Magnetic field, activity and companions of V410 Tau
Louise F. Yu, Jean-François Donati, Konstantin Grankin, Andrew Collier Cameron, Claire Moutou, Gaitee Hussain, Clément Baruteau, Laurène Jouve, MaTYSSE collaboration
MMNRAS , 1–28 (2018) Preprint 6 September 2019 Compiled using MNRAS L A TEX style file v3.0
Magnetic field, activity and companions of V410 Tau
L. Yu (cid:63) , J.-F. Donati , K. Grankin , A. Collier Cameron , C. Moutou , G. Hussain , ,C. Baruteau , L. Jouve and the MaTYSSE collaboration All affiliations are listed at the end of the paper
Accepted 2019 September 3. Received 2019 August 28; in original form 2019 July 11
ABSTRACT
We report the analysis, conducted as part of the MaTYSSE programme, of a spec-tropolarimetric monitoring of the ∼ ∼ M (cid:12) disc-less weak-line T Tauristar V410 Tau with the ESPaDOnS instrument at the Canada-France-Hawaii Tele-scope and NARVAL at the T´elescope Bernard Lyot, between 2008 and 2016. WithZeeman-Doppler Imaging, we reconstruct the surface brightness and magnetic field ofV410 Tau, and show that the star is heavily spotted and possesses a ∼
550 G relativelytoroidal magnetic field.We find that V410 Tau features a weak level of surface differential rotation betweenthe equator and pole ∼ ∼ Key words: magnetic fields – stars: imaging – stars: rotation – stars: individual:V410 Tau– techniques: polarimetric
Investigating the birth and youth of low-mass stars ( < (cid:12) )and of their planetary systems heavily contributes to unveil-ing the origin and history of the Sun and of its planets, inparticular the life-hosting Earth. We know that stars andtheir planets form from the collapse of parsec-sized molecu-lar clouds which progressively flatten into massive accretiondiscs, until finally settling as pre-main-sequence (PMS) starssurrounded by protoplanetary discs. T Tauri stars (TTSs)are PMS stars that have emerged from their dust cocoonsand are gravitationally contracting towards the main se-quence (MS); typically aged 1-15 Myr, they are classicalTTSs (cTTSs) when they are still surrounded by a massiveaccretion disc (where planets are potentially forming), andweak-line TTSs (wTTSs) when their accretion has stoppedand their inner disc has dissipated. Large-scale magnetic (cid:63) E-mail: [email protected] fields are known to play a crucial role in the early life oflow-mass stars, as they can open a magnetospheric gap atthe center of the disc, funnel accreting disc material ontothe star, induce stellar winds and prominences, and thusimpact the angular momentum evolution of TTSs (Donati& Landstreet 2009). Observing and understanding the mag-netic topologies of TTSs is therefore a necessary endeavourto complete our understanding of stellar and planetary for-mation (e.g. Bouvier et al. 2007).Since the first detection of a magnetic field arounda cTTS nearly 20 years ago (Johns-Krull et al. 1999),the large-scale topologies of a dozen cTTSs were mapped(e.g. Donati et al. 2007; Hussain et al. 2009; Donati et al.2010a, 2013) thanks to the MaPP (Magnetic Protostarsand Planets) Large Observing Programme allocated on the3.6 m Canada-France-Hawaii Telescope (CFHT) with theESPaDOnS (Echelle SpectroPolarimetric Device for the Ob-servation of Stars) high-resolution spectropolarimeter, usingZeeman-Doppler Imaging (ZDI), a tomography technique c (cid:13) a r X i v : . [ a s t r o - ph . S R ] S e p L. Yu et al. designed for imaging the brightness features and magnetictopologies at the surfaces of active stars (eg Brown et al.1991; Donati & Brown 1997). This first exploration showedthat the topologies of cTTSs are either quite simple or rathercomplex depending on whether the stars are fully convectiveor largely radiative respectively (Gregory et al. 2012; Donatiet al. 2013). Moreover, these fields are reported to vary withtime (e.g. Donati et al. 2011, 2012, 2013) and resemble thoseof mature stars with similar internal structure (e.g. Morinet al. 2008), suggesting that they are produced through dy-namo processes within the bulk of the convective zone.The MaTYSSE (Magnetic Topologies of Young Starsand the Survival of close-in giant Exoplanets) Large Pro-gramme aims at mapping the large-scale magnetic topologiesof ∼
35 wTTSs, comparing them to those of cTTSs and MSstars, and probing the potential presence of massive close-in exoplanets (hot Jupiters/hJs) around its targets. It wasallocated at CFHT over semesters 2013a to 2016b (510 h)with complementary observations from the ESPaDOnS twinNARVAL on the T´elescope Bernard Lyot (TBL) at Pic duMidi in France and from the HARPS spectropolarimeter atthe ESO Telescope at La Silla in Chile. Up to now, about adozen wTTSs were studied with MaTYSSE for their mag-netic topologies and activity, for example V410 Tau (Skellyet al. 2010), LkCa 4 (Donati et al. 2014) and V830 Tau (Do-nati et al. 2017). These studies showed that the fields ofwTTSs are much more diverse than those of cTTSs, withfor example V410 Tau and LkCa 4 displaying strong toroidalcomponents despite being fully convective, as opposed to theresults obtained on cTTSs (see discussion in Donati et al.2014). MaTYSSE fostered the detection of two hJs aroundwTTSs, the 2 Myr-old V830 Tau b (Donati et al. 2016, 2017)and the 17 Myr-old TAP 26 b (Yu et al. 2017).This new study focuses on V410 Tau, a very young( ∼ Our spectropolarimetric data set spans from 2008 Oct to2016 Jan, totalling 144 high-resolution optical spectra, bothunpolarized (Stokes I ) and circularly polarized (Stokes V ).It is composed of 8 runs, most of which cover around 15days, taken during 4 different seasons: 2008b-2009a, 2011a,2013b and 2015b-2016a. The full journal of observations isavailable in Table A1. The 2008b data set and 4 points inthe 2009a data set were taken with the ESPaDOnS echellespectropolarimeter at CFHT, while the rest were taken withthe ESPaDOnS twin NARVAL installed at TBL.The raw frames are processed with the nominal reduc-tion package Libre Esprit as described in e.g. Donati et al.(1997, 2011), yielding a typical root-mean-square (rms) RVprecision of 20-30 m s − (Moutou et al. 2007; Donati et al.2008). The peak signal-to-noise ratios (S/N, per 2.6 km s − velocity bin) reached on the spectra range between 82 and238 for the majority (3 spectra have a S/N lower than 70 andwere rejected for ZDI and the RV analysis), with a medianof 140.Time is counted in units of stellar rotation, us-ing the same reference date and rotation period as inSkelly et al. (2010), namely BJD = , , . and P rot = . ± . d (Stelzer et al. 2003) respectively: c = ( BJD − BJD ) / P rot . (1)The stellar phase is defined as the decimal part of the cycle c . The emission core of the Ca ii infrared triplet (IRT)presents an average equivalent width (EW) of (cid:39)
13 km s − (0.37 ˚A). The He i D line is relatively weak with an averageEW of 13 km s − as well (0.25 ˚A), in agreement with thenon-accreting status of V410 Tau. The H α line has an aver-age EW of 14 km s − (0.33 ˚A) and a rms EW of 27 km s − and exhibits a periodicity of period . ± . d (see Ap-pendix C). From the He i D line, we detected small flares on2008 Dec 10 (rotational cycle -15+3.514, as per Table A1),on the night of 2013 Dec 08 to 2013 Dec 09 (rotational cy-cles 959+4.090 and 959+4.151), and on the night of 2016Jan 20 (rotational cycles 1376+0.021 and 1376+0.040). Onebig flare, on 2008 Dec 15 (rotational cycle -15+6.181), wasvisible not only in He i D (EW (cid:39) km s − ) but also in H α (EW (cid:39) km s − ) and the Ca ii IRT (core emission EW (cid:39) km s − ). We removed the 6 flare-subjected observationsfrom our data sets in order to proceed with the mapping ofthe photosphere and surface magnetic field, as well as theRV analysis.Least-squares deconvolution (LSD, see Donati et al.1997) was applied to all our spectra in order to add up infor-mation from all spectral lines and boost the resulting S/Nof both Stokes I and V LSD profiles. The spectral maskwe employed for LSD was computed from an
Atlas9
LTEmodel atmosphere (Kurucz 1993) featuring T e ff =4,500 K and log g =3.5, and involves about 7 800 spectral features (withabout 40 % from Fe i , see e.g. Donati et al. 2010b, formore details). Stokes I and Stokes V LSD profiles shown in
MNRAS , 1–28 (2018) ag. field, activity & companions of V410 Tau Section 4 display distorsions that betray the stellar activ-ity with a periodicity corresponding to the rotation of thestar. Moonlight pollution, which affects 15 of our Stokes I LSD profiles, was filtered out using a two-step tomographicimaging process described in Donati et al. (2016). The S/Nin the Stokes I LSD profiles, ranging from 1633 to 2930 (per1.8 km s − velocity bin) with a median of 2410, is measuredfrom continuum intervals, including not only the noise fromphoton statistics, but also the (often dominant) noise intro-duced by LSD (see Table A1). The S/N in Stokes V LSDprofiles, dominated by photon statistics, range from 1817 to6970 with a median value of 3584.Phase coverage is of varying quality depending on theobservation epoch. The 2008b data set, with only 6 points,covers only half the surface of the star (phases -0.20 to 0.30).The 2009a data set, although the densest with 48 points in16 days and including data from both instruments, lacksobservations between phases 0.05 and 0.20. The 2011a dataset presents a large gap between phases -0.05 and 0.15, anda smaller one between phases 0.65 and 0.80. The 2013b and2015b data sets are well sampled, and the 2016a data set,with only 9 points, lacks observations between phases 0.25to 0.45 and -0.15 to 0.05.Contemporaneous BVR J I J photometric measurements,documented in Table A2, were taken from the Crimean As-trophysical Observatory 1.25 m and 0.60 m telescopes be-tween August 2008 and March 2017, counting 420 observa-tions distributed over 9 runs at a rate of one run per year,each run covering 3 to 7 months. In each run, the visiblemagnitude presents modulations of a period ∼ V410 Tau is a very well-observed three-star system lo-cated in the Taurus constellation at d = . ± . pc fromEarth (Galli et al. 2018, we chose this value over theGaia result, . ± . pc, because it is both in agree-ment with it and more precise). V410 Tau B was esti-mated to have a mass . ± . times that of V410 Tau A,and V410 Tau C to have a mass . + . − . times that ofV410 Tau AB (Kraus et al. 2011). The sky-projected sep-aration between V410 Tau A and V410 Tau B was mea-sured at . ± . arcsec, i.e. . ± . au, and that be-tween V410 Tau AB and V410 Tau C was measured at . ± . arcsec, i.e. ± au. Given that V410 Tau Ais much brighter than V410 Tau B and V410 Tau C inthe optical bandwidth (Ghez et al. 1997), we considerthat the spectra analysed in this study characterize thelight of V410 Tau A predominantly. Applying the auto-matic spectral classification tool developped within theframe of the MaPP and MaTYSSE projects (Donati et al.2012), we constrain the temperature and logarithmic grav- ity of V410 Tau A to, respectively, T e ff = ± K and log g = . ± . .Its rotation period was previously estimated to P rot = . ± . d (Stelzer et al. 2003), a valuewhich we use throughout this paper to phase our data (seeEq. 1). Comparing both our contemporary measurements(Table A2) and those found in Grankin et al. (2008), wefind that the minimum magnitude measured on V410 Tauis . ± . , value that we use as a reference to computethe unspotted magnitude.Our photometric measurements yield a mean B − V in-dex of . ± . , and since the theoretical B − V at 4500 Kis . ± . (Pecaut & Mamajek 2013, Table 6), the amountof visual extinction is A V = . · (1 . − . = . ± . .The bolometric correction at T e ff being equal to − . ± . (Pecaut & Mamajek 2013, Table 6), and the distance mod-ulus to − · log ( d / = − . ± . , we find an absolutemagnitude of . ± . .The value of v sin i found from the spectra, . ± . km s − (see Section 4), indicates that the mini-mum radius of the star R (cid:63) sin i is equal to . ± .
007 R (cid:12) ,which implies a maximum absolute unspotted magnitudeof . ± . given the photospheric temperature. The dis-crepancy with the value found in the previous paragraphindicates the presence of dark spots on the photosphereeven when the star is the brightest. If we assume a spotcoverage at maximum brightness of ∼
25 %, typical of ac-tive stars, (like it was done in Donati et al. 2014, 2015;Yu et al. 2017), then the unspotted absolute magnitudewould be . ± . , which corresponds to an inclination of ± ◦ . However, the models best fitting our spectrahave an inclination of ± ◦ (see Sec 4), which would re-quire the spot coverage at maximum brightness to actuallybe ∼
50 %. Such a high permanent spot coverage is unusualbut not unconceivable, since another wTTS, LkCa4, was ob-served to have as much as 80 % of its surface covered withspots (Gully-Santiago et al. 2017). Assuming a spot cov-erage at maximum brightness of ± % for V410 Tau,we derive an absolute unspotted magnitude of . ± . , alogarithmic luminosity log( L (cid:63) / L (cid:12) ) = . ± . , and a stel-lar radius R (cid:63) = √ L (cid:63) / L (cid:12) · ( T (cid:12) / T (cid:63) ) = . ± . (cid:12) . This valuefor the radius, combined with the v sin i derived from thespectra, yields an inclination of ± ◦ .The position of V410 Tau on the Hertzsprung-Russelldiagram is displayed in Figure 1. According to Siess et al.(2000) stellar evolution models for pre-main sequence stars,with solar metallicity and overshooting, V410 Tau is a . ± .
15 M (cid:12) star, aged . ± . Myr and fully convec-tive. Baraffe models (Baraffe et al. 2015) disagree with theSiess models for stars as young as V410 Tau and yield anage of < . ± .
10 M (cid:12) . However, forthe sake of consistency with the other MaPP and MaTYSSEstudies, we will consider the values yielded by the Siess mod-els in this paper. Our values are in good agreement withWelty & Ramsey (1995) and Skelly et al. (2010), who hadpreviously derived masses of ∼ M (cid:12) and . ± . (cid:12) re-spectively, radii of ∼ R (cid:12) and ∼ R (cid:12) respectively, andages of − Myr . ± . Myr respectively. Moreover, Skelly line-of-sight-projected equatorial rotation velocity angle between the stellar rotation axis and the line of sightMNRAS , 1–28 (2018) L. Yu et al.
Figure 1.
Position of V410 Tau (red) in the Hertzsprung-Russelldiagram. The curves yielded by the Siess models (with solarmetallicity and overshooting) are represented in black and thoseyielded by the Baraffe models are represented in magenta. In bothcases, evolution tracks are displayed in dashed lines, except theSiess 1.4 M (cid:12) track, the one we chose to model the evolution ofV410 Tau, which is shown as a full line. Isochrones are displayedin dotted lines. The thresholds where the radiative core startsdevelopping (”Full convection”) and where it reaches 25% of thestellar radius, according to the Siess models, are marked in blue. Table 1.
Physical parameters of wTTS V410 Tau. From top tobottom: distance from Earth, effective temperature, rotation pe-riod, luminosity, minimum stellar radius, stellar radius, line-of-sight-projected equatorial velocity, inclination, mass and age.Parameter Value Reference d ± T e ff ±
100 K P rot ± log( L (cid:63) / L (cid:12) ) ± R (cid:63) sin i ± R (cid:12) R (cid:63) ± R (cid:12) v sin i ± − ZDI (Section 4) i ± ◦ ZDI (Section 4) M (cid:63) ± M (cid:12) Age 0.84 ± et al. (2010) had deduced that V410 Tau could have a ra-diative core of radius between 0.0 R (cid:63) and 0.28 R (cid:63) . Table 1sums up the stellar parameters of V410 Tau found in thisstudy. To map the surface brightness and magnetic topology ofV410 Tau, we use the tomographic technique ZDI (Brownet al. 1991; Donati & Brown 1997), which inverts simul-taneous time-series of Stokes I and Stokes V LSD profilesinto brightness and magnetic field surface maps. At eachobservation date, Stokes I and Stokes V profiles are synthe-sized from model maps by integrating the spectral contri-bution of each map cell over the visible half of the stellarsurface, Doppler-shifted according to the local RV (i.e. line-of-sight-projected velocity) and weighted according to thelocal brightness, cell sky-projected area and limb darkening.The main modifier of local RV at the surface of the star is, inZDI, the assumed rotation profile at the stellar surface, e.g. the solid-body rotation of the star or a square-cosine-typelatitudinal differential rotation. Local Stokes I and Stokes V line profiles are computed from the Unno-Rachkovsky ana-lytical solution to the polarized radiative transfer equationsin a Milne-Eddington model atmosphere (this is where thelocal magnetic field and the Zeeman effect intervene, seeLandi degl’Innocenti & Landolfi 2004). To fit the LSD pro-files of V410 Tau in this study, we chose a spectral line ofmean wavelength, Doppler width, Land´e factor and equiva-lent width of respective values 640 nm, 1.8 km s − , 1.2 and3.8 km s − .ZDI uses a conjugate gradient algorithm to iterativelyreconstruct maps whose synthetic profiles can fit the LSDprofiles down to a user-provided reduced chi-square ( χ )level. To lift degeneracy among the multiple solutions com-patible with the data at the given reduced chi square, ZDIlooks for the maximal-entropy solution, considering that theminimized information from the resulting maps is the mostreliable. While the brightness value can vary freely from cellto cell, the surface magnetic field is modelled as a combi-nation of poloidal and toroidal fields, both represented asweighted sums of spherical harmonics and projected ontothe spherical coordinate space (Donati et al. 2006, for theequations). In this study, the magnetic field was fitted withspherical harmonics of orders l = to l = .Because ZDI does not reconstruct intrinsic temporalvariability except for differential rotation, there is a limit tothe duration a fittable data set can span. At the same time,ZDI needs a good phase coverage from the data to build acomplete map. For those reasons, ZDI was not applied toruns 2008 Oct and 2013 Nov; moreover, we reconstructed adifferent set of brightness and magnetic images for each ofthe runs on which ZDI was applied.Using ZDI on our data yielded values for v sin i and i of . ± . km s − and ± ◦ respectively. We also adjustedthe systemic RV of V410 Tau with ZDI, and noticed a driftin the optimal value with time (see Table 2). Time-series of Stokes I and Stokes V LSD profiles are shownin Figure 2, both before and after removal of lunar pollu-tion, as well as synthetic profiles generated from the recon-structed ZDI maps. The corresponding maps are shown inFigure 3, with brightness maps in the first column and ra-dial, meridional and azimuthal components of the surfacemagnetic field in the second to fourth columns. Propertiesof these reconstructed maps are listed in Table 2. Since the2008 Dec data set has a phase coverage of only half the star,the derived parameters characterizing the global field topol-ogy at this epoch are no more than weakly meaningful andwere not used for the following analysis and discussion. Ourdata have been fitted down to χ = with a feature cover-age between 15 % and 18 % depending on the epochs, and alarge-scale field strength of 0.5-0.6 kG. Since ZDI is only sen-sitive to mid- to large-scale surface features, and returns themaximum-entropy solution, this amount of spot coverage isnot discrepant with the assumption made in Section 3; itfurther suggests that 30% of the star is more or less evenlycovered with small-scale dark features.Brightness maps display a complex structure with manyrelatively small-scale features, and a high contrast. At all MNRAS000
100 K P rot ± log( L (cid:63) / L (cid:12) ) ± R (cid:63) sin i ± R (cid:12) R (cid:63) ± R (cid:12) v sin i ± − ZDI (Section 4) i ± ◦ ZDI (Section 4) M (cid:63) ± M (cid:12) Age 0.84 ± et al. (2010) had deduced that V410 Tau could have a ra-diative core of radius between 0.0 R (cid:63) and 0.28 R (cid:63) . Table 1sums up the stellar parameters of V410 Tau found in thisstudy. To map the surface brightness and magnetic topology ofV410 Tau, we use the tomographic technique ZDI (Brownet al. 1991; Donati & Brown 1997), which inverts simul-taneous time-series of Stokes I and Stokes V LSD profilesinto brightness and magnetic field surface maps. At eachobservation date, Stokes I and Stokes V profiles are synthe-sized from model maps by integrating the spectral contri-bution of each map cell over the visible half of the stellarsurface, Doppler-shifted according to the local RV (i.e. line-of-sight-projected velocity) and weighted according to thelocal brightness, cell sky-projected area and limb darkening.The main modifier of local RV at the surface of the star is, inZDI, the assumed rotation profile at the stellar surface, e.g. the solid-body rotation of the star or a square-cosine-typelatitudinal differential rotation. Local Stokes I and Stokes V line profiles are computed from the Unno-Rachkovsky ana-lytical solution to the polarized radiative transfer equationsin a Milne-Eddington model atmosphere (this is where thelocal magnetic field and the Zeeman effect intervene, seeLandi degl’Innocenti & Landolfi 2004). To fit the LSD pro-files of V410 Tau in this study, we chose a spectral line ofmean wavelength, Doppler width, Land´e factor and equiva-lent width of respective values 640 nm, 1.8 km s − , 1.2 and3.8 km s − .ZDI uses a conjugate gradient algorithm to iterativelyreconstruct maps whose synthetic profiles can fit the LSDprofiles down to a user-provided reduced chi-square ( χ )level. To lift degeneracy among the multiple solutions com-patible with the data at the given reduced chi square, ZDIlooks for the maximal-entropy solution, considering that theminimized information from the resulting maps is the mostreliable. While the brightness value can vary freely from cellto cell, the surface magnetic field is modelled as a combi-nation of poloidal and toroidal fields, both represented asweighted sums of spherical harmonics and projected ontothe spherical coordinate space (Donati et al. 2006, for theequations). In this study, the magnetic field was fitted withspherical harmonics of orders l = to l = .Because ZDI does not reconstruct intrinsic temporalvariability except for differential rotation, there is a limit tothe duration a fittable data set can span. At the same time,ZDI needs a good phase coverage from the data to build acomplete map. For those reasons, ZDI was not applied toruns 2008 Oct and 2013 Nov; moreover, we reconstructed adifferent set of brightness and magnetic images for each ofthe runs on which ZDI was applied.Using ZDI on our data yielded values for v sin i and i of . ± . km s − and ± ◦ respectively. We also adjustedthe systemic RV of V410 Tau with ZDI, and noticed a driftin the optimal value with time (see Table 2). Time-series of Stokes I and Stokes V LSD profiles are shownin Figure 2, both before and after removal of lunar pollu-tion, as well as synthetic profiles generated from the recon-structed ZDI maps. The corresponding maps are shown inFigure 3, with brightness maps in the first column and ra-dial, meridional and azimuthal components of the surfacemagnetic field in the second to fourth columns. Propertiesof these reconstructed maps are listed in Table 2. Since the2008 Dec data set has a phase coverage of only half the star,the derived parameters characterizing the global field topol-ogy at this epoch are no more than weakly meaningful andwere not used for the following analysis and discussion. Ourdata have been fitted down to χ = with a feature cover-age between 15 % and 18 % depending on the epochs, and alarge-scale field strength of 0.5-0.6 kG. Since ZDI is only sen-sitive to mid- to large-scale surface features, and returns themaximum-entropy solution, this amount of spot coverage isnot discrepant with the assumption made in Section 3; itfurther suggests that 30% of the star is more or less evenlycovered with small-scale dark features.Brightness maps display a complex structure with manyrelatively small-scale features, and a high contrast. At all MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau (a) 2008 Dec (b) 2009 Jan(c) 2011 Jan (d) 2013 Dec(e) 2015 Dec (f) 2016 Jan Figure 2.
LSD profiles for observation epochs 2008 Dec (a), 2009 Jan (b), 2011 Jan (c), 2013 Dec (d), 2015 Dec (e) and 2016 Jan (f).On the right of each profile is written the corresponding rotation cycle as indicated in Table A1. The cyan, black and red lines representrespectively the profiles before removal of Moon pollution, after removal of Moon pollution and the fit obtained with Zeeman DopplerImaging. For each epoch, Stokes I profiles are on the left and Stokes V profiles on the right. 3 σ -error bars are displayed beside eachStokes V profile.MNRAS , 1–28 (2018) L. Yu et al.
Figure 3.
ZDI maps of the logarithmic relative surface brightness (first column), and the radial, meridional and azimuthal magnetic field(second to fourth columns) of V410 Tau, reconstructed from data collected in 2008 Dec, 2009 Jan, 2011 Jan, 2013 Dec, 2015 Dec and2016 Jan (top to bottom rows). Each map is shown as a flattened polar view, with the equator being represented as a full line, and 60 ◦ ,30 ◦ , and -30 ◦ latitude parallels as dashed lines, and ticks around the star mark the spectropolarimetric observations. For the brightnessmaps, cool spots are colored in brown and bright plages in blue. For the magnetic maps, red represents outwards and anti-clockwise fieldon the radial and azimuthal field maps respectively, and the direction of the visible pole on the meridional field maps.MNRAS000
ZDI maps of the logarithmic relative surface brightness (first column), and the radial, meridional and azimuthal magnetic field(second to fourth columns) of V410 Tau, reconstructed from data collected in 2008 Dec, 2009 Jan, 2011 Jan, 2013 Dec, 2015 Dec and2016 Jan (top to bottom rows). Each map is shown as a flattened polar view, with the equator being represented as a full line, and 60 ◦ ,30 ◦ , and -30 ◦ latitude parallels as dashed lines, and ticks around the star mark the spectropolarimetric observations. For the brightnessmaps, cool spots are colored in brown and bright plages in blue. For the magnetic maps, red represents outwards and anti-clockwise fieldon the radial and azimuthal field maps respectively, and the direction of the visible pole on the meridional field maps.MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau Table 2.
Characteristics of the ZDI models for V410 Tau at each observation epoch.
Column 1 : observation epoch.
Column 2 : numberof spectropolarimetric observations used for ZDI.
Column 3 : contribution of cool (”spots”) and hot (”plages”) areas on the brightnessmap.
Column 4 : average magnetic strength, defined as the square root of the average squared magnetic field over the surface of the star.
Columns 5 to 7 : normalized contribution of the poloidal field, part of the poloidal field that is dipolar and part of the poloidal fieldthat is symmetric.
Columns 8-9 : part of the toroidal field that is dipolar and part of the toroidal field that is symmetric.
Column 10 :dipole characteristics: field strength, tilt with respect to the rotation axis and phase of the pole.
Column 11 : systemic RV of the star asmeasured with ZDI, the error bar on those values is 0.20 km s − . Error bars on the magnetic field ratios are typically of 0.1.Date N obs Spot+plage B r pol r dip / pol r sym / pol r dip / tor r sym / tor Dipole strength (G), RV bulk coverage (%) (G) tilt & phase (km s − )2008 Dec 6 5.8+4.4 486 0.32 0.13 0.37 0.89 0.96 129, 23 ◦ & 0.71 16.302009 Jan 48 9.6+7.1 556 0.55 0.26 0.09 0.54 0.79 165, 54 ◦ & 0.54 16.302011 Jan 20 8.1+6.6 560 0.40 0.24 0.23 0.72 0.85 239, 44 ◦ & 0.62 16.402013 Dec 25 11.0+7.5 568 0.49 0.23 0.34 0.66 0.81 254, 18 ◦ & 0.56 16.502015 Dec 21 8.9+6.7 600 0.68 0.37 0.45 0.62 0.78 458, 30 ◦ & 0.54 16.652016 Jan 9 7.9+6.5 480 0.77 0.38 0.30 0.68 0.87 400, 44 ◦ & 0.51 16.65 epochs, a large concentration of dark spots is observed atthe pole. In 2009 Jan, 2013 Dec and 2015 Dec, the bright-ness map exhibits a strong equatorial spot, respectively atphases 0.27, 0.48 and 0.48. The presence of a strong polarspot is consistent with the maps published in Skelly et al.(2010), Rice et al. (2011) and Carroll et al. (2012) for dataset 2009 Jan. At that particular epoch, the equatorial spotat phase 0.27, and another equatorial spot at phase 0.60, arealso visible in both Skelly et al. (2010) and Rice et al. (2011)(figure 8), albeit less contrasted compared to other featuresthan they are on our map. A remnant of the 2015 Dec equa-torial spot is observed on the 2016 Jan map, where its in-tensity seems to have decreased, but this has to be takenwith caution since ZDI maps are somewhat dependent onphase coverage. Dark spots and bright plages contribute tothe feature coverage at about 9 % / 7 % respectively.Photometry curves from the ZDI brightness maps weresynthesized and a comparison to contemporary CrAO data,and WASP data in the case of 2011 Jan, is shown in Fig-ure 4. Despite a slightly underestimated amplitude at phase0.60 in 2008b-2009a, at phase 0.20 in 2011a, at phase 0.20in 2013b and at phases 0.20 and 0.80 in 2015b-2016a, ZDImanages to retrieve the measured photometric variations ofV410 Tau rather satisfyingly. We notice a small temporalevolution of the light curve in the WASP data during season2010b-2011a, where the regions around phases 0.20 and 0.70globally darken by 0.02-0.03 mag ( (cid:39) σ ) over the 4 monthsthat the data set spans.The magnetic field maps also show a high complexity,with a poloidal component that has a weak dipolar contri-bution and that is rather non-axisymmetric, and a toroidalcomponent contributing to ∼
50% of the overall magnetic en-ergy in 2009, 2011 and 2013, and decreasing towards (cid:39) %in 2015-2016, that is both strongly dipolar and highly ax-isymmetric. The dipole pole is tilted at various angles de-pending on the epoch, with a tilt as high as 54 ◦ in 2009 Jan,down to 18 ◦ in 2013 Dec. The phase of the pole is alwaysaround 0.50-0.60, and the intensity of the poloidal dipoleincreases over time, from 165 G in 2009 Jan to (cid:39) G in2015-2016. We note that the maximum emission of H α cor-responds to the phase at which the dipole is tilted (Fig. C1).For visualisation purposes, 3-dimensional potential fieldswere extrapolated from the radial components of the mag-netic maps, and displayed in Figure 5, with phase 0.50 facingthe reader. We do not observe a particular correlation between ourbrightness and our magnetic maps, meaning the areas withstrong magnetic field are not necessarily crowded with darkspots, according to the ZDI reconstruction. Without differential rotation, ZDI cannot fit an extendeddata set, such as 2008 Dec + 2009 Jan, 2013 Nov + 2013Dec or 2015 Dec + 2016 Jan (shortened in this subsectionto 08b+09a, 13b and 15b+16a respectively), down to χ =1,it only manages to reach values of 1.66, 1.20 and 2.64 re-spectively. This implies that some level of variability existsand impacts the data on time scales of a few months, whichcould come from the presence of differential rotation at thesurface of V410 Tau. We model differential rotation with thefollowing law: Ω ( θ ) = Ω eq − (cos θ ) d Ω where θ is the colatitude, Ω eq the equatorial rotation rate and d Ω the pole-to-equator rotation rate difference. We constrain Ω eq and d Ω by pre-setting the amount of information ZDI isallowed to reconstruct, and having ZDI minimize the χ inthese conditions.We performed this analysis on the three afore-mentioned extended data sets, and on Stokes I and Stokes V time-series separately, reconstructing only brightness or onlymagnetic field respectively. From the resulting χ mapsover the { Ω eq , d Ω } space, one can plot the contours of the1 σ - (68.3%) and 3 σ - (99.7%) areas of confidence for eachobservation epoch. Figure 6, which shows such contours,highlights clear minima surrounded by almost elliptic ar-eas of confidence at each epoch, and shows that each 3 σ -confidence area overlaps at least two other 3 σ -confidenceareas. Numerical results for each epoch are given in Ta-ble 3. We chose to use a unique set of parameters to recon-struct all images shown in Section 4: the weighted means ofthe six seasonal minima, Ω eq = . ± . rad d − and d Ω = . ± . rad d − .Following the method described in Donati et al. (2003),we computed, for each epoch, the colatitude at which therotation rate is constant along the confidence ellipse ma-jor axis. This value corresponds to the colatitude wherethe barycenter of the brightness/magnetic features impos-ing a correlation between Ω eq and d Ω are located. For both MNRAS , 1–28 (2018)
L. Yu et al. R e l a t i v e b r i g h t n e ss (a) 08b & 09a R e l a t i v e b r i g h t n e ss (b) 10b+11a R e l a t i v e b r i g h t n e ss (c) 13b+14a R e l a t i v e b r i g h t n e ss (d) 15b & 16a Figure 4.
Phase-folded photometry data (dots with 1 σ error bars) and ZDI models (lines) for observation epochs 08b & 09a (a),10b+11a (b), 13b+14a (c) and 15b & 16a (d). In the case of 08b & 09a and 15b & 16a, two ZDI curves are plotted for the two ZDI mapsreconstructed within each epoch. Orange, red, purple and blue colors each indicate a quarter of the total time span of the observations(photometric and spectropolarimetric together), in chronological order. Spectropolarimetric observations are marked by ticks above thelight curves. In Figure b, WASP data were added as desaturated crosses, with the size of the cross branches indicating their 1 σ errorbars. Stokes I and Stokes V , we note a slight increase with timeof the cosine of this colatitude (Table 3), i.e. an increase inthe barycentric latitude of the dominant features of ± ◦ and ± ◦ respectively.These models exclude solid-body rotation at a level of3.6 to 22 σ depending on the epoch. We note that, even withdifferential rotation, ZDI cannot fit the data of 08b+09a andof 15b+16a down to χ =1, no matter the amount of infor-mation allowed. This indicates that surface features are alsoaltered by a significant level of intrinsic variability within the2-month span of our data set. This issue is further discussedin section 5.3. Radial velocity values were derived as the first-order mo-ment of the continuum-subtracted Stokes I LSD profiles, forall spectra except the 3 with low S/N and the 6 in whichwe identified flares (see Table A1). The raw RVs we obtaincontain a contribution from the inhomogeneities on the pho-tosphere, called activity jitter, which we aim to filter out in order to access the actual RV of the star, and look for a po-tential planet signature. The activity jitter is modelled withtwo different techniques, ZDI and Gaussian Process Regres-sion. Raw RVs and jitter models are plotted in Figure 7 andlisted in Table A1. For the 2015-2016 points, a new versionof ZDI, with the logarithmic brightness of surface featuresallowed to lineary vary with time, was tested (section 5.3).The raw RVs present modulations whose amplitude vary be-tween 4 and 8.5 km s − , with a global rms of 1.8 km s − . Likewith the photometric data, the RV variations are the lowestin 2009 Jan and the strongest in 2013 Dec. The first method consists in deriving the activity jitter fromthe ZDI models (see Fig. 2), computed as the first-ordermoment of the continuum-subtracted synthetic Stokes I pro-files. Indeed, when computing the raw RV from the observedStokes I LSD profiles, this activity jitter is added on top ofthe radial motion of the star as a whole. We model the ac-tivity jitter separately for epochs 2009 Jan, 2011 Jan, 2013
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MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau Table 3.
Summary of differential rotation parameters obtained for V410 Tau on each season. All rotation rates are given in mrad d − .Column 2 gives the total number of data points used in the imaging process, then columns 3 to 7 correspond to Stokes I data while column8 to 12 correspond to Stokes V data. Columns 3 and 8 list the derived equatorial rotation rate Ω eq , with its 68% (i.e. 1 σ ) confidenceinterval, columns 4 and 9 the difference in rotation rate d Ω between the equator and pole, with its 68% confidence interval, columns 5and 10 give the reduced chi square of the ZDI model compared to the data, columns 6 and 11 give the inverse slope of the ellipsoid inthe Ω eq - d Ω plane (also equal to cos θ s , where θ s denotes the colatitude of the gravity centre of the spot distribution, see Donati et al.2000), and columns 7 and 12 give the rotation rate Ω s at colatitude θ s . | Stokes I data / brightness reconstruction | Stokes V data / magnetic field reconstructionEpoch n | Ω eq d Ω χ
2r cos2 θ s Ω s | Ω eq d Ω χ
2r cos2 θ s Ω s | ± ± ± ± | ± ± ± ± | ± ± ± ± | ± ± ± ± | ± ± ± ± | ± ± ± ± (a) 2009 Jan (b) 2011 Jan(c) 2013 Dec (d) 2015 Dec(e) 2016 Jan Figure 5.
Potential field extrapolations of the ZDI-reconstructedsurface radial field, as seen by an Earth-based observer, for ob-servation epochs 2009 Jan (a), 2011 Jan (b), 2013 Dec (c), 2015Dec (d) and 2016 Jan (e) at phase 0.50. Open/closed field linesare shown in orange/black respectively, and colours on the stellarsurface depict the local value of the radial field (in G, as shown inthe left-hand panels of Fig. 3). The source surface at which fieldlines open is set to 2.1 R (cid:63) , corresponding to the corotation radiusand beyond which field lines are expected to quickly open undercentrifugal forces given the high rotation rate of V410 Tau. Ω eq (rad/d) d Ω (r ad / d ) Figure 6.
Evolution of the differential rotation of V410 Tauas measured from Stokes I (blue) and Stokes V (red) profiles.The points corresponding to observation epoch 2008b-2009a aremarked with o symbols, then the dashed lines link the epochsin chronological order (2013b-2014a and 2015b-2016a are markedwith x symbols). 68.3% and 99.7% contours of confidence are dis-played for each observation epoch. The weighted average of the sixmeasurements, chosen to produce the maps shown in Section 4,is represented as a black +, with overlayed error bars in green.MNRAS , 1–28 (2018) L. Yu et al. -5.00.05.00 1 2 3 4 5 6 7 8 9-0.50.00.5 (a) 2009 Jan. Respective rms: 1.20, 0.13, 0.08 km s − -5.00.05.00 1 2 3 4 5 6-0.50.00.5 (b) 2011 Jan. Respective rms: 2.40, 0.14, 0.06 km s − -5.00.05.00 1 2 3 4 5 6 7 8 9 10 11-0.50.00.5 (c) 2013 Dec. Respective rms: 2.43, 0.22, 0.09 km s − -5.00.05.00 1 2 3 4 5 6 7 8 9 10-0.50.00.5 (d) 2015 Dec. Respective rms: 1.93, 0.22, 0.08 km s − -5.00.05.00 1 2 3 4 5-0.50.00.5 (e) 2016 Jan. Respective rms: 1.41, 0.09,0.01 km s − Figure 7.
Raw and filtered RVs of V410 Tau for each observation epoch. On each figure, the top plot depicts the raw RVs (red dots), theZDI reconstruction (red full line) and the GP fit (blue full line with 1- σ area of confidence marked as blue dashed lines, see Section 5.1).The bottom plot depicts the RVs filtered from the ZDI-modelled activity (red dots) and the RVs filtered from the GP-modelled activity(blue dots). The subcaptions indicate the rms of the raw RVs, the ZDI-filtered RVs and the GPR-filtered RVs respectively. All rotationalcycles are displayed as in Table A1. Dec, 2015 Dec and 2016 Jan (excluding 2008 Dec because ofthe poor phase coverage).The second method uses Gaussian Process Regression(Haywood et al. 2014; Donati et al. 2017), a numericalmethod focusing on the statistical properties of the model.In short, GPR extrapolates a continuous curve described bya given covariance function from some given data points. Todescribe the activity jitter here, we use a pseudo-periodiccovariance function: K ( t , t (cid:48) ) = θ . exp − ( t − t (cid:48) ) θ − sin (cid:16) π ( t − t (cid:48) ) θ (cid:17) θ (2) where t and t (cid:48) are the dates of the two RV points betweenwhich the covariance is computed, θ is the amplitude of theGP, θ the recurrence time scale (expected to be close to P rot ), θ the decay time scale (i.e., the typical spot lifetimein the present case) and θ a smoothing parameter (within[0, 1]) setting the amount of high frequency structures thatwe allow the fit to include. The modelling process thereforeconsists in optimizing the 4 parameters θ , θ , θ and θ ,called hyperparameters. To do so, we use a Markov ChainMonte-Carlo algorithm, and allocate to each point of the hy-perparameter space a likelihood value, which takes into ac-count both the quality of the fit and some penalizations on MNRAS000
Raw and filtered RVs of V410 Tau for each observation epoch. On each figure, the top plot depicts the raw RVs (red dots), theZDI reconstruction (red full line) and the GP fit (blue full line with 1- σ area of confidence marked as blue dashed lines, see Section 5.1).The bottom plot depicts the RVs filtered from the ZDI-modelled activity (red dots) and the RVs filtered from the GP-modelled activity(blue dots). The subcaptions indicate the rms of the raw RVs, the ZDI-filtered RVs and the GPR-filtered RVs respectively. All rotationalcycles are displayed as in Table A1. Dec, 2015 Dec and 2016 Jan (excluding 2008 Dec because ofthe poor phase coverage).The second method uses Gaussian Process Regression(Haywood et al. 2014; Donati et al. 2017), a numericalmethod focusing on the statistical properties of the model.In short, GPR extrapolates a continuous curve described bya given covariance function from some given data points. Todescribe the activity jitter here, we use a pseudo-periodiccovariance function: K ( t , t (cid:48) ) = θ . exp − ( t − t (cid:48) ) θ − sin (cid:16) π ( t − t (cid:48) ) θ (cid:17) θ (2) where t and t (cid:48) are the dates of the two RV points betweenwhich the covariance is computed, θ is the amplitude of theGP, θ the recurrence time scale (expected to be close to P rot ), θ the decay time scale (i.e., the typical spot lifetimein the present case) and θ a smoothing parameter (within[0, 1]) setting the amount of high frequency structures thatwe allow the fit to include. The modelling process thereforeconsists in optimizing the 4 parameters θ , θ , θ and θ ,called hyperparameters. To do so, we use a Markov ChainMonte-Carlo algorithm, and allocate to each point of the hy-perparameter space a likelihood value, which takes into ac-count both the quality of the fit and some penalizations on MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau Table 4.
Priors for our GP-MCMC run on our raw RVs. For themodified Jeffreys prior, the knee value is given, for the Gaussianprior we give the mean and standard deviation, and for the Jef-freys and the uniform priors we give the lower and upper bound-aries. Hyperparameter Prior θ (km s − ) Modified Jeffreys ( σ RV ) θ ( P rot ) Gaussian (1.0000, 0.1000) θ ( P rot ) Jeffreys(0.1, 500.0) θ Uniform (0, 1)
Figure 8.
Phase plot of the MCMC-GPR run on the raw RVs,model without planet. The yellow, red and blue colors indicaterespectively the 1 σ -, 2 σ - and 3 σ -areas of confidence, and the opti-mal values for the hyperparameters are marked with black dashedlines, with 1 σ -intervals marked with black dotted lines. GP am-plitude ( θ ): . + . − . km s − , Cycle length ( θ ): . ± . P rot ,Decay time ( θ ): + − P rot , Smoothing ( θ ): . ± . . the hyperparameters (for example we penalize high ampli-tudes, low decay times and low smoothings). The priors arelisted in Table 4. The phase plot of the MCMC is displayedin Figure 8 and the best fit is shown in Figure 7, togetherwith the ZDI fits. We note that, contrary to ZDI, GPR, be-ing capable of describing intrinsic variability in a consistentway, is able to fit our whole 8-year-long data set with onemodel. We obtain θ = . + . − . km s − , θ = . ± . P rot , θ = + − P rot and θ = . ± . P rot .The rms of the filtered RVs for each epoch and eachmethod are summarized in Table 5. The RV curve filteredfrom the ZDI model presents a global rms of 0.167 km s − ,i.e. ∼ < σ RV > (see Table A1). The epoch where the filteringis most efficient is 2009 Jan, although the rms of the filteredRVs is only at 1.5 < σ RV > , and it goes up to 3 < σ RV > in2011 Jan and 2013 Dec. On the other hand, the GPR modelfilters the RV out down to 0.076 km s − = 0.94 < σ RV > . Table 5.
Rms of RVs. All rms RVs are given in km s − .Epoch 2009 2011 2013 2015 2016 AllRaw 1.200 2.392 2.429 1.932 1.411 1.8ZDI filt. 0.131 0.141 0.215 0.222 0.094 0.167GP filt. 0.084 0.064 0.087 0.075 0.009 0.076 Lomb-Scargle periodograms for both raw and filtered RVs,for both methods (Fig. 9 for each individual epoch, 10 forthe whole data set), show that the stellar rotation period orits first harmonic are clearly present in 2009 Jan and 2011Jan, but not well retrieved in 2013 Dec, 2015 Dec and 2016Jan. However the periodogram for the whole RV raw data setpresents neat peaks at P rot and its first two harmonics. P rot and its first harmonic are well filtered out by both modellingmethods, and the second harmonic is well filtered out in theGP residuals. A weak signal remains at P rot /3 in the ZDIresiduals but looking at a phase-folded plot does not revealany particularly obvious tendency, leading us to suspect thatit mostly reflects the contribution of a few stray points. Noother period stands out with a false-alarm-probability lowerthan 5%, which allows us to conclude that no planet signa-ture is found in this data set with our filtering methods. Seeing that the filtered RVs when using GPR have a rmstwice lower than when using ZDI (Table 5), we try to im-prove our ZDI filtering process by implementing a new fea-ture: instead of only having one brightness value in eachcell, we give it a brightness value and an evolution parame-ter, so that ZDI brightness maps are allowed to evolve withtime to better fit time-series of LSD profiles with variability.Thus we reconstruct two maps for the brightness: the bright-ness at time 0 and the map of the evolution parameter. Wechoose, for now, a simple model where the logarithmic rel-ative brightness of each cell k is allowed to evolve linearilywith time: log Q k ( t ) = log Q k (0) + m k t , (3)where Q k ( t ) is the local surface brightness and m k is the evolu-tion parameter. Applying this new method to the 2015-2016extended data set, we manage to fit the whole data set downto a χ of 1 where classical ZDI, even with differential ro-tation, could not reach lower than χ =2.5 (see Section 4.2).Maps associated to this reconstruction are shown in Fig. 11,and derived RVs are plotted in Fig. 12 and 13, to be com-pared with RVs derived from classical ZDI maps. The rms ofthe filtered RVs here, 0.194 km s − , does not decrease com-pared to when using classical ZDI, which means our modelis still too simple and cannot fully account for the observedvariability. However, Fig. 13 shows that global trends in thetemporal evolution of the RV curve are well-reproduced bythis new ZDI model, such as the jitter maximum movingfrom phase 0.37 to 0.32, or the local minimum at phase 0.54in 2015 Dec moving to 0.50 in 2016 Jan. MNRAS , 1–28 (2018) L. Yu et al. (a) 2009 Jan(b) 2011 Jan(c) 2013 Dec
Figure 9.
Periodograms of the raw RVs (top), of the ZDI-filteredRVs (bottom, red full line) and of the GP-filtered RVs (bottom,blue dashed line), for observation epochs 2009 Jan (a), 2011 Jan(b), 2013 Dec (c), 2015 Dec (d) and 2016 Jan (e). False-alarmprobability levels of 1% and 0.1% are represented as horizontalcyan dashed lines, and P rot and its first two harmonics as verticalcyan dashed lines (continuing next page). This paper reports the analysis of an extended spectropo-larimetric data set on the ∼ (d) 2015 Dec(e) 2016 Jan Figure 9. (Continued from previous page). last three of which were observed as part of the MaTYSSEobservation programme. Contemporaneous photometric ob-servations from the CrAO and from the WASP programmecomplemented the study. ESPaDOnS, NARVAL and CrAOobservations are documented in Appendix A.V410 Tau is composed of an inner close binary(V410 Tau A-B) around which orbits a third component (CGhez et al. 1997), with V410 Tau A being much brighterthan the other two in the optical domain, and thus the starthat our data inform. The stellar parameters derived in thiswork are summed up in Table 1: at ∼ . ± .
15 M (cid:12) and . ± . (cid:12) wTTS. Applying LSD then ZDI on our data set, we estimated the v sin i and inclination of V410 Tau at . ± . km s − and ± ◦ respectively. Considering the well-determined rota-tion period of . ± . (Stelzer et al. 2003) andthe minimal observed visible magnitude of 10.52 (Grankinet al. 2008), this implies a relatively high level ( ∼ MNRAS000
15 M (cid:12) and . ± . (cid:12) wTTS. Applying LSD then ZDI on our data set, we estimated the v sin i and inclination of V410 Tau at . ± . km s − and ± ◦ respectively. Considering the well-determined rota-tion period of . ± . (Stelzer et al. 2003) andthe minimal observed visible magnitude of 10.52 (Grankinet al. 2008), this implies a relatively high level ( ∼ MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau Figure 10.
Periodograms of the raw RVs (top), of the RVs filtered from ZDI-modelled activity (middle) and of the RVs filtered fromGP-modelled activity (bottom, blue dashed line), for observation epochs 2009 Jan (a), 2011 Jan (b), 2013 Dec (c), 2015 Dec (d) and2016 Jan (e). Periodograms of the whole data set raw RVs (top), RVs filtered from ZDI-modelled activity (middle) and RVs filtered fromGP-modelled activity (bottom). False-alarm probability levels of 1% and 0.1% are represented as horizontal cyan dashed lines, and P rot and its first two harmonics as vertical cyan dashed lines. Figure 11.
Brightness map and evolution rate reconstructed byZDI on data set Dec 2015-Jan 2016. Pole-on view with the equatorbeing represented as a full line, and 60 ◦ , 30 ◦ , and -30 ◦ latitudeparallels as dashed lines. Cool spots are colored in brown andbright plages in blue, and ticks around the star mark the spec-tropolarimetric observations. around 7 percent at all epochs. Since ZDI mostly recov-ers large non-axisymmetric features and misses small onesevenly distributed over the star, the spot and plage cover-age is underestimated, which makes this result compatiblewith the spot coverage obtained from the aforementionedV magnitude measurements. We note that V410 Tau beingheavily spotted makes it difficult to pinpoint its age. We fita 2-temperature model (photosphere at 4500 K and fixed- temperature spots with a varying filling factor) into our B-Vand V magnitude data, and found an optimal spot temper-ature of around 3750 K, which implies a contrast of ∼
750 Kbetween dark spots and the photosphere (see Fig. B7). Thiscontrast is slightly lower than the one retrieved for the 2 MyrwTTS LkCa 4 in Gully-Santiago et al. (2017). V410 Tau al-ways presents a high concentration of dark spots around thepole, and several big patches of dark spots on the equator.V410 Tau has a relatively strong large-scale magneticfield, with an average surface intensity that is roughly con-stant over the years at ± G. Its radial field reacheslocal values beyond -1 kG and +1 kG in several epochs. Thebrightness and magnetic surface maps both present somevariability from epoch to epoch (Fig. 3, Table 2), whichpoints to a dynamo-generated magnetic field rather thana fossil one. The magnetic energy is, at all epochs, equallydistributed between the poloidal and toroidal componentsof the field, with the poloidal component being rather non-dipolar and non-axisymmetric, whereas the toroidal com-ponent is mostly dipolar and axisymmetric. The poloidaldipole, tilted towards a phase that stays within . ± . dur-ing the whole survey, but at an angle varying between 20 ◦ and 55 ◦ depending on the epoch, sees its intensity increasealmost monotonously from 165 G to 458 G over 8 years, andthe dipolar contribution to the poloidal field also increasesfrom ∼
25% to ∼
40% (see Table 2).
MNRAS , 1–28 (2018) L. Yu et al. -5.00.05.0 R a w R V s ( k m . s ) F il t e r e d R V s ( k m . s ) Figure 12.
Comparison between the GP model, the new ZDI model and the classical ZDI models for V410 Tau RVs in season 2015b-2016a. Rotation cycles are offset to concur with Table A1. Top: raw RVs (black dots) with σ -error bars, GP model (purple full line),new ZDI model (cyan full line) and classical ZDI models for both observation epochs 2015 Dec and 2016 Jan (red full lines). Bottom:RVs filtered from the GP model (purple dots), from the new ZDI model (cyan dots) or from the classical ZDI models (red dots). Therms of the filtered RVs with GP, new ZDI and classical ZDI are respectively 0.065, 0.194 and 0.193 km s − . R a w R V s ( k m . s − ) Figure 13.
Raw RVs of V410 Tau in the 2015b-2016a season,between cycles 1349 and 1381 as referenced in Table A1, plottedagainst stellar rotation phase. The GPR and new ZDI modelsare represented by full lines colored in gradients, from earliest tolatest cycle, respectively pink to purple and green to blue, whilethe classical ZDI models for 2015 Dec and 2016 Jan are plottedin orange and red respectively. Observations are plotted as dotswith 1 σ -error bars, orange for 2015 Dec and 2016 Jan. The toroidal component, which displays a constant ori-entation throughout our data set, is unusually strong com-pared to other fully convective rapidly-rotating stars (e.g.V830 Tau is 90 percent poloidal, see Donati et al. 2017).A similarly strong toroidal field was observed on one otherMaTYSSE target, LkCa 4 (Donati et al. 2014). The ori-gin of this strong toroidal field is still unclear: could it bemaintained by an α dynamo, like in the simulations of low-Rossby fully convective stars by Yadav et al. (2015)? Theremnants of a subsurfacic radial shear between internal lay-ers accelerating due to contraction, and disc-braked outerlayers? Or would the even earlier toroidal energy, from rightafter the collapse of the second Larson core (as found in thesimulations of Vaytet et al. 2018), somehow not have entirelysubsided yet? Would the early dissipation of the disc, a com-mon factor between LkCa 4 and V410 Tau, have somethingto do with this?At ∼ Latitude θ ( ° ) Ω (r ad / d ) Ω ( θ ) Ω s,I ( θ s,I ) Ω s,V ( θ s,V )RV raw (GPR)H- α (GPR)B L (GPR) Figure 14.
Differential rotation curve of V410 Tau(black full line) with 1 σ uncertainty in gray, with Ω eq = . ± . rad d − and d Ω = . ± . rad d − .The stellar rotation rate chosen to phase our data is representedas a dashed horizontal line. The rotation rates derived from theRVs (red), from the H α equivalent widths (magenta) and fromthe longitudinal field measurements (cyan) are positioned onthe differential rotation curve as triangles with 1 σ error bars,thus yielding the barycentric latitude of the features determiningthe period. The dots represent couples { − θ s , Ω s } derived inour epoch-wise differential rotation measurements, those comingfrom Stokes I / Stokes V data being plotted in blue / redrespectively. wTTSs (Kraus et al. 2012, Fig. 3). Assuming that, when thedisc was present, V410 Tau was magnetically locked to it ata rotation period of ∼ ∼ ∼ R (cid:12) when the disc dissipated, to match theangular momentum that we measure today (Bouvier 2007).According to the Siess models (Siess et al. 2000), this cor-responds to an age of ∼ ∼ R (cid:12) ,V410 Tau would have needed a magnetic dipole barely above G to maintain the assumed magnetospheric cavity, even
MNRAS000
MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau with an accretion rate of ∼ − M (cid:12) yr − just before disc dissi-pation. That value is compatible with the 200-400 G dipolewe measure on the ∼ R (cid:12) star today. Kraus et al. 2012(Fig. 1) shows a correlation between the presence of a closecompanion and the early depletion of the accretion disc,which indicates that V410 Tau B, observed at a projectedseparation of . ± . au (Ghez et al. 1995), could havebeen responsible for the early depletion of the disc.In our H α dynamic spectra, we observe a conspicuousabsorption feature in the second part of the 2009 Jan runaround phase 0.95 (Fig. C1), that could be the signatureof a prominence (see e.g. Collier Cameron & Woods 1992).Fitting a sine curve in the absorption feature yields an ampli-tude of ∼ v sin i , corresponding to a prominence ∼ R (cid:63) awayfrom the center of V410 Tau, confirming that the promi-nence is located close to the corotation radius. Plotting the3D potential field extrapolation of the reconstructed sur-face radial field for 2009 Jan, at phases 0.95, 0.20, 0.45 and0.70, we observe the presence of closed field lines reaching ∼ R (cid:63) at phase 0.95 (Fig. 15), which may be able to supportthe observed prominence. We also observe similar absorp-tion features in 2009 Jan around phase 0.8 and in 2011 Janaround phase 0.35, but they are less well-covered by our ob-servations. We however found corresponding field lines atthe right phase for each (see Fig. 15 for 2009 Jan).We also constrained the differential rotation ofV410 Tau with ZDI: we obtained six values for theequatorial rotation rate Ω eq and for the pole-to-equatorrotation rate difference d Ω , by using separately ourStokes I and Stokes V LSD profiles from each of thethree data sets 2008b+2009a, 2013b and 2015b+2016a.Overall mean values are Ω eq = . ± . rad d − and d Ω = . ± . rad d − . The differential rotation ofV410 Tau is thus relatively weak, with a pole-to-equator ro-tation rate difference 5.6 times smaller than that of the Sun,and a lap time of ± d. Compared to other wTTSs pre-viously analyzed within the MaTYSSE programme, the dif-ferential rotation of V410 Tau is similar to that of V830 Tau(Donati et al. 2017) but much smaller than that of TAP 26,which is almost of solar level, consistent with the fact thatTAP 26 is no longer fully convective and has developped aradiative core (of size 0.6 R (cid:63) , Yu et al. 2017). Even with differential rotation, it is impossible for our cur-rent version of ZDI to model data sets spanning a fewmonths down to noise level, which shows that the surfaceof V410 Tau undergoes significant instrinsic variability, cor-roborating the hypothesis of a dynamo-generated field. Thevariations of the photosphere and of the surface magneticfield over the years might be the manifestation of a mag-netic cycle, whose existence has been suggested by previousstudies (Stelzer et al. 2003; Hamb´alek et al. 2019). No clearchange in d Ω is observed while the dipole grows in intensity(Table 3), which could indicate a time lag in the dynamointeraction between the magnetic field and the rotation pro-file. The bulk RV of V410 Tau exhibits a drift throughoutour 8-year campaign, from . ± . km s − in 2008b-2009ato . ± . km s − in 2015b-2016a. One explanation couldbe a variation in the suppression of convective blueshift in re- gions of strong magnetic field (Haywood et al. 2016; Meunieret al. 2010), which could further support a secular evolutionof the magnetic topology. It could also be a manifestation ofthe binary motion of V410 Tau A-B. The central binary ofV410 Tau was observed twice, with a sky-projected separa-tion of . ± . au in 1991 Oct and . ± . au in 1994 Oct( . ± . arcsec and . ± . arcsec resp. in Ghez et al.1995), and a mass ratio of . ± . (Kraus et al. 2011).Assuming a mass ratio of 0.2 and an edge-on circular orbit,we find that an orbit of the primary star of radius 6.0 au,i.e. binary separation 36.0 au and period 166 a, fits our bulkRVs and the sky-projected separations at a level of σ (seeFig. 16). No binary motion was detected in the 2013 to 2017astrometry measurements of Galli et al. (2018), which isconsistent with our model where the sky-projected velocityvaries by only 0.13 mas a − over these 3.5 years (roughly a50th of the orbital period). More measurements would en-able to estimate the eccentricity and potentially fit the sky-projected separations to a better level, as well as to decidewhether the binary motion can explain the RV drift observedin this study.The rotation period derived from our V magnitude mea-surements, in each observing season, also displays long-termvariations. Placing the periods found from the photometricdata on a period-latitude diagram representing the modeleddifferential rotation (Fig. B5), we observe that the latitudescorresponding to the successive periods tend to increase from0 in 2008 to ∼ ◦ in 2016. We note that this trend is observedwith both the periods derived from sine fits to the photomet-ric data and those derived from GPR (see B). This impliesthat the largest features, ie those with the biggest impact onthe photometric curve, underwent a poleward migration ,reminiscent of the Solar butterfly diagram (albeit reversed).This would suggest that the dynamo wave, if cyclic, has aperiod of at least 8 a and likely much longer (16 a if our datacovers only one half of a full cycle). Previous studies usingdifferent data have suggested the existence of an activity cy-cle on V410 Tau, with periods of 5.4 a and 15 a respectively(Stelzer et al. 2003; Hamb´alek et al. 2019). We further notethat our differential rotation measurements confirm that thebarycenter of surface features migrates to higher latitudesover time (see Fig 6).Applying GPR with MCMC parameter exploration toour H α equivalent widths and longitudinal magnitude fieldmeasurements ( B (cid:96) , first-order moment of the Stokes V LSDprofiles, Donati & Brown 1997), we also found rotationperiods from which we derive mean barycentric latitudesof features constraining the modeling of each quantity (seeFig. 14). The period found from H α is equal within errorbars to the one derived in Stelzer et al. 2003 from pho-tometry, whereas the period found from B (cid:96) seems tied toequatorial features. It is worth mentioning that we also findlong decay times for these two activity proxies: + − dand + − d respectively, which suggests, with the cautionneeded with such high error bars, that the H α and B (cid:96) mod-ulations are particularly sensitive to large, long-lasting fea-tures. The phase plots are displayed in Appendix C. We modeled the activity RV jitter from line profiles syn-thetized from our ZDI maps, and filtered it out from the RV
MNRAS , 1–28 (2018) L. Yu et al. (a) Phase 0.95 (b) Phase 0.15 (c) Phase 0.35 (d) Phase 0.55 (e) Phase 0.75(f) Phase 0.35 (g) Phase 0.55 (h) Phase 0.75 (i) Phase 0.95 (j) Phase 0.15
Figure 15.
Potential field extrapolations of the ZDI-reconstructed surface radial field, as seen by an Earth-based observer, for observationepochs 2009 Jan (top) and 2011 Jan (bottom) at different phases. Open/closed field lines are shown in orange/black respectively,and colours at the stellar surface depict the local value of the radial field (in G, as shown in the left-hand panels of Fig. 3). Thesource surface at which field lines open is set to 2.18 R (cid:63) . The field lines that would carry the potential observed prominences (phase0.95 and 0.8 in 2009, phase 0.35 in 2011) are colored in magenta. Animated versions with the star rotating are available at http://userpages.irap.omp.eu/~lyu/jan09a.gif and http://userpages.irap.omp.eu/~lyu/jan11n.gif . Table 6.
Various evolution time scales.Quantity Time scale (d)RV decay time + − V mag decay time + − H α decay time + − B (cid:96) decay time + − Differential rotation lap time ± curve of V410 Tau. From a rms of 1.802 km s − in the rawRVs, we get residuals with a rms of 0.167 km s − . We also ap-plied GPR to our raw RVs and found a jitter of periodicity . ± . d and decay time + − d, with residualsof rms 0.076 km s − . The period derived from the GPR onour raw RVs is shorter than the period we used to phase ourdata, and corresponds to a latitude of 5.5 ◦ . This period ismuch closer to the period derived with GPR from B (cid:96) thanto the period derived from H α , showing that in this case, B (cid:96) is a better activity proxy than H α (for a more system-atic study of the correlation of B (cid:96) with stellar activity, seeH´ebrard et al. 2016). The decay time associated to RVs ismuch shorter than the differential rotation lap time and thedecay times of the V magnitude, H α and B (cid:96) (see Table 6),which suggests that RVs are more sensitive to small-scaleshort-lived features while the photometry, H α and B (cid:96) aremore sensitive to large-scale long-lasting features. Through both processes, the residual RVs present nosignificant periodicity which would betray the presence ofa potential planet. To estimate the planet mass detectionthreshold, GPR-MCMC was run on simulated data sets,composed of a base activity jitter (our GP model from Sec-tion 5), and a circular planet signature, plus a white noiseof level 0.081 km s − . Various planet separations and masseswere tested, and for each case, GPR-MCMC was run severaltimes with different randomization seeds, to mitigate statis-tical bias. For every randomization seed, GPR-MCMC wasrun with a model including a planet and a model includingno planet, and the difference of logarithmic marginal likeli-hood between them (hereafter ∆ L ) was computed. Finally,the detection threshold was set at ∆ L = and the minimumdetectable mass at each separation was interpolated fromthe mass/ ∆ L curve. Fig. 17 shows the planet mass detectionthreshold as a function of planet-star separation: we thus ob-tained a detectability threshold of ∼ M Jup for a < . au and ∼ M Jup for a = . au. The figure also shows the param-eters of V830 Tau b and TAP 26 b, showing that we wouldlikely have detected a planet like TAP 26 b but not one likeV830 Tau b. Planets beyond a = . au are difficult to de-tect due to the temporal coverage of our data, that neverexceeds 19 d at any given epoch. The early depletion of thedisc may have prevented the formation and/or the migrationof giant exoplanets. Kraus et al. 2016 outlines a correlationbetween the presence of a close companion and a lack ofplanets, in a sample of binary stars with mass ratios q > . ,which could support the hypothesis that V410 Tau B, al-though having a slightly lower mass ratio ( q = . ± . , MNRAS000
Various evolution time scales.Quantity Time scale (d)RV decay time + − V mag decay time + − H α decay time + − B (cid:96) decay time + − Differential rotation lap time ± curve of V410 Tau. From a rms of 1.802 km s − in the rawRVs, we get residuals with a rms of 0.167 km s − . We also ap-plied GPR to our raw RVs and found a jitter of periodicity . ± . d and decay time + − d, with residualsof rms 0.076 km s − . The period derived from the GPR onour raw RVs is shorter than the period we used to phase ourdata, and corresponds to a latitude of 5.5 ◦ . This period ismuch closer to the period derived with GPR from B (cid:96) thanto the period derived from H α , showing that in this case, B (cid:96) is a better activity proxy than H α (for a more system-atic study of the correlation of B (cid:96) with stellar activity, seeH´ebrard et al. 2016). The decay time associated to RVs ismuch shorter than the differential rotation lap time and thedecay times of the V magnitude, H α and B (cid:96) (see Table 6),which suggests that RVs are more sensitive to small-scaleshort-lived features while the photometry, H α and B (cid:96) aremore sensitive to large-scale long-lasting features. Through both processes, the residual RVs present nosignificant periodicity which would betray the presence ofa potential planet. To estimate the planet mass detectionthreshold, GPR-MCMC was run on simulated data sets,composed of a base activity jitter (our GP model from Sec-tion 5), and a circular planet signature, plus a white noiseof level 0.081 km s − . Various planet separations and masseswere tested, and for each case, GPR-MCMC was run severaltimes with different randomization seeds, to mitigate statis-tical bias. For every randomization seed, GPR-MCMC wasrun with a model including a planet and a model includingno planet, and the difference of logarithmic marginal likeli-hood between them (hereafter ∆ L ) was computed. Finally,the detection threshold was set at ∆ L = and the minimumdetectable mass at each separation was interpolated fromthe mass/ ∆ L curve. Fig. 17 shows the planet mass detectionthreshold as a function of planet-star separation: we thus ob-tained a detectability threshold of ∼ M Jup for a < . au and ∼ M Jup for a = . au. The figure also shows the param-eters of V830 Tau b and TAP 26 b, showing that we wouldlikely have detected a planet like TAP 26 b but not one likeV830 Tau b. Planets beyond a = . au are difficult to de-tect due to the temporal coverage of our data, that neverexceeds 19 d at any given epoch. The early depletion of thedisc may have prevented the formation and/or the migrationof giant exoplanets. Kraus et al. 2016 outlines a correlationbetween the presence of a close companion and a lack ofplanets, in a sample of binary stars with mass ratios q > . ,which could support the hypothesis that V410 Tau B, al-though having a slightly lower mass ratio ( q = . ± . , MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau −20 0 20x (au)−30−20−100102030 z ( a u ) V z ( k m / s ) P r o j e c t e d s e p ( a u ) Figure 16.
Circular model for the binary motion of V410 Tau Aand V410 Tau B: edge-on orbit, separation 36.0 au, period 166 aand systemic radial velocity 16.06 km s − . Top : top-view of themodel orbit, with the z-axis parallel to the line-of-sight, wherethe positions of V410 Tau A and B according to the model aremarked by red and black stars at the times of the separation mea-surements and of our spectropolarimetric seasons (2008b-2009a,2011a, 2013b and 2015b-2016a) respectively.
Middle : RV bulk ofV410 Tau A with time, as measured by us in black dots with σ error bars and as derived from the model orbit in blue. Thepredicted RV bulk at the times of the separation measurements arerepresented by red stars. Bottom : Sky-projected binary separa-tion as a function of time, as measured by Ghez et al. 1995 in reddots with σ error bars, and as derived from the model orbit inblue. The predicted sky-projected separations at the dates of ourobserving seasons are marked in black stars. Kraus et al. 2011), played a role in the early disc dissipa-tion, which in turn prevented the formation of a hot Jupiter.In terms of methodology, GPR fits the data down to asignificantly lower χ than ZDI because it is capable of ac-counting for most of the mid-term variability, contrarily toZDI, which for now only integrates differential rotation anda simplistic description of intrinsic variability. Small stru-tures evolve on time scales of ∼ few weeks, so we need tobe able to model their temporal evolution in a more elabo-rate way to be able to match the capability of GPR to fittime-variable RV curves. Self-consistent methods that com-bine the physical faithfulness of ZDI and the flexibility ofGPR will be developped in the near future and applied tomore MaTYSSE data, as well as to data from the SPIRou M P s i n i ( M J u p ) Figure 17.
Detectability threshold in terms of M sin i for plan-ets at various a , with the RV filtering technique involving GPR.V830 Tau b is plotted in red (parameters from Donati et al. 2017)and TAP 26 b in blue (parameters from Yu et al. 2017). (Spectropolarimetre InfraRouge) Legacy Survey (SLS). Fi-nally, observing V410 Tau and other wTTSs with SPIRouwill yield spectra in the near infrared, where we expect asmaller jitter than in the optical bandwidth, and will of-fer an opportunity to benchmark our activity jitter filteringtechnique performances. ACKNOWLEDGEMENTS
This paper is based on observations obtained at the CFHT,operated by the National Research Council of Canada(CNRC), the Institut National des Sciences de l’Univers(INSU) of the Centre National de la Recherche Scientifique(CNRS) of France and the University of Hawaii, and atthe TBL, operated by Observatoire Midi-Pyr´en´ees and byINSU / CNRS. We thank the QSO teams of CFHT andTBL for their great work and efforts at collecting the high-quality MaTYSSE data presented here, without which thisstudy would not have been possible. MaTYSSE is an in-ternational collaborative research programme involving ex-perts from more than 10 different countries (France, Canada,Brazil, Taiwan, UK, Russia, Chile, USA, Ireland, Switzer-land, Portugal, China and Italy).JFD also warmly thanks the IDEX initiative at Univer-sit´e F´ed´erale Toulouse Midi-Pyr´en´ees (UFTMiP) for fund-ing the STEPS collaboration program between IRAP/OMPand ESO. JFD acknowledges funding from the EuropeanResearch Council (ERC) under the H2020 research & inno-vation programme (grant agreements
Mat-plotlib python module (Hunter 2007).
MNRAS , 1–28 (2018) L. Yu et al.
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MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau APPENDIX A: OBSERVATIONS
This appendix informs all the observations, both spectropo-larimetric (Table A1) and photometric (Table A2), that weused in this study, excluding the WASP data. The spectropo-larimetric data are spread over 8 runs (2008 Oct, 2008 Dec,2009 Jan, 2011 Jan, 2013 Nov, 2013 Dec, 2015 Dec and 2016Jan) and the photometric data are spread over 9 seasons:08b+09a (short for 2008b + 2009a; all the other seasons fol-low the same naming convention), 09b+10a, 10b, 11b+12a,12b+13a, 13b+14a, 14b, 15b+16a and 16b+17a.The instruments with which the spectropolarimetricdata was taken, ESPaDOnS and NARVAL, are twin spec-tropolarimeters and cover a 370 to 1000 nm wavelength do-main, with respective resolving powers of 65 000 (i.e. re-solved velocity element of 4.6 km s − ) and 60 000 (resolvedvelocity element of 5.0 km s − ). Each polarization exposuresequence consists of four subexposures of 600 s each, takenin different polarimeter configurations to allow the removalof all spurious polarization signatures at first order (Donatiet al. 1997), except three observations comprised of only twosubexposures of 600 s (2008 Dec 05 at phase 0.827, 2009 Jan05 at phase 0.602, and 2013 Nov 07 at phase 0.541), andthree observations comprised of four subexposures of 800 s(2009 Jan 10 at phases 0.229, 0.251 and 0.272).The telescopes used for photometry at CrAO are AZT-11, a 1.25 m telescope with a five-channel photometer-polarimeter, and T60Sim, a 0.60 m telescope with a four-channel photometer. MNRAS , 1–28 (2018) L. Yu et al.
Table A1.
Information on the V410 Tau spectropolarimetric data. The first three columns contain the time at which the observationswere taken: Coordinated Universal Time in the 1st column, Barycentric Julian Date in the 2nd and corresponding rotational cycle ofV410 Tau, c , in the 3rd, as is defined in equation 1. The 4th column indicates the instrument used for the observation (E: ESPaDOnS,N: NARVAL) and column 5 the spectrum S/N. Column 6 indicates rejected spectra or the presence of moon pollution ( a : not used inZDI, b : not used in GPR, c : not used for period retrieval from H α EW). Columns 7 and 8 contain the S/N in the Stokes I and Stokes V LSD profiles respectively. Columns 9 to 12 show the raw RVs, the RVs filtered with ZDI, the RVs filtered with GPR and the 1 σ RV errorbar respectively, column 13 lists the equivalent width of the H α line (with a typical error bar of 10 km s − ) and column 14 informs thelongitudinal projection of the magnetic field integrated over the visible surface (with a typical error bar of 50 G). UTC BJD Cycle Instr. S/N Comment S/N I S/N V RV raw RV filt / ZDI RV filt / GP σ RV EWH α B long − ) (km s − ) (km s − ) (km s − ) (km s − ) (G)15 11:42:34 54.992 0.552 E 228 Isolated a a a a a , b i D flare a , b a , b , c a , b a , b (a) Observations of the late 2008 and early 2009 runs. MNRAS000
Information on the V410 Tau spectropolarimetric data. The first three columns contain the time at which the observationswere taken: Coordinated Universal Time in the 1st column, Barycentric Julian Date in the 2nd and corresponding rotational cycle ofV410 Tau, c , in the 3rd, as is defined in equation 1. The 4th column indicates the instrument used for the observation (E: ESPaDOnS,N: NARVAL) and column 5 the spectrum S/N. Column 6 indicates rejected spectra or the presence of moon pollution ( a : not used inZDI, b : not used in GPR, c : not used for period retrieval from H α EW). Columns 7 and 8 contain the S/N in the Stokes I and Stokes V LSD profiles respectively. Columns 9 to 12 show the raw RVs, the RVs filtered with ZDI, the RVs filtered with GPR and the 1 σ RV errorbar respectively, column 13 lists the equivalent width of the H α line (with a typical error bar of 10 km s − ) and column 14 informs thelongitudinal projection of the magnetic field integrated over the visible surface (with a typical error bar of 50 G). UTC BJD Cycle Instr. S/N Comment S/N I S/N V RV raw RV filt / ZDI RV filt / GP σ RV EWH α B long − ) (km s − ) (km s − ) (km s − ) (km s − ) (G)15 11:42:34 54.992 0.552 E 228 Isolated a a a a a , b i D flare a , b a , b , c a , b a , b (a) Observations of the late 2008 and early 2009 runs. MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau Table A1. (Continued from previous page).
UTC BJD Cycle Instr. S/N Comment S/N I S/N V RV raw RV filt / ZDI RV filt / GP σ RV EWH α B long − ) (km s − ) (km s − ) (km s − ) (km s − ) (G)14 19:01:45 76.297 0.291 N 145 2789 4035 -2.162 -0.114 -0.117 0.069 -15.778 -4314 19:46:58 76.328 0.308 N 152 2757 4175 -2.052 0.363 0.172 0.070 -24.729 -14714 20:32:12 76.360 0.324 N 155 2789 4172 -2.511 0.181 0.011 0.069 -28.375 -4114 21:17:25 76.391 0.341 N 150 2765 4113 -2.923 -0.070 -0.071 0.069 -25.398 -10315 19:09:27 77.302 0.828 N 138 2710 3877 -2.672 -0.127 -0.023 0.071 6.283 -3015 19:54:41 77.334 0.845 N 144 Moon 2739 3915 -2.552 -0.043 -0.002 0.070 2.985 -2415 20:39:55 77.365 0.861 N 133 Moon 2693 3584 -2.287 0.085 0.097 0.071 0.295 -6115 21:25:07 77.396 0.878 N 143 2761 3909 -2.226 -0.084 -0.078 0.069 -3.177 -4816 19:27:12 78.314 1.369 N 146 Moon 2710 3867 -3.130 -0.327 -0.069 0.071 0.040 -2916 20:12:25 78.346 1.385 N 148 Moon 2735 4044 -2.706 -0.140 0.085 0.070 3.591 3016 20:57:37 78.377 1.402 N 151 Moon 2802 4122 -2.236 -0.063 -0.018 0.069 10.411 -3917 18:20:45 79.268 1.878 N 135 2706 3739 -2.165 -0.028 -0.033 0.071 -0.629 -3317 19:05:56 79.299 1.895 N 130 2671 3441 -1.782 0.041 0.030 0.071 -1.757 -2517 19:51:09 79.331 1.912 N 132 2698 3503 -1.406 0.039 0.009 0.071 -6.538 7020 18:49:53 82.288 3.491 N 86 2194 1903 1.561 0.186 0.003 0.084 19.874 12620 19:35:05 82.320 3.508 N 82 2173 1817 1.979 -0.008 -0.003 0.085 27.037 1822 21:18:44 84.391 4.615 N 93 Moon 2249 2087 3.210 0.100 0.025 0.083 34.575 1522 22:03:58 84.423 4.632 N 106 Moon 2484 2539 2.815 0.024 0.014 0.076 29.770 -7023 21:53:42 85.415 5.162 N 147 2714 3773 1.397 0.027 0.008 0.070 6.158 -4024 18:46:11 86.285 5.627 N 140 Moon 2779 3766 2.885 0.000 -0.023 0.069 37.130 -104 (b) Observations of the early 2011 run. UTC BJD Cycle Instr. S/N Comment S/N I S/N V RV raw RV filt / ZDI RV filt / GP σ RV EWH α B long − ) (km s − ) (km s − ) (km s − ) (km s − ) (G)07 22:44:14 4.453 0.528 N 118 Isolated a a i D flare a , b i D flare a , b (c) Observations of the late 2013 run. UTC BJD Cycle Instr. S/N Comment S/N I S/N V RV raw RV filt / ZDI RV filt / GP σ RV EWH α B long − ) (km s − ) (km s − ) (km s − ) (km s − ) (G)01 22:44:05 58.453 0.313 N 121 1776 2974 2.464 0.058 0.050 0.102 -6.540 -6002 02:00:59 58.590 0.386 N 109 1690 2561 3.011 0.461 0.122 0.107 1.045 -23902 22:07:10 59.427 0.833 N 101 1699 2360 -1.825 -0.080 0.008 0.107 0.196 -26203 01:30:49 59.569 0.909 N 114 1807 2794 -2.520 0.226 0.066 0.101 -5.859 -10803 21:39:29 60.408 1.357 N 109 1743 2543 3.040 0.203 -0.115 0.104 -10.697 -11204 02:23:31 60.605 1.462 N 119 1786 2804 0.070 -0.168 -0.009 0.102 19.800 -16506 22:53:07 63.459 2.987 N 152 1875 3764 -2.324 -0.086 -0.002 0.097 9.991 -1707 01:15:19 63.558 3.040 N 152 1886 3853 -1.401 0.091 0.004 0.097 -0.787 -6807 23:02:53 64.466 3.525 N 94 1633 2122 -0.690 0.173 0.093 0.110 30.595 -7008 01:31:49 64.569 3.580 N 142 1844 3566 -0.195 -0.186 0.008 0.099 34.071 -25109 21:05:04 66.384 4.549 N 103 1713 2409 -0.775 -0.116 -0.083 0.105 37.568 -12811 22:05:07 68.426 5.640 N 129 2147 3344 1.638 -0.028 0.028 0.086 71.742 -18712 00:56:44 68.545 5.704 N 136 2124 3424 2.077 0.065 -0.159 0.087 28.515 -6512 22:13:39 69.432 6.177 N 113 2077 2927 -0.153 0.050 0.015 0.089 -29.140 -5813 01:11:33 69.555 6.243 N 135 2144 3458 0.521 -0.283 -0.032 0.086 -16.825 -2313 22:12:08 70.431 6.711 N 106 1962 2553 2.337 0.433 0.186 0.093 9.573 -9314 01:29:54 70.568 6.784 N 134 2095 3343 -0.445 -0.204 -0.035 0.088 9.052 -14518 01:04:45 74.550 8.912 N 124 2036 3124 -2.902 -0.225 -0.046 0.090 13.281 -14418 21:36:31 75.406 9.369 N 138 2087 3452 2.918 0.224 -0.000 0.089 -21.444 -17119 01:09:10 75.553 9.447 N 124 1972 3016 0.091 -0.426 -0.039 0.093 17.394 -18919 20:09:16 76.345 9.870 N 106 1918 2497 -2.441 -0.020 -0.023 0.095 21.741 -162016 Jan 2457400+ 1376+ (km s − ) (km s − ) (km s − )20 22:42:30 8.450 0.021 N 153 He i D flare a , b i D flare a , b (d) Observations of the late 2015 and early 2016 runs.MNRAS , 1–28 (2018) L . Y u e t a l . Table A2.
Information on the V410 Tau photometric data. For each table, the first and second columns indicate the time at which the observations were taken, in UTC and BJDformat respectively. The third column contains the measured visible magnitude, then columns 4 to 6 list color indexes B-V, V-R J and V-I J provided in the Johnson UBVRI system, andcolumn 7 indicates the name of the telescope used for the observation.Date HJD (2454000+) V (mag) B-V V-R J V-I J Telescope | Date HJD (2455000+) V (mag) B-V V-R J V-I J Telescope03-Aug-2008 682.5282 10.878 1.182 1.075 - AZT-11 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | M N R A S , ( ) a g . fi e l d , a c t i v i t y & c o m pa n i o n s o f V T a u Table A2. (Continued from previous page).Date HJD (2455000+) V (mag) B-V V-R J V-I J Telescope | Date HJD (2455000+) V (mag) B-V V-R J V-I J Telescope09-Oct-2009 114.4547 10.734 1.136 1.025 - AZT-11 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | (b) Photometric measurements of the second half of the set 09b+10a (left) and of the set 10b (right). M N R A S , ( ) L . Y u e t a l . Table A2. (Continued from previous page).Date HJD (2455000+) V (mag) B-V V-R J V-I J Telescope | Date HJD (2455000+) V (mag) B-V V-R J V-I J Telescope29-Jul-2011 772.5187 10.801 1.154 1.049 1.767 AZT-11 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | (c) Photometric measurements of the set 11b+12a. M N R A S , ( ) a g . fi e l d , a c t i v i t y & c o m pa n i o n s o f V T a u Table A2. (Continued from previous page).Date HJD (2456000+) V (mag) B-V V-R J V-I J Telescope | Date HJD (2456000+) V (mag) B-V V-R J V-I J Telescope13-Aug-2012 153.4989 10.801 1.166 1.036 1.762 AZT-11 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | M N R A S , ( ) L . Y u e t a l . Table A2. (Continued from previous page).Date HJD (2456000+) V (mag) B-V V-R J V-I J Telescope | Date HJD (2456000+) V (mag) B-V V-R J V-I J Telescope15-Aug-2013 520.5078 10.786 1.155 1.041 1.785 AZT-11 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | (e) Photometric measurements of the set 13b+14a (left) and of the set 14b (right). M N R A S , ( ) a g . fi e l d , a c t i v i t y & c o m pa n i o n s o f V T a u Table A2. (Continued from previous page).Date HJD (2457000+) V (mag) B-V V-R J V-I J Telescope | Date HJD (2457000+) V (mag) B-V V-R J V-I J Telescope15-Aug-2015 250.5211 10.864 1.166 1.109 1.779 AZT-11 | | | | | | | | | | | | | | | (f) Photometric measurements of the set 15b+16a (left) and of the set 16b+17a (right). M N R A S , ( ) L. Yu et al.
APPENDIX B: PHOTOMETRY ANALYSIS
From our photometric data, we retrieved the stellar rotationperiod at each epoch and derived the photosphere contrast.To retrieve the stellar rotation period, we applied twotypes of models to our V magnitude curves: a periodic fitinvolving the fundamental frequency and the first two har-monics to each of the 9 datasets individually (as well asa periodic fit involving the fundamental frequency and thefirst four harmonics to the whole data set), and GPR (seeSection 5). Since the data sets 15b+16a and the 16b+17aare particularly small (15 and 13 points respectively) andconsecutive, we grouped them together for the GPR.The results of the sine fits are listed in Table B1, andplotted in Figures B1 and B2. All observation epochs yield amodulation period within 1 σ of the value we use throughoutthis paper for the stellar rotation period. We note that theerror bar recovered on the whole data set is underestimatedsince it was measured on the curvature of the χ ( P rot ) curvearound the minimum, curve which presents many aliasedlocal minima due to the observation sampling.For the GPR, we made a first run on the global dataset (phase plot in Fig. B3) and used its result to freeze thedecay time for the modelling of the individual data sets, toavoid degeneracy. The retrieved hyperparameters are givenin Table B2. The phase plots of the individual data sets aredisplayed in Figure B4. Again, a neat period around 1.87 isoutlined for each data set. The periods found with GPR andwith sine fits are generally consistent, but the error bar forthe rotation period on the whole data set is more trustworthywhen computed statistically from GPR-MCMC than fromthe local curvature of the sine fit aliased χ curve.All derived rotation periods, from sine fits and GPR,are plotted against their corresponding latitude using theZDI-retrieved differential rotation in Figure B5), and thethus-derived latitudes are plotted against time in Figure B6,showing a global increasing trend of that latitude, regardlessof the period retreval method.We computed B-V(V) models from the Kurucz mod-els for colors of main sequence stars with log(g)=3.5,T e ff =4500 K and E(B-V)=0.10 mag (Kurucz 1993): we fita two-temperature model with a photospheric temperatureof 4500 K and different values for the spot temperature.Then, for each tested spot temperature, for all values ofspot coverage from 0 too 100 %, we computed the resultingB and the resulting V using the following formulas, fromwhich we derived the B-V. The resulting models are plottedin Figure B7. We find that a spot temperature of 3750 K fitsour B-V measurements well, from which we deduce that theextension of our data imply a spot coverage on V410 Taubetween 50 and 75%, in agreement with the assumption inSection 5.3. V ( r ) = − . ( r − V spot2 . + (1 − r )10 − V star2 . ) B ( r ) = − . ( r − B spot2 . + (1 − r )10 − B star2 . ) APPENDIX C: ACTIVITY PROXIES
This section shows the line profiles of H α , He i and Ca ii , aswell as some results on B (cid:96) . Table B1.
Sinfit results on photometric data.
Data Period (d) Amplitude (mag) Dispersion (mag)08b+09a 1.8695 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± C1 H α H α dynamic spectra are plotted in Figure C1, with the 2009Jan data set being split in half to better see the absorptionfeature around phase 0.95. We also see two other absorp-tion features in 2009 Jan around phase 0.80 and in 2011 Janaround phase 0.35, and we fit a sine curve in each to deter-mine the potential altitude of a prominence or cloud thatcould be the origin of these absorption features. We find asine semi-amplitude of ∼ v sin i for each of them.Lomb-Scargle periodograms for individual epochs areplotted in Figure C2, and the periodogram for the wholedata set is shown in Figure C3, showing a neat peak at therotation period. C2 He i D The He i line profiles are shown in Fig. C5. We can clearlysee the flares at the dates marked in Table A2. C3 Ca ii The Ca ii line profiles are shown in Fig. C6. C4 B (cid:96) We derived longitudinal magnetic field values as first-ordermoments of our Stokes V LSD profiles, and applied a GPR-MCMC run on them. The phase plot is shown in Figure C7.
This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000
This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau (a) 08b+09a (b) 09b+10a (c) 10b(d) 11b+12a (e) 12b+13a (f) 13b+14a(g) 14b (h) 15b+16a (i) 16b+17a Figure B1.
Fits of the V magnitude measurements at each epoch by a sine curve and two harmonics. Each data set was folded accordingto the corresponding period listed in Table B1. As a consequence, the rotation phase used in these plots does not correspond to therotation phase in Table A1, but rather to the phase in the model sine curves.
Figure B2.
Fit of the V magnitude in the whole data set bya sine curve and four harmonics. The color-to-dataset correspon-dency is as follows: red=08b+09a, orange=09b+10a, yellow=10b,green=11b+12a, turquoise=12b+13a, cyan=13b+14a, blue=14b,purple=15b+16a, pink=16b+17a.MNRAS , 1–28 (2018) L. Yu et al.
Table B2.
Results of the GPR-MCMC runs on our V magnitude measurements.Data set GP Period (d) Decay time (d) Smoothing (d) Amplitude (mag)(best) (best) (best) (best)08b+09a 1.8693 ± ± + . − . (0.02)09b+10a 1.8714 ± ± + . − . (0.11)10b 1.8723 ± ± + . − . (0.03)11b+12a 1.8704 ± ± + . − . (0.05)12b+13a 1.8718 ± ± + . − . (0.04)13b+14a 1.8721 ± ± + . − . (0.08)14b 1.8735 ± ± + . − . (0.09)15b+16a 1.8727 ± ± + . − . (0.06)+16b+17aAll V mag 1.8715 ± + − (311.1333) 0.74 ± + . − . (0.070) Figure B3.
GPR-MCMC phase plot for the entire data set ofV magnitudes. GP amplitude θ = . + . − . mag, cycle length θ = . ± . d, decay time θ = + − d, smoothing param-eter θ = . ± . d. MNRAS000
GPR-MCMC phase plot for the entire data set ofV magnitudes. GP amplitude θ = . + . − . mag, cycle length θ = . ± . d, decay time θ = + − d, smoothing param-eter θ = . ± . d. MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau (a) 08b+09a (b) 09b+10a (c) 10b (d) 11b+12a(e) 12b+13a (f) 13b+14a (g) 14b (h) 15b+16a+16b+17a Figure B4.
MCMC phase plots for GPR applied to each of our V magnitude data sets.
Latitude θ ( ° ) Ω (r ad / d ) Ω ( θ ) θ V (sinfit) θ V (GPR) Figure B5.
Differential rotation curve in blue, with parameters Ω eq and d Ω as defined in the introduction of Section 4. Red: H α rotation rates, green: B (cid:96) rotation rates, circles: derived from 2013Dec data set, triangles: derived from 2015 Dec data set, x symbols:derived from the whole data set (143 points for H α and 135 for B (cid:96) ). Photometry rotation rates are displayed, those derived withsinfit (Table B1) in green and those derived with GPR (Table B2)in magenta. Year (2000+) La t i t ude θ ( ° ) θ V (sinfit) θ V (GPR)Vmag all B-V all
Figure B6.
Colatitude found for the V magnitude, for each epochand for the whole data set with sinfit (x-coordinate: 20) as wellas for B-V with sinfit (x-coordinate: 21).MNRAS , 1–28 (2018) L. Yu et al.
V (mag) B - V ( m ag ) T=3500KT=3750KT=4000KT=4250K
Figure B7.
Fit of the B-V(V) curve with Kurucz models, with aphotosphere temperature of 4500 K, log g of 3.5, E(B-V) of 0.10.Each full line corresponds to a particular value of the spot tem-perature, and dots mark the spot coverage with steps of 10% (thedot at V=10.0 and B-V=1.08 corresponding to a 0% spot cov-erage). The extension of our data correspond to a spot coverageconstantly between 50% and 75%. MNRAS000
Fit of the B-V(V) curve with Kurucz models, with aphotosphere temperature of 4500 K, log g of 3.5, E(B-V) of 0.10.Each full line corresponds to a particular value of the spot tem-perature, and dots mark the spot coverage with steps of 10% (thedot at V=10.0 and B-V=1.08 corresponding to a 0% spot cov-erage). The extension of our data correspond to a spot coverageconstantly between 50% and 75%. MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau (a) 2008 Dec (b) 2009 Jan cycles 0-4 (c) 2009 Jan cycles 5-8 (d) 2011 Jan(e) 2013 Dec (f) 2015 Dec (g) 2016 Jan Figure C1. H α dynamical spectra for epochs 2008 Dec (a), 2009 Jan (b,c), 2011 Jan (d), 2013 Dec (e), 2015 Dec (f) and 2016 Jan (g).(a) 2009 Jan (b) 2011 Jan Figure C2.
Periodograms for the H α line EW, for observation epochs 2009 Jan (a), 2011 Jan (b), 2013 Dec (c) and 2015 Dec (d).False-alarm probability levels of 1% and 0.1% are represented as horizontal cyan dashed lines, and P rot and its first harmonic as verticalcyan dashed lines.MNRAS , 1–28 (2018) L. Yu et al. (c) 2013 Dec (d) 2015 Dec
Figure C2. (Continued from the previous page).
Figure C3.
Periodogram of the equivalent width of the H α line. The maximum power is found at 3.3636 rad/d, or 1.8680 d.MNRAS000
Periodogram of the equivalent width of the H α line. The maximum power is found at 3.3636 rad/d, or 1.8680 d.MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau Figure C4.
GPR-MCMC phase plot for our H α equiva-lent width data. Amplitude θ = . + . − . km s − , decay time θ = + − P rot , Cycle length θ = . ± . P rot , Smoothing θ = . ± . P rot .MNRAS , 1–28 (2018) L. Yu et al. (a) 2008 Oct (b) 2008 Dec (c) 2009 Jan (d) 2011 Jan(e) 2013 Nov (f) 2013 Dec (g) 2015 Dec (h) 2016 Jan
Figure C5. . He i D Oct 2008 (ref cycle: -42), Dec 2008 (ref cycle: -15), Jan 2009 (ref cycle: 0), Jan 2011 (ref cycle: 397) Nov 2013 (refcycle: 946), Dec 2013 (ref cycle: 959), Dec 2015 (ref cycle: 1349) and Jan 2016 (ref cycle: 1376) MNRAS000
Figure C5. . He i D Oct 2008 (ref cycle: -42), Dec 2008 (ref cycle: -15), Jan 2009 (ref cycle: 0), Jan 2011 (ref cycle: 397) Nov 2013 (refcycle: 946), Dec 2013 (ref cycle: 959), Dec 2015 (ref cycle: 1349) and Jan 2016 (ref cycle: 1376) MNRAS000 , 1–28 (2018) ag. field, activity & companions of V410 Tau (a) 2008 Oct (b) 2008 Dec (c) 2009 Jan (d) 2011 Jan(e) 2013 Nov (f) 2013 Dec (g) 2015 Dec (h) 2016 Jan Figure C6. Ca ii D Oct 2008 (ref cycle: -42), Dec 2008 (ref cycle: -15), Jan 2009 (ref cycle: 0), Jan 2011 (ref cycle: 397) Nov 2013 (refcycle: 946), Dec 2013 (ref cycle: 959), Dec 2015 (ref cycle: 1349) and Jan 2016 (ref cycle: 1376)MNRAS , 1–28 (2018) L. Yu et al.
Figure C7.
GPR-MCMC phase plot for B (cid:96) . GP amplitude θ = + − G, cycle length θ = . ± . P rot , decay time θ = + − P rot , smoothing θ = . ± . P rot . MNRAS000