Investigation of the magnetic fields of a young Sun-like...

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UPTEC F 20002 Examensarbete 30 hp January 2020 Investigation of the magnetic fields of a young Sun-like star π 1 UMa Lawen Ahmedi 1

Transcript of Investigation of the magnetic fields of a young Sun-like...

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UPTEC F 20002

Examensarbete 30 hp January 2020

Investigation of the magnetic fields of a young Sun-like star π1 UMa

Lawen Ahmedi

1

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Abstract

Investigation of the magnetic fields of a young Sun-like star π1 UMa

Lawen Ahmedi

In astronomy, the Sun has an important role. It keeps the solar-system together and is the source for life, heat, light and energy to Earth. As any other star or planet, the Sun has a magnetic field. The magnetic field of the Sun has a great impact on the Sun itself as well as its surrounding. The magnetic field shapes solar wind, causes flares and drives coronal mass ejections radiating towards the Earth (and other planets). The Sun’s magnetic field is still not fully understood, and therefore it is useful to study other stars with properties similar to the Sun. So by studying young solar-type stars, the evolution of the Sun can be more easily understood. The aim of this project is to study the surface magnetic field in a young solar-type star, π1 UMa to see how the magnetic field is distributed and if there are any patterns like polarity reversals. Magnetic field generates polarisation and with Stokes vector I and V, polarisation can be described. Earlier measurements from two time-epochs (2014 and 2015) of Stokes I and V have been obtained from the spectropolarimeter NARVAL. To get the desired mean polarisation profiles of the star, a technique called least square deconvolution was applied which increases the signal-to-noise level. To reconstruct the magnetic topology the Zeeman-Doppler imaging technique was used. Then we obtained the surface magnetic field maps of both measurements. No change of the polarity of magnetic field at the visible stellar pole was found. Most of the magnetic field energy was contained in the spherical harmonic modes with angular degrees l=1-3. The star shows dominance in the toroidal component so the study seem to agree with the previously established trend that younger and faster rotating stars have predominantly toroidal magnetic fields and older stars with slower rotation rate, like the Sun, have predominantly poloidal field. Looking at the magnetic field plots, the star show dominance in the azimuthal field component, and the mean magnetic field strength is similar to one found in the previous study. The results of the surface magnetic field in our study thus agrees with previous study of the same star. With this we can conclude that the Sun’s magnetic field probably been different when it was younger, and possibly similar to the star analyzed in this study.

Handledare: Oleg Kochukhov Ämnesgranskare: Eric Stempels Examinator: Tomas Nyberg ISSN: 1401-5757, UPTEC F 20002

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POPULÄRVETENSKAPLIG SAMMANFATTNING Solen är en viktig del inom astronomin och är källan till värme, ljus och liv på jorden. Likt andra planeter och stjärnor har solen ett magnetfält som har en stor inverkan på solen och dess omgivning. Från solens magnetfält bildas bland annat starka solvindar och emitterar partiklar som åker mot jorden och andra planeter. Aktiviteten hos solen varierar regelbundet i en 11-årscykel då magnetfältet byter polaritet. Men än idag finns fler oklarheter kring hur solens magnetfält utvecklats från sitt yngre stadie då solen var flera miljoner år yngre. Ett sätt att analysera solens utveckling och magnetfält på är genom att studera andra stjärnor som har liknande egenskaper som solen men som är flera miljoner år yngre. I denna studie har den 300 Myr gamla stjärnan π1 UMas magnetfält studerats. Med hjälp av analytiska metoder, som Zeeman Doppler Imaging (ZDI) är det möjligt att rekonstruera stjärnans polariseringsprofiler till kartor över magnetfältet på ytan. Med Stokes vektorer I och V är det möjligt att beskriva dessa polariseringsprofiler, som givits från två olika mätningar tagna 2014 respektive 2015 från spektropolarimetern NARVAL. För att beräkna polariseringsprofilerna applicerades tekniken “least square deconvolution analysis” som höjer signal-brusnivån, och därefter användes ZDI-metoden för att rekonstruera magnetfältet, vilket resulterade i mappar på magnetfältets distribution på stjärnans yta. Stjärnans magnetfält visar dominans hos toroidala magnetfältet samt ett starkare fältstyrka i azimutala komponenten. Vid jämförelse med en tidigare studie [7] av samma stjärna som använt mätningar från 2007 verkar resultatet överensstämma och det verkar som att stjärnan har hållit samma magnetiska aktivitetsnivå mellan dessa två epoker. Ingen förändring hos stjärnans polaritet kunde påvisas, men eftersom mätningarna i denna studie var tätt inpå varandra (6 månader emellan) och stjärnan haft samma polaritet år 2007 från förra studien, är det troligt att skiftningen hos polariteten missats om det nu skett. Med detta kan vi konstatera att solens magnetfält möjligtvis sett annorlunda ut när den varit yngre, och möjligtvis likt stjärnan som analyserats i denna studie.

Contents

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Abstract 2

POPULÄRVETENSKAPLIG SAMMANFATTNING 3

1.Introduction 5

2.Theoretical background 7 2.1 Magnetic fields 7 2.2 Polarisation 8 2.3 Stokes parameter spectra 9 2.4 Mapping of stellar magnetic fields 10 2.5 π1 UMa 12

3. Method 12 3.1 Observational data 12 3.2 Least squares deconvolution analysis 13 3.3 Zeeman Doppler Imaging 14

4.Results 16 4.1 Observed LSD profiles 16 4.2 Reconstructed magnetic field maps 20

5. Summary and discussion 28

6.References 30

1.Introduction

Stars are fascinating objects covering the sky. The Sun is a star in our Galaxy, the Milky Way and the closest one to the Earth. The Sun plays an important role in determining properties of

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planetary atmospheres. The magnetic field of the Sun has a great impact on the Sun itself as well as its surrounding, for example the Earth. The magnetic field shapes solar wind, causes flares and drives coronal mass ejections radiating towards the Earth (and other planets), which are storms consisting of high-energy particles. The Earth has a magnetic field that is partly protecting it from the Sun’s magnetic activity by deflecting the particles from the solar wind. But still, the most powerful eruptions are affecting us since they affect the outer layers of the terrestrial atmosphere, can harm electronics, the satellites in outer space, and could also be harmful for the astronauts. Thus, the solar magnetic activity has an important role. The Sun’s magnetic field is still not yet fully understood, and therefore it is useful to study other stars, for example Sun-like stars of younger ages, to understand long-term evolution of the solar magnetic activity. The focus in this project will be on such young solar-type star,

UMa.π1 Solar-type stars are defined as objects within the mass range 0.6M⊙≤M≤1.5M⊙, where M⊙ is the mass of the Sun. These stars belong to the spectral classes from mid-F to late-K. They have similar pattern of “magnetic activity”, believed to be the outcome of the dynamo process operating in the stellar interiors. The presence of the field at the stellar surface is revealed by several types of indirect activity proxies, such as cool starspots, X-ray emission or enhanced emission in the chromospheric lines. Solar-type stars are born rapid rotators. The rapid rotation makes the generation of magnetic field more efficient, and therefore young and/or rapidly rotating stars are more active and have therefore stronger fields and the magnetic spots are larger on their surfaces. Magnetic fields of most of the solar-type active stars have a strong azimuthal component usually arranged in azimuthal rings on the stellar surface[3]. With age, the stars will become slower rotators and their magnetic activity will decrease. The Sun rotated much faster when it was younger because the stellar rotation is slowed down by the magnetic field, which also implies that the magnetic activity was higher. Previous studies have suggested that the level of activity of cool stars decreases with age and therefore the Sun is currently different from its younger state[7]. The aim of this project is to study the magnetic field of a young solar-analogue star to gain more knowledge about stellar magnetism in the context of understanding the history of the Sun and its magnetic activity. Understanding the Sun’s activity history would also enhance our knowledge of the effect of the activity of the Sun on earlier atmospheres of the terrestrial planets such as Earth and Mars. Earlier studies have shown that the Earth in its younger stages have always had liquid water, which must indicate that its surface was warmer than could be sustained by fainter young Sun. Probably Earth was heated by greenhouse gases such as methane and carbon dioxide. Previous observations of Mars have shown convincible signs of liquid water on its surface, which means that its atmosphere has been more compact and warmer at earlier evolutionary stages[2]. These are some examples of what information investigation of stellar magnetism of solar-analogue stars can provide us, which is a part of the purpose of this study. It is also interesting to study the pattern of magnetic polarity reversal, which is a change of magnetic field to the opposite polarity. This occurs in intervals, and by studying the magnetic

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field over different time-periods, the polarity reversal can be observed. For example, the Sun has a 11-year cycle where the Sun’s magnetic field reverses polarity[15]. Also, in reference [7] one has studied several different Sun-like stars and a polarity reversal have been seen for one of the stars (χ1 Ori) with a periodical cycle of either 2,6 or 8 years, which might suggest that the Sun had a different cycle when it was younger. Magnetic field generation is an important process in stars. Due to magnetic fields dark spots are formed at the surface of the Sun. These dark spots cause variability, flares and short-wavelength emission that affects the immediate stellar environment and the entire planetary system. It is possible to study magnetic fields and spots of stars other than the Sun using computer tomography techniques to convert time variability of polarisation profiles of stellar spectral lines into two-dimensional maps of spots and vector magnetic fields on the stellar surface. In this project, an analysis of the young star named UMa, which is similar π1 to what our Sun was a few hundred million years after its formation, will be performed using previously collected observations, obtained 2014 and 2015. The analysis begins with detecting a weak polarisation signal (which is a direct signature of magnetic field) in stellar spectral lines by applying multi-line analysis technique. The study is performed with computer codes written in IDL and Fortran. The detected signal will be modeled with a tomographic code and the magnetic fields will be reconstructed followed by an analysis of the obtained results. This surface magnetic field mapping will be performed using Zeeman-Doppler imaging (ZDI) analysis of Stokes I and V spectropolarimetric observations with a code developed by [3]. We aim to obtain maps of the magnetic field at the stellar surface. This allows us to study the magnetic field distribution of the star in detail for two different sets of observations obtained with the NARVAL spectropolarimeter in 2014 and 2015 and investigate how the field change over time. Also a comparison with the results of the study from reference [7] will be carried out, where the magnetic field of UMa was π1 studied using observations from 2007. This project will hopefully improve our understanding of the magnetic field of π1 UMa in particular and magnetism of young Sun-like stars in general.

2.Theoretical background

2.1 Magnetic fields In stars, magnetic fields have an important role during many stages of stellar formation and evolution. The motion of the conductive plasma in a stellar interior creates magnetic fields. The magnetic energy is derived from the kinetic energy of stellar rotation and convection. Interaction between the magnetic field and stellar wind is the main mechanism of angular momentum loss in young stars. Magnetic fields produces phenomena such as star spots, X-ray emission, flares, which are observed at the surface of our Sun and other stars.

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The presence of a magnetic field leads to changes in the atomic energy levels resulting in changes of the properties of spectral lines. The magnetic field splits spectral lines in a number of components and introduces circular and linear polarisation within these components. These effects make it possible to detect stellar magnetic fields. In a magnetic field, the atomic Hamiltonian is given by (1)− ∇ (r) (r)L (L S) (B )H = ħ

2m2 + V + ξ · S + eħ

2mc + 2 · B + e2

8mc2 × r 2 where m and e correspond to the electron mass and charge, c and ħ are the speed of light and the Planck constant respectively, L and S are the orbital and spin angular momentum operators and B corresponds to the magnetic field vector. Three different regions are defined depending on the relative strength of the spin-orbit interaction and the magnetic field terms. The linear Zeeman effect occurs when the quadratic field term is smaller than the linear field term which in turn is smaller than the spin-orbit term. The Paschen-Back effect occurs if the quadratic field term and the spin-orbit term are smaller than the linear field term. The quadratic Zeeman effect occurs when the quadratic field term is larger than the linear field term and larger than the spin-orbit term.

Figure 1. Representation of atomic energy levels and corresponding spectral lines when there is no magnetic field in comparison to the case when a magnetic field is present. In figure (1) we can see an illustration of the typical effect of magnetic field on a spectral line. Without magnetic field, the transition between the upper and lower atomic levels gives rise to a single spectral line. When a magnetic field is present the line splits into three groups of Zeeman components ( [3]. Figure (1) shows the simplest type of splitting , π, σ )σb r (normal Zeeman triplet) when each group is represented by a single Zeeman component.

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2.2 Polarisation

Electromagnetic waves have a property to oscillate in a certain way, which is called polarisation. If the direction of the electric field vector within the electromagnetic wave varies randomly in time the light is said to be unpolarized. A non-random behaviour of the electric field determines what type of polarisation is present in electromagnetic wave. Linear polarisation is a plane electromagnetic wave, i.e the electric field vector is moving in a single plane along the direction of propagation. Circular polarisation is when the electric field vector rotates in a circle, i.e the electric field in the wave has two orthogonal linear components which are equal in amplitude and vary with a phase shift of . Elliptical polarisation is /2π when the light consists of two perpendicular components with any amplitude and shift [3]. The Stokes vector describes the polarisation of an electromagnetic wave and it is given by: T (2)I = {I , , , }Q U V Each term is defined as: Stokes I - Total intensity of radiation, which is equal to the sum of two beams with orthogonal polarisation, i.e . (3)I = I0 + I90 = I45 + I135 = I↻ + I↺ Stokes Q - Difference in the intensity of (4).Q = I0 − I90 Stokes U - Difference in the intensity of (5).U = I45 − I135 Stokes V - Difference in the intensity of . (6)V = I↻ − I↺ where (3) corresponds to unpolarised light, (4) and (5) describes linear polarisation and (6) describes circular polarisation [3].

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Figure 2. An illustrations of the properties for polarisation of the radiation that is emitted in the Zeeman components ( and ) for different orientations of the magnetic field vectorπ σ relative to the line of sight. The Zeeman components in the split spectral line have distinct polarisation properties. The polarisation depends on the angle between the magnetic field vector and the direction of the emitted light, and changes according to that, as can be seen in figure (2). When light emitted is parallel to the field vector, the π components disappears and the σb and σr components have opposite circular polarisation. If the line of sight is perpendicular to the field vector, the π components are linearly polarised parallel to the field and the σb and σr components are linearly polarised perpendicular to the field. This means that the π components can only be linearly polarised and the σ components can have both circular and linear polarisation[3]. Due to weakness of surface magnetic fields in stars it is usually challenging to detect them directly. Nevertheless, it is possible to study and detect these fields using polarisation in spectral lines. This will be presented in section 2.4.

2.3 Stokes parameter spectra To interpret polarisation spectra of stars we need to know how to theoretically compute shapes of spectral lines in four Stokes parameters. This is accomplished by solving the radiative transfer equation in the thin outer layer of the star - the atmosphere. This equation describes the interaction between matter and radiation. When a magnetic field is present, a single scalar equation for the intensity is replaced by the analogous transfer equation for the Stokes I vector as following: (7)− Idz

dI = K + J where z is the height in the stellar atmosphere, is the Stokes vector. The I = {I , , , }Q U V T parameter K is a matrix which describes the absorption of light and attenuation of its polarisation characteristics and J is the emission vector. In order to solve the polarised radiative transfer (PRT) equation several input parameters are required, listed below:

- the magnetic field vector B as a function of z, - the temperature and pressure as a function of z, - the data of relative concentrations of chemical elements whose lines we want to

model - a database that contains information about the continuum opacity coefficients of

relevant absorbers, and a line list with information about the position of spectral lines, their transition probabilities, broadening parameters and parameters and Ju,l gu,l needed in order to compute the Zeeman splitting patterns

With all of these input parameters applied, one can numerically solve the PRT equation obtaining the Stokes vector I as a function of wavelength for each layer in the stellar atmosphere.

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Another way of solving the PRT equation with less complication than the previous one, is to make use of approximate analytical solutions. One such solution corresponds to the Milne-Eddington (ME) atmosphere, which assumes that the magnetic field, the ratio of the line and continuum opacity as well as the absorption and anomalous dispersion profiles which enter the matrix K are all constant in the line formation region, and that the source function (which enters the definition of the emission vector J) is linearly dependent on the optical depth .τ The preceding discussion concerned the problem of calculating the local Stokes vector from local properties. But studying stellar magnetic fields is more complex. Stars are unresolved objects, and the magnetic field vector changes from one region to another on the surface. Every surface zone creates its own Stokes vector. Because of stellar rotation these local Stokes vectors are Doppler-shifted and weighted according to the local brightness and projected surface area. The contribution of the surface zones on the stellar hemisphere and adding them all together produces disk integrated Stokes profiles, which approximates the real observations of stars. These disk integrated profiles are time-dependent since the star is rotating and we are observing the field structure from different angles.

2.4 Mapping of stellar magnetic fields The main tools for investigating stellar magnetic fields and surface structures are high-resolution spectroscopy and spectropolarimetry. The spectral line profile shapes are distorted due to inhomogeneities on stellar surface, which create detectable signatures in the line profiles. As described above, magnetic field generates polarisation in spectral lines through the Zeeman effect. This allows detection of stellar magnetic fields and reconstruction of their topologies. To accomplish the latter task, spectropolarimetric observations have to be obtained several times to resolve rotational modulation. Reconstruction of two-dimensional maps of stellar surface is carried out using Doppler imaging (DI) and Magnetic/Zeeman Doppler imaging (MDI/ZDI), which are the highest resolution indirect imaging methods in astronomy. These techniques use the fact that distortions generated by magnetic fields and star spots move across Doppler-broadened intensity and polarisation line profiles. Any point in the line profile represents an interval of Doppler shifts corresponding to a vertical stripe on the stellar surface. Features in a single Stokes profile can be used to determine longitudinal position of a magnetic or cool spot on the stellar surface. The latitudinal information can be obtained from a times series of Stokes profiles recorded at different rotational phases. For example, if the star has an inhomogeneity at the surface near the pole, its line signature is visible only near the centre of the profile and persists during more than half of the rotation period. On the other hand, a spot near the equator travels through the whole profile and is visible during only half of the rotation cycle. By bringing the latitudinal and longitudinal information together the whole stellar surface can

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be reconstructed. The magnetic field topology can be reconstructed using the same principle applied to polarised spectra, and this technique is known as Zeeman Doppler Imaging. Starting from some input parameters (brightness, magnetic field, temperature, chemical abundance) and by using a times series of the line profiles, a reconstruction of a two-dimensional map of the stellar surface can be accomplished. DI is mathematically an ill-posed problem, meaning that infinite number of solutions can fit a given set of observations. DI needs an additional constraint in order to have an unique solution, which is called regularization.[3] Zeeman Doppler Imaging (ZDI) or the inverse method, is the only technique for reconstructing stellar magnetic field topologies with the ability to extract a quantitative information about stellar magnetic field for stars with complex fields[3]. Since many stars are too far away from the observer, the technique uses the rotation of the star to reconstruct the magnetic field distribution at the surface of the star. Doppler Imaging is most efficient for fast rotating stars but can also be applied for slow rotating stars. It is not optimal to use only circular polarisation. Only information about the line of sight component of the magnetic field vector can be obtained from Stokes V since it is independent of the azimuth angle. For that reason the same Stokes V profile can represent different field configurations [7]. However, in practice, recordings of the Stokes Q and U spectra are very difficult to obtain because they have about 10 times smaller amplitude than the Stokes V signal. In this study we rely only on Stokes V.

2.5 π1 UMa The young solar-analogue star π1 UMa is considered to be a member of the Ursa Major Moving Group, which includes a group of stars that has the same velocity vectors in space and is believed to have a common origin in space and time. All of the stars belonging to this group have an age of about 300 Myr[7]. In comparison, the Sun has an age of 4.6 Gyr. We will compare our results with reference [7] who also studied the star π1 UMa.

3. Method

3.1 Observational data In order to perform magnetic field analysis of the star UMa two sets of previously π 1 observed data are used. The first set of observed data is taken April 2014 and the second set of data is taken January 2015. The first dataset contains 14 observations. The observations are taken with short intervals, typically 1-2 days, and some have been taken the same night at different times. The second dataset consists of 12 observations with a 1-day interval and two observations that have been registered after 5 days.

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The reduced spectra have been taken from a database called PolarBase. PolarBase is a database which contains all stellar data that has been obtained with the high-resolution spectropolarimeters ESPaDOnS and NARVAL, in their reduced form[4]. NARVAL is one of the few astronomical facilities around the world fully dedicated to stellar spectroscopy, located in Pic du Midi, France. It is installed at Télescope Bernand Lyot and is a “twin” of the spectropolarimeter ESPaDOnS installed at the Canada-France-Hawaii Telescope (CFHT; Mauna Kea Observatory). NARVAL permits long-term surveillance and investigation of brighter targets. It gives the opportunity to have coordinated observations with ESPaDOnS in order to achieve a continuous surveillance of rotating and variable stars, which is possible due to the 160° shift in longitude between Hawaii and France[1]. The spectrograph NARVAL has a polarimetric unit consisting of three Fresnel rhombs. This device allows one to obtain circular polarisation spectra or spectra in all four Stokes parameters. In order to prevent spurious polarisation and to decrease the number of reflections before light passes into the spectrograph, the polarimetric unit is installed at the Cassegrain focus. The resolving power of NARVAL is around 65 000 and the spectrograph covers the wavelength range from 3700 to 10 500 Å, which corresponds to the entire optical spectrum. The reduced spectral files include, in addition to Stokes I and V, the so-called “null” spectrum, which should only contain background noise created by suppressing the true stellar polarisation. The “null” spectrum is included from the data files which were downloaded from the Polarbase archive. In polarimetric mode, when a spectrum is produced, four subexposures are combined with the polarimetric optics rotated by varying the angle on the optical axis. By adding together the four subexposures, corresponding spectrum for Stokes I is obtained. Taking other combinations of the subexposures makes it possible to obtain the “null” spectra[4]. This spectrum is useful for assessing instrumental artefacts and noise in the data. The measurements analysed here were obtained only with NARVAL, a stellar spectropolarimeter, duplicated from the spectropolarimeter ESPaDOnS.

3.2 Least squares deconvolution analysis The star analysed here has a has relatively weak magnetic field. Weak magnetic fields are hard to detect, because they produce weak polarization signatures in individual lines. By using the LSD (least squares deconvolution) method it is possible to detect weak polarisation signals by increasing the signal-to-noise (S/N) level. The LSD technique adds together all lines in a spectrum into a single intensity profile. The same principle is applied to the corresponding polarization profiles, instead of studying them in individual lines. The LSD analysis is developed by [9] and is a method of extracting highly precise mean Stokes V signatures from polarisation observations with a moderate S/N. This technique was extended to all four Stokes parameters by [10]. LSD represents the entire stellar spectrum as a linear superposition of scaled mean profiles. Mathematically it can be described as: (8)X = M · ZX where M is the line pattern matrix and ZX is the mean profile which is sought for, and X=X obs corresponds to the observed polarisation spectrum with error bars .σobs

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The least squares problem of fitting the model given by equation (8) to observations is given by (9)χ X ) (X ) > min 2 = ( obs − M · ZX

T · E2obs obs − M · ZX −

and its solution is following , (10)M )ZX = ( T · E2

obs · M −1* M T · E2

obs · Xobs here is the diagonal matrix consisting of the inverse of the error bars, [3].Eobs /σ1 obs The purpose of the LSD analysis is to solve the inverse problem of equation (8), which means obtaining the mean line profile Z for a given pattern matrix M and observed spectrum Xobs. During this step the observed mean line profiles of Stokes I and V, the corresponding errors and the “null” spectra for all observations of π1 UMa spectrum are going to be obtained. The stellar parameters used for compiling the list of lines for LSD are the effective temperature

and the surface gravity [7].875 KT ef f = 5 og(g) .49 cm/s l = 4 2 For the LSD analysis a code has been developed by [8]. From the Vienna Atomic Line Database [13] we received a line list for the stellar parameters given above. The starting wavelength used was 3900 Å and ending wavelength 10000 Å. The microturbulence is set to

(typical value for Sun-like stars) and the line-depth selection threshold to 0.01, km/sξt = 1

meaning that only lines deeper than 1% of the continuum were included in the list. As a simple measure of magnetic field strength, we analysed the mean longitudinal magnetic field, Bz. Following reference [14] it is calculated from the Stokes I and V profiles with

[G] (11)− 14Bz = 7V (v)dz∫

v

λg [1−I(v)]dv∫

where λ [µm] is the mean wavelength of LSD line mask and g is the mean effective Landé factor, v [km/s] is the velocity shift with respect to the center. The mean longitudinal magnetic field corresponds to the disk-averaged line of sight magnetic field component. It is not sensitive to small-scale fields [5]. It is possible that the integral from the equation (11) becomes zero, even if there is a magnetic signature in the Stokes V profile. Such symmetry in the profile indicates that both sides of the star have magnetic structures of the same strength but opposite polarities. An example of such configuration is a ring of azimuthal field encircling the star. Thus, the longitudinal field depends on the magnetic field strength and its configuration.

3.3 Zeeman Doppler Imaging The analysis begins with the forward calculation, performed with some input parameters including the stellar parameters, the line data and magnetic field vectors , of the B , , )( r Bθ Ba local Stokes parameters. Then surface integration is carried out, and the appropriate Doppler shift is applied to each local profile according to the value of projected rotational velocity v

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sin i and location of a given surface element. The local profile is then multiplied by a weight which is the product of the projected surface area of this surface element and limb-darkening function. The limb-darkening function accounts for the fact that the stellar surface brightness decreases from the center of the stellar disk to the limb. Afterwards the local Stokes profiles obtained in previous steps are summed up resulting in the disk-integrated model Stokes profiles. Then these spectra are compared to the observed data at each rotational phase. Based on this comparison, the ZDI code adjusts the surface magnetic field distribution, recalculates the model disk-integrated profiles and compares them with observation again. This process continues until a good fit to observations is achieved. In parallel, the code calculates regularisation function and tries to ensure that solutions that it converges to also satisfy the regularisation constraint. When modelling the LSD profiles we assume that they behave similar to a single spectral line with average line parameters. By solving the polarised radiative equation using the Milne-Eddington approximation, the local model Stokes profiles were calculated. We obtained the central wavelength and the effective Landé factor from the LSD line mask. The line shape was parameterized by a combination of a depth parameter (the line depth; the parameter regulating the strength of Stokes I line in the model that we fit to LSD profiles) and a Voigt function described by the two broadening parameters. The final stellar parameters that are necessary for ZDI analysis are the stellar radial velocity vrad, rotational period Prot, the projected rotational velocity v sin i and the inclination angle i of the stellar rotational axis. The initial values of these parameters, vrad=12.35 km/s, i=60 degrees, Prot=4.9 d and v sin i=11.2 km/s, were adopted from [7]. Prot and v sin i are further optimised below to obtain better fit to the observational analysed here. The limb darkening is treated according to a linear function law, with the coefficient 0.65 taken from [6]. The radial, azimuthal and meridional magnetic field components are described with the spherical harmonic expansion:

(12)(θ, ) − Y (θ, )Br φ = ∑lmax

l=1∑l

m=−lαl,m l,m φ

(13)(θ, ) − [β Z (θ, ) X (θ, )]Bm φ = ∑lmax

l=1∑l

m=−ll,m l,m φ + γl,m l,m φ

(14)(θ, ) − [β X (θ, ) Z (θ, )]Ba φ = ∑lmax

l=1∑l

m=−1l,m l,m φ − γl,m l,m φ

where , (15)(θ, ) − P (cosθ)K (φ)Y l,m φ = C l,m l,|m| m

(16)(θ, ) K (φ)Z l,m φ = l+1C l,m

∂θ∂P (cosθ)l,|m|

m (17)(θ, ) − mK (φ)X l,m φ = l+1

C l,msinθ

P (cosθ)l,|m|−m

are the real spherical harmonic functions describing the mode with the angular degree l and azimuthal order m with the terms

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(18) C l,m = √ 4π2l+1

(l+|m|)!(l−|m|)!

and

(19) where θ and ф are the longitude and co-latitude angles at the stellar surface. Pl,m(θ) is the associated Legendre polynomial. The regularisation function applied to the magnetic field

will reduce unnecessary contribution of higher-order harmonic(α )R = ∑

l,ml2 2

l,m + β2l,m + γ2

l,m

modes. The parameters characterise the contributions of the radial, poloidal, , , γα l,m β

l,m l,m

horizontal poloidal and horizontal toroidal magnetic field components. These harmonic coefficients alpha, beta, gamma are the actual free parameters optimised by the ZDI code. The expansion is carried out to a sufficiently large . In this study we used [7].lmax 0lmax = 1

4.Results

4.1 Observed LSD profiles Figure (3) and (4) are plots of the observed LSD Stokes I and V profiles for each dataset presented with the “null” spectrum, which characterises noise and false signals. These profiles are the starting point for modelling magnetic field of π1 UMa.

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Figure 3. Observed LSD profiles Stokes I, Stokes V and “null” respectively, taken in April 2014 scaled with a factor 158. The intensity is shown on the y-axis and the velocity on the x-axis. Rotational phase is marked in blue. The profiles are offset vertically for display purposes. The grey rectangles in the figure represent the intervals within which the statistics and Bz are calculated.

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Figure 4. Observed LSD profiles Stokes I, Stokes V and “null” respectively, taken in January 2015, scaled with a factor 170. The intensity is shown on the y-axis and the rotational velocity on the x-axis. Rotational phase is marked in blue. The profiles are offset vertically for display purposes. One of the important parameters of LSD profile is the cut-off value, which determines how many lines to include when constructing LSD profile. 5% cut-off means that all lines deeper than 5% of the continuum are included, 10% means that all lines deeper than 10% of the continuum are included, etc. The larger the cut-off, the fewer lines are retained in the list, which can be seen in the last row of table (1). In principle, the more lines we include - the better. But one can also worry that including many weak lines will degrade the quality of the mean profile. This is why one of the steps of our analysis was to test S/N of LSD profile as a function of cut-off. In order to try to improve the quality in our calculations, an analysis of the cut-off value was performed to see if any change in this parameter would give any difference. We tried the cut-off with values 5%, 20%, 30% and 40%. The quality is

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determined by the ratio, of the maximum absolute Stokes V amplitude and the mean error of Stokes V profile, Q=Max(V)/Err(V). The higher this ratio, the better is the LSD profile quality. The total number of lines decreases as the cut-off is increases. One can not see any significant difference when comparing the quality factors. We concluded that changing cut-off has an insignificant effect on the results. We have chosen to use the cut-off 20 % throughout the whole project. Table 1. Table of quality factors Q for different cut-offs (%) for the observed data from April 2014. The last row shows the total number of lines for each cut-off. HJD Q (5%) Q (20%) Q (30%) Q (40%)

2456754.3504 31.2 30.9 30.8 30.0

2456756.4024 18.0 17.9 17.7 17.5

2456757.3753 30.0 31.5 31.2 31.1

2456759.4513 14.2 14.3 14.5 14.0

2456759.4961 13.5 13.5 13.5 13.0

2456760.3670 19.7 19.8 19.0 18.5

2456760.4119 17.4 17.4 17.2 16.2

2456761.3636 14.7 14.9 14.9 14.4

2456761.4084 14.7 14.7 15.0 14.9

2456763.3745 30.0 30.1 29.4 28.8

2456763.4194 31.8 31.8 31.5 31.1

2456765.3696 11.3 11.3 11.2 11.0

2456765.4144 12.0 11.7 11.4 10.6

2456794.3862 20.3 20.2 19.6 19.7

Total number of lines 7040 3597 2779 2191

The observations from April 2014 and January 2015 are summarised in tables (2ab). The mean error of Stokes V in the first dataset is smaller than for the second dataset. The error is of order 10-5 in both observations and is typically around 1.9*10-5 in table (2a) and around 3.5*10-5 in table (2b). The maximum amplitude is of order 10-5 in both observations and differs slightly in both sets. The maximum amplitude reaches 6.1*10-4 in the first dataset and 5.6*10-4 for the second dataset. The longitudinal field Bz varies for both datasets, changing sign, which indicates that the magnetic field topology is relatively complex and definitely not axisymmetric. Bz is mostly negative for both datasets with the highest value 0.88 G and smallest value -10.83 G in the first set, in the second set the highest value is 11.62 G and the smallest value reaches -10.87 G. The error of Bz is higher for the second data set.

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Table 2a. Data from the dataset obtained in April 2014. The table gives heliocentric Julian date, error of Stokes V, maximum amplitude (by absolute value) of Stokes V LSD profile and the longitudinal field Bz in Gauss and the uncertainty. HJD Err(V) Max(V) Bz(V) [G]

2456754.3504 1.921E-05 5.945E-04 -2.34±0.68

2456756.4024 1.324E-05 2.374E-04 -5.97±0.47

2456757.3753 1.426E-05 4.491E-04 -7.40±0.51

2456759.4513 2.281E-05 3.257E-04 -2.58±0.82

2456759.4961 2.053E-05 3.778E-04 -2.75±0.74

2456760.3670 1.905E-05 3.766E-04 0.64±0.69

2456760.4119 1.932E-05 3.364E-04 0.88±0.70

2456761.3636 1.862E-05 2.779E-04 -7.08±0.67

2456761.4084 1.875E-05 2.764E-04 -6.69±0.67

2456763.3745 1.931E-05 5.815E-04 -0.22±0.69

2456763.4194 1.926E-05 6.126E-04 -1.21±0.69

2456765.3696 2.076E-05 2.354E-04 0.01±0.75

2456765.4144 2.067E-05 2.428E-04 -0.19±0.74

2456794.3862 1.983E-05 3.998E-04 -10.83±0.72

Table 2b. Same as 2a but for the dataset obtained in January 2015. HJD Err(V) Max(V) Bz(V) [G]

2457028.4817 3.852E-05 5.535E-04 -9.41±1.38

2457028.6877 3.366E-05 4.606E-04 -6.53±1.20

2457029.5907 3.892E-05 4.559E-04 0.70±1.39

2457030.4794 3.830E-05 2.427E-04 -8.54±1.37

2457031.4211 4.517E-05 4.487E-04 7.34±1.62

2457031.6776 5.936E-05 5.688E-04 11.62 ± 2.12

2457032.5984 3.451E-05 3.331E-04 -6.65±1.23

2457033.5474 3.363E-05 2.638E-04 -1.64±1.21

2457034.4654 4.617E-05 3.222E-04 0.29±1.66

2457034.7127 3.444E-05 2.009E-04 -10.87±1.23

2457040.4187 4.477E-05 2.156E-04 -4.49±1.61

2457040.5782 6.981E-05 3.805E-04 -3.36±2.51

4.2 Reconstructed magnetic field maps

Before modelling Stokes V profiles we need to adjust relevant parameters to reproduce Stokes I profile as good as possible. We adjusted two parameters, the line depth and v sin i. The lowest deviation for Stokes I (0.33547 %) we could obtain corresponded to v sin i 0.6= 1km/s and a line depth of 3.4.

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As mentioned previously, regularisation is an important ingredient of ZDI. We start by exploring impact of different choices of regularisation on the quality of fits to observational data. Figure (5) shows the fit quality (mean standard deviation of (V/Ic) in %) for 13 different values of the regularization parameter. The goal is to obtain a simple map with an acceptable fit. The plot represents which deviation Stokes V was obtained for inversions with different regularisation parameters. The deviation decreases with a decreasing value of the regularisation parameter, so for lower values of the regularization parameter, better results are obtained. The optimal value for regularization roughly corresponds to where the largest change of slope occurs in Fig. 5, i.e. around log Λ=-10.

Figure 5. The deviation of Stokes V (in %) is plotted as a function of value of regularization parameter. From the slope it is possible to determine the optimal value for the regularization. The deviation of Stokes V is decreasing when the value of regularization parameter is decreasing. The stellar rotational period is another critical parameter. The initial value used for the rotational period of the star was Prot=4.9 d. We reconsidered this value since it may not be exact, to get a better fit for Stokes V profiles corresponding to the 2014 dataset. Changing Prot several times suggests Prot=4.93 d. The mean deviation for the initial value of Prot=4.9 d was 0.00409% and for the adjusted value it changed to 0.00360%. In figure (6) it can be seen that the best-fitting rotational period value is Prot=4.93 d.

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Figure (6). Plot of the deviation of Stokes V as a function of rotational period Prot. Figures (7) and (9) present spherical projections of 2D magnetic field maps of π1 UMa maps obtained using the ZDI. We also provide plots of the observed LSD profiles in comparison to the model profiles, see figure (8) and (10). If we interpret the strength of the magnetic field from the color chart in figure (7), we can see a lot of blue areas in the radial components, which corresponds to magnetic fields of strength 0 G and below (negative, i.e. inward directed). The field strength is mostly negative at the north pole of the star, and has a positive region at the stellar equator. In the meridional field there is a positive field strength region on the north pole but it decreases approaching the equator. Overall, the star has dominant blue areas, so the meridional field is negative in most areas of the star. In the azimuthal field the red zones are dominant at the stellar equator, so the field strength is mostly positive, and stronger than the other two components. The maps show one small region at the north pole where the azimuthal field strength is negative. In figure (9) the field topology of π1 UMa is shown for the dataset taken in January 2015. The radial field is negative at the north pole but has some regions where the field is positive, for example around the low latitudes near the south pole. The meridional component has a mix of negative and positive fields all over the star, and the azimuthal components is dominant with white and red color, which corresponds to positive field strength in the 0-97 G range. The azimuthal field is positive and strong at the stellar equator, but is negative on the north pole.

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Figure 7. Reconstructed magnetic field maps for the April 2014 dataset. Br shows the radial magnetic field, Bm is the meridional magnetic field and Ba is the azimuthal magnetic field. The last row is the field orientation with blue vectors showing the inward directed field and red vectors corresponding to the outward directed field. Next to each sphere the rotational phase is indicated. The magnetic field strength is given in units Gauss (G), presented by the colorbar to the right.

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Figure 8. Plot of Stokes I profile to the left and Stokes V profile to the right for the dataset obtained in April 2014. The LSD profile intensity is on the y-axis and the velocity on the x-axis. The black lines are the observed profiles and the red lines are the fitted model profiles. The rotational phase is indicated in blue.

Figure 9. Same as figure 7 for the magnetic field maps reconstructed from the January 2015 dataset.

Figure 10. Same as figure 8 for the observed and model profiles corresponding to the January 2015 dataset.

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The spherical harmonic description of the magnetic field topology used by the ZDI code allows us to characterise contributions of different harmonic modes. The relative energies of different harmonic components were considered for different l-values, where l goes from 1 to 10. This information is given in Table 4ab, which provides the fraction of poloidal, toroidal and total energy for each data set. For the first dataset (table 4a) the poloidal energy is deposited in l=1-4, and the toroidal in l=1-3. For the second dataset (table 4b) the poloidal and toroidal energy are concentrated in l=1-3. The total toroidal energy dominates over the poloidal energy for both epochs. The total toroidal field energy is higher for the second dataset. The fraction Eall together with Ep1,2 and Et1,2 is also presented in a plot (see figure (11,12)) for each dataset to illustrate the distribution of the magnetic field energy over different l-values. Table 4a. Relative energies for different spherical harmonic components of the magnetic map derived for April 2014. Ep is the toroidal energy and Et is the toroidal energy whereas Eall is the sum of poloidal and toroidal energy, in percent.

l Ep1 Et1 Eall

1 16.7 40.3 57.0

2 9.1 7.7 16.8

3 3.4 6.2 9.6

4 10.9 0.9 11.8

5 2.0 0.6 2.6

6 0.7 0.4 1.1

7 0.3 0.1 0.3

8 0.5 0.0 0.5

9 0.2 0.0 0.2

10 0.1 0.0 0.1

All 43.7 56.3 100

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Table 4b. Same as Table 4a but for January 2015.

l Ep2 Et2 Eall

1 9.1 41.0 50.2

2 18.9 3.9 22.7

3 3.8 13.7 17.5

4 1.3 3.3 4.6

5 1.0 0.6 1.7

6 1.8 0.6 2.4

7 0.4 0.1 0.5

8 0.2 0.0 0.2

9 0.1 0.0 0.1

10 0.1 0.0 0.1

All 36.7 63.3 100

Figure 11. Plot of poloidal energy in blue, toroidal energy in orange, and total magnetic energy in red as a function of spherical harmonic angular degree l for the April 2014 magnetic map.

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Figure 12. Same as Fig. 11 for the January 2015 magnetic map. As an additional characteristic of the field, one can consider average strengths of different field vector components.The mean absolute strengths the radial, azimuthal and meridional field components have been listed in table (5). From the table it can be noted that the mean field is higher for the radial and meridional components in 2014 than in 2015, but the azimuthal field is almost the same. The total mean field is somewhat higher for the dataset taken in 2014, but the difference is only 3.8 G (13% of the total mean field). Table 5. The average strength of the radial field, meridional field, azimuthal field and the total field for the magnetic field maps derived from April 2014 and January 2015 datasets. The mean field modulus is defined as

.< B = √B2r + B2

a + B2m >

Field [G] April 2014 January 2015

mean |Br| 12.9 8.8

mean |Bm| 13.2 8.4

mean |Ba| 20.9 21.8

mean field modulus 31.6 27.8

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5. Summary and discussion

This study has presented analysis of the surface magnetic field of the young solar analogue star π1 UMa. This work contributes to understanding evolution of the Sun’s magnetic field by looking at a star that represents a proxy of the Sun when it was just a few hundred million years old. We started the analysis by downloading spectropolarimetric observations collected in 2014 and 2015 from a public database. We then applied the LSD technique to derive spectra of the mean Stokes I and V line profiles. We assessed detection of the circular polarisation signatures and measured the mean longitudinal magnetic field from these profiles. Then, these LSD Stokes profiles were modelled with the ZDI method to reconstruct maps of stellar magnetic field topology and provide detailed information about the stellar magnetic field. The topology of the surface magnetic field of π1 UMa seems to be dominated by the azimuthal field component. Since we worked with only two sets of observations taken within 7 months of each other, it is hard to make far-reaching conclusions about long-term variability of the stellar field. However, comparing to the earlier study in [7] of the same star, the results of the magnetic field topology seem to agree. Also, the mean magnetic field strengths were calculated from similar ZDI modelling of the observations obtained in 2007 in their study. The field components strength values obtained in [7] were Br =9 G, Bm=4 G and Ba=21 G. These results agree very well with the mean magnetic field strengths derived in our study based on observations from 2014 and 2015. Thus, it can be concluded that π1 UMa maintained approximately the same level of magnetic activity at these two epochs 7 years apart. Looking at figure (4) from reference [7] the distribution of Eall is plotted as a function of L for π1 UMa. Their result doesn’t differ much from ours (see figure (11,12)), because almost all of the total magnetic field energy is found in l=1-3 in both studies. In addition, the energy decreases as l increases from 1 to 3 in both plots. The dipole component l=1 contains most of the field energy. The results of the both ZDI studies show a dominantly toroidal field. This agrees with the trend found in study [12] that younger/faster rotating stars tend to show dominance of the toroidal component, while stars with increasing age or slower rotation rate, like the Sun, tend to have a dominant poloidal component. The difference in the mean characteristics of the global field of π1 UMa between April 2014 and January 2015 is not so significant. The radial and meridional fields are somewhat weaker in the later epoch and the azimuthal field is a little stronger. Perhaps we are observing long-term activity cycle of this star with magnetic energy exchanging between the toroidal (mainly azimuthal) and poloidal (mainly radial and meridional) components. In the radial component the field is predominantly negative at the visible rotational pole for both epochs, the meridional field is a combination of positive and negative polarities for both epochs, and

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the azimuthal field components remains dominantly positive for both epochs. Perhaps, it is not surprising that the change of the magnetic topology between the two epochs is not significant because the two datasets were taken merely 7 months apart. Our analysis thus suggests that the main properties of the global field topology of π1 UMa remain stable on this time scale. In reference [7] the magnetic field of UMa was obtained with ZDI using observations π1 from 2007. The radial field was found to be negative at the north pole but has positive regions around the equator and low latitudes close to the south pole. The meridional field is negative in most areas, especially at the north pole. The azimuthal field is positive over the whole star, and has the strongest spots at the stellar equator. This is very similar to the results obtained in this project. There is no evidence that polarity switching has occurred because there is no clear, systematic change in the sign of any of the magnetic field components in the three epochs. However, since there is a big gap between the observational data from 2007 and 2014, several polarity reversals could have been missed between these years. The 2014 and 2015 data are 7 months apart and show the same field polarity, suggesting that, the magnetic cycle might be one year or more. If no polarity switch has really occurred between 2007 and 2014 the magnetic cycle for the star might be 8 years or more. Moreover, the azimuthal component is the strongest for all three epochs (2007, 2014 and 2015) and corresponds to a larger mean field strength than radial and meridional field. This indicates persistent dominance of the horizontal fields likely associated with toroidal global field component. Finally, this study has thereby presented analysis of the surface magnetic field of the young solar analogue star π1 UMa. This work contributes to understanding evolution of the Sun’s magnetic field (which today have a much weaker global magnetic field of ~1 G) by looking at a star that represents a proxy of the Sun when it was just a few hundred million years old.

6.References [1] Petit, P., Dintrans, B., Solanki, S. K., et al. Magnetic geometries of Sun-like stars : impact of rotation, Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics Eds.: C. Charbonnel, F. Combes and R. Samadi, 388, 80, 2008 [2] H.Lammer et al. Variability of solar/stellar activity and magnetic field and its influence on planetary atmosphere evolution. Earth Planets Space, 64, 179–199, 2012 [3] Kochukhov, O. Stellar Magnetic Fields. In J. Sánchez Almeida & M. Martínez González (Eds.), Cosmic Magnetic Fields. (Canary Islands Winter School of Astrophysics, pp. 47-86). Cambridge: Cambridge University Press. 2018 [4] Petit, P.; Louge, T.; Théado, S.; Paletou, F.; Manset, N.; Morin, J.; Marsden, S. C.; Jeffers, S. V. PolarBase: A Database of High-Resolution Spectropolarimetric Stellar

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Observations, Publications of the Astronomical Society of the Pacific, Volume 126, Issue 939, pp. 469, 2014 [5] J.-F. Donati and J. D. Landstreet. Magnetic Fields of Nondegenerate Stars. Annual Review of Astronomy & Astrophysics, 47:333–370, 2009. [6] Claret,A. A new non-linear limb-darkening law for LTE stellar atmosphere models. Calculations for -5.0 <= log[M/H] <= +1, 2000 K <= Teff <= 50000 K at several surface gravities. A&A, 363, 1081-1190, 2000 [7] L. Rosén, O. Kochukhov, T. Hackman, and J. Lehtinen. Magnetic fields of young solar twins. A&A 593, A35, 2016 [8] O. Kochukhov, T. Lüftinger, C. Neiner, E. Alecian, and the MiMeS collaboration. Magnetic field topology of the unique chemically peculiar star CU Virginis. A&A 565, A83, 2014. [9] J.-F. Donati, M. Semel, B. D. Carter, D. E. Rees, and A. Collier Cameron. Spectropolarimetric observations of active stars. Mon. Not. R. Astron. Soc, 291, 658, 1997. [10] Wade, G. A,. Donati, J-F,. Landstreet, J.D,. & Shorlin, S. L. S. High-precision magnetic field measurements of Ap and Bp stars. Mon. Not. R. Astron. Soc,. 313, 823, 2000 [11] Kochukhov, O. & Wade, G. A, Magnetic Doppler imaging of α2 Canum Venaticorum in all four Stokes parameters. Unveiling the hidden complexity of stellar magnetic fields, A&A, 513, A13, 2010 [12] Petit, P.; Dintrans, B.; Solanki, S. K.; Donati, J.-F.; Aurière, M.; Lignières, F.; Morin, J.; Paletou, F.; Ramirez Velez, J,; Catala, C.; Fares, R. Toroidal versus poloidal magnetic fields in Sun-like stars: a rotation threshold. Mon. Not. R. Astron. Soc,. Volume 388, Issue 1, pp. 80-88, 2008 [13] VALD3 (2018). Vienna Atomic Line Database [database] Retrieved from: http://vald.astro.uu.se [14] Donati, J. -F.; Semel, M.; Carter, B. D.; Rees, D. E.; Collier Cameron, A. Spectropolarimetric observations of active stars. Mon. Not. R. Astron. Soc,. Volume 291, Issue 4, pp. 658-682, 1997 [15] P. Janardhan.; K. Fujiki.; M. Ingale.; Susanta Kumar Bisoi.; and Diptiranjan Rout. Solar cycle 24: an unusual polar field reversal. A&A, 618, A148, 2015