Magnetic activity of the young solar analogue V1358 Ori
L. Kriskovics, Zs. Kővári, K. Vida, K. Oláh, T. A. Carroll, T. Granzer
AAstronomy & Astrophysics manuscript no. aa_pdf c (cid:13)
ESO 2019May 22, 2019
Magnetic activity of the young solar analogue V1358 Ori
L. Kriskovics , Zs. K˝ovári , K. Vida , K. Oláh , T. A. Carroll , and T. Granzer Konkoly Observatory, Research Center for Astronomy and Earth Sciences, Budapest, Hungarye-mail: [email protected] Leibniz Institute for Astrophysics (AIP), Potsdam, GermanyReceived ...; accepted ...
ABSTRACT
Context.
Young, fast rotating single stars can show dramatically di ff erent magnetic signatures and levels of magnetic activity ascompared with the Sun. While losing angular momentum due to magnetic breaking and mass loss through stellar winds, the starsgradually spin down resulting in decreasing levels of activity. Studying magnetic activity on such solar analogues plays a key role inunderstanding the evolution of solar-like stars and allows a glimpse into the past of the Sun as well. Aims.
In order to widen our knowledge of the magnetic evolution of the Sun and solar-like stars, magnetic activity of the young solaranalogue V1358 Ori is investigated.
Methods.
Fourier analysis of long-term photometric data is used to derive rotational period and activity cycle length, while spectralsynthesis is applied on high resolution spectroscopic data in order to derive precise astrophysical parameters. Doppler imaging isperformed to recover surface temperature maps for two subsequent intervals. Cross-correlation of the consecutive Doppler maps isused to derive surface di ff erential rotation. The rotational modulation of the chromospheric activity indicators is also investigated. Results.
An activity cycle of ≈ ff erential rotationwith a surface shear parameter of α = . ± .
010 which fits pretty well to our recently proposed empirical relation between rotationand di ff erential rotation. The chromospheric activity indicators showed a rotational modulation. Key words. stars: activity – stars: imaging – starspots – stars: individual: V1358 Ori
1. Introduction
Studying magnetic activity – indicators of the state of the un-derlying magnetic dynamo – plays a key role in understandingthe evolution of solar-like stars along the main sequence, sincethe magnetic dynamo strongly a ff ects not just the stellar struc-ture (Berdyugina 2005), but the spin–down of the star due tomagnetic breaking (e.g. Barnes 2003), angular momentum lossthrough stellar winds (MacGregor & Brenner 1991), etc.Strong magnetic activity could also a ff ect orbiting planetarysystems through strong stellar winds or high-energy electromag-netic or particle radiation, which may ultimately erode planetaryatmospheres as well (e.g. Vida et al. 2017).Solar magnetic fields are generated by an α Ω -type dynamo(Parker 1955). Here, the poloidal field is wind up and amplifiedby the di ff erential rotation creating the toroidal field ( Ω e ff ect),while the α e ff ect creates small-scale poloidal fields from thetoroidal field via rotationally induced convective turbulence, andturbulent di ff usion builds up the large scale poloidal field by re-connection of the small-scale field.In rapidly rotating late-type (G–K) dwarfs, the Ω -e ff ect canbe suppressed, resulting in an α Ω type dynamo (see Ossendri-jver 2003, and references therein, or e.g. K˝ovári et al. 2004 forobservational evidence). However, as these stars evolve, theyspin down mostly due to magnetic breaking (Skumanich 1972,Barnes 2003), thus their dynamo shift from the α Ω domain tothe α Ω (Ossendrijver 2003).In order to properly understand the magnetic evolution of theSun and solar-like stars along the main sequence (and its e ff ect to their vicinity), it is imperative to study the magnetic activityof young solar-type stars.V1358 Ori (HD 43989, HIP 30030) was originally identifiedas an active star and potential Doppler imaging target and classi-fied as G0IV by Strassmeier et al. (2000). Later it was reclassi-fied as an F9 dwarf by Montes et al. (2001), which was confirmedby McDonald et al. (2012). Vican & Schneider (2014) estimatedthe e ff ective temperature to be T e ff = ≈
30 Myr) solar analogue.Hackman et al. (2016) carried out a Zeeman–Doppler analy-sis of V1358 Ori. They reported a strong toroidal magnetic fieldcomponent on the Stokes V maps, and prominent polar featureson the brightness map, as well as some weaker, lower latitudespots, however their phase-coverage was poor.In this paper, we carry out a photometric and spectroscopicanalysis of the young solar-analogue V1358 Ori, based on a18 years long, homogeneous photometric data set and Dopplerimaging applied on high-resolution spectra covering two rota-tions. We also derive surface di ff erential rotation from the con-secutive Doppler maps and compare the results to similar youngactive solar analogues.
2. Observations
Photometric data of Strömgren by and Johnson-Cousins V I C datawere gathered with Wolgang and
Amadeus , the 0.75-m automatic
Article number, page 1 of 10 a r X i v : . [ a s t r o - ph . S R ] M a y & A proofs: manuscript no. aa_pdf
Fig. 1.
Strömgren y (red) and Johnson V (blue) di ff erential photometry of V1358 Ori. The overplotted curve is the combination of the two long-term cycles. For the long-term, fit we used the Johnson V data only, and the Stromgren y data were left out because of the 0.025 mag shift whichis probably due to the di ff erent transmission characteristics the two passbands. See text for details. Table 1.
Spectroscopic observing log for V1358 Ori. S1 and S2 indicatethe two subsets used for Doppler imaging,HJD − φ and S / N arethe reduced Heliocentric Julian dates, rotational phases and signal-to-noises, respectively.
S1 S2HJD-2456000 φ S / N HJD-2456000 φ S / N636.4313 0.848 314 641.4184 0.523 242636.6900 0.039 361 641.6738 0.711 289637.4258 0.581 332 642.4252 0.265 299637.6757 0.765 321 643.3967 0.981 335638.6755 0.502 342 647.3944 0.926 332639.4232 0.053 312 647.6529 0.117 347639.6681 0.233 317photoelectric telescopes operated by the Leibniz Institute for As-trophysics Potsdam, and located at Fairborn Observatory, Ari-zona (Strassmeier et al. 1997) between 4 Mar 1997 and 8 Mar2015. All di ff erential measurements were taken respect to thecomparison star HD 44517. Between 4 Mar 1997 and 16 Apr1997, the check star was HD 44019. After that, HD 45215 wasused. For details on the data reduction, see Strassmeier et al.(1997) and Granzer et al. (2001). Photometric Vy data are plot-ted in Fig. 1. We note that there is a 0.025 mag shift betweenthe Stromgren y data of the first two seasons and the rest of theobservations, which is probably due to the di ff erent transmissioncharacteristics of the two passbands.Spectroscopic observations were gathered via OPTICONwith the NARVAL high-resolution echelle spectropolarimetermounted on the 2-m Bernard Lyot Telescope of ObservatoireMidi-Pyrénées at Pic du Midi, France between 09-20 Dec 2013.A peak resolution of R = t exp = ≈
300 (hereafter S / N) at 6400 Å(see Table 1 for details).To phase the observations, the following ephemeris wasused:HJD = . + . × E (1) T was adopted from Hackman et al. (2016). For details on therotational period, see Sect. 3. Spectroscopic data reduction were carried out with the stan-dard NARVAL data reduction pipeline. ThAr arc-lamps wereused for wavelength calibration. An additional continuum fit andnormalization were applied in order to avoid erroneous contin-uum fits during the spectral synthesis and Doppler inversion.
3. Photometric analysis
The period analysis were carried out on the Johson V data.Strömgren y data were excluded as it only consist of 458 pointsand there is a long gap at the beginning of the time series (seeFig. 1).For photometric period determination, we used MuFrAn ,a code for frequency analysis based on Fourier-transformation(Kolláth 1990, Csubry & Kolláth 2004). We accepted the peakat the cycle-per-day value c / d = . P rot = . P rot = .
16 d). The reasoning behind thisis the following. We gradually pre-whitened the Fourier-spectrawith the two peaks corresponding to the two long-term changes( ≈ ≈ V and I each day). The resultingFourier-spectrum contained no signal which could reasonably beconsidered real within the precision of the photometry, and sincethe only period is the 1.3571 d which cannot be attributed to arti-ficial origin or a long-term cyclic behaviour, we accepted it as therotational period. Moreover, only P rot = . P rot = . v sin i value (see Sect. 4).Fig. 2 shows the seasonal phased V light curves to demon-strate the robustness of the derived photometric period. Solidlines indicate a two-spot analytic spot model fitted to each sea-son using SML (Ribárik et al. 2003). Shaded zones show theposition of the dominant spot, the width of the zone denotesthe error of the spot longitude. Vertical dashed lines indicatethe weaker spot (on the plot of the fourth season, both activelongitudes are weak, while in case of the seventh season, it https://konkoly.hu/staff/kollath/mufran.html Article number, page 2 of 10. Kriskovics et al.: Magnetic activity of the young solar analogue V1358 Ori
Fig. 2.
Seasonal light curves of V1358 Ori phased with P rot = . is hard to decide which is the dominant longitude). Seasonswhere the low number of points made the inspection impossi-ble were omitted from the plot: two between HJD = ≈ ≈ ≈ ≈ ≈ V lightcurve, a long-term cyclic behaviour can be suspected as well, Fig. 3.
Upper panels: Fourier spectra of the V light curve of V1358 Ori.Orange rectangles denote the strongest peaks of the subsequent steps.Consecutive plots show pre-whitened spectra with the periods shown.Bottom panel: Fourier window-function. See Sect. 3 for details. which was confirmed by the Fourier-analysis, yielding P cyc ≈ ≈ V light curvein Fig. 1. There is a 0.025 mag systematic shift between the V and y data, which is probably due to the di ff erent transmissioncharacteristics of the two bands. Therefore we decided not toinclude the Strömgren data of the first two seasons in our long-term analysis.For further discussion on the spot configuration, suspectedflip-flop and the activity cycle, see Sect. 7.
4. Fundamental parameters
Precise astrophysical parameters are fundamental for Dopplerinversion, therefore we carried out a detailed spectroscopic anal-ysis based on spectral synthesis using the code SME (Piskunov& Valenti 2017). During the synthesis, MARCS models were
Article number, page 3 of 10 & A proofs: manuscript no. aa_pdf used (Gustafsson et al. 2008). Atomic line parameters were takenfrom the VALD database (Kupka et al. 1999). Macroturbulencewas estimated using the following equation (Valenti & Fischer2005): v mac = (cid:32) . − T e ff − (cid:33) km s − (2)Astrophysical parameters were determined using the follow-ing methodology:1. Determination of v sin i using initial astrophysical parame-ters taken from Montes et al. (2001) and assuming solarabundances.2. Microturbulence ( ξ ) fit with same astrophysical parametersand v sin i from step 1 using lines with log g f < − .
5, sinceweak lines are more sensitive to the change of microturbu-lence.3. Refitting v sin i and ξ simultaneously to check robustness.4. Fitting T e ff using solar metallicity and line broadening pa-rameters from step 3.5. Fitting metallicity using the e ff ective temperature value fromstep 4.6. Fitting log g using stronger lines (log g f > T e ff , log g and metallicity simultaneously to checkrobustness.8. Refitting v sin i .Lithium abundance fit was carried out using NLTE departurecoe ffi cients (Piskunov & Valenti 2017). The fit yielded A (Li) = . ± .
05. An example of the fit can be seen in Fig A.2. Theastrophysical parameters are summarized in Table 2.
The
Gaia
DR2 parallax of π = . ± .
05 mas (Gaia Collab-oration et al. 2016, 2018) gives a distance of d = ± . V brightness of the star yieldsa bolometric magnitude of M bol = . + . − . (with extinctionfrom Schlafly & Finkbeiner 2011 and bolometric correctionfrom Flower 1996 taken into account). This results in a lumi-nosity of L / L (cid:12) = . + . − . , which is in good agreement with thevalue from Gaia
DR2 ( L / L (cid:12) = . ± . R / R (cid:12) = . ± .
03. The radiuswith the photometric period and the v sin i = ± − fromthe spectral synthesis yields an inclination of i = ± ◦ . Table 2.
Fundamental astrophysical parameters of V1358 Ori. T e ff ±
25 Klog g . ± . / H] 0 . ± . v mic . ± . − v mac (computed ) 3 . − v sin i ± − Distance 52 . ± . M bol . m + . − . L / L (cid:12) . + . − . R / R (cid:12) . ± . ± ◦ P rot . . ± . The lithium abundance from the spectral synthesis using the em-pirical correlation between age and abundance from Carlos et al.(2016) would yield t ≈ . ± . ≈
30 Myr. This is in good agreement with the computed gy-rochronological age ( t gyro = ±
5. Doppler imaging
Our Doppler imaging code iMap (Carroll et al. 2012) carries outmulti-line Doppler inversion on a list of photospheric lines be-tween 5000–6750 Å. We included 40 virtually non-blended ab-sorption lines with suitable line-depth, temperature sensitivityand well defined continuum. The stellar surface is divided into5 ◦ × ◦ segments. For each local line profile, the code utilizesa full radiative solver (Carroll et al. 2008). Then the local lineprofiles are disk integrated, and the individually modeled, disk-integrated lines are averaged. Atomic line data are taken fromVALD (Kupka et al. 1999). Model atmospheres are taken fromCastelli & Kurucz (2004) and are interpolated for the neces-sary temperature, gravity or metallicity values. Due to the highcomputational capacity requirements, LTE radiative transfer isused instead of spherical model atmospheres, but imperfectionsin the fitted line shapes are well compensated by the multi-lineapproach. Additional input parameters are micro- and macrotur-bulence, and v sin i .For the surface reconstruction, an iterative regularizationmethod based on a Landweber algorithm is used (Carroll et al.2012), meaning no additional constraints are imposed in the im-age domain. According to our tests (Appendix A in Carroll et al.2012), the iterative regularization proved to be e ff ective and in-versions based on the same datasets always converged to thesame image solution. The available 15 spectra are divided into two subsets. Thecorresponding time intervals are 2456636.43–2456639.66 and2456641.42–2456647.65. The first subset consists of 8 spectraand covers 2.4 rotations, while the other 7 spectra of the sec-ond subset covers 4.6 rotations. The phase coverages are notcompletely uniform, nevertheless both subsets are suitable forDoppler imaging.The resulting two Doppler reconstructions (henceforth S1and S2) for V1358 Ori are plotted in Fig. 4. The average pro-files are plotted along with the final profile fits (thick black andthin red lines, respectively) in Fig 5.The overall characteristics of the two individual Doppler re-constructions are quite similar. This is supported by the averageDoppler image using all the available spectra, see Fig. 6. Theresulting average map is pretty similar to the two individual im-
Article number, page 4 of 10. Kriskovics et al.: Magnetic activity of the young solar analogue V1358 Ori
S1S2
Fig. 4.
The two consequent Doppler images of V1358 Ori plotted in four di ff erent rotational phases. The corresponding average HJDs for the mapsare 2456638.1 and 2456653.9 for S1 and S2 respectively. S1 S2
Fig. 5.
Observed averaged line profiles and final fits for the two sub-sets. The dotted (black) lines represent the observations, while the solid(red) lines are the fits from the Doppler imaging. . Rotational phases aredenoted on the right side of the plots. ages in Fig. 4, indicating only minor changes in the spot config-uration.Doppler imaging reveals a strong polar cap, as well as bothcool and hot features at lower latitudes, down to the equator.Spot temperatures range from ≈ ≈ ≈
350 K higher than the temperature of the quiet photosphere.The contrast of the coolest features of ≈ ≈ ◦ latitude around 0.4 phase, another eccentricity is seen at ≈ . ◦ latitude and 0.9 phase, which almost completely disappears on S2, however thesub-equatorial spot around 0.2 phase becomes more prominent.The other cool low latitude features at ≈ . ≈ . ≈ . ff erent shape on the second image. There are alsoother much weaker equatorial bright spots on both maps, withslightly di ff erent shapes.Hot features are often considered to be of artificial origin,mostly caused by insu ffi cient phase coverage, as previous tests(e.g. Lindborg et al. 2014) have pointed out. However, decreas-ing the phase coverage usually introduces both cool and hot fea-tures on roughly the same longitude (Fig. 3 in Lindborg et al.2014). Also, the two Doppler maps both show similar hot fea-tures at the same positions, and are based on completely inde-pendent datasets with di ff erent phase coverages. These featurescan also be seen on the average map derived from all of the spec-tra, where the phase coverage is inherently better, roughly at thesame positions. The shape of the chromosperic activity indicatorcurves might also support the conclusion that the bright spots arereal (see Sect. 6). Thus, we conclude that it is more likely thatthese hot spots are indeed real features. Longitudinal spot displacements from the first series comparedto the second can be used as a tracer of surface di ff erential ro-tation. Visual inspection of the two subsequent Doppler mapsmay indicate such rearrangements (see Sect. 5.2): the longitudi-nal displacement of the subequatorial cool spot around 0.4 phaseor displacement and change of shape of the bright feature at ≈ . ff erential rotation can be measured by longitudi-nally cross-correlating consecutive Doppler images (Donati &Collier Cameron 1997), and fitting the latitudinal correlation Article number, page 5 of 10 & A proofs: manuscript no. aa_pdf
Fig. 6.
Average Doppler image of V1358 Ori derived using all of the spectra plotted in four rotational phases. peaks by an assumed quadratic rotational law (see e.g. K˝ováriet al. 2012): Ω ( β ) = Ω eq − ∆Ω sin β, (3)where Ω ( β ) is the angular velocity at β latitude, Ω eq is theangular velocity of the equator, and ∆Ω = Ω eq − Ω pole gives thedi ff erence between the equatorial and polar angular velocities.With these, the dimensionless surface shear parameter α is de-fined as α = ∆Ω / Ω eq .We cross-correlate the available two Doppler images to buildup a 2D cross-correlation function map shown in Fig. 7. Despitesome noise it is clear from the figure that the equator rotates mostrapidly and the rotation velocity decreases with increasing lati-tude angle, i.e., the correlation pattern indicates solar-type sur-face di ff erential rotation on V1358 Ori. When fitting the patternwith a solar-type di ff erential rotation law of the quadratic formabove, we get Ω eq = . ± . ◦ / day and ∆Ω = . ± . ◦ / dayresulting in α = . ± .
004 surface shear parameter. Errorsare estimated from the FWHMs and amplitudes of the Gaus-sian fits to the latitudinal bins. However, having only two con-secutive Doppler images could introduce false correlation, mak-
Fig. 7.
Cross-correlation function map obtained by cross-correlating thetwo subsequent surface maps shown in Fig. 4. Darker regions corre-spond to stronger correlation. The best fit rotation law to the correlationpeaks (dots) suggests solar-type di ff erential rotation with a surface shearof α = . ing the cross-correlation technique less powerful (K˝ovári et al.2017). Moreover, due to projection e ff ect, the actual impact ofthe strongest polar features is restricted. Also, the low latitudefeatures are less contrasted. As a result, the true errors of thederived surface shear may be somewhat larger, therefore we as-sume ± α instead of the formal value of ± ff erential rotation,see Sect. 7.
6. Chromospheric activity
We measured the Ca II R HK , H α and Ca ii IRT chromosphericactivity indices for all of the spectra individually to see if there isany rotational modulation present for the two covered rotations.For the R HK , we first calculated the non-calibrated S -indexas described in Vaughan et al. (1978). The instrumental valueswere then transformed into the original Mt. Wilson scale with thecalibration coe ffi cients for NARVAL derived by Marsden et al.(2014). To avoid color-dependence, S -index were transformedto R HK (Middelkoop 1982, Rutten 1984). To subtract the pho-tospheric adjunct, we applied the correction formula of Noyes(1984). For the H α , we used the indicator defined by Kürsteret al. (2003). The IRT index was calculated using the formula ofMarsden et al. (2014).The apparent average surface temperatures were also com-puted for the two Doppler images in the same phases for compar-ison. The R HK , H α and the IRT indices are plotted in Fig. 8 alongwith average temperatures. The values itself are summarized inTable 3. The errors were estimated using error propagation, andare in the order of 0.004, 0.003 and 0.001 for R HK , H α and IRT,respectively.All of the indices clearly show some change with the rota-tional phases which could be interpreted as rotational modula-tion. Apart from a few outlying points, the R HK H α and IRT in-dices show roughly the same behaviour. The overall shape ofthe curves are similar, however, there is no clear correlation be-tween the position of the maximal chromospheric activity andthe highest spot coverage (i.e., the lowest average surface tem-perature). One might argue that the largest di ff erence betweenthe chromospheric activity and the average surface temperatureis around 0.2–0.4 phase, where, on the Doppler images, thestrongest hot spot becomes gradually visible, which might indi-cate that the hot structure has a chromospheric counterpart (andfurther strengthen the suspicion that these features are indeed notartifacts of the Doppler imaging process). Article number, page 6 of 10. Kriskovics et al.: Magnetic activity of the young solar analogue V1358 Ori
Fig. 8.
Calcium R HK (top panels), H α (second row) and Ca ii IRT (thirdrow) curves for the two rotations compared to the average projectedsurface temperatures for the the same rotational phases from Dopplerimaging (bottom panels).
7. Discussion
Both visual inspection and Fourier analysis suggest a possibleactivity cycle of ≈ ≈ . y photometry (see Fig. 1) also seems to confirm this cycle,but these data were not included in the Fourier analysis, as theStrömgren y filter is much narrower than the Johnson V (23 vs88 nm) and there is also a large gap in those observations. Therotational period and the length of the activity cycle are two im-portant observables of magnetically active stars. Their ratio isrelated to the dynamo number which is an indicator of magneticactivity. The derived rotational period and the long-term cycles(including the suspected ≈ = − ≈ ff erential rotation. They found that the reported flip-flopperiods of several years ( ≈ −
9) are found in the range of | α | = | ∆Ω / Ω eq | ≈ . − .
15 (see Elstner & Korhonen 2005 andreferences therein). Our values ( α = . ± .
01 and P ff ≈ ff erence in latitudinal spot distributions betweenthe Sun and stars. They found that as the rotation rate increases,magnetic flux emerges at higher latitudes, and a quiet regionopens around the equator. They also pointed out that at 8 timesof the rotational rate of the Sun ( P rot ≈ ff erential rota-tion fits well to the observation that active young solar-typestars exhibit weak solar-type di ff erential rotation, e.g: LQ Hya α = . P rot = .
597 d, K˝ovári et al. 2004) or AB Dor( α ≈ .
006 with P rot = . ff ers et al. 2007). Moreover,K˝ovári et al. (2011) derived α = .
009 for V889 Her, with a ro-tational period of 1.337 days and Marsden et al. (2005) reporteda surface shear of 0.012 on V557 Car ( P rot = .
557 d). Our resultis also in good agreement with the empirical relation betweenthe rotational period and the surface shear parameter of | α | ≈ . P rot [days] (4)for single stars suggested by K˝ovári et al. (2017), see Fig. A.1.The rotational modulation of the chromospheric activity in-dicators suggest that the positions of the chromospheric struc-tures (plages?) more-or-less coincide with the positions of thephotospheric nests. The most apparent di ff erence in the shape ofthe curves is in the region of 0.2–0.4 phase, which coincides withthe phase where the most prominent hot feature on the Dopplerimages of both rotation becomes visible do to the rotation of thestar. This may mean that the photospheric hot features extend tothe chromosphere as well, and contribute to the overall chromo-spheric emission.
8. Summary – Based on a 14 years long photometric dataset we derive arotational period of P rot = . Article number, page 7 of 10 & A proofs: manuscript no. aa_pdf
Table 3.
Chromospheric activity indices of V1358 Ori in the observed rotational phases. See Sect. 6 for more details.
S1 S2 φ log R HK log I H α log I IRT φ log R HK log I H α log I IRT – An activity cycle with the period of roughly 1600 days isdetected, which is consistent with the findings of Oláh et al.(2016). A flip-flop time scale of 6 years may also be present. – By a spectral synthesis technique we determine precise as-trophysical parameters for V1358 Ori. – We perform Doppler imaging to map the surface tempera-ture distribution for two subsequent epochs separated by twoweeks. The surface structure is dominated by a large polarcap accompanied with weaker features at low latitudes, con-sistent with previous observations of young solar analoguesand recent dynamo models as well. Hot features are alsopresent on both maps. – Surface di ff erential rotational is derived by cross-correlatingthe two subsequent Doppler images. The resulting surfaceshear parameter α = . ± .
01 fits to the rotational period-surface shear empirical relationship proposed recently byK˝ovári et al. (2017). – Chromospheric activity indicators are calculated and com-pared to the average apparent surface temperatures. Rota-tional modulation is present on the activity indicator curvesin both rotations. The shapes of the curves are similar. Themost prominent di ff erence between the activity indicatorcurves and the average surface temperatures may indicatethat the hot spots contribute to the chromospheric emission. Acknowledgements.
The authors acknowledge the Hungarian National Re-search, Development and Innovation O ffi ce grant OTKA K-113117, and supportsthrough the Lendület-2012 Program (LP2012-31) of the Hungarian Academy ofSciences, and the ESA PECS Contract No. 4000110889 / / NL / NDe. KV is sup-ported by the Bolyai János Research Scholarship of the Hungarian Academyof Sciences. The authors thank A. Moór for the useful conversations on stellarages. Finally, the authors would like to thank the anonymous referee for her / hisvaluable insights. References
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Appendix A:
Article number, page 9 of 10 & A proofs: manuscript no. aa_pdf
Fig. A.1.
The absolute value of the dimensionless surface shear parameter of single stars plotted against their rotational period in days (see K˝ováriet al. 2017 and references therein). Circles denote results from the sheared image method, while squares indicate values obtained with the cross-correlation technique. The blue filled square indicates V1358 Ori. The dotted line represents a linear fit with a steepness of ≈ . Fig. A.2.
An example NLTE Li i A NLTE (Li) = . ± ..