# Magnetic fields on young, moderately rotating Sun-like stars II. EK Draconis (HD 129333)

Ian Waite, Stephen Marsden, Brad Carter, Pascal Petit, Sandra Jeffers, Julien Morin, Aline Vidotto, Jean-François Donati, The Bcool Collaboration

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Ian Waite, Stephen Marsden, Brad Carter, Pascal Petit, Sandra Jeffers, et al. Magnetic fields on young, moderately rotating Sun-like stars II. EK Draconis (HD 129333). Monthly Notices of the Royal Astronomical Society, 2017, 465 (2), pp.2076-2091. 10.1093/mnras/stw2731. hal-01529764

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# Magnetic fields on young, moderately rotating Sun-like stars – II. EK Draconis (HD 129333)

I. A. Waite,<fnref n="1" />* S. C. Marsden,<fnref n="1" />* B. D. Carter,<fnref n="1" />* P. Petit,<fnref n="2" />,<fnref n="3" /> S. V. Jeffers,<fnref n="4" /> J. Morin,<fnref n="5" />
A. A. Vidotto,<fnref n="6" /> J.-F. Donati<fnref n="2" />,<fnref n="3" /> and the BCool Collaboration

Accepted 2016 October 20. Received 2016 October 20; in original form 2016 March 6

## ABSTRACT

The magnetic fields, activity and dynamos of young solar-type stars can be empirically studied using time series of spectropolarimetric observations and tomographic imaging techniques such as Doppler imaging and Zeeman-Doppler imaging. In this paper, we use these techniques to study the young Sun-like star EK Draconis (SpType: G1.5V, HD 129333) using *ESPaDOnS* at the Canada-France-Hawaii Telescope and *NARVAL* at the Télescope Bernard Lyot. This multi-epoch study runs from late 2006 until early 2012. We measure high levels of chromospheric activity indicating an active, and varying, chromosphere. Surface brightness features were constructed for all available epochs. The 2006/2007 and 2008 data show large spot features appearing at intermediate latitudes. However, the 2012 data indicate a distinctive polar spot. We observe a strong, almost unipolar, azimuthal field during all epochs, which is similar to that observed on other Sun-like stars. Using magnetic features, we determined an average equatorial rotational velocity, $\Omega_{eq}$, of $\sim 2.50 \pm 0.08$ rad d$^{-1}$. High levels of surface differential rotation were measured with an average rotational shear, $\Delta\Omega$, of $\sim 0.27_{-0.26}^{+0.24}$ rad d$^{-1}$. During an intensively observed 3-month period, from 2006 December until 2007 February, the magnetic field went from predominantly toroidal ($\sim 80$ per cent) to a more balanced poloidal-toroidal ($\sim 40–60$ per cent) field. Although the large-scale magnetic field evolved over the epochs of our observations, no polarity reversals were found in our data.

**Key words:** line: profiles–stars: activity–stars: individual: HD 129333–stars: magnetic fields–stars: solar-type–starspots.

## 1 INTRODUCTION

In Sun-like stars, the generation of the magnetic field is via a dynamo process, with differential rotation being one of the key drivers. The classical dynamo model for the Sun is believed to be operating at the interface between the radiative zone and the convective zone and is known as a $\alpha$-$\Omega$ or ‘shell’ dynamo (e.g. Parker 1955; Charbonneau 2010). While the large-scale toroidal magnetic field is understood to be buried in the sub-surface layers of the Sun, it is observed at the surface of a range of rapidly rotating solar-type stars through the presence of strong unipolar surface azimuthal magnetic fields (e.g. Donati et al. 2003a; Petit et al. 2004a; Marsden et al. 2006). One explanation is that these stars distribute the

dynamo action closer to the surface of the star (e.g. Brandenburg et al. 1989; Moss et al. 1995; Brown et al. 2010). Detailed three-dimensional magnetohydrodynamic modelling using anelastic spherical harmonic code produces models of azimuthal field wreaths that are similar to the ring-like surface field structures observed on rapidly rotating solar-type stars (e.g. Brown et al. 2011; Nelson et al. 2013).

Given the key role of differential rotation on magnetic field generation with increasing stellar rotation, long-term measurements of magnetic field topologies and differential rotation for individual stars potentially provide a way to survey the emerging stellar magnetic cycles of young stars, and so to indirectly study the origins of the solar cycle (e.g. Charbonneau 2010). This makes differential rotation and magnetic topologies key tools for the study of the activity of Sun-like stars. These provide a deeper understanding of stellar evolution and, in particular, the

<footnote n="1">¹Computational Engineering and Science Research Centre, University of Southern Queensland, Toowoomba, QLD 4350, Australia</footnote>

<footnote n="2">²Université de Toulouse, UPS-OMP, Institut de Recherche en Astrophysique et Planétologie, F-31400 Toulouse, France</footnote>

<footnote n="3">³CNRS, Institut de Recherche en Astrophysique et Planétologie, 14 Avenue Edouard, Belin, F-31400 Toulouse, France</footnote>

<footnote n="4">⁴Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany</footnote>

<footnote n="5">⁵LUPM, Université de Montpellier, F-34095 CNRS, France</footnote>

<footnote n="6">⁶School of Physics, Trinity College Dublin, University of Dublin, Dublin-2, Ireland</footnote>

<footnote>* E-mail: ian.waite@usq.edu.au (IAW); stephen.marsden@usq.edu.au (SCM); brad.carter@usq.edu.au (BDC)</footnote>
operation and evolution of stellar magnetic fields, and the likely impact on any emergent planetary systems (e.g. do Nascimento et al. 2016).

The focus of this investigation is the young Sun-like star EK Draconis (HD 129333, HIP 71631). It is an ideal proxy of the infant Sun at a near-zero-age main-sequence age and, as a result, has been the subject of many campaigns using a range of telescopes such as NASA's *Hubble Space Telescope* (e.g. Ayres & France 2010; Linsky et al. 2012), the *Far Ultraviolet Spectroscopic Explorer* (e.g. Guinan, Ribas & Harper 2003), the *Extreme Ultraviolet Explorer* (e.g. Audard, Güdel & Guinan 1999), the *Röntgensatellit* X-ray observatory (e.g. Güdel et al. 1995) and ESA's *X-ray Multi-Mirror Mission* (e.g. Scelsi et al. 2005) to name a few. Additionally, a range of techniques have been utilized such as speckle interferometry (König et al. 2005), direct imaging (Metchev & Hillenbrand 2004), pho-tometry (both broad-band and Strömgren, e.g. Fröhlich et al. 2002; Zboril 2005), Doppler imaging (DI, e.g. Strassmeier & Rice 1998; Järvinen et al. 2007; Rosén et al. 2016), and now in this paper, we reconstruct its large-scale magnetic field geometry using Zeeman-Doppler imaging (ZDI).

The application of ZDI (Semel 1989; Donati & Semel 1990; Donati & Brown 1997; Donati et al. 2003a) enables us to indi-rectly observe the large-scale surface magnetic field geometry. The mapping of the magnetic fields on EK Draconis is important as pre-viously only brightness maps have been developed for this young Sun-like star (Strassmeier & Rice 1998; Järvinen et al. 2007). Mag-netic imaging will assist with the long-term goal of understanding the early magnetic life of our Sun and its influences on the young emerging planetary systems (e.g. do Nascimento et al. 2014, 2016). This paper continues a series of papers that investigates the mag-netic field topologies of moderately rotating, young Sun-like stars with HD 35296 and HD 29615 being the subjects of paper I of this series (Waite et al. 2015, hereinafter, Paper I). This study of EK Draconis is part of the BCool<fnref n="1" /> collaboration investigating the magnetic activity of low-mass stars (e.g. Marsden et al. 2014).

## 2 PREVIOUS STUDIES OF EK DRACONIS

EK Draconis is a G1.5V star (Montes et al. 2001) and is consid-ered a good proxy for a young Sun (e.g. Dorren & Guinan 1994; Strassmeier & Rice 1998; Järvinen et al. 2007). The *HIPPAR-COS* space mission measured a parallax of 29.30 ± 0.37 mas (van Leeuwen 2007). Montes et al. (2001) suggest that EK Draconis is a member of the Local Association based on its space motion, high levels of activity and strong Li II with an equivalent width of 189 mÅ. Granzer et al. (2000) estimated its age to be between 30 and 50 Myr.

For some time, EK Draconis was considered an infant Sun (Dorren & Guinan 1994). Järvinen et al. (2007) observed the wings of the 866.2 nm Ca II infrared triplet (IRT) line and concluded that the photosphere is very similar to that of the Sun. Paletou et al. (2015) estimated an effective temperature of 5561 K for the pri-mary component while Järvinen et al. (2007) determined the mi-croturbulence to be $\xi_t = 1.6 \pm 0.1$ km s$^{-1}$ with a metallicity of [M/H] = 0.0 ± 0.05. Duquennoy & Mayor (1991) suggested that EK Draconis was a binary with a secondary component of mass ≥0.37 M☉. Metchev & Hillenbrand (2004) used adaptive optics on the 5-m Palomar Telescope to directly image EK Draconis to con-firm the existence of the secondary component. They defined the primary component to be 0.9 ± 0.1 M☉ with the secondary being

0.5 ± 0.1 M☉ in a highly eccentric ($e = 0.82 \pm 0.03$) orbit. Addi-tionally, they speculated on the possibility of EK Draconis being a triple system but König et al. (2005) used speckle interferometry to rule out the possibility of a third companion. They also determined the orbital period of this binary system to be ~45 ± 5 yr making it unlikely that the two components interact. Thus, EK Draconis is a wide binary whose primary is akin to a single Sun-like star and may be considered an excellent proxy for a young Sun.

EK Draconis has been the focus of many longitudinal photomet-ric studies, some spanning ~45 yr. For example, Dorren & Guinan (1994) found that its activity underwent cyclic variations with an activity cycle of ~12 yr. Baliunas et al. (1995) reported variability, with no apparent periodicity, in the Mount Wilson Ca II H&K mea-surements. Järvinen, Berdyugina & Strassmeier (2005) found peri-odicities in the total spot area on a range of time-scales longer than 45 yr, with additional variations with a period of ~10.5 yr. Fröhlich et al. (2002) have observed a dimming of 0.0057 ± 0.0008 mag yr$^{-1}$ since 1975, that has even been more pronounced in recent times. Messina & Guinan (2003) noted the rotational period ranged from 2.551 to 2.886 d which they interpret in terms of surface differential rotation. This all shows EK Draconis to be an active young Sun.

DI maps have been produced by both Strassmeier & Rice (1998) and Järvinen et al. (2007). Strassmeier & Rice (1998) were the first to use DI to map the spot topography on the surface of EK Draconis. These authors were able to recover high-latitude spots, ≈ 70–80°, with a photosphere-to-spot temperature of $\Delta T \approx 1200$ K and mid-latitude spots with $\Delta T \approx 400$ K. Järvinen et al. (2007) produced DI maps that show high-latitude spot features, with $\Delta T \sim 500$ K. The mean spot latitude of these features varied during the 1 yr time frame of their observations, drifting towards the equator at an annual rate of ≈15°–25°, depending on the longitude studied.

## 3 OBSERVATIONS AND ANALYSIS

The fundamental parameters used in this study for EK Draconis are shown in Table 1. EK Draconis was observed using the Canada-France-Hawaii Telescope (CFHT – Mauna Kea, Hawaii) and the Télescope Bernard Lyot (TBL – Observatoire du Pic du Midi, France). Observations commenced in 2006 November 30,

<table><caption>**Table 1.** The parameters used to reconstruct the surface brightness and magnetic field distribution of EK Draconis. Except otherwise stated, these values were determined by this work.</caption><thead><tr><th>Parameter</th><th>EK Draconis</th></tr></thead><tbody><tr><td>Spectral type</td><td>G1.5V<sup>a</sup></td></tr><tr><td>Rotational period</td><td>2.766 ± 0.002 d</td></tr><tr><td>Inclination angle</td><td>60 ± 5°</td></tr><tr><td>v sin i</td><td>16.4 ± 0.1 km s<sup>-1</sup></td></tr><tr><td>T<sub>phot</sub></td><td>5561 K<sup>b</sup></td></tr><tr><td>ΔT<sub>phot-spot</sub></td><td>1700 K</td></tr><tr><td>Radial velocity, v<sub>rad</sub></td><td>-20.28 ± 0.04 km s<sup>-1</sup></td></tr><tr><td>Stellar radius</td><td>0.94 ± 0.07 R<sub>☉</sub></td></tr><tr><td>Mass</td><td>0.95 ± 0.04 M<sub>☉</sub><sup>c</sup></td></tr><tr><td>log g</td><td>4.47 ± 0.08</td></tr><tr><td>Stokes V: Ω<sub>eq</sub></td><td>~2.50 ± 0.08 rad d<sup>-1</sup></td></tr><tr><td>Stokes V: ΔΩ</td><td>~0.27<sup>+0.12</sup><sub>-0.14</sub> rad d<sup>-1</sup></td></tr></tbody></table>

Notes. <sup>a</sup>Montes et al. (2001);

<sup>b</sup>Paletou et al. (2015);

<sup>c</sup>based on the theoretical models of Siess, Dufour & Forestini (2000).

<footnote n="1">¹ http://bcool.ast.astr.mip.fr</footnote>
<table><caption>**Table 2.** The journal of spectropolarimetric observations of EK Draconis using the CFHT.</caption><thead><tr><th>UT date</th><th>UT<br>middle</th><th>Exp. time<sup>a</sup><br>(s)</th><th>Stokes V<br>SNR</th><th>Stokes I<br>reject<sup>b</sup></th></tr></thead><tbody><tr><td>2006 Nov 30</td><td>16:17:34</td><td>4×60</td><td>4310</td><td>1,2,3,4</td></tr><tr><td>2006 Dec 05</td><td>16:10:53</td><td>4×60</td><td>5384</td><td>3,4</td></tr><tr><td>2006 Dec 06</td><td>16:12:30</td><td>4×100</td><td>8291</td><td>3,4</td></tr><tr><td>2006 Dec 07</td><td>16:09:35</td><td>4×180</td><td>10850</td><td>4</td></tr><tr><td>2006 Dec 07</td><td>16:21:37</td><td>4×60</td><td>6415</td><td>1,2,3,4</td></tr><tr><td>2006 Dec 08</td><td>16:12:36</td><td>4×150</td><td>8254</td><td>2,3,4</td></tr><tr><td>2006 Dec 09</td><td>16:15:40</td><td>4×120</td><td>8649</td><td>2,3,4</td></tr><tr><td>2006 Dec 10</td><td>16:12:58</td><td>4×150</td><td>9157</td><td>3,4</td></tr><tr><td>2006 Dec 11</td><td>16:12:43</td><td>4×120</td><td>8063</td><td>3,4</td></tr></tbody></table>

Notes. <sup>a</sup>Each sequence consists of four sub-exposures with each sub-exposure being 60 s (for example).

<sup>b</sup>The frame number of each Stokes *I* sub-exposure that was rejected due to solar contamination, as explained in Section 3.3.

and the star was periodically revisited until 2012 February 9. A journal of observations is shown in Tables 2 and 3. The data were obtained from POLARBASE<fnref n="2" /> (Petit et al. 2014).

## 3.1 High-resolution spectropolarimetric observations from the CFHT and the TBL

High-resolution spectropolarimetric data were obtained using ESPaDOnS (Donati et al. 2006a) at the CFHT and NARVAL (Aurière 2003), the twin of ESPaDOnS, at the TBL. Each instrument has a mean pixel resolution of 1.8 km s⁻¹ per pixel. The spectral coverage was from ~370 to 1048 nm with a resolution of ~68 000. The grating has 79 gr mm⁻¹ with a 2k × 4.5k CCD detector covering 40 orders (orders no. 22 to no. 61). This extends to the Ca II H&K lines and also to the Ca II IRT lines. Each instrument consists of one fixed quarter-wave retarder sandwiched between two rotating half-wave retarders and coupled to a Wollaston beamsplitter. Each polarimetric sequence consists of four sub-exposures. After each sub-exposure, the rotating half-wave retarder of the polarimeter was rotated so as to remove instrumental polarization signals from the telescope and the polarimeter. Silvester et al. (2012) demonstrated the stability of both instruments over a lengthy period of time; hence these are ideal instruments for multi-epoch studies spanning several years.

## 3.2 Spectropolarimetric analysis

The initial data reduction was completed using the dedicated pipeline LIBREESPRIT (Échelle Spectra Reduction: an Interactive Tool) software package (Donati et al. 1997, 2003a). Preliminary processing involved removing the bias and using a nightly master flat. Each stellar spectrum was extracted and wavelength calibrated against a thorium-argon lamp. After using LIBREESPRIT, least-squares deconvolution (LSD) was applied to the reduced spectra. LSD is a multiline technique that combines several thousand spectral lines into a single line profile with greatly improved signal-to-noise ratio. LSD can be applied to both Stokes *I* and *V* spectra. A G2 line mask created from the Kurucz atomic data base and ATLAS9 atmospheric models (Kurucz 1993a,b) was used during the LSD process. In order to correct for the minor instrumental shifts in wavelength space as a result of small atmospheric temperature and pressure variations, each spectrum was shifted to match the Stokes *I* LSD profile of the telluric lines contained in the spectra, as was done by

<table><caption>**Table 3.** The journal of spectropolarimetric observations of EK Draconis using the TBL.</caption><thead><tr><th>UT date</th><th>UT<br>middle</th><th>Exp. time<sup>a</sup><br>(s)</th><th>Stokes V<br>SNR</th><th>Stokes I<br>reject<sup>b</sup></th></tr></thead><tbody><tr><td>2007 Jan 26</td><td>6:39:37</td><td>4×300</td><td>6446</td><td>2,3,4</td></tr><tr><td>2007 Jan 27</td><td>5:49:08</td><td>4×300</td><td>7058</td><td>—</td></tr><tr><td>2007 Jan 28</td><td>5:18:08</td><td>4×300</td><td>7603</td><td>—</td></tr><tr><td>2007 Jan 29</td><td>5:16:09</td><td>4×300</td><td>9244</td><td>—</td></tr><tr><td>2007 Jan 30</td><td>5:07:46</td><td>4×300</td><td>6215</td><td>—</td></tr><tr><td>2007 Feb 01</td><td>4:44:47</td><td>4×300</td><td>6240</td><td>—</td></tr><tr><td>2007 Feb 02</td><td>5:54:59</td><td>4×300</td><td>8026</td><td>—</td></tr><tr><td>2007 Feb 03</td><td>5:32:38</td><td>4×300</td><td>8522</td><td>—</td></tr><tr><td>2007 Feb 04</td><td>5:49:48</td><td>4×300</td><td>9129</td><td>—</td></tr><tr><td>2007 Feb 15</td><td>6:20:12</td><td>4×300</td><td>7531</td><td>3,4</td></tr><tr><td>2007 Feb 18</td><td>4:16:06</td><td>4×300</td><td>925</td><td>4<sup>c</sup></td></tr><tr><td>2007 Feb 19</td><td>3:10:57</td><td>4×300</td><td>7079</td><td>—</td></tr><tr><td>2007 Feb 22</td><td>2:27:53</td><td>4×300</td><td>8469</td><td>—</td></tr><tr><td>2007 Feb 23</td><td>3:10:56</td><td>4×300</td><td>8484</td><td>—</td></tr><tr><td>2007 Feb 28</td><td>1:59:43</td><td>4×300</td><td>1517</td><td>3,4<sup>c</sup></td></tr><tr><td>2008 Jan 23</td><td>6:34:03</td><td>4×300</td><td>9772</td><td>3,4</td></tr><tr><td>2008 Jan 24</td><td>6:27:32</td><td>4×300</td><td>7065</td><td>2,3,4</td></tr><tr><td>2008 Jan 26</td><td>6:07:26</td><td>4×400</td><td>10997</td><td>—</td></tr><tr><td>2008 Jan 27</td><td>6:22:54</td><td>4×300</td><td>10978</td><td>—</td></tr><tr><td>2008 Jan 28</td><td>6:41:25</td><td>4×300</td><td>11937</td><td>—</td></tr><tr><td>2008 May 26</td><td>20:39:17</td><td>4×300</td><td>10512</td><td>—</td></tr><tr><td>2008 May 26</td><td>21:08:42</td><td>4×400</td><td>11809</td><td>—</td></tr><tr><td>2008 May 29</td><td>21:04:03</td><td>4×300</td><td>14546</td><td>—</td></tr><tr><td>2008 May 29</td><td>21:54:48</td><td>4×550</td><td>14468</td><td>—</td></tr><tr><td>2009 Jan 3</td><td>5:53:33</td><td>4×600</td><td>17250</td><td>—</td></tr><tr><td>2009 Jan 4</td><td>5:24:57</td><td>4×600</td><td>14155</td><td>—</td></tr><tr><td>2009 Jan 5</td><td>5:22:01</td><td>4×600</td><td>15814</td><td>—</td></tr><tr><td>2009 Jan 11</td><td>4:51:47</td><td>4×600</td><td>12926</td><td>—</td></tr><tr><td>2010 Feb 15</td><td>5:53:00</td><td>4×300</td><td>9510</td><td>—</td></tr><tr><td>2012 Jan 12</td><td>6:11:54</td><td>4×300</td><td>9959</td><td>—</td></tr><tr><td>2012 Jan 13</td><td>5:37:09</td><td>4×300</td><td>9502</td><td>—</td></tr><tr><td>2012 Jan 14</td><td>5:20:28</td><td>4×300</td><td>8206</td><td>—</td></tr><tr><td>2012 Jan 15</td><td>5:20:07</td><td>4×300</td><td>8149</td><td>—</td></tr><tr><td>2012 Jan 16</td><td>5:01:48</td><td>4×300</td><td>7770</td><td>—</td></tr><tr><td>2012 Jan 17</td><td>4:36:18</td><td>4×300</td><td>6175</td><td>—</td></tr><tr><td>2012 Jan 18</td><td>4:28:34</td><td>4×300</td><td>8276</td><td>—</td></tr><tr><td>2012 Jan 23</td><td>5:59:36</td><td>4×300</td><td>8930</td><td>—</td></tr><tr><td>2012 Jan 24</td><td>5:06:37</td><td>4×300</td><td>8127</td><td>—</td></tr><tr><td>2012 Feb 09</td><td>1:25:37</td><td>4×300</td><td>7797</td><td>—</td></tr></tbody></table>

Notes. <sup>a</sup>Each sequence consists of four sub-exposures with each sub-exposure being 60 s (for example).

<sup>b</sup>The frame number of each Stokes *I* sub-exposure that was rejected due to solar contamination, as explained in Section 3.3.

<sup>c</sup>In these cases, LSD profiles were rejected due to poor signal to noise of the spectrum.

Donati et al. (2003a), Marsden et al. (2006) and others. Further information on LSD can be found in Donati et al. (1997) and Kochukhov, Makaganiuk & Piskunov (2010).

## 3.3 Solar contamination

There was some severe solar contamination in the red wing and core of the Stokes *I* LSD profile for the CFHT data (most nights) and some contamination in the TBL data due to the observations being taken close to sunrise. To determine the influence of sunrise on the contamination, the first Stokes *I* profile of the sequence of four sub-exposures was used as the template with successive profiles individually subtracted, producing a difference profile. If

<footnote n="2">² http://polarbase.irap.omp.eu</footnote>
<table><caption>**Table 4.** The coefficients, listed in equation (1), as determined by Marsden et al. (2014).</caption><thead><tr><th>Coefficient</th><th>ESPaDOnS</th><th>NARVAL</th></tr></thead><tbody><tr><td>C<sub>1</sub></td><td>7.999</td><td>12.873</td></tr><tr><td>C<sub>2</sub></td><td>-3.904</td><td>2.502</td></tr><tr><td>C<sub>3</sub></td><td>1.150</td><td>8.877</td></tr><tr><td>C<sub>4</sub></td><td>1.289</td><td>4.271</td></tr><tr><td>C<sub>5</sub></td><td>-0.069</td><td>1.183 × 10<sup>-3</sup></td></tr></tbody></table>

the deviation of this difference profile exceeds the average noise level of the first LSD profile in the areas of concern, this Stokes *I* profile was excluded from the brightness mapping process. Those profiles excluded are listed in Tables 2 and 3. All Stokes *I* profiles taken on 2006 November 30 were rejected because the first profile was severely affected. Additionally, the first profile of the second sequence of exposures taken on 2006 December 07 was severely affected hence all four profiles were rejected. We searched for evidence of contamination in the Stokes *V* profiles by examining the null profile. The null profile is used as a check for spurious magnetic signatures and is generated by pair-processing sub-exposures corresponding to the identical positions of the half-wave Fresnel Rhomb of the polarimeter during each sequence of four sub-exposures (see Donati et al. 1997). There appeared no contamination in the Stokes *V* data exceeding the typical noise level of the data.

## 4 CHROMOSPHERIC ACTIVITY

Chromospheric activity was measured using the Ca II H&K, Ca II IRT and Hα spectral lines. The two Ca II H&K absorption lines are the most widely used optical indicators of chromospheric activity as their source functions are collisionally controlled and hence they are very sensitive to electron density and temperature. This leads to emission in the core of these lines. The Ca II IRT lines share the upper levels of the H&K transitions and are formed in the lower chromosphere (e.g. Montes et al. 2004). The Hα spectral line is also collisionally filled in as a result of higher temperatures, is formed in the middle of the chromosphere and is often associated with

<figure><img src="image_3.png" /><figcaption>**Figure 1.** The left series of panels show the long-term variations of the activity indices from 2006 December until 2012 January. Each data point was an average of all the respective indices for a particular data set, while the error bars provide the range of the activity indices. The right series of panels show the variation in the chromospheric activity over the intensely observed 2006/2007 observing run. The data were phased to the ephemeris using equation (4). Diamonds represent CFHT, circles TBL from January to early February observations and squares TBL 2007 late February. The error bars indicate the range of the measurement during four sub-exposures.</figcaption></figure>

plages and prominences (e.g. Thatcher & Robinson 1993; Montes et al. 2004).

The Ca II H&K index for EK Draconis was determined using the procedure as explained in Wright et al. (2004). This was converted to match the Mount Wilson S-values (Duncan et al. 1991) using

$$S\text{-index} = \frac{C_1 F_H - C_2 F_K}{C_3 F_{VHK} + C_4 F_{RHK}} - C_5, \quad (1)$$

where $F_H$ and $F_K$ were the fluxes determined in the line cores, centred on 393.3663 and 396.8469 nm, respectively, from the two triangular bandpasses with a full width at half-maximum (FWHM) of 0.218 nm. Two 2 nm-wide rectangular bandpasses, $F_{RHK}$ and $F_{VHK}$, centred on 400.107 and 390.107 nm, respectively, were used for the continuum flux in the red and blue sides of the H and K lines. The transformation coefficients used to calibrate these data to the Mt Wilson Survey were determined by Marsden et al. (2014). The coefficients for equation (1) are listed in Table 4.

Two further activity indices were used: the Hα spectral line and the Ca II IRT lines. The Hα-index was determined using

$$N_{H\alpha} - \text{index} = \frac{F_{H\alpha}}{F_V + F_R}, \quad (2)$$

where $F_{H\alpha}$ is the flux determined in the line core, centred on 656.285 nm, using a triangular bandpass with an FWHM of 0.36 nm. Two 0.22 nm wide rectangular bandpasses $F_V$ and $F_R$, centred on 655.885 and 656.730 nm, respectively, were used for the continuum flux in the blue and red sides of the Hα line (Gizis, Reid & Hawley 2002). The Ca IRT index was determined using

$$N_{\text{Ca} \text{IRT}} - \text{index} = \frac{\sum F_{\text{IRT}}}{F_V + F_R}, \quad (3)$$

where $\sum F_{\text{IRT}}$ is the total flux determined in the line cores of the three spectral lines, 849.8023, 854.2091 and 866.2141 nm, using triangular bandpasses with an FWHM of 0.2 nm. Two 0.5 nm wide rectangular bandpasses $F_V$ and $F_R$, centred on 847.58 and 870.49 nm, respectively, were used for the continuum flux in the blue and red sides of the Ca II IRT spectral lines (Petit et al. 2013).

Fig. 1(a) shows the variation in the Ca II H&K lines, Hα and Ca II IRT spectral lines during the 5 yr interval that these data cover.

<figure><img src="image_4.png" /></figure>
<table><caption>**Table 5.** Average activity indices found for EK Draconis. Also listed is the range of the respective index for the observing season. Additionally, the number of exposures included varies, as listed in Tables 2 and 3.</caption><thead><tr><th>Date</th><th>S-index<sup>a</sup></th><th>N<sub>Hα</sub></th><th>N<sub>CalIRT</sub></th></tr></thead><tbody><tr><td>2006 Dec</td><td>0.669 ± 0.116</td><td>0.386 ± 0.010</td><td>1.054 ± 0.025</td></tr><tr><td>2007 Jan</td><td>0.629 ± 0.068</td><td>0.392 ± 0.010</td><td>1.081 ± 0.023</td></tr><tr><td>2007 Feb</td><td>0.620 ± 0.199</td><td>0.389 ± 0.007</td><td>1.055 ± 0.029</td></tr><tr><td>2008 Jan</td><td>0.637 ± 0.064</td><td>0.388 ± 0.007</td><td>1.076 ± 0.015</td></tr><tr><td>2008 May</td><td>0.633 ± 0.013</td><td>0.388 ± 0.001</td><td>1.083 ± 0.005</td></tr><tr><td>2009 Jan</td><td>0.648 ± 0.034</td><td>0.392 ± 0.008</td><td>1.075 ± 0.017</td></tr><tr><td>2010 Feb</td><td>0.648 ± 0.010</td><td>0.393 ± 0.001</td><td>1.062 ± 0.002</td></tr><tr><td>2012 Jan</td><td>0.652 ± 0.053</td><td>0.392 ± 0.012</td><td>1.075 ± 0.019</td></tr></tbody></table>

*Notes.* <sup>a</sup>The resulting N<sub>CaH</sub>-index was converted to match the Mount Wilson S-values (Duncan et al. 1991) using the transformation shown in equation (1).

Using the values from Schröder, Reiners & Schmitt (2009), log R'_{HK} was determined to be -4.08 ± 0.043. Table 5 shows the average S-index, N<sub>Hα</sub> and N<sub>CalIRT</sub> indices for the respective data sets. Each value is the average over the observing run with the range showing the variation during this time, most likely as a result of modulation due to the rotation of the star. Additionally, the number of exposures included varies, as listed in Tables 2 and 3. The phases, φ, of the observations were calculated using the ephemeris in equation (4):

$$ \phi = 2454 070.174 98 + 2.766E, \quad (4) $$

where E is the epoch of each observation.

Focusing on the 3-month period extending from 2006 December until 2007 February, the indices show variations based on the rotation of EK Draconis. This is shown in Fig. 1(b). There are variations due to spot evolution; for example, there is enhanced activity in the 2007 January observing season when compared with the 2006 December season, especially the N<sub>CalIRT</sub> index (bottom panel) at phase ~0.5.

The average S-index from the Mount Wilson sample was 0.4714 ± 0.0156 and log R'_{HK} was -4.245 ± 0.088 (Duncan et al. 1991). Between 1984 and mid-2003, Lockwood et al. (2007) measured an average S-index of 0.5475 and a log R'_{HK} of -4.148. At the same time, they observed a decrease in the brightness of the star using Strömgren b-y photometry. From late 2006 until early 2012, our project measured an average S-index of 0.644 ± 0.06, and the log R'_{HK} of -4.08 ± 0.043. This is substantially higher than that of the Mount Wilson Ca H&K project but closer to that of Lockwood et al. (2007). We infer that the activity on EK Draconis has increased since 1991 matching the reduction in brightness observed by Fröhlich et al. (2002) and Lockwood et al. (2007). EK Draconis continues to exhibit both short- and long-term variations, as shown in Fig. 1, and alluded to by Baliunas et al. (1995) during the Mount Wilson Ca II H&K project. However, we cannot confirm the 12 yr cyclic period reported by Dorren & Guinan (1994) due to the sparseness of observations during our 5 yr time frame.

# 5 IMAGE RECONSTRUCTION

## 5.1 Doppler imaging: brightness maps

Surface brightness images of this moderately rotating star were produced for five epochs ranging from 2006 December until 2012 January. Each of these maps, assuming solid-body rotation, was generated through the inversion of time series Stokes I LSD profiles. The imaging code used was that of Brown et al. (1991) and Donati

& Brown (1997). This inversion process is an ill-posed problem where an infinite number of solutions are possible. Therefore, this code implements the Skilling & Bryan (1984) maximum-entropy optimization that produces an image with the minimum amount of information. Synthetic Gaussian profiles are often used to produce the initial model (Unruh & Collier Cameron 1995), although these are not so effective for slow and moderate rotators. Hence, the initial modelled profiles of the photosphere and spots were generated using the local line profiles from slowly rotating G2 and K5 stars, respectively. This allowed for more effective fitting of the wings of the LSD profile.

The imaging code was used to establish the values of a number of basic parameters, including the star's projected rotational velocity (v sin i), inclination angle, continuum level and radial velocity. A grid of plausible values was generated and full DI maps produced. The key parameters that were systematically changed were the continuum level (in 0.0001 steps), v sin i and v<sub>rad</sub> (both in 0.1 km s<sup>-1</sup> increments) and inclination angle (±5°). The average LSD profile was then compared with the average modelled LSD profile and deviations between the two were measured across the full profile. By minimizing the deviations at each point on the profile, the best set of parameters was identified. These parameters are listed in Table 1.

The imaging code assumes a two-temperature model: one being the quiet photosphere while the other being the temperature of the spot. The estimate of the effective temperature of the photosphere is usually derived from line-depth ratio measurements (Gray 1994), photometric colours (V - I or similar, e.g. Bessell, Castelli & Plez 1998) or more recently, principal component analysis (Paletou et al. 2015). However, the determination of the temperature of starspots is more problematic. All spots are assumed to be at the same temperature; hence, penumbral influences are neglected. This is necessary to overcome the limitations of rotational blurring and finite signal-to-noise factors (Collier Cameron 1992). There has been some debate in the literature regarding the temperature of the spots occurring on EK Draconis. Järvinen et al. (2007) used DI to determine that the spots were only 500 K cooler than the surrounding photosphere. Strassmeier & Rice (1998) also used DI and found spot temperatures that were 400–1200 K cooler than the surrounding photosphere. Scheible & Guinan (1994) also noted that the spot temperature was approximately 460 K less than the surrounding photosphere using photometric light curves. Using the most recent photospheric temperature estimate, of Paletou et al. (2015), of 5561 K, we started with a relatively warm starspot temperature (T<sub>spot</sub> = 5161 K) and systematically increased the difference between the spot temperature and the surrounding photosphere (ΔT) to determine the best fit of the modelled data, in a minimum χ<sup>2</sup> sense. We found that ΔT = 1700 K provided the best fits to the data. Any further reduction in the spot temperature made little difference to the resulting fits. This is consistent with the spot temperatures expected of early G-type stars, and supports the relationship found by Berdyugina (2005, fig. 7 on page 27 of that work). Further support comes from the work of O'Neal et al. (2004), who used molecular band modelling, in particular, the TiO bands at 705.5 and 886.0 nm. They found that the spot temperatures on EK Draconis were ~3800 K; closer to the typical minimum sunspot umbral temperatures (e.g. Penn et al. 2003). Still, spot temperatures are extremely difficult to estimate based solely on DI in the absence additional information such as the inclusion of simultaneous photometry (e.g. Waite et al. 2011b) or more in-depth analysis (e.g. Strassmeier 2009).

There has been a range of rotational periods found in the literature for EK Draconis. Many authors use a period of ~2.6–2.8 d.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Maps of EK Draconis for data from 2006 December until 2012 January. Solid-body rotation was assumed. The larger tick marks (in red) show the phase each observation was taken. The spot filling factors are listed in Table 6. The colour scale at the bottom of the maps indicates the gradation of spottedness with 0 (absent) to 1 (most intense). The associated fits between the modelled LSD profiles that were used to generate these maps and stellar LSD profiles are shown in Fig. 3.</figcaption></figure>

König et al. (2005) used radial velocity variations in their speckle observations to infer a period of $2.767 \pm 0.005$ d. Messina & Guinan (2003) used long-term photometry to investigate surface differential rotation and found variations in the period. They found two rotational periods in several seasons, both showing monotonical decreases along with each starspot cycle with solar-like behaviour. Güdel et al. (1995) reported a period of $2.6_{-0.3}^{+0.4}$ d in X-rays and a $2.75 \pm 0.05$ d period in the optical spectrum. They also observed periodicity in the X-ray light curve and, using auto-correlation analysis, found a 'broad secondary maximum around $P = 2.53$ d'. Assuming solid-body rotation and using an inclination angle of $60^\circ$ (Strassmeier & Rice 1998; Järvinen et al. 2007) and a $v \sin i$ of $16.4$ km $s^{-1}$ (as determined by this work), two separate maps were produced: one for 2006 December and the other for 2007 January. Using the ephemeris in equation (4), the rotational period was systematically modified so that the location of the predominant high-latitude spot was at the same phase in both maps. This is shown in the top two panels of Fig. 2.<fnref n="3" /> The rotational period

was found to be $2.766 \pm 0.002$ d; consistent with that of König et al. (2005). However, this period determination did not take into consideration differential rotation or spot evolution. Therefore, this may be considered the period at $70^\circ$ latitude and is more analogous to the photometric period obtained using broad-band photometry. Further data were also obtained in 2008 and in 2012. The solid-body rotation maps are shown in Fig. 2.

Fig. 3 shows the residual fits to the data. These are normalized profiles as each stellar LSD profile was subtracted from an average stellar LSD profile; likewise, each of the modelled profiles was subtracted from the average modelled profile. Each normalized stellar profile was compared with the associated modelled normalized profile. This has the advantage of accentuating the deviations between the stellar and modelled fits. The $\chi_r^2$ values for each map are listed in Table 6. This modelling assumed solid-body rotation, so small deviations observed between the normalized stellar profiles and the modelled profiles are taken to be the result of differential rotation and spot evolution. Differential rotation is discussed in Section 5.3.

## 5.2 Zeeman-Doppler imaging: magnetic mapping

The magnetic topology was reconstructed using ZDI. The modelling strategy of Donati & Brown (1997) was used to construct the radial, azimuthal and meridional fields. The mapping procedure uses the spherical harmonic expansions of the surface magnetic field, as implemented by Donati et al. (2006b). The maximum spherical harmonic expansion $\ell_{max} = 12$ was selected as any further increase in $\ell$ did not produce any significant difference in the magnitude and topology of the magnetic field recovered. Differential rotation was measured on all data sets, using the technique explained in Section 5.3.

The magnetic maps for EK Draconis are shown in Fig. 4 while the associated fits between the modelled data and the actual LSD profiles are shown in Fig. 5. The $\chi_r^2$ values of the magnetic models are listed in Table 6. The magnetic maps produced for EK Draconis show complex, and evolving, magnetic topologies from 2006 to 2012 and, in particular, during the intensely observed 3-month period in 2006/2007.

## 5.3 Differential rotation

Surface differential rotation on solar-type stars has been measured using a range of techniques: the Fourier transform of spectral lines (e.g. Reiners 2006), cross-correlation between images (e.g. Donati et al. 2000), spot tracking, particularly using KEPLER data (e.g. Reinhold & Reiners 2013; Reinhold & Gizon 2015) and Ca II H&K (e.g. Baliunas et al. 1995). Incorporating a solar-like differential rotation law, as defined in equation (5), into the modelling process has been successfully applied to a number of late F-/early-G stars (e.g. Petit, Donati & Collier Cameron 2002; Barnes, James & Collier Cameron 2004; Petit et al. 2004b; Barnes et al. 2005; Marsden et al. 2006):

$$ \Omega(\theta) = \Omega_{eq} - \Delta\Omega \sin^2 \theta, \quad (5) $$

where $\Omega(\theta)$ is the rotation rate at latitude $\theta$ in rad $d^{-1}$, $\Omega_{eq}$ is the equatorial rotation rate and $\Delta\Omega$ is the rotational shear between

2006–2007 data set was split into three separate maps: the CFHT data from 2006 November 30 to December 11, the TBL data from 2007 January 25 to February 4, while the third data set was taken from 2007 February 15 to February 28.

<footnote n="3">³ When attempting to model the combined 2006/2007 data set, the minimum reduced $\chi_r^2$ value was 1.9. This is higher than the typical $\chi_r^2$ values achieved (e.g. Petit et al. 2004b; Marsden et al. 2011a). Thus the 3-month long</footnote>
the equator and the pole, both in rad d⁻¹. This technique utilizes
a grid search for the two differential parameters, by systematically
adjusting Ωₐeq and ΔΩ, and determining χ²ᵣ for a fixed number
of iterations. This grid search produces a non-uniform χ²ᵣ land-
scape from which we fit a two-dimensional paraboloid to determine
the Ωₐeq-ΔΩ combination that best fits the data along with 1σ
errors.

Morin et al. (2008) demonstrated the possibility of applying
differential rotation modelling to relatively long time-scales pro-
vided that the magnetic topology is relatively stable. Differential
rotation measurements were attempted on the complete, 3-month
data set in 2006/2007 using the Stokes I LSD profiles. How-
ever, no differential rotation was found. Differential rotation mea-
surements were attempted for the three individual data sets: the
CFHT data from 2006 November 30 to December 11, the TBL
data from 2007 January 25 to February 4 and the TBL data from
2007 February 15 to February 28. Additionally, differential ro-
tation measurements were attempted on the data sets from 2008
to 2012. Despite these attempts, no measurable result was found
due to the inability to find a uniquely located minimum value on

the χ² landscape. We conclude that differential rotation is likely
(as shown next using Stokes V) although not measurable using
Stokes I.

Like for the brightness maps, Ω<sub>eq</sub> and ΔΩ pairs were generated
using the Stokes V data and tested using the χ<sup>2</sup><sub>r</sub> minimization tech-
nique. Whereas the determination of differential rotation was not
possible with the Stokes I data, the Stokes V data produced differ-
ential rotation measurements in all data sets. One hypothesis is that
the brightness maps were dominated by one high-latitude feature
whereas the magnetic maps have features spread over a range of lat-
itudes. A typical χ<sup>2</sup><sub>r</sub> minimization landscape is shown in Fig. 6. This
landscape was produced using the optimal parameter set for 2007
January Stokes V data. The Ω<sub>eq</sub> − ΔΩ measurement was deter-
mined by fitting a two-dimensional paraboloid to the χ<sup>2</sup><sub>r</sub> landscape.
This determines a 1σ error estimate of that fit. The value of this
measurement, coupled with the error estimate, was superimposed
on χ<sup>2</sup><sub>r</sub> landscape.

Several more χ² landscapes were produced by systematically
varying stellar parameters including the star’s inclination angle
(±5°), v sin i (±0.1 km s⁻¹) and the global magnetic field strength

<figure><img src="image_6.png" /><figcaption>**Figure 3.** The residual maximum-entropy fits to the Stokes *I* LSD profiles for EK Draconis during the 2006 December 4 to 2007 February 27 observing run. Differential rotation was not incorporated into the mapping process. These are normalized profiles with the black line showing the normalized raw profile while the red line shows the normalized modelled profile. The error bars on the left indicate the average error, 1σ. The numbers on the right represent the phase of each observation, based on the ephemeris in equation (4).</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 3 – (Continued)</figcaption></figure>

<table><caption>**Table 6.** The mapping parameters used for EK Draconis. Column 2 shows the timespan over which the data were taken. Column 3 shows the number of epochs, $\phi_I$, (number of observations) used in the mapping process; Columns 4 and 5 list the $\chi_r^2$ achieved and the spot coverage used in producing the brightness maps from the Stokes I profiles (see Fig. 2). The brightness maps assume solid-body rotation. Columns 6–10 are the number of epochs, $\phi_V$, $\chi_r^2$, average magnetic field strength ($|B|$), $\Omega_{eq}$ and $\Delta\Omega$ used in producing the magnetic maps from the Stokes V profiles (see Fig. 4). The final column is the laptime which is the time the equatorial regions need to lap the pole. Variations (errors) are based on the systematic recalculation of the models based upon varying stellar parameters: $\Omega_{eq}$, $\Delta\Omega$, $v \sin i$, global magnetic field and inclination.</caption><thead><tr><th rowspan="2">Year</th><th>Timespan</th><th colspan="5">Stokes I</th><th colspan="3">Stokes V</th></tr><tr><th>d</th><th>φ<sub>I</sub></th><th>χ<sub>r</sub><sup>2</sup></th><th>Spot coverage</th><th>φ<sub>V</sub></th><th>χ<sub>r</sub><sup>2</sup></th><th>&lt;|B|&gt; (G)</th><th>Ω<sub>eq</sub> (rad d<sup>-1</sup>)</th><th>ΔΩ (rad d<sup>-1</sup>)</th><th>Laptime<sup>a</sup> (d)</th></tr></thead><tbody><tr><td>2006 Dec</td><td>7</td><td>7(13)</td><td>0.70</td><td>3.7 per cent</td><td>8</td><td>1.35</td><td>91.8</td><td>2.42±0.1</td><td>0.19±0.18</td><td>~33</td></tr><tr><td>2007 Jan</td><td>9</td><td>9(33)</td><td>1.00</td><td>3.3 per cent</td><td>9</td><td>1.40</td><td>78.3</td><td>2.52±0.05</td><td>0.38±0.13</td><td>~16</td></tr><tr><td>2007 Feb</td><td>5</td><td>5(17)</td><td>0.45</td><td>3.9 per cent</td><td>5</td><td>1.25</td><td>57.7</td><td>2.51±0.06</td><td>0.13<sup>+0.2</sup><sub>-0.04</sub></td><td>~48</td></tr><tr><td>2008 Jan</td><td>5</td><td>5(15)</td><td>0.25</td><td>4.6 per cent</td><td>5</td><td>1.3</td><td>60.7</td><td>2.57±0.03</td><td>0.39±0.1</td><td>~16</td></tr><tr><td>2012 Jan</td><td>28</td><td>10(40)</td><td>0.70</td><td>2.6 per cent</td><td>10</td><td>2.0</td><td>92.3</td><td>2.44±0.02</td><td>0.25±0.06</td><td>~25</td></tr></tbody></table>

Note. $^a$The laptime is calculated using $\frac{2\pi}{\Delta\Omega}$.

(±10 per cent; Petit et al. 2002). Each minimum $\Omega_{eq} - \Delta\Omega$ pair, with the $1\sigma$ error, was calculated and superimposed on this landscape shown in Fig. 6. An ellipse was generated to encompass all of these differential rotation measurements and was used to estimate the overall error in the differential rotation measurement. For this particular (2007 January) data set, the equatorial rotational velocity, $\Omega_{eq}$, was estimated to be 2.52 ± 0.05 rad d⁻¹ with a rotational

shear, $\Delta\Omega$, of 0.38 ± 0.13 rad d⁻¹. The error quoted for the differential rotation parameters could be more appropriately called a 'variation' as it was determined by varying the parameters, as described above, and the associated ellipse known as a 'variation' ellipse (Waite et al. 2011b). However, the true error associated with this, and other differential rotation measurements listed in Table 6, may be slightly larger due to intrinsic spot evolution during each
<figure><img src="image_8.png" /><figcaption>**Figure 4.** Maps of EK Draconis for the CFHT/TBL from 2006 December until 2012 January data, with the differential rotation parameters incorporated into the imaging process. The extended tick lines (red) along the phase axes indicate the phase of the observation. The colour scale, on the bottom of the maps, is the magnetic field intensity in gauss. The associated fits between the modelled data that were used to generate these maps and the actual LSD profiles are shown in Fig. 5. The LSD profile from 2007 18 February was excluded from the mapping process due to poor signal to noise.</figcaption></figure>

data collection time frame (e.g. Morgenthaler et al. 2012). When considering all available differential rotation measurements across the 6 yr of observations, the average equatorial rotational velocity, $\Omega_{eq}$, was $\sim 2.50 \pm 0.08$ rad d$^{-1}$ with an average rotational shear, $\Delta\Omega$, of $\sim 0.27_{-0.26}^{+0.24}$ rad d$^{-1}$.

# 6 DISCUSSION

EK Draconis is a young, very active Sun-like star that underwent significant photospheric and chromospheric changes during our observations. This is particularly evident during the intense time sampling in 2006 December, 2007 January and 2007 February.

## 6.1 Brightness maps

The brightness maps, in Fig. 2, were generated from the Stokes *I* LSD profiles. EK Draconis displays spot features at low- to mid-latitudes. This is similar to other young, early G-type stars such as He 699 ($v \sin i = 93.5$ km s$^{-1}$), in the $\alpha$ Persei cluster (e.g. Barnes et al. 2001; Jeffers, Barnes & Collier Cameron 2002) and HD 141943 ($v \sin i = 35$ km s$^{-1}$; Marsden et al. 2011a). Additionally, EK Draconis displays a relatively large, intermediate-latitude feature at approximately $\sim 40^\circ - 70^\circ$ latitude in the 2006/2007

observing epochs, similar to that observed by Järvinen et al. (2007). The extensive high-latitude spot feature shown in Fig. 2 appears to coincide with the enhanced chromospheric activity shown in Fig. 1(b).

This study extends the original 2001 and 2002 observations of Järvinen et al. (2007) until 2012; an interval of approximately 10 yr. During this 10 yr time frame, the mid-to-high-latitude spot features appeared to migrate poleward to form a giant polar spot on EK Draconis. This is the first time a polar spot has been recorded on this star since the observations made by Strassmeier & Rice (1998) in 1995, although their analysis at that time was inconclusive regarding the existence of a polar spot. A hypothesis for this poleward migration involves the transport of magnetic flux through the process of meridional circulation (e.g. Kitchatinov & Rüdiger 1999; Schrijver & Title 2001). It is yet to be determined whether the polar spots on young solar-type stars are formed at high latitudes by the strong Coriolis effect in these rapidly rotating stars, or formed at low latitudes and pushed poleward by subsurface meridional flows. Weber, Strassmeier & Washuetl (2005) have reported tentative evidence for large poleward meridional circulation on giant stars. Zhao et al. (2013) report that poleward meridional flow on the Sun has a speed of $15$ m s$^{-1}$. If spot migration is indeed the mechanism, then the drift from 2007 until 2008 was approximately $5^\circ$ in latitude, indicating
<figure><img src="image_9.png" /><figcaption>**Figure 5.** The maximum-entropy fits to the Stokes V LSD profiles for EK Draconis with surface differential rotation incorporated into the analysis. The thick (red) lines represent the modelled lines produced by the ZDI process whereas the thin (black) lines represent the actual observed LSD profiles. Each successive profile has been shifted down by 0.002, or 0.003 when there were fewer profiles, for graphical purposes. The error bar on the left of each profile is plotted to $1\sigma$. The rotational phases are indicated to the right of each profile based on the ephemeris in equation (4). Each column represents an observation run. The LSD profile from 2007 18 February was excluded from the mapping process due to poor signal to noise.</figcaption></figure>

that the drift rate would have to be approximately 2.0 m s⁻¹. More massive stars such as KIC 8366239 (R = 5.30 ± 0.08 R☉; Beck et al. 2012) have a meridional flow speed of 26 m s⁻¹ or ~13° per year (Küker & Rüdiger 2012), and Arcturus (R = 25.4 ± 0.2 R☉; Ramírez & Allende Prieto 2011) has a meridional flow speed of 170 m s⁻¹ or ~17° per year Küker & Rüdiger (2011). Alternatively, if a spot group disappeared and was replaced by another emerging at the surface our maps could signify intrinsic spot evolution on a time-scale less than the year between the 2007 and 2008 observations, so the present data cannot be used to decide between spot migration and evolution.

As the Stokes *I* data did not enable differential rotation to be measured, a solid-body model was able to fit down to the noise level, with the reduced χ² values of less than one obtained, as shown in Table 6. Nevertheless, some systematic residuals remained, and so individual model and observed profiles were normalized by subtracting them from average profiles before being compared in Fig. 3. The advantage of this approach was to accentuate any mis-fitting during the modelling process. This was particularly evident for phases

$\phi \sim 0.26–0.3$ in the 2012 data. For example, removing the first four profiles (phases $\phi = 0.285–0.290$) had the effect of constraining the significant spot feature from phase ~0 to a slightly lower latitude. We attribute the mis-fits between the stellar and modelled profiles as due to differential rotation and spot evolution. Nevertheless, these maps, assuming solid-body rotation, clearly show the location of large intermediate-latitude features during 2006/2007 and 2008 with a distinctive polar spot appearing in 2012.

Constraining spot features using small data sets is problematic in Stokes *I* mapping. For example, the data set in 2007 late February and again in 2008 January only had five phases to reconstruct the brightness map. The DI code could not effectively recover both the latitude and phase of these features. This is shown in Fig. 2 with the smearing of the high-latitude feature (~70°) and the finger-like features crossing several degrees of latitude. For this reason, no DI (or ZDI) was attempted for 2009 January data as only four phases of observations were recorded.

EK Draconis' $v \sin i$ is at the extreme lower limit that permits constructing robust DI brightness maps. For example, Paper I and
<figure><img src="image_10.png" /><figcaption>**Figure 6.** Surface differential rotation $\chi_{\tau}^2$ minimization landscape for the Stokes V profiles for EK Draconis using the data set from 2007 January. The coloured landscape shows the $\chi_{\tau}^2$ values obtained systematically changing the $\Omega_{eq}$ and $\Delta\Omega$ pairs for the optimum parameter set. This is a 2$\sigma$ projection, with the darker colours indicating a lower $\chi_{\tau}^2$ value. Superimposed on this landscape is the variation ellipse showing the differential rotation measurements for the Stokes V. This variation ellipse was generated by varying some of the stellar parameters, including the star's inclination (±5°), $v \sin i$ (±0.1 km s⁻¹) and the global magnetic field strength (±10 per cent). The error bars on each individual measurement are 1$\sigma$ errors in the parabolid fit.</figcaption></figure>

Mengel et al. (2016) unsuccessfully attempted DI on HD 35296 and $\tau$ Boötis, respectively, with both stars having a $v \sin i \sim 15.9$ km s⁻¹. Rosén et al. (2016) produced brightness maps of EK Draconis using the same data as presented here from 2007 to 2012. Their brightness maps show different mean spot latitudes when compared to our maps. One reason could be that they used the published $v \sin i = 16.8$ km s⁻¹ (Valenti & Fischer 2005) whereas we determined the $v \sin i = 16.4$ km s⁻¹ using our complete profile fitting approach. The estimation of $v \sin i$ for slow and moderate rotators is particularly challenging. Collier-Cameron & Unruh (1994) found that overestimating $v \sin i$ produces excess equatorial spots while underestimating $v \sin i$ produces an excess of high-latitude spot features. This is the reason why we fitted the entire profile instead of merely fitting the wings of the LSD profile. Thus, our approach aims to use the observations we have obtained to measure $v \sin i$ rather than rely on previously published values.

## 6.2 Magnetic maps and configurations

The magnetic maps in Fig. 4 show a strong, almost unipolar, azimuthal field with complex and varying radial and meridional field structures. This is similar to other BY Draconis-type stars such as HN Pegasi<fnref n="4" /> (Boro Saikia et al. 2015) and HD 171488⁵ (Marsden et al. 2006; Jeffers et al. 2011), which show complex

and variable magnetic field geometry. During the intensive 3-month investigation, the magnetic field evolved from predominantly toroidal (~80 per cent) to a more balanced poloidal–toroidal (~40–60 per cent) field. This reorganization resulted in differing levels of activity observed on EK Draconis. The Ca II H&K and Hα spectral lines are formed in the mid-levels of the chromosphere, while the Ca II IRT lines are formed in the lower levels of the chromosphere. Fig. 1(b) shows the variation in these indices. Variation of all indices due to stellar rotation is evident, as explained in Section 4. What is more interesting is that the Hα and Ca II IRT indices hint that EK Draconis was more active during the 2007 January series of observations when the magnetic field was undergone a reorganization to a more balanced configuration. This is less obvious when considering the Ca II H&K index with the scatter much higher in the data. This could be due to the spectral lines appearing on the edge of the detector where the continuum is not so easily determined. Even removing the overlapping order, as explained in Section 4, the continuum was not as well defined as with the other spectral lines. Nevertheless, it may be concluded that the variation in the level of the activity occurring in the lower- to mid-levels of the chromosphere was due to the reorganization of the magnetic field.

The dynamo operating in the Sun is known as a $\alpha$–$\Omega$ dynamo (e.g. Parker 1955; Charbonneau 2010). The $\alpha$-effect is associated with the twisting of the magnetic fields by helical convection while the $\Omega$-effect produces the toroidal field by shearing the poloidal field due to differential rotation. Hubbard, Rheinhardt & Brandenburg (2011) argue that the $\alpha$-effect also has a role to play in the production of the toroidal field as opposed to the toroidal field being produced entirely by the $\Omega$-effect. In the Sun, the tachocline, or shear, layer is at the base of the convection zone (Thompson et al. 2003); hence, the toroidal component is in the sub-surface layers of the Sun. However, this might not be the case on young Sun-like stars. The toroidal component of the large-scale dynamo field may manifest itself in the form of strong azimuthal magnetic fields on, or near, the surface of rapidly rotating solar-type stars (e.g. Donati et al. 2003a; Petit et al. 2004b). Brown et al. (2010) used three-dimensional magnetohydrodynamic modelling in their anelastic spherical harmonic code to show that persistent wreathes of azimuthal magnetic fields can be produced. The azimuthal magnetic field observed on EK Draconis is practically unipolar and strongly negative in all epochs although the 2007 January map does show some small and very weak positive regions in this configuration as shown in Fig. 4. This may be due to the reorganization from a strongly toroidal field (~83 ± 1 per cent) to a weaker toroidal configuration (~59 ± 2 per cent) as shown in Fig. 7. The errors listed in these values, in Fig. 7 and Table 7, are based on the systematic recalculation of the models based upon varying stellar parameters: $\Omega_{eq}$, $\Delta\Omega$, $v \sin i$, $v_{rad}$, global magnetic field and inclination. However, the true error bars could be larger due to the intrinsic evolution of the magnetic field (e.g. Morgenthaler et al. 2012). On the longer time-scale of 5 yr, there appears to be no polarity reversals as the strongly persistent azimuthal field remains strongly negative, with some changes in the radial and the meridional field observed, but again, no evidence for a polarity reversal (see Fig. 4). It could be that the magnetic cycle of EK Draconis is longer than 5 yr or that EK Draconis has yet to establish significant polarity reversals, unlike the much older $\tau$ Boötis (Donati et al. 2008a; Fares et al. 2009; Mengel et al. 2016). Further observations are required to determine which is true.

Donati & Landstreet (2009) suggest that stars with a Rossby number of $\le 1$, but more massive than 0.5 M$_\odot$, have a substantial toroidal component with a mostly non-axisymmetric poloidal

<footnote n="4">⁴ SpType: G0V, Gray et al. (2006).</footnote>

<footnote n="5">⁵ SpType: G2V, Cutispoto et al. (1999).</footnote>
<figure><img src="image_11.png" /><figcaption>**Figure 7.** The variation in the magnetic field as a function of time. The left series of panels focusses on the poloidal field configuration while the right series of panels focus on the toroidal field configuration. The top two panels show the variation in the respective field strength as a percentage of the total magnetic field energy. The middle two panels show the various components for dipole (ℓ = 1: black, ...), quadrupole (ℓ = 2: blue, ...), and octupole (ℓ = 3: red, —). The bottom two panels show the percentage of the poloidal (left) and toroidal (right) fields which are axisymmetric (m = 0). The error bar on each data point was generated by varying some of the stellar parameters, including the star's Ω<sub>eq</sub>, ΔΩ, inclination (± 5°), v sin i (± 0.1 km s<sup>-1</sup>) and the global magnetic field strength (±10 per cent).</figcaption></figure>

component. The Rossby number (Noyes et al. 1984) for EK Draconis was determined using

$$R_o = \frac{P_{\text{obs}}}{\tau_c}, \qquad (6)$$

where τ<sub>c</sub> is the convective turnover time. τ<sub>c</sub> was determined using the empirical formula of Wright et al. (2011) and is shown in equation (7):

$$\log \tau_c = 1.16 - 1.49 \log \frac{M}{M_{\odot}} - 0.54 \log^2 \frac{M}{M_{\odot}}. \qquad (7)$$

Equation (7) holds for stellar masses in the range of 0.09 ≤ M/M<sub>☉</sub> ≤ 1.36. EK Draconis has a convective turnover time of 12.6 d, hence, a Rossby number of ~0.2. EK Draconis conforms to this pattern as it has a substantial toroidal component, even though it was reorganizing itself from the strongly toroidal field of

83 ± 1 per cent in 2006 December to 59 ± 2 per cent by the end of 2007 February. This toroidal field remained firmly axisymmetric during the 5 yr of observations, with over 90 per cent of the toroidal field being axisymmetric, as shown in the bottom panels of Fig. 7. This is consistent with the findings of See et al. (2015) that strong toroidal fields are predominantly axisymmetric. The poloidal field, as shown in Fig. 7, has increased dominance from ~17 per cent in 2006 December to ~41 per cent by the end of 2007 February. The poloidal field was predominantly non-axisymmetric reaching a maximum of 82 ± 3 per cent during the data set from 2007 January. These observations of significant toroidal fields with non-axisymmetric poloidal fields during the 5 yr of observations support the conclusions of Donati & Landstreet (2009) for stars that have Rossby numbers less than one but are more massive than 0.5 M<sub>☉</sub>.

Our magnetic maps are broadly consistent with the magnetic maps produced by Rosén et al. (2016) of EK Draconis using the data taken in 2007 and again in 2012, although their phase is approximately 0.5 different from ours. Although the brightness maps are sensitive to the v sin i measurement, the magnetic maps are not so sensitive; hence, our maps appear very similar to those of Rosén et al. (2016). Their maps show a strong, unipolar azimuthal magnetic field with complex and varying radial and meridional field structures. Additionally, we have incorporated differential rotation into the imaging process. Differential rotation will be explored in more detail in Section 6.4.

The 2007 February and 2008 January magnetic maps were reconstructed using only five observations. Marsden et al. (2011b) found that small data sets provided difficulties in determining the magnitude and makeup of the global mean magnetic field. To examine the effect of using fewer profiles in the imaging process, we reduced the number of profiles used in the reconstruction of the data set from 2007 January to the number in the data set from late-February. Additionally, we attempted to approximate the appropriate phase coverage of the data set. This resulted in a reduction of the mean field strength from 81 to 58 G. The conclusion is that the observed variation in the mean field strength from 2007 January to February is most likely a result in the reduced number of profiles, as opposed to any actual changes in the mean magnetic field strength of the star itself. When considering the configuration of the field, the poloidal component was slightly enhanced when using fewer profiles (from 32 ± 1 to 41 ± 2 per cent), as shown in Fig. 7 and listed in Table 7. Additionally, the toroidal field remained predominantly axisymmetric and the poloidal field remained predominantly

<table><caption>**Table 7.** Summary of magnetic topology evolution of EK Draconis from 2006 December to 2012 January. The second column shows the global magnetic field strength. The third column is the number of observations (φ). The fourth and fifth columns show the percentage of the total magnetic energy contained in the poloidal and toroidal components. The next three columns show the percentage of the poloidal field that is contained in the dipole (ℓ=1), quadrupole (ℓ=2) and octupole (ℓ=3). The next three columns show the same information of the toroidal field. The final two columns show the percentage of the axisymmetric modes (modes with m = 0) in each of the poloidal and toroidal components. Variations (errors) are based on the systematic recalculation of the models based upon varying stellar parameters: Ω<sub>eq</sub>, ΔΩ, v sin i, v<sub>rad</sub>, global magnetic field and inclination.</caption><thead><tr><th>Year</th><th>⟨|B|⟩<br>(G)</th><th>No. of<br>φ</th><th>Pol.<br>per cent tot.</th><th>Tor.<br>per cent tot.</th><th>Dip.<br>per cent pol.</th><th>Quad.<br>per cent pol.</th><th>Oct.<br>per cent pol.</th><th>Dip<br>per cent tor.</th><th>Quad.<br>per cent tor.</th><th>Oct.<br>per cent tor.</th><th>Axisym.<br>per cent pol.</th><th>Axisym.<br>per cent tor.</th></tr></thead><tbody><tr><td>2006 Dec</td><td>90±2</td><td>7</td><td>17±1</td><td>83±1</td><td>41±3</td><td>16±2</td><td>14±1</td><td>90±2</td><td>7±2</td><td>1±1</td><td>35±3</td><td>99±0.5</td></tr><tr><td>2007 Jan</td><td>81±2</td><td>7</td><td>32±1</td><td>68±1</td><td>50±3</td><td>16±1</td><td>7±1</td><td>79±4</td><td>15±3</td><td>2±1</td><td>6±1</td><td>96±1</td></tr><tr><td>2007 Feb</td><td>54<sup>+3.0</sup><sub>-2.5</sub></td><td>5</td><td>41±2</td><td>59±2</td><td>24±4</td><td>10±2</td><td>14±1</td><td>79±5</td><td>13±4</td><td>2±1</td><td>17±4</td><td>95±1</td></tr><tr><td>2008 Jan</td><td>59<sup>+2.0</sup><sub>-1.7</sub></td><td>5</td><td>37±1</td><td>63±1</td><td>33±3</td><td>9±0.5</td><td>16±2</td><td>82±4</td><td>14±4</td><td>2±1</td><td>8±1</td><td>97±1</td></tr><tr><td>2012 Jan</td><td>92<sup>+3.4</sup><sub>-3.1</sub></td><td>10</td><td>43±2</td><td>57±2</td><td>7±0.5</td><td>17±2</td><td>17±1</td><td>68±4</td><td>14±2</td><td>5±1</td><td>17±2</td><td>89±2</td></tr></tbody></table><p>Using fewer profiles of the 2007 January data to match the number of profiles used in 2007 February</p><table><tbody><tr><td>2007 Jan</td><td>58<sup>+2.1</sup><sub>-1.7</sub></td><td>5</td><td>41±2</td><td>59±2</td><td>31±3</td><td>32±2</td><td>9±2</td><td>79±3</td><td>15±4</td><td>2.5±0.3</td><td>13±4</td><td>96±0.7</td></tr></tbody></table>
<figure><img src="image_12.png" /><figcaption>Figure 8. This figure shows a comparison of the average magnetic field strength (G) as a function of latitude for the radial (left), azimuthal (middle) and meridional fields (right) for the five epochs observed. The dots (red) represent the fractional magnetic fields for the CFHT run, the solid (blue) line represents the data from the TBL in January while the dashed (green) line represents the field from 2007 late February. The dot-dashed (magenta) line is for 2008 while the ^ (cyan) line is for the 2012 data.</figcaption></figure>

a complex, non-axisymmetric field. Thus, we can conclude that the global magnetic field topology has evolved with a weakening of the toroidal component during the 3-month time frame.

## 6.3 Latitude dependence of the magnetic field

Determining the fractional magnetic field strength as a function of latitude enables further quantitative analysis of the distribution of the magnetic fields on the surface of the star. We calculate the average magnetic field per latitude bin, $B_f(\theta)$, as

$$B_f(\theta) = \sum_{i=1}^{n} B_i(\theta) \frac{\cos(\theta)d\theta}{2}, \qquad (8)$$

where $\theta$ is the latitude and $d\theta$ the width of the bin. Fig. 8 shows the average magnetic field as a function of latitude for each field component.

The magnetic maps shown in Fig. 4 clearly show a strong, almost unipolar azimuthal magnetic field. This is reflected in Fig. 8 with a stable, strongly negative field at all latitudes, with this field dominating the equatorial regions. The radial and meridional fields appear to have minor changes in latitude distribution during the 5 yr time frame. The 2012 meridional field appears to dominate at higher latitudes, perhaps reflecting the poleward migration of the large spot group observed in the brightness maps. There can be some cross-talk between the radial and meridional components, particularly, at low latitudes (Donati & Brown 1997; Rosén & Kochukhov 2012; Kochukhov et al. 2013), although with an inclination angle of 60° this is less likely for EK Draconis.

## 6.4 Differential rotation

A key driver of the dynamo operating in young Sun-like stars is differential rotation. Applying a solar-like differential rotation to the Stokes V information, the equatorial region of EK Draconis was

found to rotate at $\Omega_{eq} \sim 2.50 \pm 0.08$ rad $d^{-1}$, equating to a period of $\sim 2.51 \pm 0.08$ d. Considering the size of the respective error ellipse, as shown in Fig. 6, the results are consistent with the observations of Messina & Guinan (2003) who showed that EK Draconis exhibited solar-like surface differential rotation with the equator rotating faster than the poles. Table 6 shows the variation in the differential rotation measurement. Temporal variation in differential rotation measurements has been noted in the literature, including the photometric studies listed already. Additionally, longitudinal DI studies of the K-dwarf star AB Doradus<fnref n="6" /> have observed variations in the laptimes (the time it takes for the equator to lap the polar regions) from $\sim 70$ to $\sim 140$ d (Collier Cameron & Donati 2002; Donati, Collier Cameron & Petit 2003b). The variations in the rotational shear observed on EK Draconis are potentially real; however, these variations could be due to the paucity of observations, particularly in the 2007 late-February data. The level of differential rotation is much higher for the data set from 2007 January and again the data set from 2008 January when compared with other epochs. Numerical simulations have been performed by Petit et al. (2002) which show phase coverage and/or observational cadence can effect the magnitude of the $\Omega_{eq} - \Delta\Omega$ recovered. For EK Draconis, observational cadence was certainly a factor with the lower rotational shear during the data set from 2007 February. However, the CFHT data set has similar cadence to 2007 January and again in 2008 but showed significantly lower $\Omega_{eq}$ ($2.42 \pm 0.1$ rad $d^{-1}$ compared with $2.52 \pm 0.05$ rad $d^{-1}$ using 2007 January data) and $\Delta\Omega$ ($0.19 \pm 0.18$ rad $d^{-1}$ compared with $0.38 \pm 0.13$ rad $d^{-1}$). One can speculate that the strong toroidal field of the CFHT data set might have masked the differential rotation signature. The data set from 2007 January and again from 2008 had a more balanced field configuration with both data sets showing poloidal fields of $\sim 37$ and 38 per cent, respectively. The rotational shear in both 2007 and 2008 epochs was $\sim 0.39$ rad $d^{-1}$. Further observations would be required to determine the exact nature of the relationship between the magnetic configuration and the recovered differential rotation. Nevertheless, the conclusion is that EK Draconis has a significant rotational shear that could be as high as $0.39$ rad $d^{-1}$, equating to a laptime of $\sim 16$ d.

The error in the $\Delta\Omega$ values impacts on the laptime calculated and listed in Table 6. As explained in Section 5.3, the error in $\Delta\Omega$ was found by varying certain parameters ($v \sin i$, inclination angle, $\langle B_{\text{mod}} \rangle$) and determining the minimum $\chi^2$ in each of the respective landscapes. The individual error bar on each measurement, as shown in Fig. 6, was a $1\sigma$ error in the fit to the two-dimensional paraboloid. On some occasions, such as the differential rotation measurement with the CFHT data, the $1\sigma$ error bar is relatively large thereby increasing the size of the error ellipse.

## 6.5 Moderate rotators

EK Draconis is very similar to HD 35296 and HD 29615 that were studied by Waite et al. (2015); see Paper I. All three of these Sun-like stars have similar age, mass, radius and rotational velocity. Table 8 highlights these similarities. All exhibit reasonably high levels of differential rotation, with laptimes (where the equator laps the polar regions) ranging from $\sim 13$ d for HD 29615 to $\sim 29$ d for HD 35296.

All three stars are considered as moderately rotating solar-type stars (as defined by Waite et al. 2011a), with all these stars having projected rotational velocities ($v \sin i$) ranging from 5 to 20 km s$^{-1}$. It is within this range where the strength of the magnetic dynamo is

<footnote n="6">⁶ SpType: K0V, Torres et al. (2006).</footnote>
<table><caption>**Table 8.** A comparison between the fundamental parameters of HD 35296, HD 29615 and EK Draconis.</caption><thead><tr><th>Parameter</th><th>HD 35296<sup>a</sup></th><th>HD 29615<sup>a</sup></th><th>EK Draconis</th></tr></thead><tbody><tr><td>Equatorial period, d</td><td>3.5 ± 0.2</td><td>2.34 ± 0.2</td><td>2.51 ± 0.07</td></tr><tr><td>Inclination angle, °</td><td>65 ± 5</td><td>70 ± 5</td><td>60 ± 5</td></tr><tr><td>Photospheric temperature, T<sub>phot</sub>, K</td><td>6080</td><td>5820 ± 50</td><td>5561</td></tr><tr><td>ΔTemp = T<sub>phot</sub> - T<sub>spot</sub>, K</td><td>–</td><td>1900</td><td>1700</td></tr><tr><td>Stellar radius: R<sub>☉</sub></td><td>~1.21</td><td>~1.0</td><td>~0.94</td></tr><tr><td>Stellar mass: M<sub>☉</sub></td><td>~1.10</td><td>~0.97</td><td>~0.95</td></tr><tr><td>Projected rotational velocity, v sin i, km s<sup>-1</sup></td><td>15.9 ± 0.1</td><td>19.5 ± 0.1</td><td>16.4 ± 0.1</td></tr><tr><td>log g</td><td>4.31 ± 0.03</td><td>4.43 ± 0.04</td><td>4.47 ± 0.08</td></tr><tr><td>Convective turnover time, d</td><td>10.3</td><td>16.0</td><td>17.2</td></tr><tr><td>Rossby number</td><td>0.34</td><td>0.15</td><td>0.15</td></tr><tr><td>Stokes V: Ω<sub>eq</sub>, in rad d<sup>-1</sup></td><td>1.804 ± 0.005</td><td>2.74<sup>+0.02</sup><sub>-0.04</sub></td><td>~2.50 ± 0.08</td></tr><tr><td>Stokes V: ΔΩ, in rad d<sup>-1</sup></td><td>0.22<sup>+0.04</sup><sub>-0.02</sub></td><td>0.48<sup>+0.11</sup><sub>-0.12</sub></td><td>0.27<sup>+0.24</sup><sub>-0.26</sub></td></tr><tr><td>Laptime<sup>b</sup>, d</td><td>~29</td><td>~13</td><td>~23</td></tr></tbody></table>

Notes. $^a$Data taken from Paper I, Waite et al. (2015), and the references therein.

$^b$The laptime is the time it takes for the equator to lap the polar regions.

believed to be related to the star's rotation rate (Vidotto et al. 2014). Above the rotational velocity of ~20 km s⁻¹ the strength of the magnetic dynamo is believed to be no longer dependent on stellar rotation. All three of these Sun-like stars, with rotation rates below 20 km s⁻¹, exhibit near-surface azimuthal fields indicating that the dynamo could be fundamentally different from that operating in the Sun today. The toroidal fields of all three Sun-like stars are strongly axisymmetric while the poloidal fields tend to be more non-axisymmetric; however, HD 29615 does not follow the other two stars where the poloidal field is predominantly axisymmetric as only ~33 per cent of the poloidal energy held in the non-axisymmetric configuration. The Rossby number, a measure of how strongly the Coriolis force is capable of affecting the convective eddies, is lowest in HD 29615 amongst the three stars ($R_o = 0.14$). Donati & Landstreet (2009) suggest that small $R_o$ values indicate very active stars rotating fast enough to ensure that the Coriolis force strongly impacts convection. One can speculate that the star with the lowest Rossby number also has the highest differential rotation shear, as is the case when comparing these three Sun-like stars. However, for very low Rossby numbers, $R_o \le 0.1$, in the case of M-dwarf stars, the differential rotation is very small (Barnes et al. 2005; Donati et al. 2008b). Gastine et al. (2014), using three-dimensional simulations, concluded that maximum $\Delta\Omega/\Omega$ is achieved at moderate $R_o$ ($R_o < 1.0$), and then converges towards zero at the fastest rotation rates (see fig. 4 of their work).

Surface differential rotation has a key role in the generation of the stellar magnetic field. Surface differential rotation measurements have shown variations when using brightness features compared with the value found using magnetic features (e.g. Donati et al. 2003b; Waite et al. 2015). One hypothesis is that these features are anchored at different depths within the convective zone, meaning that the radial differential rotation structure must be very different from that observed on the Sun. HD 29615 is one star that shows this variation, albeit an extreme example with $\Delta\Omega = 0.48_{-0.12}^{+0.11}$ using Stokes V and $0.07_{-0.03}^{+0.10}$ using Stokes I. In the absence of a definitive measurement using DI, we cannot confirm this hypothesis with EK Draconis.

## 7 CONCLUSIONS

Our investigations have observed strong differential rotation using the magnetic features on EK Draconis. Additionally, our observations show significant evolution occurring on EK Draconis in

a short 3-month period. These changes are possibly the result of the magnetic field reorganizing itself from a strongly toroidal field (~80 per cent) to a more balanced poloidal-toroidal field (~40-60 per cent) in only 3 months, as shown in Fig. 7. The poloidal field on EK Draconis was predominantly non-axisymmetric while the toroidal field was almost entirely axisymmetric during all epochs. This is consistent with other Sun-like stars. EK Draconis appeared to show intermediate-latitude features during earlier epochs while a distinctive polar spot was observed during 2012. Additionally, a persistent, almost unipolar azimuthal field was observed at all epochs indicating that no polarity reversals were found in our data.

## ACKNOWLEDGEMENTS

Thanks must go to the staff of the CFHT and TBL for their assistance in taking these data. The authors thank the anonymous referee for the insightful comments that have significantly enhanced this paper. The authors thank Professor Jardine, Dr See, Ms Boro Saikia and Mr Mengel on their valuable contributions to this work. This research has made use of NASA's Astrophysics Data System. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A&AS, 143, 23. This research used the High Performance Computing Facility at the University of Southern Queensland.

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