Context. EX Lupi is the prototype of EX Luptype stars which are classical T Tauri stars cTTSs with luminosity bursts and outbursts of 15 magnitudes that last for a few months to a few years. These events are ascribed to an episodic accretion that can occur repeatedly but whose physical mechanism is still debated. Aims. We aim to investigate the magnetically driven accretion of EX Lup in quiescence. We include for the first time a study of the small and largescale magnetic field. This allows us to characterise the magnetospheric accretion process of the system completely. Methods. We used spectropolarimetric times series acquired in 2016 and 2019 with the Echelle SpectroPolarimetric Device for the Observation of Stars and in 2019 with the SpectroPolarimtre InfraRouge at the CanadaFranceHawaii telescope during a quiescence phase of EX Lup. We were thus able to perform a variability analysis of the radial velocity the emission lines and the surfaceaveraged longitudinal magnetic field in different epochs and wavelength domains. We also provide a smallscale magnetic field analysis using Zeeman intensification of photospheric lines and a largescale magnetic topology reconstruction using ZeemanDoppler imaging. Results. Our study reveals that typical magnetospheric accretion is ongoing on EX Lup. A main accretion funnel flow connects the inner disc to the star in a stable fashion and produces an accretion shock on the stellar surface close to the pole of the magnetic dipole component. We also measure one of the strongest fields ever observed on cTTSs. This strong field indicates that the disc is truncated by the magnetic field close to but beyond the corotation radius where the angular velocity of the disc equals the angular velocity of the star. This configuration is suitable for a magnetically induced disc instability that yields episodic accretion onto the star.
['accretion', 'accretion disks', 'techniques: polarimetric', 'techniques: spectroscopic', 'stars: magnetic field', 'stars: individual: EX Lup', 'stars: variables: T Tauri', 'Herbig Ae/Be', 'Astrophysics - Solar and Stellar Astrophysics']
Multikilogauss magnetic field driving the magnetospheric accretion process in EX Lupi
{'A multi-kiloGauss magnetic field driving the magnetospheric accretion process in EX Lupi': 'K. Pouilly 1 , M. Audard 1 , Á. Kóspál 2,3 , and A. Lavail 4  \n- 1 Department of Astronomy, University of Geneva, Chemin Pegasi 51, CH-1290 Versoix, Switzerland e-mail: Kim.Pouilly@unige.ch\n- 2 Konkoly Observatory, HUN-REN Research Centre for Astronomy and Earth Sciences, MTA Centre of Excellence, Konkoly-Thege Miklós út 15-17, 1121 Budapest, Hungary\n- 3 Institute of Physics and Astronomy, ELTE Eötvös Loránd University, Pázmány Péter sétány 1 / A, 1117 Budapest, Hungary\n- 4 Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, CNRS, IRAP / UMR 5277, 14 avenue Edouard Belin, 31400 Toulouse, France  \nReceived 16 July 2024; accepted 04 September 2024', 'ABSTRACT': 'Context. EX Lupi is the prototype of EX Lup-type stars, meaning classical T Tauri stars (cTTSs) showing luminosity bursts and outbursts of 1 to 5 magnitudes lasting for a few months to a few years. These events are ascribed to an episodic accretion that can occur repeatedly but whose physical mechanism is still debated.  \nAims. In this work, we aim to investigate the magnetically-driven accretion of EX Lup in quiescence, including for the first time a study of the small and large-scale magnetic field. This allows us to provide a complete characterisation of the magnetospheric accretion process of the system.  \nMethods. We use spectropolarimetric times series acquired in 2016 and 2019 with the Echelle SpectroPolarimetric Device for the Observation of Stars and in 2019 with the SpectroPolarimètre InfraRouge at the Canada-France-Hawaii telescope, during a quiescence phase of EX Lup. We were thus able to perform a variability analysis of the radial velocity, the emission lines and surface averaged longitudinal magnetic field along di ff erent epochs and wavelength domains. We also provide a small-scale magnetic field analysis using Zeeman intensification of photospheric lines and large-scale magnetic topology reconstruction using Zeeman-Doppler Imaging. Results. Our study reveals a typical magnetospheric accretion ongoing on EX Lup, with a main accretion funnel flow connecting the inner disc to the star in a stable fashion and producing an accretion shock on the stellar surface close to the pole of the magnetic dipole component. We also measure one of the strongest fields ever observed on cTTSs. Such a strong field indicates that the disc is truncated by the magnetic field close but beyond the corotation radius, where the angular velocity of the disc equals the angular velocity of the star. Such a configuration is suitable for a magnetically-induced disc instability yielding episodic accretion onto the star.  \nKey words. Stars: variables: T Tauri - Stars: individual: EX Lup - Stars: magnetic field - Accretion, accretion disks - Techniques: spectroscopic - Techniques: polarimetric', '1. Introduction': 'EXLup-type objects (EXors) are classical T Tauri stars (cTTSs), meaning low-mass pre-main sequence stars surrounded by an accretion disc, that show burst and outburst events ascribed to episodic accretion. During these phases, they can increase their optical luminosity from 1 to 5 magnitudes, lasting typically for a few months to a few years, and can occur repeatedly (for a review, see, e.g., Fischer et al. 2023). These events are therefore more moderate, both in duration and luminosity increase, than the FU Orionis-type stars (FUors).  \nWhile the magnetospheric accretion of cTTSs, where the strong stellar magnetic field truncates the disc, forcing the accreted material to follow the magnetic field lines (see the review by Hartmann et al. 2016), seems to be ongoing on EXors as well, the origin of this episodic accretion is still highly debated. The di ff erent hypotheses can be gathered in three groups (see review by Audard et al. 2014): (i) the magnetospheric accretion itself, being inherently episodic when a strong magnetic field truncates the disc close to the corotation radius (D\'Angelo & Spruit 2010). (ii) The disc, showing viscous-thermal (Bell & Lin 1994), or gravitational and magneto-rotational (Armitage et al. 2001) in-  \nstabilities, or accretion clumps in a gravitationally unstable environment (Vorobyov & Basu 2005, 2006). (iii) The presence of a companion, perturbing the accretion trough tidal e ff ect (Bonnell & Bastien 1992), or thermal instabilities (Lodato & Clarke 2004).  \nThis work aims to investigate the magnetospheric accretion process of the prototypical EXors, EX Lup, using highresolution spectropolarimetric time series, as it was done for cTTSs not displaying episodic accretion (e.g., Pouilly et al. 2020, 2021). This object is a young M0.5-type star (GrasVelázquez & Ray 2005), known to have both moderate and shorttimescale variability and rare extreme episodic outbursts. It is located at 154.7 ± 0.4 pc ( Gaia DR3 parallax 6.463 ± 0.015 mas, Gaia Collaboration et al. 2023) and has a rotation period of 7.417 days, from its radial velocity modulation that was first ascribed to a low-mass companion (Kóspál et al. 2014), before being ascribed to stellar activity (Sicilia-Aguilar et al. 2015). The system has a moderate-to-low inclination of its rotation axis (between 20 · and 45 · ) according to modelling of the spectral energy distribution (Sipos et al. 2009) and emission lines analysis (Goto  \net al. 2011; Sicilia-Aguilar et al. 2015), and a projected rotational velocity v sin i = 4.4 ± 2.0 km s -1 (Sipos et al. 2009).  \nThis object is one of the most studied EXors using spectroscopy, both in quiescence (i.e., Kóspál et al. 2014; Sipos et al. 2009; Sicilia-Aguilar et al. 2012, 2015, 2023; Campbell-White et al. 2021; Wang et al. 2023) and in outburst (Sicilia-Aguilar et al. 2012; Cruz-Sáenz de Miera et al. 2023; Wang et al. 2023; Singh et al. 2024, for the last two outbursts in 2008 and 2022), but the present study is the first one including spectropolarimetry, giving access to information on the magnetic field together with the accretion diagnostics. Its spectrum contains the typical accretion-related emission lines observed in cTTSs, in addition to numerous neutral metallic emission lines superimposed to photospheric absorption in quiescence, and overwhelming any absorption feature in outburst (Sicilia-Aguilar et al. 2012). The present dataset was acquired during quiescence, meaning that we will characterise the "stable" accretion of the system, even if Sicilia-Aguilar et al. (2012, 2023) have shown that the accretion pattern seems stable in quiescence and outburst, only the amount of accreted material is a ff ected. This accretion pattern is consistent with the typical cTTS magnetospheric accretion, through accretion funnel flows connecting the disc to the stellar surface, except that "clumps" of material are also accreted through these funnel flows, detected thanks to the day-to-day variation of the emission lines\' broad component (BC) (see Fig. 17 of SiciliaAguilar et al. 2012).  \nThis article is organised as follows: we describe the observations in Sect. 2, the analysis and results are presented in Sect. 3 and discussed in Sect. 4. We conclude this work in Sect. 5.', '2. Observations': 'The spectropolarimetric time series used in this work were acquired at the Canada-France-Hawaii Telescope at two di ff erent epochs (2016 and 2019, proposals 16AF03 and 19AF50, respectively). The second data set is composed of two subsets using two di ff erent instruments: the Echelle SpectroPolarimetric Device for the Observation of Stars (ESPaDOnS, Donati 2003) in the optical and the SpectroPolarimètre InfraRouge (SPIRou, Donati et al. 2020b) in the near-infrared, both used in polarimetric mode, while the first one only used ESPaDOnS. This means that each observation is composed of four sub-exposures taken in di ff erent polarimeter configurations, which are then combined to obtain the intensity (Stokes I ), the circularly polarised (Stokes V ), and the null polarisation spectra. A complete journal of observation is provided in Table 1.', '2.1. ESPaDOnS': 'The ESPaDOnS observations, which cover the 370 to 1050 nm wavelength range and reach a resolving power of 68 000, consist of 11 nights between 2016 June 09 and 2016 June 24, with an approximately nightly cadence, and 6 nights between 2019 May 31 and 2019 June 12, the 5 latter respecting a 1-day sampling. The signal-to-noise ratio (S / N) of the 2016 (2019) observations ranges between 69 and 142 (111 and 140) for the Stokes I at 731 nm. Each observation was reduced using the Libre-ESpRIT package (Donati et al. 1997).', '2.2. SPIRou': 'The SPIRou observations are covering the 960 to 2350 nm wavelength range with R ∼ 75 000. They were acquired during 8 con-  \nTable 1. Log of EX Lup observations.  \nNotes. The S / NI corresponds to the peak S / Nbyspectral pixel at 731 nm (in H-band) for ESPaDOnS (SPIRou) observations. The S / NLSD is the e ff ective S / N of the Stokes V LSD profiles (see Sect. 3.3.1). The last column indicates which instrument was used for each observation (EESPaDOnS, S-SPIRou). The horizontal line separates the 2016 from the 2019 observations.  \nsecutive nights, between 2019 June 14 and 2019 June 21, and the S / N of the unpolarised spectra in the H-band range between 107 and 175. The observations were reduced using the APERO pipeline (Cook et al. 2022).', '3. Results': 'In this section, we present the results obtained from the analysis of the observations described in Sect. 2. They consist of the analysis of the radial velocity, emission lines, and stellar magnetic field.', '3.1. Radial Velocity': 'To determine the radial velocity of EX Lup, we cross-correlated each spectrum with a synthetic spectrum computed using the ZEEMAN code (Landstreet 1988; Wade et al. 2001; Folsom et al. 2012), with MARCS atmospheric models (Gustafsson et al. 2008) and VALD (Ryabchikova et al. 2015) line lists adapted to EX Lup stellar parameters for ESPaDOnS and SPIRou wavelengths. We computed the cross-correlation function (CCF) over 27 (14) wavelength windows of about 10 nm ranging from 441 to 890 nm (1150 to 2290 nm) for ESPaDOnS (SPIRou) observations. Then we performed a sigma-clipping across all the CCFs for each observation and used the mean and standard deviation of the remaining values as measurement of the radial velocity and its uncertainty. The results are plotted  \nFig. 1. Radial velocities curves determined for the observation summarised in Table 1. Top: radial velocity vs. HJD, the vertical dotted line marks the switch from ESPaDOnS to SPIRou, and the di ff erent colours represent di ff erent rotation cycles. Bottom: same as above but folded in phase with P = 7.417 d and T0 = 2 457 544.40981. The dotted curve shows the sinus fit.  \n<!-- image -->  \nin Fig. 1. A quick sinusoidal fit of the values for each data set allowed us roughly measure the periodicity and mean value of the radial velocity and yielded P = 7.55 ± 0.24 d and ⟨ Vr ⟩ = -0.48 ± 0.07 km s -1 (ESPaDOnS 2016), P = 7.63 ± 0.18 d and ⟨ Vr ⟩ = -0.59 ± 0.14 km s -1 (ESPaDOnS 2019), and P = 8.36 ± 0.97 d and ⟨ Vr ⟩ = -0.58 ± 0.14 km s -1 (SPIRou). These measures are consistent within the uncertainties with previous results obtained by Kóspál et al. (2014), Prot = 7.417 ± 0.001 d and ⟨ Vr ⟩ = -0.52 ± 0.07 km s -1 , we will thus adopt the latter values for our analysis. Finally, we folded all the measurements in phase using Prot = 7.417 d and an arbitrary T0, and we fitted this curve with a sinus to estimate the T0 required to set ϕ = 0.5 at the mean velocity between the maximum and minimum of the modulation (when the spot modulating the curve is facing the observer, see Fig. 7 of Sicilia-Aguilar et al. 2015). The resulting T0 is HJD 2 457 544.40981. We will thus use the following ephemeris for the rest of this work:  \nHJD(d) = 2 457 544 . 40981 + 7 . 417 E , (1)  \nwhere E is the rotation cycle. All our radial velocity measurements are in phase (Fig. 1), but the 2019 measurements show a larger amplitude of its modulation. This indicates an evolving feature modulating the radial velocity but located at the same longitude in 2016 and 2019. The values are summarised in Table 2.', '3.2. Emission line variability': 'In this section, we present the analysis of emission lines tracing either the accretion funnel flow, here the Balmer lines (Muzerolle et al. 2001), or the accretion shock such as the Ca ii infrared triplet (IRT), the He i D3 (Beristain et al. 2001), or the He i at 1083 nm lines. For each line, we analysed the profile variability, their periodicity (except for the 2019 ESPaDOnS data set which does not cover a su ffi cient time span), and the correlations of these variabilities. Most of the analyses of this section were performed using PySTEL(L)A 1 , a Python tool for SpecTral Emission Lines (variabiLity) Analysis.  \nTable 2. Radial velocities measured for each observation and their uncertainties.  \nNotes. The phases ϕ are computed using the ephemeris provided in Eq. 1. The horizontal line separates the 2016 from the 2019 observations.', '3.2.1. Balmer lines': 'The Balmer lines are partly formed in the accretion funnel flow and are thus tracing the magnetospheric accretion process. In this work, we focussed on H α , H β , and H γ , which we corrected from the photospheric contribution using the moderately active Mdwarf HD 42581 as template (Te ff = 3822 K, v sin i = 2.6 km s -1 , Manara et al. 2021), broadened to the v sin i of EX Lup. The 2016 and 2019 profiles are shown in Fig. 2 and Fig. 3, respectively. On both data sets, the profiles are composed of a broad and a narrow component, both highly variable. Furthermore we can notice a flux depletion, going below the continuum, around + 200 kms -1 , and extending up to + 300 kms -1 . This behaviour is characteristic of the so-called inverse P Cygni (IPC) profiles, the red-shifted absorption forming due to infalling material.  \nThe 2D-periodograms, consisting of a Lomb-Scargle periodogram computed in each velocity channel, of the 2016 lines are presented in Fig. 2, and show a periodic signal along the whole H β and H γ lines consistent with the rotation period of the star, with a false alarm probability (FAP, computed from Baluev 2008, prescriptions) of 0.04 and 0.01, respectively. The H α line is showing this signal in a less continuous way with a much higher FAP (0.25). The symmetric signal around 0.9 d -1 is the 1-day alias, a spectral leakage of the Fourrier transform reconstructing (with the real period) the observation sampling, meaning approximately one observation per day.  \nTo separate the various parts of the line profile undergoing di ff erent variability patterns, we computed the auto-correlation matrices of each line. This tool consists of computing a linear  \n656.28 nm  \n434.05 nm  \nFig. 2. Variability of 2016 Balmer lines of EX Lup. Top row: H α (left) , H β (middle) , and H γ (right) residual lines profiles. Each colour represents a di ff erent observation. Bottom row: 2D-periodograms of H α (left) , H β (middle) , and H γ (right) residual lines. The white dotted line marks the rotation period of 7.417 d.  \n<!-- image -->  \ncorrelation coe ffi cient (here a Pearson coe ffi cient) between the velocity channels of the line. A correlated region (close to 1) indicates a variability dominated by a given physical process. An anti-correlated region (close to -1) indicates a variability dominated by a given physical process or, at least, linked physical processes. The auto-correlation matrices of 2019 H α , H β , and H γ lines are shown in Fig 3. Here again, H α is showing a di ff erent behaviour than H β and H γ . H α shows three main correlated regions between -100 kms -1 and + 100 kms -1 , corresponding to the core of the line, around + 150 kms -1 , corresponding to a slight emission excess in the IPC profile (occurring around HJD 24588634.95 and 2 458 641.95), and around + 250 km s -1 , corresponding to the IPC profile itself. The two latter regions are anti-correlated between them, but the ∼ + 150 kms -1 region is also slightly anti-correlated with the line center. This means that this region might be a broadening of the BC invoked by Sicilia-Aguilar et al. (2012), causing a global decrease of the line (and so a anti-correlation with the whole line profile). H β and H γ are showing the same correlation around the line centre and + 350 kms -1 , without significant anti-correlation around + 150 km s -1 , and a moderate anti-correlation of the profile with a region around + 350 km s -1 which seems to be an artefact as it is located in the continuum.', '3.2.2. He i D3 587.6 nm': "The He i D3 lines of 2016 and 2019 data sets are presented in Fig. 4 and are composed of a narrow component (NC) only, extending from approximately -35 to + 50 km s -1 , without significant BC. The NC is formed in the post-shock region of the accretion spot (Beristain et al. 2001), and can thus trace the accretion  \nclose to the stellar surface. One can note the strong variability of this NC with a maximum reached around ϕ = 0.6 2 .  \nThe auto-correlation matrices shown in Fig. 4 confirm that this region is formed by one physical process, and the periodicity of the 2016 NC's variation, consistent with the stellar rotation period (see Fig. 5, FAP = 0.06), indicates that this component is tracing an accretion shock at the stellar surface.  \nWe thus performed a fit of the He i D3 NC's radial velocity following the method described in Pouilly et al. (2021) to recover the emitting region's location. The results are shown in Fig. 6 and summarised below:  \n- -Vflow = 7.091 + 2 . 00 kms -1\n- -Vrot = 3.85 ± 5.0 km s -1 ,\n- -d ϕ = 0.10 + 0 . 15\n- -θ = 12.05 + 48 . 00 -6 . 7 ·\n- -α = 55.40 .  \n-1 . 03 , -0 . 11 , , + 20 00 -16 . 36 · ,  \nwhere Vflow is the velocity of the material in the post-shock region, Vrot the equatorial velocity, 0.5 + d ϕ the phase where the emitting region is facing the observer, θ the colatitude of the spot, and α = 90 · -i, i being the inclination of the rotation axis. This means that the emitting region is located at ϕ = 0.6, 70 · latitude. These results are consistent within uncertainties with the measurements by Campbell-White et al. (2021), who give for the He i emitting region 3 a latitude of 60 ± 25 · , longitude 40 ± 5 · , meaning ϕ = 0.1 ± 0.01. However, the authors are using the first date of observation as T0, translated to our ephemeris (Eq. 1), this yields ϕ = 0.7.  \n656.28 nm  \nFig. 3. Variability of 2019 Balmer lines of EX Lup. Top row: H α (left) , H β (middle) , and H γ (right) residual lines profiles. Each colour represents a di ff erent observation. Bottom row: H α (left) , H β (middle) , and H γ (right) residual lines auto-correlation matrices. The colour code is scaling the correlation coe ffi cient. Light yellow represents a strong correlation and dark purple a strong anti-correlation. Please note that the strong anticorrelation around + 350 km s -1 on H β and H γ matrices is probably an artefact as located in a poorly varying part of the continuum.  \n<!-- image -->", '3.2.3. Ca ii infrared triplet': "As the He i D3 NC, the Ca ii IRT NC is formed in the post-shock region, we thus studied these lines as well. The three components of this triplet show identical shape and variability, we thus focussed on one of them located at 854.209 nm. The 2016 and 2019 residual line profiles are shown in Fig. 7 and Fig. 8, respectively. The two sets of lines show an IPC profile around 200 kms -1 , which is, for the 2016 line, periodic with the stellar rotation period (see the 2D-periodogram in Fig. 7, FAP = 0.05).  \nThe 2016's auto-correlation matrix (Fig. 7) exhibits several correlated regions: from -130 to -60, -50 to -10, + 50 to + 100, + 110 to + 160, and + 170 to + 200 km s -1 . However, these regions are less numerous on the 2019 matrix (Fig. 8), with only three correlated regions from -40 to + 40, + 80 to + 130, and + 150 to + 200 kms -1 , but we retrieve the correlated region around the IPC profile in both matrices, which is anti-correlated with the NC in 2019. This goes with the smaller (larger) variability of the NC (BC) observed in 2016 compared to 2019, showing the di ff erent origins of the two components and a small change in the accretion pattern between the two epochs.", '3.2.4. He i 1083 nm': "The only accretion tracer in emission in EX Lup's SPIRou observation is the He i line at 1083 nm. The profiles, the 2Dperiodogram and the auto-correlation matrix are shown in Fig. 9. The profiles seem composed of two peaks blue- and red-shifted around -50 and + 100kms -1 , and two absorptions, blue-and redshifted at higher velocity ( -150 and + 200kms -1 ).  \nThe red-shifted peak and the two absorptions display significant variability and seem modulated on the stellar rotation pe-  \nriod with FAPs reaching 0.001, 0.01, and 0.02 for the red-shifted peak, the blue- and the red-shifted absorption, respectively.  \nThe auto-correlation matrix revealed a more complex decomposition. Indeed, if the four substructures seen in the profiles are represented, it seems that the two absorptions can both be separated in two regions, from -250 to -160 and -160 to -110 kms -1 for the blue-shifted absorption, and from + 130 to + 170 kms -1 and + 210 to + 270 kms -1 for the red-shifted absorption. Furthermore, the main peak at ∼ + 100 kms -1 is anticorrelated with the most blue- and red-shifted regions only. This can be interpreted as follows: the blueshifted absorption, probably a P-Cygni profile traditionally ascribed to a wind, is also associated with a redshifted emission excess, producing the two substructures seen in the redshifted absorption. The IPC profile, as the opposite physical phenomenon, is also associated with a blueshifted emission excess, responsible for the two substructures in the blueshifted absorption.", '3.2.5. Correlation matrices ESPaDOnS': "As the several lines studied are tracing di ff erent regions of the accretion, we can compute correlation matrices between two di ff erent lines to analyse the link between the di ff erent regions identified from the auto-correlation matrices. The correlation matrices of the 2016 and 2019 lines are presented in Appendix A.  \nIn 2016, the H α line centre is correlated with the He i D3 and the Ca ii IRT NCs, and slightly anti-correlated with the region of Ca ii IRT IPC profile. The latter is also strongly anti-correlated with the He i D3 NC. The region of the H α 's IPC profile is also anti-correlated with the He i D3 NC, and correlated with the region of the Ca ii IRT IPC profile. In 2019 matrices, we observe  \nFig. 4. EX Lup He i D3 line variability. Top row: 2016 (left) and 2019 (right) lines profiles. Bottom row: Auto-correlation matrices of 2016 (left) and 2019 (right) lines.  \n<!-- image -->  \nFig. 5. He i D3 2016 2D-periodogram.  \n<!-- image -->  \nthe same behaviour between the NCs and IPC regions of the var-  \nFig. 6. Radial Velocity fit of He i D3 2016 NC (top) and its version folded in phase (bottom) following the ephemeris given in Eq. 1  \n<!-- image -->  \nious lines, but both the correlation and anti-correlation coe ffi -cients are stronger.  \nFig. 7. ESPaDOnS 2016 Ca ii IRT (854.2 nm) residual line profiles (left) , 2D-periodogram (middle) , and auto-correlation matrix (right) .  \n<!-- image -->  \n854.21 nm  \nFig. 8. ESPaDOnS 2019 Ca ii IRT (854.2 nm) residual line profiles (top) and auto-correlation matrix (bottom) .  \n<!-- image -->", '3.3. Magnetic field': 'In this section, we present the magnetic analysis of EX Lup. This was done at two scales: the large scale using the ZeemanDoppler Imaging technique (ZDI, Donati et al. 2011), and the small scale using the Zeeman intensification of photospheric lines.', '3.3.1. Large-scale': "In this section, we used the Least-Squares Deconvolution method (LSD, Donati et al. 1997; Kochukhov et al. 2010) to study the large-scale magnetic field. This method allows us to increase the S / N of the Stokes I (unpolarised) and Stokes V (circularly polarised) profiles, by using as many photospheric lines as possible. To compute the LSD profiles we used the LSDpy 4 Python implementation. We normalised our LSD weights using an intrinsic line depth, a mean Landé factor and a central wavelength of 0.2, 1.2, and 500 nm (respectively) for ESPaDOns, and 0.1, 1.2, and 1700 nm for SPIRou observations. The photospheric lines were selected from a mask produced using the same VALD line list and MARCS atmospheric models as in Sect. 3.1, and removing the emission lines, the telluric and the heavily blended lines regions using the SpecpolFlow 5 Python package. About 12 000 lines were used for ESPaDOnS, and 1600 for SPIRou observations. The LSD profiles of ESPaDOnS 2016, 2019, and SPIRou observations are presented in Figs. 10, 11, and 12, respectively, and the S / N of the profiles are available in Table 1. Please note that the observation at HJD 2 457 551.87135 was remove from this analysis because of its low S / N.  \nAclear Stokes V signature is detected for all ESPaDOnS observations and most of the SPIRou observations. Furthermore, one can note that the SPIRou Stokes V signatures are much weaker than ESPaDOnS and the signal at ϕ ∼ 0.6-0.7 almost vanishes on SPIRou when maximal on ESPaDOnS.  \nDirectly from the LSD profiles, one can estimate the surface averaged longitudinal magnetic field (B ℓ , Donati et al. 1997; Wade et al. 2001):  \nB ℓ = -2 . 14 × 10 11 × R vV ( v ) dv λ gc R (1 -I ( v )) dv , (2)  \nwhere B ℓ is in Gauss, v the velocity relative to the line centre, and λ and g the central wavelength and the mean Landé factor used for the LSD computation. The integration was performed on a ± 25 kms -1 ( ± 35 kms -1 ) velocity range around the stellar rest frame to minimise the uncertainties without losing any magnetic information on ESPaDOnS (SPIRou) observations. The B ℓ curves are shown in Fig. 13. The three curves are modulated with the stellar rotation period and show a maximum around ϕ = 0.6. In the optical frame, the modulation's amplitude is slightly larger in 2019 than in 2016, which is reminiscent of the radial velocity  \n<!-- image -->  \n<!-- image -->  \nFig. 9. SPIRou He i (1083 nm) line profiles (left) , 2D-periodogram (middle) , and auto-correlation matrix (right) .  \n<!-- image -->  \n<!-- image -->  \n<!-- image -->  \n<!-- image -->  \nFig. 10. ESPaDOnS 2016 LSD Stokes I (left) and V (right) profiles. The rotation phase and the HJD are indicated at the left and right, respectively, of each profile. The scale is indicated at the bottom left of each plot.  \n<!-- image -->  \nbehaviour (see Sect. 3.1). Finally, as expected from the Stokes V signatures, the SPIRou measurements are far weakest, and mostly negatives.  \nThe same analysis can be performed on the NC of the He i D3 line (on a ± 50 km s -1 velocity range), formed close to the accretion shock and thus gives access to the magnetic field strength at the foot of the accretion funnel flow. The B ℓ obtained are shown in Fig. 13 and range between -1.4 and -3.7 kG in 2016 and between -0.5 and -4.5 kG in 2019. As for the radial velocity (see Sect. 3.1), the amplitude of the modulation is larger in 2019, but both curves are in phase, with a minimum reached at ϕ = 0.6, which is also consistent with the emitting region's position obtained from the radial velocity modulation of the He i D3 NC (see Sect. 3.2.2). These measurements are also opposed in phase and sign with the LSD B ℓ . This is a common behaviour on cTTSs (e.g., see the studies of S Cra N, TWA Hya, CI Tau by Nowacki  \net al. 2023; Donati et al. 2011, 2020a, respectively) reflecting that LSD and He i D3 NC are two diagnostics probing di ff erent regions of di ff erent polarities.  \nFinally, we performed a complete ZDI analysis of the three data sets using ZDIpy 6 (described in Folsom et al. 2018) with a recent implementation of Unno-Rachkovsky's solutions to polarised radiative transfer equations in a Milne-Eddington atmosphere (Unno 1956; Rachkovsky 1967; Landi Degl'Innocenti & Landolfi 2004) presented in Bellotti et al. (2023) to used a more general description than the weak-field approximation used initially in ZDIpy . The magnetic topology is reconstructed in two steps: (i) the building of a Doppler image (DI), starting from a uniformly bright stellar disk and iteratively adding dark and bright features to fit the observed LSD Stokes I profiles, and (ii) fitting the LSD Stokes V profiles to derive the magnetic  \nFig. 11. Same as Fig. 10 for ESPaDOnS 2019 observations.  \n<!-- image -->  \n<!-- image -->  \nI  \n/  \nV  \ns  \ne  \nk  \no  \nt  \nS  \ntopology by adjusting its spherical harmonic components (Donati et al. 2006), here with a maximum degree of harmonic expansion ℓ max = 15. We used as input parameter Prot = 7.417 d, v sin i = 4.4 km s -1 . Then we ran a grid of ZDI Stokes V reconstruction over the inclination range derived in the literature (2045 · ) and used the minimal χ 2 obtained to set our input inclination value (i = 30 · ). The resulting maps are presented in Fig. 14 and 15 and the corresponding LSD profile fits are shown in Appendix B.  \nThe ESPaDOnS 2016's DI map shows a main dark spot around ϕ = 0.55 and extending between 70 and 20 · latitude, which is consistent with He i D3 emitting region, and a bright structure from ϕ = 0.95 to 0.20 around the equator which is certainly a plage as even the accretion shock is dark at the photospheric level. The magnetic topology, mostly toroidal (61%) is dominated by the dipolar (60%) and the quadrupolar (17%) components, with a mean magnetic field strength of 1.00 kG (Bmax = 3.12 kG), and a magnetic dipolar positive pole of 0.728 kG located at about 30 · latitude and 166 · longitude ( ϕ = 0.46).  \nThe brightness map obtained from SPIRou revealed a main dark feature extended from ϕ = 0.0 to 0.25, and a bright plage around ϕ = 0.6, which is perfectly consistent with the 2016 map. However, the magnetic topology seems less complex than in 2016, almost fully poloidal (99%) and more dominated by the dipole component (77%). As expected, the recovered field strength is much smaller ( ⟨ B ⟩ = 0.131 kG, Bmax = 0.389 kG). The dipole negative pole is located at about 23 · latitude, 329 · longitude ( ϕ = 0.91), with a -0.231 kG-strength.  \nGiven the poor rotational phase coverage of the ESPaDOnS 2019's data set ( ϕ ∈ [0.03,0.64]), we needed to guide the reconstruction instead of starting from a uniform map to avoid a too strong extrapolation on the missing phases. We do not expect  \nTable 3. Ti i lines used for the Zeeman intensification analysis.Notes. The wavelengths in the optical frame are given in the air, the ones in the infrared frame are given in vacuum.  \na similar brightness contrast between ESPaDOnS and SPIRou, but given the very similar spectroscopic behaviour between the 2016 and 2019 ESPaDOnS data sets (see Sect. 3.1 and 3.2), we could expect similar features, we thus used the ESPaDOnS 2016 brightness map as input to guide its reconstruction. To check this assumption, we reproduce the Stokes I profiles resulting from the 2016 reconstruction on the 2019 phases, yielding a consistent behaviour (reduced χ 2 = 1.1). The obtained final brightness reconstruction shows a main dark feature, slightly shifted in phase compared to 2016 ( ϕ ≈ 0.5) and less extended in latitude. A polar bright feature is also located around ϕ = 1, which is reminiscent of the SPIRou's maps. Concerning the magnetic reconstruction, we expect a similar topology of the magnetic field between SPIRou and ESPaDOnS 2019, with di ff erent magnetic strengths as pointed out by the B ℓ analysis. We thus used the SPIRou magnetic maps as input to guide the reconstruction. The resulting topology is less poloidal-dominated than SPIRou (91%), and the dipolar component occupies a smaller fraction of this poloidal field (65%) with a similar contribution of the quadrupolar and octupolar components (about 15%). Surprisingly the mean magnetic field strength is similar to 2016 (1.08 kG) but the maximum field strength is much higher (4.79 kG), as well as the strength of the dipolar pole Bdip = 1.87 kG which is located at the same position ( ϕ = 0.43 and 30 · latitude).", '3.3.2. Small-scale': "Even if the ZDI analysis gives access to the magnetic topology, it neglects the small-scale magnetic field, which contains a major part of cool stars' magnetic energy. This is analysed by studying the change it induces in the shape of magnetically sensitive lines (Kochukhov et al. 2020), a technique called Zeeman intensification. To perform this analysis, we used the algorithm of Hahlin et al. (2021), performing a Markov Chain Monte Carlo (MCMC) sampling, using the SoBaT library (Anfinogentov et al. 2021) on a grid of synthetic spectra produced by the SYNMAST code, a polarised radiative transfer code described by Kochukhov et al. (2010). This grid is computed from the VALD line lists used in Sect. 3.1, and MARCS atmospheric models (Gustafsson et al. 2008). We parametrized a uniform radial magnetic field as a sum of magnetic field strength ranging from 0 to 6 kG, with a 2 kG step, weighted by filling factors representing the amount of stellar surface covered by this magnetic field. For ESPaDOnS observations, as in previous studies (Hahlin & Kochukhov 2022; Pouilly et al. 2023, 2024), we used the 963.5-981.2 nm region.  \nThis region contains a group of Ti i lines with di ff erent magnetic sensitivity (ge ff summarised in Table 3), allowing us to disentangle the e ff ect of the magnetic field on the equivalent widths  \nFig. 12. Same as Fig. 10 for SPIRou observations.  \n<!-- image -->  \n<!-- image -->  \n<!-- image -->  \nI  \n/  \nV  \ns  \ne  \nk  \no  \nt  \nS  \nfrom the e ff ect of any other parameters, such as the Ti i abundance. However, this region also contains many telluric lines which are superimposed on the Ti i lines from the stellar spectra and which need to be removed from the observed spectrum to perform the magnetic analysis. To do so, we used the molecfit package (Smette et al. 2015), developed to model and remove telluric lines from spectra obtained with instruments at the European Southern Observatory, and which can be used on spectra from any instrument.  \nFinally, as EX Lup is an accreting star, showing a signature of an accretion shock, the veiling might disturb the inference results, in particular the abundance, the v sin i and the radial tangential macroturbulent velocity, vmac. We thus estimated the veiling using the magnetic null line of the Ti i multiplet at 974.36 nm by performing a χ 2 minimisation using SYNMAST synthetic spectra letting the abundance, the v sin i , and the vmac as free parameters and adding a fractional veiling defined as:  \nIveil = I + r 1 + r , (3)  \nwhere Iveil is the veiled spectrum, I the spectrum without veiling, and r the fractional veiling. We stress to the reader that deriving a precise value of the veiling is outside the scope of the present study , our aim was only to get an estimation of the EX Lup mean spectrum at a wavelength close to the Ti i lines in order to minimise its e ff ect on the other inferred parameters. For the 2016's ESPaDOnS mean spectrum, the minimal χ 2 is reached for r = 0.5, v sin i = 4.9 km s -1 , vmac = 1.9 km s -1 , and a Ti i abundance of -7.2. For the mean spectrum of the 2019's ESPaDOnS observations, we obtained r = 0.48, v sin i = 5.0 km s -1 , vmac = 2.5km s -1 , and a Ti i abundance of -7.2. We will thus assume the values obtained for r and vmac, and use the others as initial guesses for the MCMC sampling.  \nWe assumed a multicomponent model given by:  \nS = X fiS i , (4)  \nwhere fi are the filling factors, meaning the fraction of the stellar surface covered by a field strength Bi , and Si the synthetic spectra of the corresponding magnetic field strength. The averaged magnetic field is thus given by:  \n⟨ B ⟩ = X fi Bi . (5)  \nTo set the number of filling factor to use, we iteratively added filling factors, with a 2-kG step in the corresponding magnetic field strength, and used the Bayesian information criterion (BIC, Sharma 2017) to only include the filling factors that significantly improve the fit. The suitable solutions are components of 0, 2, 4 kG for ESPaDOnS 2016 and SPIRou observations, and 0, 2, 4, 6, 8 kG for ESPaDOnS 2019. The free parameters of the analysis are thus the following: fi , v sin i , v r , and the Ti i abundance, for which uniform prior were adopted. Finally, we used an e ff ective sample size of 1000 (Sharma 2017).  \nThe resulting line fit and magnetic field strength posterior distributions are presented in Fig. 16. The 2016's magnetic field strength (3.08 ± 0.04 kG) is consistent with 2019 within uncertainties (3.16 ± 0.05 kG). The inferences of all parameters are summarised in Table 4.  \nFor SPIRou observations, we used here again a set of Ti i lines around 2200 nm (see Table 3). Unfortunately, this wavelength region does not contain any magnetically null line deep enough at our S / N to perform the veiling study we have done on ESPaDOnS observations. Only a visual inspection of the line at 974.4 nm is possible, indicating that at this wavelength the parameters obtained for ESPaDOnS are consistent with the SPIRou observations. We thus used the veiling values obtained for the 2019 ESPaDOnS data set, fixed the v sin i at the literature  \n<!-- image -->  \nFig. 13. B ℓ curves from LSD profile (top two panels) and from He i D3 line (bottom two panels) .  \n<!-- image -->  \nvalue and let the inference compensate for the eventual error with the non-magnetic parameters. The inferred parameters are summarised in Table 4, and the line fit and inferred magnetic field strength are shown in Fig. 17. The magnetic field strength recovered (2.00 ± 0.03 kG) is significantly lower than the values found for ESPaDOnS observations. As highlighted by Hahlin et al. (2023), the small-scale field might be overestimated when using optical wavelength. In our case, a second explanation might come from the high vmac obtained, probably needed to compensate for an underestimated veiling, which lowers the e ff ect of the magnetic field.", '4. Discussion': "This work aimed at characterising the accretion process of the prototypical EXor, EX Lup, in quiescence, together with its magnetic field at small and large scales. This study confirms that the typical magnetospheric accretion process of cTTSs is ongoing on this system, which seems stable between the two epochs studied (2016 and 2019), with a main accretion funnel flow connecting the disc to the stellar surface. This produces the IPC profile observed in the H lines studied and their modulation with the stellar rotation period. The accretion shock at the stellar surface produces the emission of the NC of the He i D3 and Ca ii IRT lines, which are modulated with the stellar period and  \nanti-correlated with the IPC profiles as expected in the magnetospheric accretion scheme. This is consistent with the maximum IPC profile occurring at the same phase as the He i D3 emitting region ( ϕ ≈ 0.6), indicating a funnel flow aligned with the accretion shock and inherently meaning that the truncation radius is located at the stellar corotation radius. As expected, this phase is also associated with an extremum of the B ℓ , for both LSD and He i D3 measurements, and with the ESPaDOnS ZDI brightness and magnetic topology reconstructions, showing the connection between accretion and magnetic field. To investigate the truncation radius r mag, we used the expression given by Bessolaz et al. (2008):  \nr mag R ⋆ = 2 m 2 / 7 s B 4 / 7 ⋆ ˙ M -2 / 7 acc M -1 / 7 ⋆ R 5 / 7 ⋆ , (6)  \nwhere the Mach number m s ≈ 1, B ⋆ is the equatorial magnetic field strength (given from the dipole strength and the relation of Gregory 2011) in units of 140 G, ˙ Macc is the mass accretion rate in units of 10 -8 M ⊙ .yr -1 , M ⋆ the stellar mass in units of 0.8 M ⊙ , and R ⋆ the stellar radius in units of 2 R ⊙ . As ˙ Macc in quiescence, we used the value before the 2022 outburst given by Wang et al. (2023), 1.8 × 10 -9 M ⊙ .yr -1 . The stellar mass and radius are given by Gras-Velázquez & Ray (2005) (0.6 M ⊙ and 1.6 R ⊙ ), and magnetic obliquity is obtained from ZDI analysis (30 · for ESPaDOnS, 23 · for SPIRou, see Sect. 3.3.1). The dipole strengths needed to obtain r mag = r corot = 8.5 ± 0.5 R ⋆ are thus 2.10 ± 0.46 kG and 1.98 ± 0.43 kG for ESPaDOnS and SPIRou, respectively. Only an estimate of the truncation radius can be obtained because we do not have a precise value of the dipole strength. Indeed, the B ℓ measurements in the He i D3 line and Zeeman intensification values also contain the higher-order components of the magnetic field, and the ZDI spherical harmonic decomposition from LSD might include flux cancellation lowering the large-scale strength obtained. Even if the ZDI results on the 2019 datasets indicate a topology largely dominated by the dipole component, using these values should yield an overestimation of r mag. The results using our magnetic field measurements are summarised in Table 5. The values obtained from the B ℓ and the small-scale field are not consistent with the corotation radius, as expected, except for the SPIRou measurement which can be explained by the smaller recovered field in the infrared domain. However, the optical ZDI values yield truncation radii consistent with the corotation radius, due to the dipole-dominant magnetic topology of the system allowing us to recover a good estimate of the dipole pole strength using ZDI. Here again, the SPIRou value yields a lower truncation radius due to the lower magnetic field strength obtained.  \nThe di ff erence in the results of the various magnetic analyses between the optical and infrared frames in 2019 needs to be discussed. From the B ℓ computed from the LSD profiles, the maximum values obtained are 911 ± 26 and 64 ± 20 G, for ESPaDOnS and SPIRou observations, respectively. These maxima both occurred around ϕ = 0.6, where the Stokes V signature is maximal with ESPaDOnS but almost vanishes with SPIRou, it is thus not surprising to have a large discrepancy here. Concerning the minimum B ℓ values, they reach -155 ± 11 and -178 ± 13 G, for ESPaDOnS and SPIRou observations, respectively, both around ϕ = 0.1 and are thus consistent. This behaviour is also visible on the ZDI reconstruction, where the strong positive radial magnetic field region at phase 0.6 on ESPaDOnS map completely vanishes on SPIRou map. The two qualitative explanations we can provide are the following: (i) this maximum value is located in an optically dark region of the photosphere, which is consistent with simultaneous optical photometry (Kóspál et al., in  \n<!-- image -->  \n<!-- image -->  \nFig. 14. Brightness maps from ESPaDOnS 2016, 2019, and SPIRou data sets ( left , middle , right , respectively) on a flattened polar view. The central dot is thus the pole, the two dotted circles are latitude 60 and 30 · , and the solid circle represents the equator. The black ticks are the rotation phases going clockwise, and the red ticks represent the observed phases. The colour code indicates the brightness on a linear scale, where a value of 1.0 represents the quiet photosphere, values less than one are darker regions and values greater than one are bright.  \n<!-- image -->  \n<!-- image -->  \n<!-- image -->  \nFig. 15. Radial (top row) , azimuthal (middle row) , and meridional (bottom row) magnetic maps from ESPaDOnS 2016, 2019, and SPIRou data sets ( left , middle , and right columns, respectively) on the same flattened polar view as Fig. 14. The colour code is scaling the magnetic field strength, going from dark blue for the strongest negative value to dark red for the strongest positive value.  \n<!-- image -->  \nTable 4. Inference results on the small-scale magnetic field on Ti i lines.  \n.  \n.  \nNotes. The fi denote the filling factor for a i kG magnetic field strength.  \nI  \n/  \nI  \nESPaDOnS 2019  \nFig. 16. Results of the ESPaDOnS Zeeman intensification analysis. Top: Distribution of the average magnetic field of EX Lup for 2016 (left) and 2019 (right) data sets. The solid red line represents the median and the dashed lines represent the 68 % confidence regions. Bottom: Fit to 2016 (left) and 2019 (right) optical Ti i multiplet of EX Lup. The solid red line shows the best fit to the observations in black, while the blue dashed line shows the non-magnetic spectra with otherwise identical stellar parameters. The shaded area marks the region used for the fit.  \n<!-- image -->  \nTable 5. Magnetospheric truncation radius obtained using the various magnetic field measurements of this work.  \nprep.), and thus less contrasted in the SPIRou domain. The magnetic contribution of this region to the Stokes V signal might be thus lowered in the infrared domain if the smaller-scale negative field is obscured in the optical domain. And (ii) the two wavelength domains allowing to trace di ff erent heights in the photosphere, that such a di ff erence might be an indication of a vertical structure of the magnetic field.  \nConcerning the two ZDI reconstructions obtained from the two ESPaDOnS epochs, they seem to point a similar brightness and magnetic radial maps even if the overall topology has evolved from a toroidal- to a poloidal-dominated state, as seen on  \nprevious objects (e.g., DQ Tau A, Pouilly et al. 2023, 2024). The strong positive radial field spot, associated with the dark spot of the stellar brightness and with the dipole pole, is also consistent with the modulation of He i D3 NC radial velocity and B ℓ . However, the latter is pointing to an accretion shock associated with a region with a strong negative field. Given the large amplitude of the He i D3 NC radial velocity variation compared to EX Lup v sin i , the emitting region has probably a very small extent, meaning that its magnetic field information is lost at the photospheric level, explaining this disparity (Yadav et al. 2015).  \nDespite this stable pattern between 2016 and 2019, we observed some disparities in some parameters. Even if the radial velocity and the B ℓ (from LSD and He i D3) modulations are well in phase, their amplitudes are slightly larger in 2019. If the stronger magnetic field obtained in the optical frame, at smalland large-scale, explains the larger amplitude of the B ℓ modulation, the accompanying e ff ect on the radial velocity points to a modulation by the hotspot. This is not a surprising behaviour, but the consistency between the ESPaDOnS and SPIRou radial velocity measurements is. Indeed, the stellar activity e ff ect on the photospheric lines, producing the apparent radial velocity modulation, is a wavelength-dependent phenomenon, an amplitude consistency between the radial velocity measurements in the optical and infrared frames is thus not expected.  \nFig. 17. Same as Fig. 16 for SPIRou IR Ti i multiplet.  \n<!-- image -->  \nTo investigate in our data sets the relation between the stellar activity and the radial velocity modulation, we compared the latter to an activity indicator, the bisector inverse slope (BIS, Queloz et al. 2001). The BIS was computed from the LSD profiles presented in Fig. 10, 11, and 12, and is defined as the difference between the mean velocity of the bisector at the top and at the bottom of the line. On ESPaDOnS profiles, the first 15%, which contains the continuum and the wings, were ignored, as well as the last 15%, where the noise or an activity signature splitting the profile into two parts can a ff ect the computation. The top and bottom regions used to compute the BIS are the 25% top and bottom parts of the remaining profile. For SPIRou profiles, the same conditions were used except that we had to ignore the first 25% of the profiles as it was a ff ected by stronger wings. In the case where the radial velocity modulation is only induced by the stellar activity, the line deformation, indicated by the BIS, is completely responsible for this modulation. This means that a strong linear correlation should appear between the BIS and the radial velocity with, in a perfect situation, a -1-slope and a BIS = 0 kms -1 at the mean velocity.  \nThe BIS versus radial velocity plots are presented in Appendix C. For each data set, the Pearson correlation coe ffi cient indicates a strong anti-correlation (2016: r = -0.81, p-value = 0.005; ESPaDOnS 2019: r = -0.95, p-value = 0.004; SPIRou: r = -0.72, p-value = 0.05). However, the optical measurements show much shallower slopes ( -0.51 ± 0.13 and -0.43 ± 0.07 in 2016 and 2019, respectively), and an intersect far from the mean velocity expected ( -0.34 ± 0.10 and  \n-0.08 ± 0.06). Only the infrared measurements are consistent (slope: -0.68 ± 0.27, intersect: -1.37 ± 0.25), but only thanks to the larger uncertainties, probably due to the lower correlation. Furthermore, the much lower BIS obtained (exclusively negative) in the infrared compared to the optical frame, indicates chromatic e ff ects that are not observed in the radial velocity modulation. Stellar activity is thus probably dominating the radial velocity modulation, but another e ff ect, such as Doppler shift induced by a companion, can not be completely excluded and needs further investigations that are unfortunately beyond the scope of the present work.  \nFinally, we would like to address the strong magnetic field recovered that drives the accretion process on EX Lup. With the B ℓ reaching 4.2 kG in the accretion shock, a dipole strength of 1.9 kG within a large-scale field of 1 kG reaching 4.8 kG locally, and a small-scale field exceeding 3 kG in the optical domain, EX Lup has one of the strongest magnetic field among cTTSs, and the strongest among those with dipole-dominated topology. Such a configuration can set the suitable condition invoked by D'Angelo & Spruit (2010) for their hypothesis of an episodic accretion due to the magnetospheric accretion process itself. Indeed, with such a dipolar strength, the magnetospheric radius is outside but close to the corotation radius (see Table 5). In such a situation, instability arises due to the fact that angular momentum is transferred from the star to the disc (the so-called propeller regime), without being enough to drive an outflow. This magnetic interaction only prevents accretion, pilling up the gas in the inner disc and increasing its pressure, thus forcing the inner edge of the disc to move inward and cross the corotation radius allowing the accretion to occur. Once the gas reservoir has been accreted, the inner edge of the disc moves outward and another cycle starts (see D'Angelo & Spruit 2010, 2011, 2012). This phenomenon is di ff erent from the magnetospheric inflation reported for other cTTSs (e.g., see the studies of AA Tau or V807 Tau by Bouvier et al. 2003; Pouilly et al. 2021, respectively), where a significant di ff erence between the truncation and corotation radii induces a torsion of the magnetic field lines producing a toroidal field that inflates the whole magnetosphere. This inflation goes up to an opening of the magnetic field lines that produces a magnetospheric ejection (Zanni & Ferreira 2013; Pantolmos et al. 2020) before reconnecting to the disc. Such a phenomenon is recurrent and produces an accretion variability as well. Still, the time scale of such a cycle is of the order of the stellar rotation period, far shorter than the gas pilling up invoked by D'Angelo & Spruit (2010), and no signature of magnetospheric inflation, nor ejection, were detected in EX Lup. However, we would like to stress to the reader that no accumulation of matter at the magnetospheric radius was detected either, EXLup seems only to be in the same initial conditions as the latter authors' theory. In addition, the BIS suggest another source for radial velocity variation, such as a companion. Tidal interactions (Bonnell & Bastien 1992) or thermal instabilities in the disc (Lodato & Clarke 2004) induced by a companion are also existing hypotheses for EXor behaviour. Moreover, recent works by Nayakshin et al. (2024b,a) favour the latter instability as the origin of the episodic accretion of FUor objects. Finally, the disc itself is not investigated in this work, instabilities in the disc are thus hypotheses that cannot be excluded (Bell & Lin 1994; Armitage et al. 2001).", '5. Conclusions': "EX Lup is the prototypical EXor-type object whose recurrent bursts and outbursts were previously studied in detail using spec-  \ntroscopy and photometry. However, until this work, no information about its magnetic field was derived, despite its key role in the accretion process of cTTS and its possible origin for episodic accretion. Here we provide the first spectropolarimetric time series, over two epochs (2016 and 2019) and two wavelength domains (optical and infrared), study of EX Lup, the first EXor whose magnetic field is studied.  \nWe confirmed an ongoing magnetospheric accretion process as seen on many cTTSs. It is represented by an accretion funnel flow and an accretion shock corotating with the stellar surface and driven by a kG dipolar magnetic field. The funnel flow seems aligned with the accretion shock, itself located near the magnetic dipole pole, which is consistent with the stable pattern observed between the epochs studied.  \nThe magnetic field of EX Lup has shown some disparities between wavelength domains, much weaker in the infrared. This can be understood as a wavelength dependency of the parameters studied, but can also point to a vertical structure of the magnetic field, as di ff erent wavelengths are tracing di ff erent heights in the photosphere. An expected small-to-large scale e ff ect is also observed, by the di ff erent field strengths recovered using ZDI and Zeeman intensification, but also by the opposite polarity of the field associated with the accretion shock and the one recovered using LSD. This indicates a very small accretion shock, as expected from the low mass accretion rate in quiescence and the large radial velocity variation of the emitting region.  \nFinally, the multi-kG field recovered for EX Lup is pointing to a magnetospheric radius being close but outside the corotation radius. This configuration is suitable for disc instabilities induced by the magnetic field that yield accretion cycles. These cycles might explain the accretion bursts observed on EX Lup, suggesting an inherently episodic magnetospheric accretion process. However, a definite identification of the origin of EXor behaviour is beyond the scope of this paper, and other theories implying the disc itself or a companion cannot be excluded.  \nAcknowledgements. We would like to warmly thanks Oleg Kochukhov for useful discussions about the magnetic field of EX Lup, as well as Colin P. Folsom for his help in using the new version of his ZDIpy package. The SpecpolFlow package is available at https://github.com/folsomcp/ specpolFlow . The PySTEL(L)A package is available at https://github. com/pouillyk/PySTELLA This research was funded in whole or in part by the Swiss National Science Foundation (SNSF), grant number 217195 (SIMBA). For the purpose of Open Access, a CC BY public copyright licence is applied to any Author Accepted Manuscript (AAM) version arising from this submission. Based on observations obtained at the Canada-France- Hawaii Telescope (CFHT) which is operated from the summit of Maunakea by the National Research Council of Canada, the institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The observations at the Canada-France-Hawaii Telescope were performed with care and respect from the summit of Maunakea which is a significant cultural and historic site. 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R., Morin, J., et al. 2015, ApJ, 813, L31\n- Zanni, C. & Ferreira, J. 2013, A&A, 550, A99", 'Appendix A: Cross-correlation matrices of optical emission lines': 'Here we present the cross-correlation matrices of the ESPaDOnS emission lines studied in this work. These matrices are discussed in Sect. 3.2.5.  \nFig. A.1. He i D3 vs. H α (top) , Ca ii IRT (854.2 nm) vs. H α (middle) , and Ca ii IRT (854.2 nm) vs. He i D3 (bottom) correlation matrices for ESPaDOnS 2016 emission lines.  \n<!-- image -->  \nFig. A.2. Same as Fig. A.1 for ESPaDOnS 2019 emission lines.  \n<!-- image -->', 'Appendix B: Stokes I and V fit from ZDI reconstruction': 'In this section, we show the fit of the LSD profiles by the ZDI reconstruction for the three data sets studied. The Stokes I profiles are shown in Fig. B.1 and the Stokes V profiles in Fig. B.2.  \nFig. B.1. Fit (red) of the observed (black) Stokes I profiles from the ZDI reconstruction for ESPaDOnS 2016 (left) , 2019 (middle) and SPIRou (right) observations. The grey-shaded area corresponds to the uncertainties in the observation, and the number at the right of each profile indicates its rotation cycle.  \n<!-- image -->  \nFig. B.2. Same as Fig. B.1 for Stokes V profiles.  \n<!-- image -->', 'Appendix C: BIS vs. radial velocity': 'This appendix presents the analysis of the BIS and radial velocity correlation. These results are discussed in Sect. 4.  \nFig. C.1. BIS vs. RV of ESPaDOnS 2016 (top) , 2019 (middle) , and SPIRou (bottom) observations. The colour code scales the HJD of observations. The red dotted line shows the best linear regression with slope "a" and intercept "b" indicated in legend. The uncertainty on the regression is shown as the grey-shaded area The green dotted line has a slope of -1 and an intercept equal to the mean radial velocity.  \n<!-- image -->'}