Measuring the magnetic fields in the chromospheres of low-mass stars

Domain: physics
Published: 2025-11-29
Editorial outcome: major_revision

Abstract
Magnetic fields in the upper atmospheres of solar-like stars are believed to provide an enormous amount of energy to power the hot coronae and drive large-scale eruptions that could impact the habitability of planetary systems around these stars. However, these magnetic fields have never been routinely measured on stars beyond the solar system. Through decade-long spectropolarimetric observations, we have now achieved the measurements of magnetic fields in the lower and middle chromospheres of three M-dwarfs. Our results indicate that the line-of-sight component of the chromospheric magnetic fields can reach up to hundreds of Gauss, whose sign frequently opposes that of the photospheric field. The measurements highlight the magnetic field complexity and the variation with height close to the surface of these M-dwarfs. They provide critical constraints on the energy budget responsible for heating and eruptions of stellar upper atmospheres, and enable assessments of how stellar magnetic activity may affect exoplanet environments.

Introduction
As a layer of the stellar upper atmosphere, the chromosphere lies above the photosphere and beneath the corona1. It serves as the locus of thermal inversion, where the temperature begins to rise after decreasing from the stellar core to the photosphere. This characteristic demarcates it as more than a transitional layer, transforming it into a site with various types of activity2–5, which significantly influences the space weather conditions surrounding orbiting planets6,7. The magnetic field predominantly governs such activity, storing and releasing energy, thereby significantly affecting the heating processes and efficiently driving large-scale eruptions in the chromosphere and corona8. Consequently, measuring the magnetic field of the upper atmosphere, including the chromosphere, is essential to understanding the physical mechanisms responsible for stellar coronal heating and eruptions9,10.

However, magnetic field measurements in the stellar upper atmosphere are very limited because of the weak signal, even for the Sun11–16. Although there have been attempts at magnetic field measurements in the chromospheres of accreting pre-main-sequence stars17,18, these signals are intertwined with the accretion process, which has a different energy-transfer mechanism compared to a solar-like chromosphere. In the absence of measurements, our knowledge about magnetic fields in the upper atmospheres of distant main-sequence stars other than the Sun relies on extrapolations19,20 of the measured photospheric magnetic fields21,22. However, comparisons between the measured and extrapolated magnetic fields in the solar upper atmosphere often reveal distinct discrepancies, suggesting that the assumptions of extrapolations are invalid at many locations on stars4,13,23. Furthermore, the majority of magnetic energy in stellar atmospheres is stored in small-scale fields24,25, which are crucial for understanding stellar activity mechanisms. Unlike the photosphere, where magnetic structures are more concentrated, chromospheric fields typically exhibit more diffuse patterns26. Solar observations demonstrate that this structural complexity can lead to varying measurement outcomes across the chromosphere27, with magnetic vectors sometimes appearing in opposite directions between chromospheric and photospheric layers in the same region28.

M-dwarfs are low-mass stars with surface temperatures lower than those of the Sun and extended convective envelopes. They are the most common type of star in the solar neighborhood, making up over 60% of all stars, and they host the majority of known exoplanets29. With a mass less than 0.6 M⊙, M-dwarfs have a lifetime longer than that of the Sun, and the habitable zones of their orbiting exoplanets are closer, often less than 0.3 astronomical unit (AU) away from the stars30. The probability of detecting habitable planets is higher around M-dwarfs compared to earlier-type stars. However, the habitability is more likely to be influenced by the magnetic activity originating primarily from the host stars20,31,32. Many M-dwarfs exhibit greatly elevated levels of magnetic activity compared to our Sun, exhibiting frequent superflares with total energies of 1033−38 erg33,34. While the surface magnetic fields of M-dwarfs have been measured over the past decades21,22,35, the energy driving this magnetic activity primarily originates from the magnetic fields in the upper atmospheres, which have not yet been measured36,37. In addition, many active M-dwarfs are fully convective stars, but it is still debated whether the lack of a tachocline will result in a dynamo different from the solar one. Understanding the key physical processes in these dynamos requires constraints from observations of small-scale magnetic field structures in stellar atmospheres, which are still lacking35.

Spectral lines originating at different atmospheric heights provide diagnostic windows into distinct stellar layers. While most absorption lines in stellar spectra form within the photosphere, the unique physical conditions in the chromosphere selectively enhance specific spectral features, particularly the Balmer lines and Calcium II lines. These enhanced emission features serve as valuable probes of chromospheric conditions and magnetic structures1. We analyzed a decade-long compilation of polarized spectra obtained with two high-resolution spectropolarimeters, ESPaDOnS and NARVAL (Methods). The data include the unpolarized normalized intensity (Stokes I, represented by normalization to the continuum Ic) and circular polarization (Stokes V, represented by V/Ic). This study focuses on three active mid to late M-dwarfs38: AD Leo, YZ CMi, and EV Lac, which have masses of 0.31–0.42 solar mass and rotation periods of 2.2–4.4 days (see Supplementary Information, SI hereafter, Table S1). We simultaneously measured the mean longitudinal magnetic field (detailed in Method) of these stars in the photosphere 〈Bp〉 from mean photospheric lines, the lower chromosphere 〈BcL〉 from the Hα line, and the middle chromosphere 〈BcM〉 from the cores of the Calcium II infrared triplet (CaIRT). Figure 1 show examples of Stokes I and V profiles for each star. The anti-symmetric features in the Stokes V profiles are typical signatures of the Zeeman effect39. Figure 2 presents all measurements for the three targets, along with the corresponding rotational phases of the observations40,41. The mean longitudinal magnetic fields in the chromospheres of the three targets can reach several hundred Gauss, comparable to the photospheric fields.

Results
Large-scale magnetic fields
The variation of the measured magnetic fields at different rotational phases can be used to characterize large-scale magnetic field structures40,42. We typically obtained one spectrum per observation night for each star, resulting in low sampling coverage within one rotation cycle. Given that the large-scale photospheric magnetic field structure can vary yearly but generally remains stable within an observing epoch (the few-month window each year during which the targets are visible)43,44, we grouped the data by epochs for further analysis. The time duration of one epoch was at least one month, which is much longer than the stellar rotational period. Therefore, our measurements of the magnetic fields were taken over multiple rotation cycles within each epoch. For each target, we focus on measurements from a marked epoch with full rotational phase coverage: AD Leo observed in 2008, YZ CMi observed in late December 2007 and early 2008, and EV Lac in 2007. We neglect the effect of differential rotation, since it is not detected for the photospheric field in the three targets40. In these epochs, we see that observations at different times form relatively stable patterns in a phase-folded diagram, as shown in Fig. 2 (SI 1B & Fig. S3), which can be fitted by a two-harmonic Fourier series. This stability suggests that the large-scale chromospheric fields co-rotating with the stars were nearly stable within an epoch, with a timescale of at least one month.

Layer-dependent magnetic fields
The different patterns of layers in the same epoch indicate that the large-scale magnetic fields vary with height. As shown in Fig. 2, while the sign of 〈BcL〉 is generally consistent with 〈Bp〉, frequent opposite signs in 〈BcM〉 highlight the different magnetic topologies with height. Among the three stars, AD Leo has the highest percentage of observations where 〈BcM〉 and 〈Bp〉 have opposite signs. Example spectra with the opposite sign between photosphere and middle chromosphere can be found in Fig. 1 for AD Leo and EV Lac, showing the same orientation of the anti-symmetric structure for V profile and opposite orientation of I profile. Disk-integrated spectral profiles average the magnetic field contributions over the entire visible surface of the star, potentially masking localized mixed-polarity magnetic features but providing a comprehensive view of the star’s overall magnetic structures. Considering that the chromospheric layer is close to the surface of a star (0.3% R⊙ for Sun), the variation in polarity and strength at different heights can be attributed to the variation of corresponding large-scale magnetic field structures with height, resulting in different contributions of positive- or negative-polarity fields to the observed signals. The sign change can be reproduced by extrapolation of the photospheric field (Fig. S4 & S5), but detailed variation between the patterns should be attributed to more complex effects close to the surface, like small-scale fields or more diffuse large-scale magnetic loops than predicted. This scenario is different from that on the Sun, where the polarity and topology of the magnetic field from the photosphere to the chromosphere are generally similar27,45.

Complexity of magnetic structure
The correlation between the large-scale magnetic fields, as shown in Table 1 (more details in Table S2), can characterize the complexity of magnetic structure in the upper layers of the star during the marked epoch, or even considering the decade-long observations (see SI 1B and Figs. S6–S9). The difference between the large-scale fields is larger, as indicated by correlation coefficients approaching zero. YZ CMi displays distinct sinusoidal rotational modulation and exhibits the strongest correlation between photospheric and chromospheric fields among our observations, suggesting a more organized and stable magnetic field topology. The modulation patterns and correlations among different layers in the other two stars, EV Lac and AD Leo, exhibit more complex characteristics. In EV Lac, 〈BcL〉 also shows a substantial correlation with 〈Bp〉, similar to YZ CMi. However, there is a notable negative correlation between 〈BcM〉 and the fields in the lower layers. AD Leo exhibits weaker correlations between the layers; magnetic fields in both chromospheric layers appear to be anti-correlated with the photospheric field during the marked epoch, while 〈BcL〉 and 〈BcM〉 are highly correlated. These observations suggest that YZ CMi has a more organized large-scale magnetic field structure than the other two stars (SI 1C). Previous theoretical investigations showed that more complex magnetic fields can support less prominence mass on M-dwarfs, resulting in weaker stellar winds19,46. AD Leo and YZ CMi are both believed to be fully convective stars with a strong dipole magnetic field43,47. Although the topologies are similar, YZ CMi has >30 times stronger stellar wind than AD Leo48. The difference in stellar winds from these M-dwarfs appear to be consistent with our evaluation of magnetic field complexity.

Relationship to flare occurrence and energetics
The occurrence frequency of solar flares has been found to be related to the complexity of magnetic field structures in active regions49. From observations of the Transiting Exoplanet Survey Satellite (TESS), we found that the flare frequency distributions on these stars likely relate to the complexity of the magnetic field (SI 1D and Fig. S10). The correlation analysis described above suggests that AD Leo has the most complex large-scale magnetic field among the three stars. We notice that, compared to the other two stars, AD Leo has fewer low-energy flares and more high-energy flares (especially superflares). The difference in stellar winds from these M-dwarfs appears to be consistent with our evaluation of magnetic field complexity. This suggests that the occurrence frequency of superflares might be related to the complexity of large-scale magnetic fields on M-dwarfs. The potential explanation is that a richer small-scale field on the surface may produce a more complex field, which can lead to increased magnetic energy dissipation (thus more large flares and fewer small flares) and prune the flare frequency distribution.

Methods
Spectropolarimetric observations
The Echelle SpectroPolarimetric Device for the Observation of Stars (ESPaDOnS) is an instrument that includes an achromatic polarimeter. This device has been stationed at the Cassegrain focus of the 3.6 m Canada-France-Hawaii Telescope at the top of Mauna Kea in Hawaii, USA, since early 200553. Meanwhile, NARVAL (not an acronym) is a stellar spectropolarimeter that was designed based on ESPaDOnS and was adapted to the specifics of the 2 m Télescope Bernard Lyot (TBL), situated atop Pic du Midi in southwest France since 200654. Under polarimetric mode, ESPaDOnS and NARVAL provide complete coverage of the 370 to 1050 nm wavelength range and reach a resolution of R=65,000 in a single exposure. During this run, polarimetric exposures are divided into four consecutive subexposures. Each subexposure collects data at different angles of the two half-wave rotatable Fresnel rhombs in the polarimetric module. This sequence is designed to eliminate spurious polarization signatures at first-order55.

In the standard procedures to extract the spectropolarimetric data, “Null” spectra are obtained by pair-processing sub-exposures corresponding to identical azimuths of the retarding plate or Cassegrain bonnette42, in parallel with polarized spectra. These “Null” spectra are not expected to contain any signal and should be pure noise if there is no spurious polarized signature in the V profile. We checked our V profile if any significant signal can be seen in the “Null” profile and whether it is similar to the V profile in order to exclude spurious signatures. The total efficiency of the instrument, including the spectrograph and CCD detector, is around 12%. The software package LibreESpRIT42,56 is used for extracting the normalized, reduced data for unpolarized and polarized spectra corresponding to each observing sequence. All reduced spectra used in this work can be extracted from the PolarBase database57.

We only used the data from 2005-2019 observed with polarimetric mode, with the peak S/N of the Stokes V signal over 60. The basic stellar parameters of the selected M-dwarfs used in this work are taken from the study dedicated to their large-scale surface magnetic topologies40. The observation time is phased with the ephemeris: HJD=HJD0+P rot E, where the zero point of heliocentric Julian dates is chosen equal to HJD0=2,453,950.0 and the rotational periods have been estimated from Zeeman-Doppler Imaging (ZDI) inversion40. Due to the Earth’s orbit, ground-based observations for our targets have only a few-month window per year, known as an epoch. A detailed observation log, including the information, can be found in Source Data.

Magnetic fields measurement from the photosphere to the middle chromosphere
Using the Least Squares Deconvolution (LSD, detailed below) method42, we extracted average photospheric information from thousands of photospheric spectral lines and produced mean Stokes I and V profiles for all the unpolarized and polarized spectra, respectively. The longitudinal (line-of-sight projected) component of the stellar-disk integrated photospheric magnetic field 〈Bp〉 was estimated from the first-order moment of the Stokes V profile based on the center-of-gravity (COG) method42,58 decribed below. The cores of the Calcium II infrared triplet (CaIRT) at wavelengths 849.8 nm, 854.2 nm, and 866.2 nm, typically formed in the middle chromosphere1,59, were analyzed. We applied the LSD method to these lines and subtracted the photospheric components to obtain the pure chromospheric Stokes I and V profiles (Fig. S1). The mean longitudinal magnetic field in the middle chromosphere 〈BcM〉 was estimated from these profiles using the COG method. Simultaneously, we measured the lower chromospheric longitudinal magnetic field 〈BcL〉 from the pure emitting hydrogen Hα line at 656.3 nm (Details in SI 1A & Fig. S2)45,60.

Mean line profiles derived by the Least-Squares Deconvolution method
Least-squares deconvolution (LSD) is a cross-correlation method, enabling the acquisition of average unpolarized and polarized line profiles with an improved signal-to-noise ratio. This enhancement uses spectral features formed in approximately the same disk region and height42. Line lists for photospheric analysis are generated from spectrum synthesis via model atmospheres extracted from the Vienna Atomic Line Database (VALD)61. We predominantly employed stronger lines corresponding to the photospheric absorption of an early-to-mid M spectral type star, aligning with the characteristics of our targets. Lines that deviate from the average behavior of the photosphere, particularly chromospheric emissions and telluric lines, are excluded from the list. Approximately 5000 intermediate to strong atomic absorption spectral lines are simultaneously utilized to extract the average polarization information in line profiles. This approach yields typical noise levels of about 0.06% (relative to the unpolarized continuum level) per 1.8 km/s velocity bin, resulting in a multiplex gain in Stokes V S/N of ~10 for the photospheric mask. We applied the same approach to the three lines of CaIRT and obtained an S/N gain of ~1.6. LSD is not applied to the Hα line. We analyze Hα individually since its broad Balmer wings break the LSD narrow-line approximation.

Mean longitudinal magnetic field
In observations of these low-mass stars, we obtain only disk-integrated Stokes profiles. Consequently, the quantity we derive with the center-of-gravity (COG) technique is the longitudinal field, 〈Bℓ〉, i.e., the first moment of the circular polarization (Stokes V) profile that represents the line-of-sight (LOS) integral over the visible stellar hemisphere. Because it is an LOS projection, 〈Bℓ〉 can take positive or negative values depending on whether the net signed flux points toward or away from the observer; its magnitude should not be interpreted as the absolute field strength at a specific atmospheric height. COG method is applicable for both the spatially resolved and unresolved profiles, and 〈Bℓ〉 can be obtained from the first moment of the normalized Stokes V39,42,58:

〈Bℓ〉 = (2.14 × 10−11 λ0 g eff)−1 ∫ υ V(υ) dυ / ∫ [Ic(υ) − I(υ)] dυ, (1)

where υ is the velocity to the line center, λ0 is the wavelength of the spectral line, c is the speed of light, and geff is the effective Landé factor. Note that this formula works for a magnetic field (kilo-Gauss) that is even stronger than that in the case of the weak field approximation (WFA), which assumes that the Zeeman splitting is significantly smaller than the line’s Doppler width. Under WFA, circular polarization signatures are proportional to 〈Bℓ〉 in the first order and can have the same estimation of 〈Bℓ〉 as the COG method for weak fields with a similar expression58. In a strong field, WFA breaks. However, the COG method is still valid. The corresponding uncertainty mainly results from the photon noise of Stokes I and V profiles. This formula is used to estimate the mean photospheric longitudinal field from the mean photospheric line profiles obtained using the LSD method, with typical geff = 1.24 and λ0 = 650 nm. In the chromosphere, the COG method is still valid and can be applied to chromospheric spectral lines62. In our work, we take geff = 1 and λ0 = 656.28 nm for the Hα line. For the LSD profiles of CaIRT formed in the middle chromosphere, we have typical values of geff = 0.968 and λ0 = 858.56 nm, taking from the LSD process, and the geff of each line is taken from the VALD.

Subtract the photospheric components from CaIRT line profiles
Information about the middle chromosphere is extracted from the Ca IRT. Both the Stokes I and V signals of these three lines can be contaminated by photospheric signals from the stellar disk17. Considering the weak signal and similar profile shapes of the triplet, we applied the LSD method to the CaIRT and obtained the mean CaIRT Stokes I and V profiles. Assuming the wing of the broad absorption band (slightly outside the emission core) anchors to the photosphere in unpolarized Stokes I profiles, we use a Lorentzian model to fit the photospheric component17. This component has a broad enough Doppler profile, where the Zeeman splitting is negligible, and WFA is applicable (see Methods: Mean longitudinal magnetic field). Under the first-order, the corresponding photospheric model in circular polarized Stokes V profile is a derivation of the Lorentzian model multiplied by a fixed coefficient of −4.67 × 10−12 geff 〈Bp〉 λ0 c, where 〈Bp〉 is the mean longitudinal photospheric field estimated from the LSD. Figure S1 shows an example for modeling the photospheric components in Stokes I & V. The residual profiles for both Stokes I & V are used to estimate the mean longitudinal middle chromospheric field 〈BcM〉. The emission core of CaIRT should be purely from the chromosphere, and we measured its magnetic fields with the COG method.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "I was asked by the editor to review the spectropolarimetric aspect of the paper. Since I do not have expertise in stellar physics, but only in spectropolarimetry, I refrain from commenting on the stellar physics aspects of this paper. That being said, I have the following comments to the spectropolarimetric observations presented in the paper. The authors use Stokes I and V data from two high-res spectropolarimeters, ESPaDOnS and NARVAL, for three active, mid to late M-dwarfs. The data come from the PolarBase data base, described in [54]. In this paper, the usage of PolarBase data for M-dwarfs is explicitly mentioned in Section 8, where the biases are discussed, which result from the selection of line masks for the used LSD models. The LSD method works especially well when many lines can contribute to the signal. For the photospheric line, the authors gained a factor of about 10 using the LSD method. For the lower chromosphere, only three spectral lines could be used, and a S/N increase by a factor of 2 could be achieved (although I would have expected at most an increase by a factor sqrt(3), and not 2). For the middle chromosphere, the LSD method obviously could not be applied, although not mentioned explicitly in the paper. Although the LSD method is useful to increase the S/N ratio, it is not efficiently reducing instrumental effects when used with very few spectral lines only. The small signals in Stokes V for the chromospheric lines make a careful consideration of the cross-talk from Stokes I to V mandatory. Semel et al. [52] describe this problem, and the approach of removing spurious polarimetric signals. In their Sect. 2 they state that one \"... should be careful about crosstalk from intensity to circular polarization.\" This careful consideration is not present in the paper. I lack in the paper a clear description of how the authors dealt with the issue of possible crosstalk from I to V. The shape of the Stokes V signals presented in Figure 1 for the lower and middle chromosphere could also be a result of this crosstalk. This suspicion is supported by the fact that the magnetic field strength derived for the middle chromosphere for all three stars is almost indistinguishable. Before a discussion about the magnetic field configuration on the three M dwarfs is possible, the reliability of the underlying data must be confirmed. The method section needs to be expanded to include a description of the crosstalk. An analysis similar to Sect 2.4. in [52] needs to be presented with the result clearly demonstrating that the presented chromospheric Stokes V profiles are not affected by the crosstalk from I to V.", "reviewer_2": "The magnetic activity of M-dwarfs is an important astronomical question for two major reasons. Firstly, it aids the understanding of the stellar activity, as these stars cover a different region in the stellar parameter space than the Sun. Any general dynamo theory should be able to explain the activity of both the Sun and other late-type stars. Secondly, most potentially habitable exoplanets are expected to orbit late-type dwarfs. Stellar magnetic activity is known to have a great impact on habitability. This is especially for M-type dwarfs, as the habitable zones are much nearer to the host star, than in the case of the Sun\nThe manuscript \"Measuring the magnetic fields in the upper atmospheres of low-mass stars\" is a study of the changes in the large-scale magnetic field of three M-type dwarfs from the photosphere to the mid chromosphere. The authors apply a novel approach, where they study the V-spectropolarimetric signal of lines originating in the photosphere, lower and mid-chromosphere. This approach reveals how the large-scale magnetic field changes through the atmosphere of these stars. In this respect, the study is truly novel, and represents a major step forward in understanding the magnetic activity of M-dwarfs. The reviewers are not aware of any previous studies, where such methods have been applied on these kinds of stars. The analysis is largely based on studies of the solar chromosphere. This is, of course, the only reasonable approach, although there are indications that the solar analogy cannot always be applied on stellar magnetic activity of more active stars (e.g., Kochukhov et al., 2023, A&A 680, L17).\nWe do not find any flaws in the analysis, other than the missing significances of the correlations (see below). The fact that the photosphere contributes to the Ca II IRT lines is challenging, but this seems to be properly addressed. However, we suggest changes in some of the conclusions.\nOur recommendation is that this study should be published in Nature Communications, but after some significant revision detailed below.\nI. Major issues:\nA major critic on the manuscript, is that the authors do not enough address the small-scale magnetic field in the objects. The small-scale field is actually only mentioned once in the manuscript (line 108). This is, of course, partly outside the scope of the study, since measuring the small-scale field vector is way beyond the achievable resolution. Still, it is a well-known fact, that most of the magnetic energy will reside in the small-scale field (see, e.g. Kochukhov et al., 2020, A&A 635, A142). Furthermore, estimates of the percentages of the large-scale vs. total magnetic field are available for the three targets in this study (see Kochukhov, 2021). The small-scale field of the Sun is particularly important in understanding the formation of sunspots, active regions, flares and CME:s. The same would be expected for M-type dwarfs. Addressing this question could easily be done by reformulating relevant sections in the manuscript. Furthermore, the ratio between the large-scale and total magnetic field for the targets could be listed in a relevant table.\nIn many places in the manuscript, the authors write about polarity reversal and changing magnetic topology with height. On the other hand, at least some of the observations can be explained by reasonably simple models (S8) involving large-scale magnetic field loops viewed from different angles. Thus, the different field topologies would rather be an apparent phenomenon depending on geometry and viewing angle. Usually in the context of active late-type stars, polarity reversals refer to possibly cyclic changes in the large-scale radial field, and changing topology is used to describe, e.g., the variations in the percentages of poloidal vs. toroidal and axisymmetric vs. non-axisymmetric field. Thus, the authors should clarify this difference and consider using other wording.\nConcerning the correlation between the magnetic fields in different atmospheric layers, the situation is not as clear, as the text in the manuscript seems to indicate. For YZ CMi the case looks quite clear, and the strong correlation can be seen in the plot S6. For AD Leo, the correlations between the chromosphere and photosphere (S5 panels a and b) are rather mixed, and it is hard to visually judge how significant the correlations within the different epochs are. The case is the same for EV Lac (S7, panels b and c). Especially in cases where the number of points is relatively small, correlations should always be given with estimates of significance. Such estimates should definitely be included in this study.\nIn most cases, the data would be sufficient for Zeeman-Doppler imaging (ZDI). At least for some of this data, ZDI maps have been calculated (e.g., Morin et al. 2008). We acknowledge the point (line 90), that extrapolating the magnetic fields from ZDI maps to the chromosphere will probably not give valid result. This would be best demonstrated by using a ZDI map of one of the targets, extrapolating the field and comparing this with the current measurements. Did the authors consider doing this?\nThe question of differential rotation is not discussed at all in the manuscript. This is apparently because in previous studies, none of these stars show indications of any strong differential rotation (Morin et al., 2008). However, on scales of months, even a very small differential rotation may have a significant effect. Did the authors consider differential rotation? The possible shear could easily be tested using, e.g., the estimates (with error bars) derived by Morin et al. (2008). In the case of YZ CMi the measurements follow quite well a double sinusoidal fit (S3). Here the stability (against, e.g., differential rotation) of the large-scale magnetic field could be tested by checking whether there is a correlation between the deviations of the points and time difference from the mean HJD.\nA small, but principally important question is whether the dynamos of fully convective stars differ that much from the solar dynamo (lines 107-8). Here, there are competing paradigms, which lead to different claims. In the flux transport dynamo paradigm, the tachocline plays an important role, which would mean that fully convective stars would have a different type of dynamo. However, the distributed dynamo paradigm does not rely on a tachocline. There are, as far as the reviewers know, no observations indicating a significant change in dynamos between fully convective stars, and stars with a convective envelope and radiative core. This would support the latter paradigm. Here, we do not demand the authors to choose a side, but rather request that this sentence should be reformulated to accommodate both possibilities.\nII. Other comments and suggestions for the main paper:\nAge estimates for the three stars are not given anywhere in the manuscript. If estimates exist, we suggest they should be included in the manuscript. As for the stellar parameters, these are taken from the previous study by Morin et al. (2008). We note that there are some discrepancies in the values of v sin i, P_rot and i for EV Lac and YZ CMi: It seems that the inclination for these two targets should be larger than 60 degrees. Did the authors consider this? On the other hand, we do not expect that this would alter the results.\nThe exposure times are not given for the observations. We urge the authors to report these, or at least their range. It would also be good to give a range for the S/N, rather than the typical noise level (line 588) for all measurements, i.e. the photospheric LSD lines, the CaII IRT and Hα.\nIt is often a custom to present null profiles of Stokes V-spectropolarimetry. We would encourage to include examples of these for each star in a similar way as in Fig. 1.\nIn the abstract (line 53) and elsewhere in the manuscript, the concept of “direct” measurement is used. We would refrain from using this wording here, as the methods used involve modelling. Still, this would not reduce the importance of this study, as any estimate of the magnetic field above the photosphere for these targets is novel.\nOn lines 72-73 we suggest replacing “through the photosphere” with “to the photosphere”.\nOn lines 78-80: “… measuring the magnetic field of the upper atmosphere, including the chromosphere, is essential to understanding the physical mechanisms responsible for stellar coronal heating and eruptions.” This is certainly true, but in the Sun the small-scale field is crucial for understanding these phenomena. One can expect that the same holds for other late-type stars.\nOn line 91 there are three references. Comparisons between extrapolation and measurements are included in Yang et al. (2020). But are the other references also relevant here?\nOn line 96: In “M-dwarfs generally have a lifetime longer”, the word “generally” seems unnecessary, since M-dwarfs always have orders of magnitude longer lifetimes than the Sun.\nOn lines 98-99: “These conditions …” should be rephrased. This is not only a question of the number and ages of M-dwarfs, as each M-dwarf in the solar neighbourhood seem to have more planets than earlier type stars.\nOn line 107: “believed to be fully convective” should be rephrased, as this is an established fact.\nParagraph starting on line 111: We suggest considering to first explain the theory, i.e. the formation of the spectral lines in the different layers, and then the observations.\nOn line 140, reference [41] should probably be [40].\nOn line 153, the wording “the stars were nearly stable within an epoch, with a timescale of at least one month” raises the question on what ground this epoch length was chosen. Maybe this could be described with a sentence?\nOn lines 164-167 the wording “the variation in polarity and strength at different heights can be attributed to the drastic variation of corresponding large-scale magnetic field structures with height” should be rephrased. Do the observations really support “drastic variations”, or can they be explained by magnetic field loops of different heights and viewed from different angles?\nOn lines 215-218, the sentence “This scenario also indicates … intense UV and X-ray radiation” is, of course, true. However, again most of the magnetic energy is known to be contained in the small-scale field, which is also probably contributing much more to flares and CMEs.\nWe suggest adding “LSD profile”, “Hα”, and “Ca II IRT” to the plot titles in Fig. 1.\nOn lines 571-572, it says “rotational periods have been estimated from the modulation of the photospheric Zeeman signatures”. However, in the paper by Morin et al. (2008) P_rot was determined in the ZDI inversions.\nOn lines 617-618. The g_eff for the chromospheric lines are given. References for these values would be useful.\nIII. Comments and suggestions for the supplement:\nOn line 15, the sentence “exhibited a small projected rotational velocity, resulting in low v sin i values” is a tautology and should be rephrased.\nOn line 40-43, the claim “generally form one repeating pattern in the phase-folded diagram” is only half the truth. For YZ CMi, this is clearly the case. But for AD Leo, there does not seem to be any clear pattern. For EV Lac, the pattern is quite vague especially for the lower chromosphere. So, this claim should be rephrased.\nOn lines 49-50, the statement “large-scale photospheric structure appears more axisymmetric (axisymmetric modes > 90%) in AD Leo than the other two stars” may have to be rephrased. At least in Morin et al. (2008), AD Leo and YZ CMi have nearly the same axisymmetry.\nOn lines 88-89, the sentences with “with m giving the order of the spherical harmonics mode and describe. In practice, αℓm is complex coefficients corresponding to the radial poloidal field” seem to have some words missing. Correct also the “m” to the right format.\nOn line 95, correct the format of B_l.\nOn lines 116-118, there is a clear need for references. With what method were the flares detected and their energies determined?\nOn lines 123-124, the sentence “It is still unclear why there are fewer low-energy flares in the situation of more complex fields”, should be rephrased to take into account the small-scale field.\nIn Fig. S1 axis titles and units are missing in the plot.\nIn Fig. S3 the units (d) are missing from the scale on the right side.\nIn the extended data table S1, references for the stellar masses would be useful. Also, the parameter ß could be explained also here.\nSeveral corrections should be made in Tables S2-S4. Firstly, the HJD and phase-values should be given with consistent accuracy. Here the possible effect of the exposure times should be considered. Furthermore, using the value “-999.0” when the number of measurements is insufficient is not a good choice, better to use a blank or “N/A” in these cases.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "(1) The authors report measurements of intensity profiles and circular polarization profiles for spectral lines which are formed in principle at three different altitudes in the atmosphere of a flare star. They interpret those line profiles in terms of magnetic field strength (including algebraic sign of the longitudinal component) at the three different altitudes. The reported field strengths are probably to be interpreted as \"mean\" fields at the various altitudes, although the word \"mean\" can hardly be interpreted as a simple arithmetic mean over the visible disk of the star at the time the data were obtained: with 3 different vector components of the field present at all locations of the stellar surface, the \"mean\" must involve a complicated process.\n(2) The authors claim that, because they have used three lines which are formed at different altitudes, their results allow them to state confidently how the field strength varies as a function of height. But this seems improbable: the 3 lines are formed at heights which are separated by no more than (perhaps) 0.3% of the stellar radius: given that the fields in the star are generated by a global dynamo, it is highly unlikely that any significant changes in \"physically real\" field strength can be expected to occur over such a tiny range of radial distance.\n(3) The authors report (with great confidence) in Tables S2-S4 that the field actually changes sign between photosphere and low chromosphere, or between lower and mid chromosphere. But I find this hard to believe: their confidence, it seems to me, is not warranted when I see the serious \"noise\" level in the V profiles which they show in Figure 1 in the text (for Halpha and also for CaIRT) and also in Figure S2 (for Halpha)."}, "author_response": "- Reviewer #1’s comments -\n1.1 Reviewer #1: The authors use Stokes I and V data from two high-res spectropolarimeters, ESPaDOnS and\nNARVAL, for three active, mid to late M-dwarfs. The data come from the PolarBase data base, described in\n[54]. In this paper, the usage of PolarBase data for M-dwarfs is explicitly mentioned in Section 8, where the\nbiases are discussed, which result from the selection of line masks for the used LSD models.\nThe LSD method works especially well when many lines can contribute to the signal. For the photospheric\nline, the authors gained a factor of about 10 using the LSD method. For the lower chromosphere, only three\nspectral lines could be used, and a S/N increase by a factor of 2 could be achieved (although I would have\nexpected at most an increase by a factor sqrt(3), and not 2). For the middle chromosphere, the LSD method\nobviously could not be applied, although not mentioned explicitly in the paper.\nResponse:\nWe thank the referee for carefully examining the spectropolarimetric part of our work. We appreciate the\nthoughtful concerns about potential cross-talk in polarimetry detection.\nIn the Method section, we added more explanation to tell the difference between the lines we used for the\nmeasurements of the three layers. We used a more accurate value of ~1.6 for the signal-to-noise ratio\nimprovement for Ca IRT. We also added a sentence explaining that we use H-alpha individually.\nIn SI, we added the S/N of all three lines (LSD, H-alpha, Ca IRT) for the magnetic field measurement for each\nobservation in the observation journal Table S2-S4.\n1.2 Reviewer #1: Although the LSD method is useful to increase the S/N ratio, it is not efficiently reducing\ninstrumental effects when used with very few spectral lines only.\nThe small signals in Stokes V for the chromospheric lines make a careful consideration of the cross-talk from\nStokes I to V mandatory. Semel et al. [52] describe this problem, and the approach of removing spurious\npolarimetric signals. In their Sect. 2 they state that one \"... should be careful about crosstalk from intensity to\ncircular polarization.\" This careful consideration is not present in the paper.\nI lack in the paper a clear description of how the authors dealt with the issue of possible crosstalk from I to V.\nThe shape of the Stokes V signals presented in Figure 1 for the lower and middle chromosphere could also be\na result of this crosstalk. This suspicion is supported by the fact that the magnetic field strength derived for the\nmiddle chromosphere for all three stars is almost indistinguishable.\nBefore a discussion about the magnetic field configuration on the three M dwarfs is possible, the reliability of\nthe underlying data must be confirmed.\nThe method section needs to be expanded to include a description of the crosstalk. An analysis similar to Sect\n2.4. in [52] needs to be presented with the result clearly demonstrating that the presented chromospheric Stokes\nV profiles are not affected by the crosstalk from I to V.\nResponse:\nWe agree that the crosstalk should be carefully considered and presented in the paper. For brevity, we omitted\nthese details from our initial manuscript. We now give the \"Null\" profiles of each V profile in Fig 1. They are\ncomparison profiles that should include no real signal but only noise and crosstalk. We believe this new\ninformation can address the concerns about the reliability of the signal.\nThe apparent similarity in magnetic field strength in the middle chromosphere of the original Fig. 1 was\ncoincidental. Subsequent analysis (see Fig. 2) confirms that the field strength varies significantly across\nepochs. In the revised manuscript, we replace Fig.1 with the observations of higher S/N and different field\nstrength measurements.\nBelow, we provide more detailed theoretical explanations and observational evidence to show why the\nantisymmetric V profiles detected in H-alpha and Ca IRT are unlikely to be attributed solely to crosstalk.\n- Theoretical aspect:\nsymmetric or more complex shape of the V spectra generated by the Zeeman effect of the magnetic field from\nthe star.\nSect. 2.1. in Semel et al.[58] gives the expression of the error in V profile as a function of I profile:\n𝛿𝑉 = 𝑀 ∗𝐼\n𝑜𝑏𝑠 41 𝑖𝑛\nThe chromospheric lines H-alpha and Ca IRT can be seen as a single line within a short wavelength range, and\n𝑀 can be treated as a constant. Therefore, the crosstalk from I-to-V should be a linear superposition of I\n41\nprofile on V. In this scenario, if the V signal includes I-to-V crosstalk, the V profile should closely resemble\nthe I profile, which can only be in a symmetric shape in our observation.\nAdditionally, Sect. 3.4 of Donati et al. 1997 [40] gives the optimal extraction of polarization spectra of\nspectropolarimeters ESPaDOnS and NARVAL, including the analysis and solution to address the concern of\nthe reviewer. According to [40], one should check the “Null” profile to see if any potential signature in the\npolarization profile is indeed real. Details are described in the “Observational aspect” below.\n- Observational aspect:\nIn the standard procedures to extract the spectropolarimetric data of ESPaDOnS and NARVAL, “Null” spectra\nare “obtained by pair-processing sub-exposures corresponding to identical azimuths of the retarding plate or\nCassegrain bonnette” (Donati et al. 1997, Sect. 3.4), in parallel with polarized spectra. These “Null” spectra\nare not expected to contain any signal and should be pure noise if there is no spurious polarized signature in\nthe V profile (PolarBase database, Petit et al. 2014 [60]). In practice, we can check if any significant signal\ncan be seen in the “Null” profile and whether it is similar to the V profile in order to exclude spurious\nsignatures. We checked all “Null” spectra of H-alpha, Ca IRT LSD, and photospheric LSD in our work, finding\nno significant spurious signal, and that is why we use these three lines to measure the magnetic field. We added\nthe Null profile in Fig. 1 of the manuscript, and added sentences in the Method part.\nIn the left panels of Fig.A1 below, we provide the “Null” profiles (N hereafter) of the AD Leo data in Fig.1 in\nthe manuscript. Additionally, there are many other chromospheric lines in the spectra, and some of them can\nbe strong in I. We didn’t use them to measure the magnetic field because their V profiles are affected by I-to-\nV crosstalk. As shown in the right panels of Fig. A1, we illustrate an example of how crosstalk appears in the\nV profile of Ca II K 393.37 nm (note that it is not Ca IRT), and an unexpected signal can be clearly seen in the\nNull profile.\nSignal without crosstalk\nSignal with crosstalk\nNoise\nCrosstalk\nFigure A1. (Left panels) The I, V, and N profiles (green lines, N/Ic) of three layers (data identical to AD Leo in Fig. 1 of the manuscript). The N\nprofiles should show random noise without any significant signal, and anti-symmetric shapes in the V profile can be detected even when the data is\nnoisy. Therefore, the inferred uncertainty of the magnetic field can reflect the noise level. (Right panels) The I, V, and N profiles of the Ca II K 393.37\nnm line from the same observation. A significant signal can be detected in the N profiles, indicating the presence of cross-talk in this line. A similar\nsymmetric profile can also be observed in the V profile, which confirms that the V profile results from spurious signatures (including crosstalk).\n2\n2. Reviewer #2 (Remarks to the Author):\nThe magnetic activity of M-dwarfs is an important astronomical question for two major reasons. Firstly, it aids\nthe understanding of the stellar activity, as these stars cover a different region in the stellar parameter space\nthan the Sun. Any general dynamo theory should be able to explain the activity of both the Sun and other late-\ntype stars. Secondly, most potentially habitable exoplanets are expected to orbit late-type dwarfs. Stellar\nmagnetic activity is known to have a great impact on habitability. This is especially for M-type dwarfs, as the\nhabitable zones are much nearer to the host star, than in the case of the Sun\nThe manuscript \"Measuring the magnetic fields in the upper atmospheres of low-mass stars\" is a study of the\nchanges in the large-scale magnetic field of three M-type dwarfs from the photosphere to the mid\nchromosphere. The authors apply a novel approach, where they study the V-spectropolarimetric signal of lines\noriginating in the photosphere, lower and mid-chromosphere. This approach reveals how the large-scale\nmagnetic field changes through the atmosphere of these stars. In this respect, the study is truly novel, and\nrepresents a major step forward in understanding the magnetic activity of M-dwarfs. The reviewers are\nnot aware of any previous studies, where such methods have been applied on these kinds of stars. The analysis\nis largely based on studies of the solar chromosphere. This is, of course, the only reasonable approach, although\nthere are indications that the solar analogy cannot always be applied on stellar magnetic activity of more active\nstars (e.g., Kochukhov et al., 2023, A&A 680, L17).\nWe do not find any flaws in the analysis, other than the missing significances of the correlations (see below).\nThe fact that the photosphere contributes to the Ca II IRT lines is challenging, but this seems to be properly\naddressed. However, we suggest changes in some of the conclusions.\nOur recommendation is that this study should be published in Nature Communications, but after some\nsignificant revision detailed below.\nResponse:\nWe thank Reviewer #2 for the thoughtful and encouraging comments. We have carefully revised the\nmanuscript accordingly: (1) small-scale fields are carefully considered in the explanation of the results; (2) a\nPFSS extrapolation based on an existing ZDI map has been added to illustrate why simple extrapolations\ncannot account for our chromospheric measurements; and (3) wording and minor issues have been clarified\nthroughout. We believe these changes fully address the reviewer’s concerns and strengthen the paper.\nI. Major issues:\n2.1.1 Reviewer #2 : A major critic on the manuscript, is that the authors do not enough address the small-scale\nmagnetic field in the objects. The small-scale field is actually only mentioned once in the manuscript (line\n108). This is, of course, partly outside the scope of the study, since measuring the small-scale field vector is\nway beyond the achievable resolution. Still, it is a well-known fact, that most of the magnetic energy will\nreside in the small-scale field (see, e.g. Kochukhov et al., 2020, A&A 635, A142). Furthermore, estimates of\nthe percentages of the large-scale vs. total magnetic field are available for the three targets in this study (see\nKochukhov, 2021). The small-scale field of the Sun is particularly important in understanding the formation\nof sunspots, active regions, flares and CME:s. The same would be expected for M-type dwarfs. Addressing\nthis question could easily be done by reformulating relevant sections in the manuscript. Furthermore, the ratio\nbetween the large-scale and total magnetic field for the targets could be listed in a relevant table.\nResponse:\nWe thank the reviewer for highlighting the importance of the small-scale magnetic field. In the revised\nmanuscript, we include the small-scale magnetic field as a potential explanation of magnetic field topology at\ndifferent heights.\nWe expand the introduction (l. 88-90) and discussion (l.222-225, 236) to explicitly highlight the importance\nof small-scale magnetic fields in stellar magnetic activity. In the SI, we include the percentages of the large-\nscale vs. total magnetic field from Kochukhov (2021) in Table S1.\nThese additions clarify the role of the small-scale magnetic field and its connection to our chromospheric\nresults.\n3\n2.1.2 Reviewer #2: In many places in the manuscript, the authors write about polarity reversal and changing\nmagnetic topology with height. On the other hand, at least some of the observations can be explained by\nreasonably simple models (S8) involving large-scale magnetic field loops viewed from different angles. Thus,\nthe different field topologies would rather be an apparent phenomenon depending on geometry and viewing\nangle. Usually in the context of active late-type stars, polarity reversals refer to possibly cyclic changes in the\nlarge-scale radial field, and changing topology is used to describe, e.g., the variations in the percentages of\npoloidal vs. toroidal and axisymmetric vs. non-axisymmetric field. Thus, the authors should clarify this\ndifference and consider using other wording.\nResponse:\nWe appreciate the suggestions for using more precise terminology to describe our observations and\ninterpretations. We agree that clarifying our use of terms like \"polarity reversal\" and \"changing magnetic\ntopology\" will enhance the clarity and accuracy of our discussion.\nWe replaced instances of \"polarity reversal\" with \"opposite sign\" to more accurately describe the observed\nchanges in the direction of the magnetic field. This term better conveys the idea that the magnetic field switches\nfrom one polarity to another without implying a dynamic or time-dependent reversal.\nAs the reviewer mentioned, \"changing magnetic topology\" should relate to the change of energy distribution\nof the magnetic map. We did a preliminary test on the ZDI map of AD Leo for Ca IRT (middle chromosphere),\nshowing that the main variation between photospheric and chromospheric fields is azimuthal and meridional\nfields, with a lower percentage of poloidal and axisymmetric fields. Since the magnetic field should be\ncontinuous from the photosphere to the chromosphere, we believed that it is reasonable to use 'topology change'\nto describe different patterns of magnetic field variation at various heights.\nWe won’t include this ZDI result because it is outside the scope of this work, and the different magnetic field\nvariation patterns can lead to a similar conclusion. We think these results can address this concern.\n2.1.3 Reviewer #2: Concerning the correlation between the magnetic fields in different atmospheric layers,\nthe situation is not as clear, as the text in the manuscript seems to indicate. For YZ CMi the case looks quite\nclear, and the strong correlation can be seen in the plot S6. For AD Leo, the correlations between the\nchromosphere and photosphere (S5 panels a and b) are rather mixed, and it is hard to visually judge how\nsignificant the correlations within the different epochs are. The case is the same for EV Lac (S7, panels b and\n4\nc). Especially in cases where the number of points is relatively small, correlations should always be given\nwith estimates of significance. Such estimates should definitely be included in this study.\nResponse:\nWe replace the less clear description with a more neutral formulation (l. 189-191). We added a new table\n(Table S5) about the correlation's performance with p and r, where the p-value gives the significance of\ncorrelations and r is the correlation coefficient. The p-values of all data indicate a high significance (0-0.05)\nof the correlation for EV Lac and YZ CMi and moderate significance for AD Leo (0.05-0.17).\n2.1.4 Reviewer #2: In most cases, the data would be sufficient for Zeeman-Doppler imaging (ZDI). At least\nfor some of this data, ZDI maps have been calculated (e.g., Morin et al. 2008). We acknowledge the point (line\n90) that extrapolating the magnetic fields from ZDI maps to the chromosphere will probably not give a valid\nresult. This would be best demonstrated by using a ZDI map of one of the targets, extrapolating the field, and\ncomparing this with the current measurements. Did the authors consider doing this?\nResponse:\nWe thank the referee for the suggestion. Since the chromosphere is very close to the photosphere (0.3% R*),\nthe extrapolated field with the PFSS method is almost the same as the ZDI map of the photosphere. Below,\nwe provide a comparison between the photospheric radial field of AD Leo and PFSS at ~1.003R*,\nand this\nfigure is included in the revised manuscript as Figure S5.\nWe can see a significant difference between ⟨𝑩 ⟩ at different heights in (c), and the opposite sign can be\n𝒍\nreproduced with a larger inclination in (d). However, this extrapolation shows almost no variation in the\npattern, indicating a correlation coefficient close to 1.\nHere is an estimation of the change of three components of the magnetic field from the photosphere to the\nchromosphere for AD Leo. Consider the PFSS equations from Lang et al. 2014 :\n𝑩 = −∑𝑵 ∑𝒍 [𝒍𝜶 𝒓𝒍−𝟏−(𝒍+𝟏)𝜷 𝒓−(𝒍+𝟐)]𝑷 (𝒄𝒐𝒔𝜽)𝒆𝒊𝒎𝝓,\n𝒓 𝒍=𝟏 𝒎=𝟏 𝒍𝒎 𝒍𝒎 𝒍𝒎\n𝑩 = −∑𝑵 ∑𝒍 [𝜶 𝒓𝒍−𝟏+𝜷 𝒓−(𝒍+𝟐)] 𝒅 𝑷 (𝒄𝒐𝒔𝜽)𝒆𝒊𝒎𝝓,\n𝜽 𝒍=𝟏 𝒎=𝟏 𝒍𝒎 𝒍𝒎 𝒅𝜽 𝒍𝒎\n𝑩 = −∑𝑵 ∑𝒍 [𝜶 𝒓𝒍−𝟏+𝜷 𝒓−(𝒍+𝟐)]𝑷 (𝒄𝒐𝒔𝜽) 𝒊𝒎 𝒆𝒊𝒎𝝓.\n𝝓 𝒍=𝟏 𝒎=𝟏 𝒍𝒎 𝒍𝒎 𝒍𝒎 𝒔𝒊𝒏𝜽\nTaking the ZDI result of Morin et al. (2008), ~60% energies in 𝓵 = 𝟏, ~10% energies in 𝓵 = 𝟐, and ignoring\nhigher orders, we can roughly estimate the field change from r=1 to r=1.003 by less than 1% for each\ncomponent. Since the variation is more or less the same for each (𝜽,𝝓), meaning that we cannot have a\n5\nsignificant change in the topology by extrapolating the field, especially the radial field. As shown in the ZDI\nof chromosphere result shown in the preview comment, we should expect a substantial change of 𝜷 and\n𝒍𝒎\n𝜸 , which is not considered in the extrapolation.\n𝒍𝒎\n2.1.5 Reviewer #2: The question of differential rotation is not discussed at all in the manuscript. This is\napparently because in previous studies, none of these stars show indications of any strong differential rotation\n(Morin et al., 2008). However, on scales of months, even a very small differential rotation may have a\nsignificant effect. Did the authors consider differential rotation? The possible shear could easily be tested using,\ne.g., the estimates (with error bars) derived by Morin et al. (2008). In the case of YZ CMi the measurements\nfollow quite well a double sinusoidal fit (S3). Here the stability (against, e.g., differential rotation) of the large-\nscale magnetic field could be tested by checking whether there is a correlation between the deviations of the\npoints and time difference from the mean HJD.\nResponse:\nAs suggested by the reviewer, we tested the differential rotation of YZ CMi in our work with the uncertainty\nlist in Morin et al. (2008) of 𝒅𝜴 = 𝟎±𝟏.𝟒𝒎𝒓𝒂𝒅𝒅−𝟏 and 𝑷 = 𝟐.𝟕𝟕𝟗±𝟎.𝟎𝟎𝟒𝒅. By setting different\nHJD0 from 2453950 (we used) to 2454800 for the 2008 epoch (around 2454500), 𝒅𝜴 at 𝝌𝟐 minimum\nslightly increase with HJD0, with a maximum 𝒅𝜴 < 𝟎.𝟕𝒎𝒓𝒂𝒅𝒅−𝟏, which is smaller than the uncertainty.\nApplying this value is not significant enough to impact the modeled Stokes V profile. Furthermore, the\ndouble sinusoidal fit also works well for AD Leo and EV Lac, suggesting that this fitting is unlikely to be\ncaused by the sheared map (we add a double sinusoidal fitting for all three stars in Fig. 2). As the differential\nrotation of the photospheric field is not in the scope of this work, and exploring its potential effect on the\nchromospheric field would be complex, we tend to neglect the impact of differential rotation, but add a\nsentence to mention it in the revised manuscript.\n2.1.6 Reviewer #2: A small, but principally important question is whether the dynamos of fully convective\nstars differ that much from the solar dynamo (lines 107-8). Here, there are competing paradigms, which lead\nto different claims. In the flux transport dynamo paradigm, the tachocline plays an important role, which would\nmean that fully convective stars would have a different type of dynamo. However, the distributed dynamo\nparadigm does not rely on a tachocline. There are, as far as the reviewers know, no observations indicating\na significant change in dynamos between fully convective stars, and stars with a convective envelope and\nradiative core. This would support the latter paradigm. Here, we do not demand the authors to choose a side,\nbut rather request that this sentence should be reformulated to accommodate both possibilities.\nResponse:\nThank you for pointing out the debate about whether the dynamo of M-dwarfs is similar to the Sun. We have\nreformulated the sentence to mention both possibilities (l.108-110):\n\"many active M-dwarfs are fully convective stars, but it is still debated whether their dynamos rely on a\ntachocline, which is different from the solar dynamo.\"\nII. Other comments and suggestions for the main paper:\n2.2.1 Reviewer #2: Age estimates for the three stars are not given anywhere in the manuscript. If estimates\nexist, we suggest they should be included in the manuscript. As for the stellar parameters, these are taken from\nthe previous study by Morin et al. (2008). We note that there are some discrepancies in the values of v sin i,\nP_rot and i for EV Lac and YZ CMi: It seems that the inclination for these two targets should be larger than\n60 degrees. Did the authors consider this? On the other hand, we do not expect that this would alter the results.\nResponse:\nWe add the age estimation listed in Engle 2024 (ApJ, 960, 1, id.62) in Table S1.\nOther parameters can calculate inclination larger than the given value in Morin et al. 2008 since they used 60\ndegrees to prevent mirroring effects in ZDI. We did a Chi2 minimum test to check the inclination of EV Lac\nand YZ CMi. The exact value of inclination is not sensitive to the results between 50 and 80 degrees.\nwould result in a large uncertainty estimation of inclination. The vsin i value of YZ CMi and EV Lac\nis >12% smaller than that of Morin et al. (2008) with CARMENES observations (Tal-Or et al. 2018), which\nresults in a smaller estimation of inclination.\nAs commented by the reviewer, the inclination value would not alter the results. Considering the larger\nuncertainty of inclination estimation, we tend to keep the value from Morin et al. (2008) to have a consistent\nstellar parameter source.\n2.2.2 Reviewer #2: The exposure times are not given for the observations. We urge the authors to report these,\nor at least their range. It would also be good to give a give a range for the S/N, rather than the typical noise level (line\n588) for all measurements, i.e. the photospheric LSD lines, the CaII IRT and Hα.\nIt is often a custom to present null profiles of Stokes V-spectropolarimetry. We would encourage to include\nexamples of these for each star in a similar way as in Fig. 1.\nResponse:\nWe updated the observational log Table S2-S4 of three stars, including information on exposure times and\nsignal-to-noise ratio of Stokes V observations. We included the Null profile in Fig.1 and some explanations\nof it in the Method section.\n2.2.3 Reviewer #2: In the abstract (line 53) and elsewhere in the manuscript, the concept of “direct”\nmeasurement is used. We would refrain from using this wording here, as the methods used involve modelling.\nStill, this would not reduce the importance of this study, as any estimate of the magnetic field above the\nphotosphere for these targets is novel.\nResponse:\nWe used \"direct\" since the polarimetric signal is the most direct method to detect the magnetic field in solar\nand stellar magnetic fields. We acknowledge that it may not be precise for the measured values. We rephrase\nthe sentences and remove \"direct\".\n2.2.4 Reviewer #2: On lines 72-73 we suggest replacing “through the photosphere” with “to the photosphere”.\nResponse:\nDone\n2.2.5 Reviewer #2: On lines 78-80: “… measuring the magnetic field of the upper atmosphere, including the\nchromosphere, is essential to understanding the physical mechanisms responsible for stellar coronal heating\nand eruptions.” This is certainly true, but in the Sun the small-scale field is crucial for understanding these\nphenomena. One can expect that the same holds for other late-type stars.\nResponse:\nWe added more discussion about small-scale fields in the revised manuscript as mentioned in Comment\n2.1.1.\n2.2.6 Reviewer #2: On line 91 there are three references. Comparisons between extrapolation and\nmeasurements are included in Yang et al. (2020). But are the other references also relevant here?\nResponse:\nThe other two references, Ishikawa et al. (2021) and Solanki et al. (2003) do not confirm this result directly\nbut provide indirect results to support the variation of magnetic loops. We change the reference here for more\naccuracy.\n2.2.7 Reviewer #2: On line 96: In “M-dwarfs generally have a lifetime longer”, the word “generally” seems\nunnecessary, since M-dwarfs always have orders of magnitude longer lifetimes than the Sun.\nResponse:\nWe remove \"generally\" here.\n2.2.8 Reviewer #2: On lines 98-99: “These conditions …” should be rephrased. This is not only a question of\nthe number and ages of M-dwarfs, as each M-dwarf in the solar neighbourhood seem to have more planets\nthan earlier type stars.\nResponse:\nhabitable planets around M-dwarfs than earlier type stars.\"\n2.2.9 Reviewer #2: On line 107: “believed to be fully convective” should be rephrased, as this is an established\nfact.\nParagraph starting on line 111: We suggest considering to first explain the theory, i.e. the formation of the\nspectral lines in the different layers, and then the observations.\nResponse:\nWe followed the suggestions, rephrased the sentences (l.108-110), and added information on line formation\n(l. 113-116).\n2.2.10 Reviewer #2: On line 140, reference [41] should probably be [40].\nResponse:\nIt is right, we have corrected this.\n2.2.11 Reviewer #2: On line 153, the wording “the stars were nearly stable within an epoch, with a timescale\nof at least one month” raises the question on what ground this epoch length was chosen. Maybe this could be\ndescribed with a sentence?\nResponse:\nWe modified l.151-152 as suggested.\n2.2.12 Reviewer #2: On lines 164-167 the wording “the variation in polarity and strength at different heights\ncan be attributed to the drastic variation of corresponding large-scale magnetic field structures with height”\nshould be rephrased. Do the observations really support “drastic variations”, or can they be explained by\nmagnetic field loops of different heights and viewed from different angles?\nResponse:\nWe acknowledge that, according to the simulations of PFSS results, even a small variation can result in our\nobservation. We have now reformulated the sentences.\n2.2.13 Reviewer #2: On lines 215-218, the sentence “This scenario also indicates … intense UV and X-ray radiation” is, of course, true. However, again most of the magnetic energy is known to be contained in the\nsmall-scale field, which is also probably contributing much more to flares and CMEs.\nResponse:\nWe add the possibility of small-scale fields in l. 218-221.\n2.2.14 Reviewer #2: We suggest adding “LSD profile”, “Hα”, and “Ca II IRT” to the plot titles in Fig. 1.\nResponse:\nWe followed the suggestion and modified the plot titles in Fig.1\n2.2.15 Reviewer #2: On lines 571-572, it says “rotational periods have been estimated from the modulation of\nthe photospheric Zeeman signatures”. However, in the paper by Morin et al. (2008) P_rot was determined in\nthe ZDI inversions.\nResponse:\nThanks for pointing out this unclear expression. We tried to avoid the word \"ZDI\" in the original manuscript\nfor potential readers who are not familiar with this method. Now that we add an explanation about this\n2.2.16 Reviewer #2: On lines 617-618. The g_eff for the chromospheric lines are given. References for these\nvalues would be useful.\nResponse:\nWe added explanations of the g_eff of Ca ITR at l. 651. More detailed information can be found in the\ncorresponding .lsdout of each observation in the data code.\nIII. Comments and suggestions for the supplement:\n2.3.1 Reviewer #2: On line 15, the sentence “exhibited a small projected rotational velocity, resulting in low v\nsin i values” is a tautology and should be rephrased.\nWe have removed \"projected rotational velocity, resulting in low\" to simplify the expression.\n2.3.2 Reviewer #2: On line 40-43, the claim “generally form one repeating pattern in the phase-folded diagram”\nis only half the truth. For YZ CMi, this is clearly the case. But for AD Leo, there does not seem to be any clear\npattern. For EV Lac, the pattern is quite vague especially for the lower chromosphere. So, this claim should be\nrephrased.\nResponse:\nWe have rephrased the expression by adding 'For AD Leo EV Lac, their patterns are less clearer than YZ\nCMi, but they can be fitted by double sine function as shown in Figure 2'.\n2.3.3 Reviewer #2: On lines 49-50, the statement “large-scale photospheric structure appears more\naxisymmetric (axisymmetric modes > 90%) in AD Leo than the other two stars” may have to be rephrased. At\nleast in Morin et al. (2008), AD Leo and YZ CMi have nearly the same axisymmetry.\nResponse:\nWe agree the comment of reviewer. We have removed this paragraph since it is not helpful to support the\nexplanation of the large-scale magnetic field with height. New Figure S5 and l.57-62 have provided related\ninformation.\n2.3.4 Reviewer #2: On lines 88-89, the sentences with “with m giving the order of the spherical harmonics\nmode and describe. In practice, αℓm is complex coefficients corresponding to the radial poloidal field” seem\nto have some words missing. Correct also the “m” to the right format.\nResponse:\nThanks for pointing out the typo, and we have added more details about αℓm.\n2.3.5 Reviewer #2: On line 95, correct the format of B_l.\nResponse:\nWe now use uniform symbols to represent the values of B_l.\n2.3.6 Reviewer #2: On lines 116-118, there is a clear need for references. With what method were the flares\ndetected and their energies determined?\nResponse:\nWe detected the flares from TESS data using the method described by Xing et al. 2024. We added the\nreferences and provided more information on these flare detections.\n2.3.7 Reviewer #2: On lines 123-124, the sentence “It is still unclear why there are fewer low-energy flares in\nthe situation of more complex fields”, should be rephrased to take into account the small-scale field.\nResponse:\nWe remove old expressions and add the possibility that small-scale fields can convert the energy of low-\nenergy flares to magnetic dissipation (l. 124-127).\n2.3.8 Reviewer #2: In Fig. S1 axis titles and units are missing in the plot.\nResponse:\nWe added the missed titles and units.\n2.3.9 Reviewer #2: In Fig. S3 the units (d) are missing from the scale on the right side.\nResponse:\nFixed.\n2.3.10 Reviewer #2: In the extended data table S1, references for the stellar masses would be useful. Also, the\nparameter ß could be explained also here.\nResponse:\nWe have reformed Table S1, and these values have been included.\n2.3.11 Reviewer #2: Several corrections should be made in Tables S2-S4. Firstly, the HJD and phase-values\nshould be given with consistent accuracy. Here the possible effect of the exposure times should be considered.\nbetter to use a blank or “N/A” in these cases.\nResponse:\nWe used '-' to represent an invalid value. The statistical information has been moved to a new table, Table S5.\n2.4 Reviewer #2 (Remarks on code availability):\nReviewer #2: The codes worked fine when downloading. However, some comments are written with Chinese\nletters, making them challenging to comprehend.\nResponse:\nWe now use English everywhere in the code and have added more comments. The new version of the code,\nincluding the scripts for new Fig.1 and Fig.2, has been uploaded.\n3. Reviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the\nNature Communications initiative to facilitate training in peer review and to provide appropriate recognition\nfor Early Career Researchers who co-review manuscripts.\n10\n4. Reviewer #4 (Remarks to the Author):\n4.1 Reviewer #4: The authors report measurements of intensity profiles and circular polarization profiles for\nspectral lines which are formed in principle at three different altitudes in the atmosphere of a flare star. They\ninterpret those line profiles in terms of magnetic field strength (including algebraic sign of the longitudinal\ncomponent) at the three different altitudes. The reported field strengths are probably to be interpreted as \"mean\"\nfields at the various altitudes, although the word \"mean\" can hardly be interpreted as a simple arithmetic mean\nover the visible disk of the star at the time the data were obtained: with 3 different vector components of the\nfield present at all locations of the stellar surface, the \"mean\" must involve a complicated process.\nResponse:\nWe appreciate the reviewer's thoughtful comments about the interpretation of \"mean\" fields in our\nmeasurements. We would like to clarify the established nature of the \"mean longitudinal magnetic field\"\nconcept in stellar magnetic field studies.\nThe term \"mean longitudinal magnetic field\" (Bl in stellar physics), Solar Mean Magnetic Field (SMMF), and\nGeneral Magnetic Field (GMF) have been standard tools for characterizing stellar magnetic fields (Babcock\n1949) and solar magnetic fields (Hale 1913) for over a century. This approach remains widely used in studying\nmagnetic fields of stars (Kochukhov 2021) and the Sun (Rescigno et al. 2024).\nOur measurements follow established procedures in stellar polarimetry (Donati & Collier Cameron 1997,\ndetailed in the Method section). The mean longitudinal field represents a \"disk-integrated\" quantity (as\ndiscussed in Kochukhov 2021, Sect. 2.3). For stars with minimal spatial resolution in spectral observations,\nthis is often the only accessible magnetic field parameter. While cancellation effects are inherent to this\napproach, it effectively characterizes stellar (include Sun-as-star) magnetic strength levels (Saar 1996, Donati\n& Landstreet 2009, Reiners 2012, Bose & Nagaraju 2018), and interplanetary magnetic fields (Scherrer et al.\n1972, Bilenko 2020, Antonucci 2020).Multiple measurements at different rotational phases further allow\ncharacterization of the star's large-scale magnetic topology (Donati & Landstreet 2009, Reiners 2012,\nKochukhov 2021).\nWe have clarified the measurement methodology and physical interpretation of this quantity in the revised\nmanuscript (lines 622-630).\n4.2 Reviewer #4: The authors claim that, because they have used three lines which are formed at different\naltitudes, their results allow them to state confidently how the field strength varies as a function of height. But\nthis seems improbable: the 3 lines are formed at heights which are separated by no more than (perhaps) 0.3%\nof the stellar radius: given that the fields in the star are generated by a global dynamo, it is highly unlikely that\nany significant changes in \"physically real\" field strength can be expected to occur over such a tiny range of\nradial distance.\nResponse:\nThere might be a misinterpretation of the physical aspect of the term “mean longitudinal measured field,\"\nas a\ncontinuing concern regarding comment 4.1. To our knowledge, although the magnetic field is generated by\nstellar dynamo, it can present a complex structure close to the surface, even for a small height (complexity of\nchromospheric field can refer Lagg et al. 2017, Sect. 5.2). Below, we would like to emphasize the observational\nevidence from the Sun that this quantity can always vary significantly from the photosphere to the\nchromosphere.\nFor instance, Ishikawa et al. (2021) used observations from the space spectropolarimeter CLASP2 to map the\nvariation of the magnetic field from the photosphere to the upper chromosphere of the Sun. This observation\nreveals significant variations in the longitudinal magnetic field at different altitudes, similar to our work.\nMoreover, the SMMF of the chromosphere is, on average, around 0.6 SMMF of the photosphere, and it also\nchanges significantly with solar rotation (Vishnu et al. 2023). The polarity reversal of the longitudinal magnetic\nfield between the photosphere and the chromosphere is also revealed by observations of the solar active region\n(Mathur et al. 2023).\nExplanations of these phenomena in the solar context typically relate to the rapid spread of magnetic field lines\nin the chromosphere (Solanki 1992), which does not conflict with the concept of “global dynamo.” This\nphenomenon can be modeled through a high-resolution 3D radiative MHD simulation (Martínez-Sykora et al.\n11\n2019) and is in line with the Fontenla-Avrett-Loeser (FAL) atmospheric model (Pietrow et al. 2020). And\nsmall-scale magnetic field simulation in fully convective stars (Yadav et al. 2021).\nTo further clarify this problem, we provided the PFSS modeling of the AD Leo observation in the selected\nepoch, as new information in SI 1B and Figure S5 (also shown below). The radial field of the photosphere and\nthe PFSS field at (1.003R ) can be similar (as commented by the reviewer), but the measured LOS field can\n*\nhave an opposite sign. The performance of this result is similar to what we observe with YZ CMi, meaning\nthat even if we consider a \"global dynamo\", we can still observe the significant change of the mean longitudinal\nfield.\n4.3 Reviewer #4: The authors report (with great confidence) in Tables S2-S4 that the field actually changes\nsign between photosphere and low chromosphere, or between lower and mid chromosphere. But I find this\nhard to believe: their confidence, it seems to me, is not warranted when I see the serious \"noise\" level in the V\nprofiles which they show in Figure 1 in the text (for Halpha and also for CaIRT) and also in Figure S2 (for\nHalpha).\nResponse:\nWe appreciate Reviewer #4’s concern regarding the apparent noise in the Stokes V profiles and its impact on\nthe accuracy of our longitudinal field measurements. We acknowledge that Figure 1 and S2 do not necessarily\nrepresent our highest signal-to-noise (S/N) data, and we have used figures with above-mean S/N in the new\nFigure 1, as well as the corresponding comparison profile (Null profile).\nWe want to stress that our measurements of field strength apply the center-of-gravity (COG) method, and\nrandom noise in individual data points is minimized through an integration across the entire line profile. The\nuncertainty of the measured longitudinal field is then derived using standard error propagation principles.\nTherefore, the uncertainty we present with the measurements correctly reflects the noise level.\nFor further confirmation of the observational evidence of opposite sign of fields, even in cases where the S/N\nis weak, the antisymmetric shape of the Stokes V profile provides a robust indicator of the field’s polarity via\nthe Zeeman effect (Degl’Innocenti & Landolfi 2004). Practically, we assign a value of a=1 emission line in\nStokes I (and a=-1 for absorption); while b=1 denotes a sine-like shape in the V profile (blue side low, red side\nhigh) and b=-1 indicates its inverse. The product a*b then determines the sign of the magnetic field (+1 for\npositive polarity and -1 for negative). Figure 4.3.1 illustrates this reliable yet straightforward procedure,\n12\nunderlining how we can accurately identify polarity—if we can recognize the antisymmetric shape in V—and\nthereby establish confidence in any observed “opposite sign”.\nBelow, we provide two spectra in Figure 4.3.2 with lower apparent noise, which could better address the\nconcern of sign reversal between the photosphere and lower chromosphere (AD Leo - 2007-06-25); between\nthe lower chromosphere and middle chromosphere (EV Lac - 2005-09-18). These examples provide additional\nconfirmation of the polarity reversals inferred by our measurements.\nFigure 4.3.1. Sketch for the simple method to determine\nthe sign of the magnetic field. The samples are modified\nfrom Degl'Innocenti & Landolfi (2004) Fig 9.18. The\nupper panels show Stokes I profiles, while the lower\npanels display the corresponding Stokes V profiles with\nthe Zeeman effect. Solid and dashed lines represent\ndifferent magnetic field situations, which do not affect\nthe sign determination. Green arrows highlight the\nantisymmetric shape of V profiles. By taking a = -1 for\nthe absorption line of Stokes I and b = -1 for the negative\nsine-like V profile (left), we can determine a positive\nsign for the magnetic field using a * b = 1. Conversely,\nwe obtain a negative sign with a = -1 and b = 1 in the\nright panels.\nFigure 4.3.2. Evidence of sign reversals between layers,\ndetermined by the simple method shown in Fig. A2. Grey\nlines are Stokes I profiles, colored lines are V profiles, red\nhorizontal lines mark the zero flux of V profiles. Green\narrow highlight the antisymmetric shape of V profiles.\n(Upper panels) Polarity reversal between the lower\nchromosphere (negative field) and the middle\nchromosphere (positive field). (Lower panels) Polarity\nreversal between the photosphere (negative field) and\nlower chromosphere (positive field).\npleased that several of the major concerns have been addressed, and the manuscript is\nnow recommended for publication pending minor revisions.\nREVIEWER COMMENTS\nReviewer #1:\n1.1 I thank the authors for addressing my criticism about the reliability of the Stokes signals.\nThe Null-spectra analysis is indeed a good method to confirm that the measurements are not\naffected by crosstalk. I agree with the authors and find my point (1.2 Reviewer #1)\nsufficiently addressed.\nHowever, I was surprised to see the big changes in Figure 1 between the original and the\nrevised submission. The authors say that the \"apparent similarity in magnetic field strength\nin the middle chromosphere of the original Fig. 1 was coincidental\", and present\n\"observations of higher S/N and different field strength measurements\".\nWhere do these higher S/N observations come from? How is it possible that the new values\nfor the field strengths are far outside the one-sigma values given in the original version?\nI did not find a mention in the paper about these new observations. The magnetic field\nvalues in the new version are vastly different from the old version. This is not raising\nconfidence in the results, and I expect a clear explanation for this discrepancy.\nIn Figure 2 the authors added a Fourier series fit to the data, \"showing the strong periodical\nsignal of the magnetic field variation\". I see at most one period, therefore one cannot make a\nstatement about how 'strong' the periodicity is.\nAs in the previous report, I refrain from commenting on the stellar physics aspects of this\npaper.\nResponse:\nThank you for the evaluation of Figures 1&2 and for raising the questions.\nIn this work, we have used observations of these three stars from 2005 to 2019. In the\nSupplement Materials, Tables S2-S4 give the Journal of observations, including the\nsignal-to-noise ratio (S/N) of line profiles of each observation and corresponding\nmagnetic field measurements. For each star at each atmosphere layer, we simply show\none line profile as an example in Figure 1.\nIn the original version of the manuscript, we set mainly four criteria on selecting the\nprofiles shown in Figure 1, in order to include as much information as possible for\nreaders:\n1) The profile should be observed in the selected epochs (2008 for AD Leo and YZ\nCMi, 2007 for EV Lac).\n2) The polarity signals should be simple enough to recognize the positive and negative\npolarities.\n3) The photospheric field and the lower chromospheric field should have the same\npolarity.\n4) The lower and middle chromospheric fields should have opposite polarities.\nUnder these conditions, we finally chose the line profiles of AD Leo observed on 2008-\n02-10, YZ CMi on 2008-02-11, and EV Lac on 2007-07-29 to show in Fig.1. The S/N of\nthese profiles is moderate due to the exposure time and weather conditions. Across all\nof our measurements, only a few observations reveal a middle chromospheric field\nclose to 195G, but we have coincidentally selected three of them.\nIn the current version, we think that there is no need to meet the above four criteria\nwhen selecting example profiles for Fig. 1. So we decide to show line profiles with\nhigher S/N in Figure 1 (AD Leo on 2007-06-25, YZ CMi on 2007-01-27, and EV Lac on\n2006-08-09), for better illustration of the observed polarity signals in the\nchromosphere. The dates of observations are mentioned in the caption of Figure 1.\nWe agree that \"strong\" may not be suitable in the sentence, so we have removed it.\nReviewer #1 (Remarks on code availability):\n1.2 I tried to run the code in Docker Desktop for MacOS (M2, 15.5 (24F74)). Unfortunately,\nthe code did not produce any output and stopped with many error messages. I attached the\nlogfile of my run (output.log).\nResponse:\nThanks for evaluating the reproducibility of the code. The\ncode works on Visual Studio Code and MacOS (M1Max,\n15.6), which should not create much difference between\n(M2, 15.5). We tested the code with Python 11.\nWe reviewed the output.log provided by the reviewer, and\nit is identical to the 'results/output’ file in the CodeOcean\nframework (as shown in the figure on the right). The\nmessages are all normal output information and warnings\n(mostly related to missing fonts). By default, the result\nfigures should be generated under the ‘result’ folder\n(/fig1.jpg and /fig2.png), as shown in the right figure.\nPlease let us know if these files are generated on your\ncomputer.\nReviewer #2:\nAs stated in the first review, the manuscript presents a novel approach to estimate the\nmagnetic field and its polarity in different atmosphere layers of three M-type dwarfs: The\nphotosphere, the lower and the mid chromosphere. This represents a significant contribution\nto our knowledge of stellar magnetic activity in M-type dwarfs.\nWe thank the authors for taking into account the main critics from our previous report and\nimproving the manuscript accordingly. We thus recommend that the manuscript would be\naccepted for publication in Nature Communications with minor revisions. The suggested\nrevisions are listed below.\nResponse:\nWe are very grateful to the reviewers for their positive scientific assessment of our\nwork. Their careful review of the manuscript has greatly helped us improve its quality.\n2.1. Line 111-112:\n“… whether their dynamos rely on a tachocline…”\nThis should be rephrased, as fully convective stars do not have a tachocline. It could, e.g.,\nbe “… whether the lack of a tachocline will result in a dynamo different from the solar one.\"\nResponse:\nWe agree that a fully convective star does not have a tachocline, and we rephrased the\nsentence as suggested.\n2.2. Line 219-221:\n\"We notice that, compared to the other two stars, AD Leo has fewer low-energy flares and\nmore high-energy flares (especially superflares), but not a general weakening of all kinds of\nflares.\"\nThis should be rephrased to clarify what the authors mean with \"weakening\".\nResponse:\nThe 'weakening' in our manuscript means a decrease in the frequency of all flares\nacross the entire energy spectrum. We now think this half-sentence is redundant and\nhave removed it from the manuscript.\n2.3. Lines 700-703:\n” …we take g_eff = 1 and λ0 = 656.28nm for the Hα line. For the LSD profiles of Ca IRT\nformed in the lower chromosphere. We have typical values of g_eff = 0.968 and λ_0 =\n858.56nm, taking from the LSD process, and the g of each line is taken from the VALD.”\nThe text \"chromosphere. We\" should be \"chromosphere, we\". The last “g” should be\nchanged to “g_eff”. The λ_0 and g_eff for the photospheric LSD lines should be given here.\nWere all the g_eff values taken from VALD? (If not then give the references.)\nResponse:\nThanks for the suggestions. We have included the typical g_eff=1.24 and λ_0 =650 nm\nfor photospheric lines. All values are taken from the VALD. Other corrections have\nbeen done as suggested.\n2.4. SI line 015:\n“…exhibit a small…”\nMaybe change to “… have small …”\nResponse: Done as suggested.\n2.5. SI line 081:\n\"r\" and \"p\" should be in math font.\nResponse: Done as suggested.\n2.6. SI lines 107, 108 and 111:\nThe term \"polarity reversal\" is still used here. The authors should consider using another\nterminology.\nResponse:\nThank you for pointing out this. We have now replaced 'polarity reversal' by 'opposite\npolarity'.\n2.7. Author's response, page 4:\n\"We did a preliminary test on the ZDI map of AD Leo for Ca IRT (middle chromosphere),\nshowing that the main variation between photospheric and chromospheric fields is\nazimuthal and meridional fields.\"\nThe authors show a ZDI map of the lower chromosphere using only the three Ca IRT lines.\nThe small number of lines, compared to the number used for the photosphere, raises the\nquestion of how reliable this map is. Furthermore, a well-known fact is that while the\nradial field can be reasonably well reconstructed using just Stokes V-profiles, especially the\nmeridional field is not well reconstructed. But since this is outside the scope, there is no\nneed to further discuss this.\nResponse:\nThank you for sharing ideas about the ZDI map for Ca IRT. We agree that the\nmeridional field in our ZDI reconstruction cannot provide useful information. At this\nstage, the preliminary ZDI can only offer a rough conclusion that the magnetic\nstructure in the chromosphere should have a visible difference from the photosphere,\nwhich can be directly obtained from the B pattern variation. Due to the small number\nl\nof available chromospheric lines, we agree that the ZDI reconstruction of the\nchromospheric field requires more careful consideration and testing. We have briefly\nmentioned this limitation in l.85-89 of the supplementary materials, “It is unclear\nwhether the small number of available chromospheric lines can be used to create a\nreliable map. Nevertheless, some ideas from ZDI can still be useful for testing the field\ncomplexity.”\nReviewer #2 (Remarks on code availability):\nWe reviewed the code when the first version was submitted, and did not see it necessary to\nrepeat.\nResponse:\nThanks for evaluating the reproducibility of the code.\nReviewer #3:\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports.\nThis is part of the Nature Communications initiative to facilitate training in peer review and\nto provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nResponse:\nThanks for evaluating the manuscript.\nReviewer #4:\nThe authors of the paper have considered in detail the objections which I noted in my\noriginal report. The authors have provided cogent replies to my objections.\nI therefore now consider the paper acceptable for publication.\nResponse:\nWe are very grateful to the reviewer for recommending publication of our paper. We\nwould like to thank the reviewer again for the previous comments, which helped us\nimprove the quality of the manuscript.\nReviewer #1:\nI thank the authors for answering my questions. I agree with the publication of the manuscript in its\ncurrent form.\nI was still not able to run the software without errors, but this may have to do with my specific\nconfiguration on my laptop, and is not relevant to my positive evaluation of the manuscript.\nResponse:\nWe sincerely thank Reviewer #1 for the feedback and positive assessment of our manuscript. Regarding\nthe difficulty in running the software, we appreciate the reviewer’s feedback and patience. We have\nrechecked our code and suspect that the errors may stem from differences in Python and SciPy versions\nacross environments. To facilitate reproducibility, we have added a comprehensive README to the Code\nOcean repository detailing the exact environment (Python, SciPy, and related dependencies) used in our\ntests. We hope this will help mitigate configuration issues, and we remain happy to assist further if any\nproblems persist."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I thank the authors for addressing my criticism about the reliability of the Stokes signals. The Null-spectra analysis is indeed a good method to confirm that the measurements are not affected by crosstalk. I agree with the authors and find my point (1.2 Reviewer #1) sufficiently addressed.\nHowever, I was surprised to see the big changes in Figure 1 between the original and the revised submission. The authors say that the \"apparent similarity in magnetic field strength in the middle chromosphere of the original Fig. 1 was coincidental\", and present \"observations of higher S/N and different field strength measurements\".\nWhere do these higher S/N observations come from? How is it possible that the new values for the field strengths are far outside the one-sigma values given in the original version?\nI did not find a mention in the paper about these new observations. The magnetic field values in the new version are vastly different from the old version. This is not raising confidence in the results, and I expect a clear explanation for this discrepancy.\nIn Figure 2 the authors added a Fourier series fit to the data, \"showing the strong periodical signal of the magnetic field variation\". I see at mose one period, therefore one cannot make a statement about how 'strong' the periodicity is.\nAs in the previous report, I refrain from commenting on the stellar physics aspects of this paper.\n(Remarks on code availability)\nI tried to run the code in Docker Desktop for MacOS (M2, 15.5 (24F74)). Unfortunately, the code did not produce any output and stopped with many error messages. I attach the logfile of my run (output.log).", "reviewer_2": "As stated in the first review, the manuscript presents a novel approach to estimate the magnetic field and its polarity in different atmosphere layers of three M-type dwarfs: The photosphere, the lower and the mid chromosphere. This represents a significant contribution to our knowledge of stellar magnetic activity in M-type dwarfs.\nWe thank the authors for taking into account the main critics from our previous report and improving the manuscript accordingly. We thus recommend that the manuscript would be accepted for publication in Nature Communications with minor revisions. The suggested revisions are listed below.\n1. Line 111-112:\n“… whether their dynamos rely on a tachocline…”\nThis should be rephrased, as fully convective stars do not have a tachocline. It could, e.g., be “… whether the lack of a tachocline will result in a dynamo different from the solar one.\"\n2. Line 219-221:\n\"We notice that, compared to the other two stars, AD Leo has fewer low-energy flares and more high-energy flares (especially superflares), but not a general weakening of all kinds of flares.\"\nThis should be rephrased to clarify what the authors mean with \"weakening\".\n3. Lines 700-703:\n” …we take g_eff = 1 and λ0 = 656.28nm for the Hα line. For the LSD profiles of Ca IRT formed in the lower chromosphere. We have typical values of g_eff = 0.968 and λ_0 = 858.56nm, taking from the LSD process, and the g of each line is taken from the VALD.”\nThe text \"chromosphere. We\" should be \"chromosphere, we\". The last “g” should be changed to “g_eff”. The λ_0 and g_eff for the photospheric LSD lines should be given here. Were all the g_eff values taken from VALD? (If not then give the references.)\n4. SI line 015:\n“…exhibit a small…”\nMaybe change to “… have small …”\n5. SI line 081:\n\"r\" and \"p\" should be in math font.\n6. SI lines 107, 108 and 111:\nThe term \"polarity reversal\" is still used here. The authors should consider using another terminology.\n7. Author's response, page 4:\n\"We did a preliminary test on the ZDI map of AD Leo for Ca IRT (middle chromosphere), showing that the main variation between photospheric and chromospheric fields is azimuthal and meridional fields.\"\nThe authors show a ZDI map of the lower chromosphere using only the three Ca IRT lines. The small number of lines, compared to the number used for the photosphere, raises the question of how reliable this map is. Furthermore, a well-known fact is that while the radial field can be reasonably well reconstructed using just Stokes V-profiles, especially the meridional field is not well reconstructed. But since this is outside the scope, there is no need to further discuss this.\n(Remarks on code availability)\nWe reviewed the code when the first version was submitted, and did not see it necessary to repeat.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The authors of the paper have considered in detail the objections which I noted in my original report. The authors have provided cogent replies to my objections.\nI therefore now consider the paper acceptable for publication."}, "author_response": "I thank the authors for answering my questions. I agree with the publication of the manuscript in its current form.\nI was still not able to run the software without errors, but this may have to do with my specific configuration on my laptop, and is not relevant to my positive evaluation of the manuscript."}, {"round": 3, "reviewer_comments": {"reviewer_1": "I thank the authors for answering my questions. I agree with the publication of the manuscript in its current form.\nI was still not able to run the software without errors, but this may have to do with my specific configuration on my laptop, and is not relevant to my positive evaluation of the manuscript."}, "author_response": ""}]

Decision letter