aim for the development of a technique allowing to denoise the originally low S/N spectroscopic data, making observations of faint targets with small telescopes possible and effective. We present a generalization of the original Least-Squares Deconvolution (LSD) method by implementing a multicomponent average profile and a line strengths correction algorithm. The method was successfully tested on the high resolution spectra of Vega and a Kepler star, KIC04749989. Application to the two pulsating stars, 20 Cvn and HD189631, showed that the technique is also applicable to intrinsically variable stars: the results of frequency analysis and mode identification from the LSD model spectra for both objects are in good agreement with the findings from literature. Depending on S/N of the original data and spectral characteristics of a star, the gain in S/N in the LSD model spectrum typically ranges from 5 to 15 times. The restored LSD model spectra contain all the information on line profile variations present in the original spectra. The method is applicable to both high- (>30 000) and low- (<30 000) resolution spectra, though the information that can be extracted from the latter is limited by the resolving power itself.

Discovery of the magnetic field in the pulsating B starβCephei

Although the star itself is not He enriched, the periodicity and the variability in the UV wind lines of the pulsating B1 IV star beta Cep are similar to what is observed in magnetic He-peculiar B stars, suggesting that beta Cep is magnetic. We searched for a magnetic field using spectropolarimetry. From UV spectroscopy, we analysed the wind variability and investigated the correlation with the magnetic data. Using 130 time-resolved circular polarisation spectra, obtained with the MuSiCoS spectropolarimeter at the 2m TBL from 1998 until 2005, we applied the least-squares deconvolution method on the Stokes V spectra and derived the longitudinal component of the integrated magnetic field over the visible hemisphere of the star. We performed a period analysis on the magnetic data and on EW measurements of UV wind lines obtained over 17 years. We also analysed the short- and long-term radial velocity variations, which are due to the pulsations and the 90-year binary motion. beta Cep hosts a sinusoidally varying magnetic field with an amplitude 97(4) G and an average value -6(3) G. From the UV wind line variability, we derive a period of 12.00075(11) days, which is the rotation period of the star, and is compatible with the observed magnetic modulation. Phases of maximum and minimum field match those of maximum emission in the UV wind lines, strongly supporting an oblique magnetic-rotator model. We discuss the magnetic behaviour as a function of pulsation behaviour and UV line variability. This paper presents the analysis of the first confirmed detection of a dipolar magnetic field in an upper main-sequence pulsating star. Maximum wind absorption originates in the magnetic equatorial plane. Maximum emission occurs when the magnetic north pole points to the Earth. Radial velocities agree with the ~90-y orbit around its Be-star binary companion.

Discovery of a magnetic field in the early B-type starσLupi

Magnetic early B-type stars are rare. To identify them, we use the strongest indirect indicator of a magnetic field in B stars, which is periodic variability of UV stellar wind lines occurring symmetric about the approximate rest wavelength. Probable magnetic candidates are targets for follow-up spectropolarimetry to search for a magnetic field. From the UV wind line variability the B1/B2V star sigma Lupi emerged as a new magnetic candidate star. AAT spectropolarimetric measurements with SEMPOL were obtained. The longitudinal component of the magnetic field integrated over the visible surface of the star was determined with the Least-Squares Deconvolution method. The UV line variations of sigma Lupi are similar to what is known in magnetic B stars, but no periodicity could be determined. We detected a varying longitudinal magnetic field with amplitude of about 100 G with error bars of typically 20 G, which supports an oblique magnetic-rotator configuration. The EW variations of the UV lines, the magnetic and the optical-line variations are consistent with the photometric period of 3.02 d, which we identify with the rotation period of the star. Additional observations with ESPaDOnS attached to the CFHT confirmed this discovery, and allowed the determination of a precise magnetic period. Analysis revealed that sigma Lupi is a He-strong star, with an enhanced N abundance and an underabundance of C, and has a chemically spotted surface. Like in other magnetic B stars the UV wind emission appears to originate close to the magnetic equatorial plane, with maximum emission when a magnetic pole points towards the Earth. The 3.01972(43) d magnetic rotation period is consistent with the photometric period, with maximum light corresponding to maximum magnetic field.

New measurements of rotation and differential rotation in A-F stars: are there two populations of differentially rotating stars?

We present projected rotational velocities and new measurements of the rotational profile of some 180 nearby stars with spectral types A-F. The overall broadening profile is derived analysing spectral line shape from hundreds of spectral lines by the method of least-squares deconvolution. Rigid and differential rotation can be distinguished in 56 cases. Ten stars with significant differential rotation rates are identified. As of now, 33 differential rotators detected by line profile analysis have been confirmed. The frequency of differential rotators decreases towards high effective temperature and rapid rotation. There is evidence for two populations of differential rotators with a gap in between at spectral type early-F. The gap can only partly be explained by an upper bound found for the horizontal shear of F stars. Apparently, the physical conditions of differential rotation change at early-F spectral types.

Revisiting the Rigidly Rotating Magnetosphere model for Ori E - I. Observations and data analysis★

We have obtained 18 new high-resolution spectropolarimetric observations of the B2Vp star sigma Ori E with both the Narval and ESPaDOnS spectropolarimeters. The aim of these observations is to test, with modern data, the assumptions of the Rigidly Rotating Magnetosphere (RRM) model of Townsend & Owocki (2005), applied to the specific case of sigma Ori E by Townsend et al. (2005). This model includes a substantially offset dipole magnetic field configuration, and approximately reproduces previous observational variations in longitudinal field strength, photometric brightness, and Halpha emission. We analyze new spectroscopy, including H I, He I, C II, Si III and Fe III lines, confirming the diversity of variability in photospheric lines, as well as the double S-wave variation of circumstellar hydrogen. Using the multiline analysis method of Least-Squares Deconvolution (LSD), new, more precise longitudinal magnetic field measurements reveal a substantial variance between the shapes of the observed and RRM model time-varying field. The phase resolved Stokes V profiles of He I 5876 A and 6678 A lines are fit poorly by synthetic profiles computed from the magnetic topology assumed by Townsend et al. (2005). These results challenge the offset dipole field configuration assumed in the application of the RRM model to sigma Ori E, and indicate that future models of its magnetic field should also include complex, higher-order components.

A dominant magnetic dipole for the evolved Ap star candidate EK Eridani

EK Eri is one of the most slowly rotating active giants known, and has been proposed to be the descendant of a strongly magnetic Ap star. We have performed a spectropolarimetric study of EK Eri over 4 photometric periods with the aim of inferring the topology of its magnetic field. We used the NARVAL spectropolarimeter at the Bernard Lyot telescope at the Pic du Midi Observatory, along with the least-squares deconvolution method, to extract high signal-to-noise ratio Stokes V profiles from a timeseries of 28 polarisation spectra. We have derived the surface-averaged longitudinal magnetic field Bl. We fit the Stokes V profiles with a model of the large-scale magnetic field and obtained Zeeman Doppler images of the surface magnetic strength and geometry. Bl variations of up to about 80 G are observed without any reversal of its sign, and which are in phase with photometric ephemeris. The activity indicators are shown to vary smoothly on a timescale compatible with the rotational period inferred from photometry (308.8 d.), however large deviations can occur from one rotation to another. The surface magnetic field variations of EK Eri appear to be dominated by a strong magnetic spot (of negative polarity) which is phased with the dark (cool) photometric spot. Our modeling shows that the large-scale magnetic field of EK Eri is strongly poloidal. For a rotational axis inclination of i = 60{\deg}, we obtain a model that is almost purely dipolar. In the dipolar model, the strong magnetic/photometric spot corresponds to the negative pole of the dipole, which could be the remnant of that of an Ap star progenitor of EK Eri. Our observations and modeling conceptually support this hypothesis, suggesting an explanation of the outstanding magnetic properties of EK Eri as the result of interaction between deep convection and the remnant of an Ap star magnetic dipole.

Least-squares deconvolution of the stellar intensity and polarization spectra

Least squares deconvolution (LSD) is a powerful method of extracting high-precision average line profiles from the stellar intensity and polarization spectra. Despite its common usage, the LSD method is poorly documented and has never been tested using realistic synthetic spectra. In this study we revisit the key assumptions of the LSD technique, clarify its numerical implementation, discuss possible improvements and give recommendations how to make LSD results understandable and reproducible. We also address the problem of interpretation of the moments and shapes of the LSD profiles in terms of physical parameters. We have developed an improved, multiprofile version of LSD and have extended the deconvolution procedure to linear polarization analysis taking into account anomalous Zeeman splitting of spectral lines. This code is applied to the theoretical Stokes parameter spectra. We test various methods of interpreting the mean profiles, investigating how coarse approximations of the multiline technique translate into errors of the derived parameters. We find that, generally, the Stokes parameter LSD profiles do not behave as a real spectral line with respect to the variation of magnetic field and elemental abundance. This problem is especially prominent for the Stokes I variation with abundance and Stokes Q variation with magnetic field. At the same time, the Stokes V LSD spectra closely resemble profile of a properly chosen synthetic line for the magnetic field strength up to 1 kG. We conclude that the usual method of interpreting the LSD profiles by assuming that they are equivalent to a real spectral line gives satisfactory results only in a limited parameter range and thus should be applied with caution. A more trustworthy approach is to abandon the single-line approximation of the average profiles and apply LSD consistently to observations and synthetic spectra.

Imaging of stellar surfaces with the Occamian approach and the least-squares deconvolution technique

Context. We present in this paper a new technique for the indirect imaging of stellar surfaces (Doppler imaging, DI), when low signalto-noise spectral data have been improved by the least-squares deconvolution (LSD) method and inverted into temperature maps with the Occamian approach. We apply this technique to both simulated and real data and investigate its applicability for different stellar rotation rates and noise levels in data. Aims. Our goal is to boost the signal of spots in spectral lines and to reduce the effect of photon noise without loosing the temperature information in the lines. Methods. We simulated data from a test star, to which we added different amounts of noise, and employed the inversion technique based on the Occamian approach with and without LSD. In order to be able to infer a temperature map from LSD profiles, we applied the LSD technique for the first time to both the simulated observations and theoretical local line profiles, which remain dependent on temperature and limb angles. We also investigated how the excitation energy of individual lines effects the obtained solution by using three submasks that have lines with low, medium, and high excitation energy levels. Results. We show that our novel approach enables us to overcome the limitations of the two-temperature approximation, which was previously employed for LSD profiles, and to obtain true temperature maps with stellar atmosphere models. The resulting maps agree well with those obtained using the inversion code without LSD, provided the data are noiseless. However, using LSD is only advisable for poor signal-to-noise data. Further, we show that the Occamian technique, both with and without LSD, approaches the surface temperature distribution reasonably well for an adequate spatial resolution. Thus, the stellar rotation rate has a great influence on the result. For instance, in a slowly rotating star, closely situated spots are usually recovered blurred and unresolved, which affects the obtained temperature range of the map. This limitation is critical for small unresolved cool spots and is common for all DI techniques. Finally the LSD method was carried out for high signal-to-noise observations of the young active star V889 Her: the maps obtained with and without LSD are found to be consistent. Conclusions. Our new technique provides meaningful information on the temperature distribution on the stellar surfaces, which was previously inaccessible in DI with LSD. Our approach can be easily adopted for any other multi-line techniques.