Symmetries of Magnetic Fields Driven by Spherical Dynamos of Exoplanets and Their Host Stars

Round 1

Reviewer 1 Report

Dear authors,

Your article is surely a very interesting topic for exoplanetary science. In fact, detecting (and modeling) magnetic fields in these planets is one of the main research line for future. Some attempts for detecting magnetism are performed by, for example, Wilson Cauley et al. 2019 -as mentioned in your paper- and Oklopčić et al. (2020). As demonstrated in the case of Earth-Sun system, magnetic fields are a key not only for searching habitability regions/planets, but also because it contributes to characterize and constrain internal properties of those planets.
In this sense, there is, for example, a final report (https://authors.library.caltech.edu/92812/1/Magnetic_final_report.pdf) where, in chapter 5 ("Roadmap and Future steps") could be placed your article.

[*] Main comment:

Sincerely, I would like to recommend to you to improve the paper by trying to do an effort, i.e. contextualizing the paper inside the exoplanet researches, I mean, for example: A first question that arises after reading the manuscript is, how can we distinguish between, let's say, a quadrupole or a dipole magnetic configuration on exoplanets? Knowing information on dynamo means that we could obtain information on the interior of the planet, i.e. iron-rich or gaseous H fluid, how this could be derived from your magnetic field configurations? What kind of data it is needed for these kind of studies in a future? Also, regarding to dynamos, they are self-sustained by convection, byoancy... what are the possibilities? Are there other mechanisms responsible for dynamos in exoplanets and/or their host stars? What are the physical properties needed to distinguish between the different configurations?

[*] Some other comments divided by sections:

1. Introduction.

Line 16. "Concerning first attempts ... " There is a paper, recently published on observations of magnetic fields in exoplanets. Here you are the reference:

Antonija Oklopcić et al. Detecting Magnetic Fields in Exoplanets with Spectropolarimetry of the Helium Line at
1083 nm, ApJ, (2020), 890,88.

It is also a way to measure magnetic fields in exoplanets as the one you mentioned through stellar-planet interactions published in Nature Astronomy (your reference [7]).

I would like to recommend you to extend a little bit the introduction discussing briefly the attempts in observing magnetic fields.

Line 24. There is an open parenthesis ( which it is not closed.

2. Spherical dynamos.

Line 38: "physic" --> physics

Line 38-41: Is there any alternative to the self-excitation? What is it more appropiated "self-excitation" or "self-sustained"?

Line 49: "vise versa" --> vice versa

Line 67: Could some reference be included regarding to the numerical investigation of $\alpha$-$\omega$ dynamo in planetary and stellar science?

3. Earth's and solar magnetism in the spherical dynamo framework

Line 75-76: I miss again some reference on the Earth magnetic field description.

Line 79: The same again, could some solar magnetic field reference be included?

Line 83: "in respect to" --> the appropiate form, I think, it is "with respect to" or "in respect of". This "in respect to" is repeted through the text, you can look for it and correct it if it is not correct.

Lines 89 - 91: In this paragraph, I miss something that introduce the objects showed in Figure 1, not only in the caption of Figure 1.

4. Parametric space for $\alpha$ $\ometa$ spherical dynamos

Here it is where I start to be a bit confused with what it is argue in the previus Section. Section 3 talks about the dipole and at the end considers other symmetries compatible with the symmetry of classical dynamo eqs., i.e. the mentioned dipole, quadrupole and mixed parity configuration. First of all, I am not sure that the title of this section is appropiated. When "parametric space" is read, generally, it is expected some numbers that characterized the models that could be compatible with observations. Here it is not the case. What means parameter space here? Is only refer to the symmetries as I guess? But, the dipole is also a solution of the $\alpha$ $\omega$ dynamo, isn't it?
In this Section, you are talking about a possible symmetry that could be present in exoplanets and/or host-stars, the quadrupole one, why do you not consider the octupole similar to Saturno's magnetic field?

In line 106: why it is difficult to verify this option? What are the implications of this symmetry from the point of view of planetary interiors, I mean, composition, or on the atmosphere of the exoplanet...?

In lines 108-110: Why higher symmetries are not compatible with symmmetry of dynamo drivers? Could you mention some reference or argue briefly?

Lines 111-117: You are arguing here mixed-parity solutions. The argument is quite clear and also the references are ok. But again, I miss some information that this kind of dynamo solutions could provide on exoplanets or stars. The description you mention on the Sun could be useful but I would like to read what implications on exoplanets interiors or stars have this solution by extrapolation for example, if there is any system that exhibits this kind of solutions, though it is questionable or just a simulation.

The rest of the section it is clear to me and it could be linked to what I recommend to you in the context of exoplanets or stars, in this case.

5. Parametric space for $\alpha$^{2} spherical dynamos.

Again, as in previous section, I am not sure if the title of the section is clear and agrees with the content. I guess that for parametric space you mean symmetries derived from $\alpha$^{2} spherical dynamos or something like that.

Line 152: Begins with "otation", I think that it is something missing.

In general this paragraph is very clear. But I miss some additional information or clarification of what you say in line 162, why this dynamos could be important to excite or sustain magnetic fields in exoplanets? What are the implications of this dynamo on exaplanets? What could sustain it? This is the kind of improvement I would like to see in this interesting paper.

6. Conclusion and discussion

This section is in general very weak. It would change if recommendations were included, but in any case, I would point out several things:

In lines 169-170: You say "The list of possibilities is composed basing on symmetry properties of exoplanets and their host stars.", I would like to recommend to write explicitly that list here in the conclusions section. And probably, it could change into "Based on symmetry properties of exoplanets and host stars -as .... - other possible magnetic field configurations are quadrupole or mixed parity." Also you can remark what are the dynamos that generate those magnetic configurations.

I also recommend to include a paragraph before Figure 2, where you can present what you are showing in that figure, just to be clear.

In line 176: when this phrase ends, I would recommend to include something about implations on the planetary-stellar interior if it is possible. Also you can include what symmetries or implications for the planetary structure/interior if a $\alpha^{2}$ dynamo takes place.

In summary, I can not accept the paper for publication at this stage, it requires some clarification as remarked at the beginning. Considering your expertise, I am pretty sure that you are able to include some of the comments and clarifications I propose to get a more valuable paper for the exoplanets researches.

Author Response

Comments and Suggestions for Authors

Dear authors, Your article is surely a very interesting topic for exoplanetary science. In fact, detecting (and modeling) magnetic fields in these planets is one of the main research line for future. Some attempts for detecting magnetism are performed by, for example, Wilson Cauley et al. 2019 -as mentioned in your paper- and Oklopčić et al. (2020). 

Dear Reviewer, thank you for your comments! We try to correct the text and give point-to-point answers below; in the text we mark corrections and explanations by bold type

As demonstrated in the case of Earth-Sun system, magnetic fields are a key not only for searching habitability regions/planets, but also because it contributes to characterize and constrain internal properties of those planets. In this sense, there is, for example, a final report (https://authors.library.caltech.edu/92812/1/Magnetic_final_report.pdf) where, in chapter 5 ("Roadmap and Future steps") could be placed your article.

Thank you! We study the final report and add discussion in the text. Also we try to discuss internal properties as best as we can!

Theoretical abilities, here, are however limited because our knowledge concerning stellar and planetary hydrodynamics required from a theoretical prediction is much more limited rather that one for the Sun and the Earth. The point is that the intensive motions of electroconducting media in form of convection or turbulence and differential rotation seems to be required for dynamo action in exoplanets and their host stars however one needs to know various delicate properties of convection to predict the shape of dynamo driven configuration. One of such fine properties is the degree of mirror asymmetry of convective flows. Quantification of such values is far to be an easy undertaking even for laboratory experiments with liquid metals. The available experiences in laboratory dynamo experiments confirm the point that moderate changes in delicate properties of the flow can result in substantial modification in dynamo driven magnetic configuration, see, e.g., \cite{sokoloff2014dynamos}. Of course, extensive numerical simulations remain a useful tool for understanding of exoplanetary dynamos however it looks unrealistic to predict the magnetic configuration basing on several integral parameters of an exoplanet. Of course, the magnetic field of an exoplanet similar to Jupiter is expected to be similar to the magnetic field of Jupiter. A non-rotating exoplanet or an exoplanet without any liquid envelope is expected to have no pronounced internal magnetic field. It is natural to expect that a larger rotational moment of a celestial body results in its larger magnetic moments. Stellar activity observations support this expectation (e.g. \cite{baliunas1996magnetic}; remarkably, this early finding in the topic is simultaneously an early paper in exoplanetary studies).

A straightforward way to obtain a dynamo generated magnetic configuration with octupolar symmetry would be to consider a spherical shell with two dynamo waves with equal in strength oppositely directed toroidal magnetic field propagating equatorwards in each hemisphere. The poloidal field produced by dynamo action from such toroidal field do has desired configuration. If the number of dynamo waves of opposite parities is even higher, the magnetic fields with even higher symmetries can be excited. Dynamo equations allow in principle such kind of configurations and they are mentioned in exploratory papers in dynamo theory (e.g. [28] and references therein). Excitation of such configurations requires dynamo drivers much more intensive rather what happens in the Sun and the Earth. There are no hints concerning such intensive dynamo action in contemporary observations. In contrast, excitation of another magnetic configurations under discussion is possible with more or less conventional dynamo drivers. Observation of exoplanets and/or host stars with pronounced octupolar magnetic field would be a message that we basically misunderstand to what extent solar or Earth’s magnetism are instructive for the problem.

Main comment:

Sincerely, I would like to recommend to you to improve the paper by trying to do an effort, i.e. contextualizing the paper inside the exoplanet researches, I mean, for example:

1) A first question that arises after reading the manuscript is, how can we distinguish between, let's say, a quadrupole or a dipole magnetic configuration on exoplanets?

The new paragraph 6 elucidate our point of view to this problem:

The structure of astrosphere as the possible indication to the dynamo symmetries

Now let us consider some aspects of observations of exoplanet/host star magnetic fields and their extension in the surrounding space. While the existence of magnetized stars is known for a quite long time, there is little data about exoplanetary magnetic fields. They were discovered and estimated recently by indirect measurements based particularly on the observations of anomalous enhanced emissions (in Radios, UV, or X-ray) that likely indicate the existence of the magnetic field of exoplanets \cite{2018SantosASSP, 2016StrugarekApJ}, see also \cite{lazio2016planetary}. 

The irradiation of the planet outer layers by the star’s wind can lead to a substantial atmospheric outflow  and therefore give the observable signatures for a distant observer (e.g., \cite{2013TremblinMNRAS, 2015KhodachenkoApJ}). This phenomenon can not be directly related to planet-host star magnetic interactions, but magnetic fields can influence and accelerate the outflows of ionized gas \cite{2011AdamsApJ}. The exact method was recently proposed \cite{Oklop_i__2020} for direct detecting of the presence of magnetic fields in the atmospheres of transiting exoplanets. This method likes one that has been initially introduced in the solar physics by \cite{2002TrujilloBuenoNatur} based on detecting radiation polarization in the helium line triplet at $1083 nm$ during transits of close-in exoplanets with extended or escaping atmospheres. This method is defined by authors as exact and extremely sensitive to the presence of magnetic fields in exoplanet atmospheres. Therefore one can expect further rapid development of investigations in this area.

The magnetic fields of extrasolar planets are of great interest because they can play the substantial role in the shielding of planetary atmospheres from cosmic and galaxy rays and in the regulation of atmospheric losses, which, in turn, are important for the estimates of their bowel and atmospheric contents and, finally, their habitability. Also magnetic fields of exoplanets can be influenced strongly by mutual exoplanets - host star interactions. As a result the investigations of impact of host star magnetic fields and corresponding environment are also the object of intensive research. In this context hot Jupiters (exoplanets with masses M greater than 13 mass of Jupiter ($M>13{{M}_{J}}$) orbiting close to their host stars at distances less than 10 star’s radii $d>10 R_{*}$, as well as brown dwarfs (a separate group of celestial bodies with $12<M<80 M_J$ moving alone or with their satellites) are the important objects for observations of magnetic activities. First observations of their magnetic fields were done in \cite{2017BerdyuginaApJ}. Recently the auroral emission for four dwarfs was detected that corresponded to magnetic fields from about 3.2 to 6.2 kG \cite{2018KaoApJS}. It was concluded that rapid dwarf rotation may be important for producing strong dipole fields in convective dynamos and it possibly contributes for driving the current systems powering auroral radio emission. 

Observations of Ly$\alpha$ line (that is a signature of neutral H atoms moving with high velocities toward and away from the star) during exoplanet HD 209458b transit in front of its host star revealed the motion of stellar wind with a velocity  $v_{sw} \approx 400 km/s$, plasma density ${{n}_{sw}}\approx 5\cdot {{10}^{2}}{{m}^{-3}}$  and finally allowed estimating the planetary magnetic moment as ${{D}_{p}}\approx 1.6\text{ }\times \text{ }{{10}^{26}}\,A/{{m}^{2}}$, that is approximately ten percent 

from the Jupiterian one \cite{2014KislyakovaSci}. 

The star–planet interactions in the form of planet-modulated chromospheric emission of the star were investigated for four hot Jupiters \cite{cauley2019magnetic}. Magnetic star–planet interactions involving the release of energy of both stellar and planetary magnetic fields allowed estimating exoplanetary magnetic fields. It was found that the value of the surface magnetic field for hot Jupiters is in range from 20 G to 120 G. That is approximately one--two order larger than it is predicted by dynamo scaling laws, but is in agreement with scaling laws relating the internal heat flux and planetary magnetic field. 

We should mention here that these first results of experimental researches of exoplanetary magnetic fields are very important, on the one side, but, on the other side, they are sometimes quite difficult for interpretation because of observational complexities of such remote planetary systems and the limited set of methods generally providing indirect data.  For comparison with above mentioned results, the dipole magnetic moment of the Earth is about $8\cdot {{10}^{22}}A\cdot {{m}^{2}}$ while the Jupiterian one is approximately $2\cdot {10^4}$ times larger \cite{Bagenal2013}. The values of magnetic induction at the surfaces are equal, correspondingly, $(0.3-0.6) G$ at the Earth and about $5 G$ at Jupiter. We see that magnetic fields of the planets of the solar system and extrasolar planets can be substantially different and the later ones sometimes could not agree with the theory. It depends on many factors that may be missing in our knowledge about extrasolar planets and their interactions with paternal stars. The fact that exoplanets have a much stronger magnetic field than Jupiter makes them suitable objects for further research. As a result the known methods of the Sun’s observations principally can 

be applied for investigations of their magnetic fields and interactions with their host stars. 

Concerning the consequences of dynamo symmetry for the formation of the astrosphere structure (by analogy with the heliosphere we name such region around the star where both magnetic field and plasma environment of the star dominate), one can highlight two problems of internal and external factors, i.e. 1) dynamo processes in the bowels of stars and planets and their influence on the shape of the magnetic field of celestial bodies in the surrounding space; 2) mutual influence of magneto-plasma shells of stars and planets, leading to the formation of sufficiently stable structures as astrospheres, magnetospheres and large-scale current systems. The large magneto-plasma structures in the surrounding of the star depend generally on its magnetic field shape, plasma density, temperature, rotation speed, tilt of the magnetic axis to the rotation and other factors. The star’s dynamo processes can indirectly lead to formation of a system of currents sheets with characteristic scales comparable with the whole astrosphere, analogous to the heliospheric current sheet in the heliosphere. This system of currents is important for the transport of ionized matter of the star and for the interaction of the stellar environment with exoplanets. Because interior dynamo processes in the bowels of stars are practically inaccessible for observations we might suppose that these large-scale current systems in astrospheres, if they can be observed and estimated, can be the good indicators of the stars magnetic configurations.

Large-scale currents in the astrosphere, as in the heliosphere, are supposed to be the extension of neutral lines in the star’s corona separating regions with oppositely directed magnetic fluxes \cite{svalgaard1974model, svalgaard1975long, balogh1998magnetic}. In the solar system during the period of minimum solar activity (i.e. the predominance of the dipole heliomagnetic field), a disk-like heliospheric current sheet is formed in the near-equatorial region of the heliosphere. During the periods of maximum of solar activity when the quadrupole field dominates two quasi-stationary conical current sheets can be formed at high latitudes as the continuation of neutral lines of quadrupole \cite{reville2015effect, maiewski2020magnetohydrodynamic}. Figure 2 illustrates the structure of possible large-scale currents in the astrosphere for different stars, including the Sun, with the dipole magnetic configuration (figure 2a) and axisymmetric quadrupole field (figure 2b). Figure 2c shows the corresponding configuration of currents in a mixed dipole-quadrupole configuration, which is characterized by the asymmetry of the magnetic field and current sheets localizations in the Northern and Southern hemispheres (see also Figure 1c with the basic magnetic configuration).

Here one can mention the remote measurements in the heliosphere of the Lyman-alpha radiation  scattered on interstellar hydrogen atoms that can be the useful tool to investigate both hydrogen atom concentration and the mass flux of the solar wind. In \cite{Petrukovich:2020} the observed by SOHO spacecraft celestial maps of the intensity of scattered Lyman-alpha radiation allowed reconstruct the dependence of the solar wind mass flux on heliolatitude. It is shown that during the periods of solar minima the dependence of solar wind concentration on heliolatitude has maximum in the Sun's equatorial region and minima at the poles. However, concentration maxima are clearly seen at mid-latitudes of ±(30 – 50)º during periods of weaker solar activity. These observational data are in good qualitative agreement with the results of MHD modeling \cite{maiewski2020magnetohydrodynamic, kislov2019quasi} indicating the formation of two quasi-stationary large-scale conic current and plasma sheets during solar maximum. Figures 2a-c illustrate our  view that two current density maxima at high heliolatitudes indicate the dominant quadrupole structure of the Sun’s magnetic field while one current sheet at equatorial region is characteristic for the dipole field. 

Let us continue our analogy between the characteristic properties of the solar system that can be useful in investigations of exoplanetary systems. It is known that large-scale current sheets as well as heliospheric current sheet can be formed at the distances beyond the Sun’s Alfven surface (the surface around the Sun where the velocity of accelerating solar wind becomes equal to the Alfven velocity), i.e. they are observed at the distances approximately larger than approximately ten of Sun’s radii. If we use the analogy of the Sun and host star with closely moving hot Jupiters (their usual separation from the host star is less than about 10 star’s radii \cite {cauley2019magnetic}), one can expect that hot Jupiters should move in the regions of subsonic plasma flows and do not cross surfaces of large-scale current sheet. 

By other words, the shape of the average star’s magnetic field seen from the hot Jupiters should be dipole or quadrupole-like as it is presented in figure 1a-c. Due to close orbiting hot Jupiters as well as Mercury in the Solar system might strongly depend on magneto-plasma structures in the star’s corona and non-stationary processes in them. Also their magnetic field, if they have, for example, dipole shape, should be like to one of Ganymede (the largest Galilean satellite of Jupiter), having its own magnetosphere and rotating in the subsonic magneto-plasma environment \cite{2009JiaGanymede}. Thus the Ganymede's magnetosphere can be reconnected with high-latitude Jupiter's magnetic field, this can result in noticeable auroral emissions in polar regions. The similar effects, but more pronounced, should be expected in the systems hot Jupiters--host stars \cite{2019ZhilkinhotJupmodel}.  

During their motion the exoplanets located at larger distances (e.g. the most common type of multiple planets system in the Galaxy having masses between 1 and 20 Earth masses are orbiting within 0.5–1 AU [Santos 2018]) from their host stars like solar system should probably cross the current sheets of the astrosphere. Therefore during their rotation in the gravitational field of the star their magnetospheres can be perturbed due to influence of both plasma and magnetic environment of the star. In \cite{Khodachenko_2011, alvarado2020earth} the characteristic shape of exoplanetary magnetosphere streamlined by star wind flow are considered with magnetopause, magnetosheath and magnetotail. In such conditions we expect the similar processes of magnetospheric perturbation as magnetic storms and substorms (e.g., \cite{zelenyi2019current}). Like the Earth the processes of atmospheric escape from exoplanets should be considered taking into account the form of planetary magnetic fields. The differences of atmospheric escape in the quadrupole and dipole magnetic fields are considered, particularly, in \cite{tsareva2020atmospheric}.”

2) Knowing information on dynamo means that we could obtain information on the interior of the planet, i.e. iron-rich or gaseous H fluid, how this could be derived from your magnetic field configurations?

3) What kind of data it is needed for these kind of studies in a future? Also, regarding to dynamos, they are self-sustained by convection, buoyancy... what are the possibilities? Are there other mechanisms responsible for dynamos in exoplanets and/or their host stars? What are the physical properties needed to distinguish between the different configurations?

Thanks! We try to work with this comments and add a lot of changes through the text

INTRODUCTION: “The point is that the intensive motions of electroconducting media in form of convection or turbulence and differential rotation seems to be required for dynamo action in exoplanets and their host stars however one needs to know various delicate properties of convection to predict the shape of dynamo driven configuration. One of such fine properties is the degree of mirror asymmetry of convective flows. Quantification of such values is far to be an easy undertaking even for laboratory experiments with liquid metals. The available experiences in laboratory dynamo experiments confirm the point that moderate changes in delicate properties of the flow can result in substantial modification in dynamo driven magnetic configuration, see, e.g., \cite{sokoloff2014dynamos}. Of course, extensive numerical simulations remain a useful tool for understanding of exoplanetary dynamos however it looks unrealistic to predict the magnetic configuration basing on several integral parameters of an exoplanet. Of course, the magnetic field of an exoplanet similar to Jupiter is expected to be similar to the magnetic field of Jupiter. A non-rotating exoplanet or an exoplanet without any liquid envelope is expected to have no pronounced internal magnetic field. It is natural to expect that a larger rotational moment of a celestial body results in its larger magnetic moments. Stellar activity observations support this expectation (e.g. \cite{baliunas1996magnetic}; remarkably, this early finding in the topic is simultaneously an early paper in exoplanetary studies). However a more deep thoughts do require an approach directed on exoplanet studies specifically!”

SECTION 4: “Observation of an exoplanet and/or host star with a magnetic structure with a pronounced octupole symmetry and negligible  dipole and quadrupole magnetic moments would be a challenge for contemporary dynamos. In principle, it might happen that a dynamo system excites a magnetic configuration with poloidal magnetic field antisymmetric in respect of the equator however its magnetic dipole moment vanishes due to some reasons and the only octupole magnetic moments survives however we never faced such example in dynamo modelling. Solar magnetic configuration do contains octupole component and even higher zonal harmonics 

(e.g. \cite{knaack2005spherical}) which are interesting for the current sheet studies (e.g. \cite{maiewski2020magnetohydrodynamic}) and can be included in corresponding modelling (e.g. \cite{reville2015effect}) however dipole component in solar magnetic field is presented as well.”

[*] Some other comments divided by sections:

1. Introduction.

Line 16. "Concerning first attempts ... " There is a paper, recently published on observations of magnetic fields in exoplanets. Here you are the reference: Antonija Oklopcić et al. Detecting Magnetic Fields in Exoplanets with Spectropolarimetry of the Helium Line at 1083 nm, ApJ, (2020), 890,88.

Thank you! We add the reference!

It is also a way to measure magnetic fields in exoplanets as the one you mentioned through stellar-planet interactions published in Nature Astronomy (your reference [7]). I would like to recommend you to extend a little bit the introduction discussing briefly the attempts in observing magnetic fields.

We add the answer in the section 6. And small comments through the text! We appreciate various specific mechanisms like compression of magnetic field frozen in stellar wind, relic magnetic field or battery effects  which may be responsible for magnetic field formation in some celestial bodies. We concentrate however on the message from dynamo theory as a main stream for explanation in stellar and planetary magnetic studies. It would be risky to insist that dynamo successfully explains any particular observation in the field however the general progress here looks remarkable.

Line 24. There is an open parenthesis ( which it is not closed.

Thank you! We correct the text.

2. Spherical dynamos.

Line 38: "physic" --> physics

Thanks!

Line 38-41: Is there any alternative to the self-excitation? What is it more appropriate "self-excitation" or "self-sustained"?

Yes, you are right, self-sustained is also appropriate, we add corrections in the text.

Line 49: "vise versa" --> vice versa

Yes, it's a mistake, we correct the text.

Line 67: Could some reference be included regarding to the numerical investigation of $\alpha$-$\omega$ dynamo in planetary and stellar science?

Yes, we add references!

3. Earth's and solar magnetism in the spherical dynamo framework

Line 75-76: I miss again some reference on the Earth magnetic field description.

Line 79: The same again, could some solar magnetic field reference be included?

We add references!

Line 83: "in respect to" --> the appropriate form, I think, it is "with respect to" or "in respect of". This "in respect to" is repeated through the text, you can look for it and correct it if it is not correct.

Yes, thank you! We look through the text and correct the text.

Lines 89 - 91: In this paragraph, I miss something that introduces the objects shown in Figure 1, not only in the caption of Figure 1.

4. Parametric space for $\alpha$ $\omega$ spherical dynamos

Here it is where I start to be a bit confused with what is argued in the previous Section. Section 3 talks about the dipole and at the end considers other symmetries compatible with the symmetry of classical dynamo eqs., i.e. the mentioned dipole, quadrupole and mixed parity configuration. First of all, I am not sure that the title of this section is appropriated. When "parametric space" is read, generally, it is expected some numbers that characterized the models that could be compatible with observations. 

Thank you! It is a very good comment. The last title was really inappropriate, so we change it to  “Magnetic configurations known for spherical dynamos” and add some comments in the text.

Here it is not the case. What means parameter space here? Is only refer to the symmetries as I guess? But, the dipole is also a solution of the $\alpha$ $\omega$ dynamo, isn't it?

Of course, conventional dipole configuration do excites by alpha-omega spherical dynamo for an appropriate choice of dynamo drivers. You are right, so we add some comments in the text and change the title about “parameter space..”, explaining about parameters below.

In this Section, you are talking about a possible symmetry that could be present in exoplanets and/or host-stars, the quadrupole one, why do you not consider the octupole similar to Saturno's magnetic field?

Note, that higher symmetries, say, octupole one, are different from the symmetry of dynamo drivers. Correspondingly, there is no reason to expect stellar or planetary magnetic fields with symmetries higher rather quadrupole one.  Observations of an exoplanet and/or host star with a pronounced octupole symmetry and negligible  dipole/quadrupole magnetic moments would be a challenge for contemporary dynamos. In principle, it might happen that a dynamo system excites a magnetic configuration with poloidal magnetic field antisymmetric in respect to the equator however its magnetic dipole moment vanishes due to some reasons and the only octupole magnetic moments survives however we never faced such example in dynamo modelling. Maybe, it gives some hint concerning possible quardupole symmetry for the solar magnetic field at these times \cite{sokoloff2009sunspot}. We also add some references in the text.

In line 106: why it is difficult to verify this option? What are the implications of this symmetry from the point of view of planetary interiors, I mean, composition, or on the atmosphere of the exoplanet...?

Variety of dynamo configuration excited by $\alpha \omega$-dynamo means that by observing exoplanetary magnetic configurations we learn more about  corresponding magnetospheres rather than hydrodynamics of planetary interiors. In contrast, if future observations will demonstrate that dipole magnetic configuration dominates in most cases and no quadrupole ones exist it will be a message that something basical is wrong in our understanding of dynamo and/or hydrodynamics of stellar and planetary interiors.

The answer is added in the text:

“We should mention here that these first results of experimental researches of exoplanetary magnetic fields are very important, on the one side, but, on the other side, they are sometimes quite difficult for interpretation because of observational complexities of such distant planetary systems and the limited set of methods generally providing indirect data.  For comparison with abovementioned results, the dipole magnetic moment of the Earth is about $8\cdot {{10}^{22}}A\cdot {{m}^{2}}$ while the Jupiterian one is $2\cdot {10^4}$ times larger \cite{Bagenal2013}. The values of magnetic induction at the surfaces are equal, correspondingly, $(0.3-0.6) G$ at the Earth and about $5 G$ at Jupiter. We see that magnetic fields of the planets of the solar system and extrasolar 

planets can be substantially different and the later ones sometimes could not agree with the theory. 

It depends on many factors that may be missing in our knowledge about extrasolar planets and their interactions with paternal star. The fact that exoplanets have a much stronger magnetic field than Jupiter makes them the suitable objects for further researches. As a result the known methods of the Sun’s observations principally can be applied for investigations of their magnetic fields and interactions with their host stars.”

Lines 111-117: You are arguing here mixed-parity solutions. The argument is quite clear and also the references are ok. But again, I miss some information that this kind of dynamo solutions could provide on exoplanets or stars. The description you mention on the Sun could be useful but I would like to read what implications on exoplanets interiors or stars have this solution by extrapolation for example, if there is any system that exhibits this kind of solutions, though it is questionable or just a simulation.

The problem is that the number of  papers discussed mixed parity solutions may be estimated as several percent of the whole bulk of papers addressing  spherical dynamo modeling. The dynamo models exciting mixed parity solution do not require however anything very specific and it is naturally to expect that they have to be presented in a sample of exoplanets and/or host stars provided it will contain hundreds of cases. We think that future observations of exoplanets and/or host stars with mixed parity solutions would be interesting as confirmation of our current understanding of dynamo nonlinear saturation and instructive concerning the possibility to get one more solar mixed parity configuration in a visible future. We add some comments in the text and try to make this question a little bit  clearer.

The rest of the section is clear to me and it could be linked to what I recommend to you in the context of exoplanets or stars, in this case.

5. Parametric space for $\alpha$^{2} spherical dynamos.

Again, as in previous section, I am not sure if the title of the section is clear and agrees with the content. I guess that for parametric space you mean symmetries derived from $\alpha$^{2} spherical dynamos or something like that.

Thank you! You are right, we change the title on “Symmetries for alpha2 dynamos”.

Line 152: Begins with "otation", I think that it is something missing.

Yes, it is a mistake in “rotation”. We fix it in the text.

In general this paragraph is very clear. But I miss some additional information or clarification of what you say in line 162, why this dynamos could be important to excite or sustain magnetic fields in exoplanets? 

What are the implications of this dynamo on exoplanets? What could sustain it? This is the kind of improvement I would like to see in this interesting paper.

We add comments about dynamo and experiments in the section 6!

6. Conclusion and discussion

This section is in general very weak. It would change if recommendations were included, but in any case, I would point out several things:

In lines 169-170: You say "The list of possibilities is composed basing on symmetry properties of exoplanets and their host stars.", I would like to recommend to write explicitly that list here in the conclusions section. And probably, it could change into "Based on symmetry properties of exoplanets and host stars -as .... - other possible magnetic field configurations are quadrupole or mixed parity." Also you can remark what are the dynamos that generate those magnetic configurations.

Thank you for this comment! We add some explanation and summarizing in the “Conclusions”. We stress again that the dynamo models based on differential rotation and mirror asymmetry which generate magnetic field with quadrupole symmetry in a spherical shell exactly symmetric in respect to the equator is just the same as that one producing magnetic field with dipole symmetry. The explanation is the following

Let us summarize the magnetic field configurations which looks relevant for exoplanet studies. The main message here is that the same dynamo mechanism based on differential rotation and mirror asymmetric convective flows which generates oscillating dipole solar magnetic field and quasi-stationary Earth's dipole with chaotic inversions in geological time scales can for relative moderate modifications of dynamo drivers excite magnetic configurations with substantially different symmetry. Thus, both the exoplanet and the host star may have a dipole/quadrupole magnetic configuration as well as mixed parity magnetic fields.

We stress again that dynamo models which generate magnetic field with quadrupole symmetry in a spherical shell exactly symmetric in respect to the equator is just the same as that one producing magnetic field with dipole symmetry. Evolution of eigenmodes with quadrupole symmetry is independent from evolution of dipole modes. The difference in corresponding growth rate is very moderate. A moderate variation of dynamo governing parameters is sufficient to make growth rate for quadrupole mode larger than that one for dipole mode. For a spherical shell which is approximately symmetric only, a growing almost quadrupole mode obtains an admixture of the mode of dipole symmetry and vice versa however the basic shape of solution remains similar to that one for an exactly symmetric shell. Symmetry in respect to the equator may be violated by magnetic field than a mixed parity solution arises. This option is expected however to be quite rare.

Dynamo can be driven by mirror asymmetry only ($\alpha^2$-dynamo). It can happen in bodies with rotation law extremely closed to the solid body one. It is difficult to expect that this kind of dynamo happens very often. $\alpha^2$-dynamo demonstrates much less pronounced symmetries in respect to the equator rather the $\alpha \omega$-dynamo.

I also recommend to include a paragraph before Figure 2, where you can present what you are showing in that figure, just to be clear.

Yes, we agree with this recommendation. We add an additional paragraph before “conclusions”, where we more accurately discuss Figure 2.

In line 176: when this phrase ends, I would recommend to include something about implications on the planetary-stellar interior if it is possible. Also you can include what symmetries or implications for the planetary structure/interior if a $\alpha^{2}$ dynamo takes place.

The question about the planetary-stellar interior is interesting, but  hard to be answered. We try to explain that in the conclusion. And the key point here is that the variety of dynamo configuration excited by $\alpha \omega$-dynamo means that observing exoplanetary magnetic configurations we learn more about  corresponding magnetospheres rather than about hydrodynamics of planetary interiors. We also add in the text a few comments about $\alpha^{2}$ dynamo. This type of dynamo  can be driven by mirror asymmetry only, demonstrating much less pronounced symmetries in respect to the equator rather than the $\alpha \omega$-dynamo. It can happen in bodies with rotation law extremely closed to the solid body one. However, it is difficult to expect that this kind of dynamo happens very often. 

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors remind us that there are other types of dynamo generated magnetic field configurations that should be considered when studying stellar and planetary magnetic fields, besides the simple centrally aligned dipole. Higher order magnetospheres (plasma-filled atmospheres) will certainly contain current sheets at various positions and angles. Fair enough.

There are no original results nor calculations in this article. There are no practical applications either, besides the recommendation that "we have to consider a substantially more rich set of possibilities".

I do recommend rejection.

Author Response

Comments and Suggestions for Authors

The authors remind us that there are other types of dynamo generated magnetic field configurations that should be considered when studying stellar and planetary magnetic fields, besides the simple centrally aligned dipole. Higher order magnetospheres (plasma-filled atmospheres) will certainly contain current sheets at various positions and angles. Fair enough. There are no original results nor calculations in this article. There are no practical applications either, besides the recommendation that "we have to consider a substantially more rich set of possibilities". I do recommend rejection.

We are grateful to the reviewer for his opinion and suggestion, however we think that this manuscript can be very useful for exoplanets specialists on the one hand and for dynamo specialists on the other hand. The point is that the intensive motions of electroconducting media in form of convection and differential rotation seems to be required for dynamo action in exoplanets and their host stars; however one needs to know various delicate properties of convection to predict the shape of dynamo driven configuration. One of such fine properties is the degree of mirror asymmetry of convective flows. Quantification of such values is far to be an easy undertaking even for laboratory experiments with liquid metals. The available experiences in laboratory dynamo experiments confirm the point that moderate changes in delicate properties of the flow can result in substantial modification in dynamo driven magnetic configuration, see, e.g., \cite{sokoloff2014dynamos}. Of course, extensive numerical simulations remain a useful tool for understanding of exoplanetary dynamos however it looks unrealistic to predict the magnetic configuration basing on several integral parameters of an exoplanet. Of course, the magnetic field of an exoplanet similar to Jupiter is expected to be similar to the magnetic field of Jupiter. A non-rotating exoplanet or an exoplanet without any liquid envelope is expected to have no pronounced internal magnetic field. It is natural to expect that a larger rotational moment of a celestial body results in its larger magnetic moments. Stellar activity observations support this expectation (e.g. \cite{baliunas1996magnetic}; remarkably, this early finding in the topic is simultaneously an early paper in exoplanetary studies). However a more deep thoughts do require an approach directed on exoplanet studies specifically! We try to rewrite text, presenting this idea more clear.

Author Response File: Author Response.pdf

Reviewer 3 Report

Dear Authors,

Please find the attached file for the review.

Comments for author File: Comments.pdf

Author Response

The paper draws on the existing knowledge of the magnetic fields in the Sun-Earth system to study the magnetic fields in exoplanets and their host stars. It is primarily based on the expected symmetries in spherical dynamos and the lead author is an internationally renowned researcher in the field of the dynamo theory. Based on the existing knowledge of possible symmetries in _! and _2 dynamo, the authors describe possible configurations of the magnetic fields in the exoplanet-host star system. They conclude that the expected magnetic field could have a dipole, quadrupole, or mixed parity. The paper presents a scientifically strong idea and certainly merits publication in the MDPI journal Symmetry but I strongly recommend making the suggested changes for paper to be accepted. I have a few major and several minor comments below.

Dear Reviewer, thank you for your comments! We try to correct the text and give point-to-point answers below; in the text we mark corrections and explanations by bold type!

Major comments

1. Magnetic field magnitude or scale?: I understand that the main focus of the paper is symmetries of the spherical dynamos, which are used to predict magnetic fields in the exoplanet-host star system but the study is incomplete without a discussion on the strength and length scale of those magnetic fields. This has to be done for both the Sun-Earth system and the exoplanet-host star system, pointing out the differences between the two and the probable reasons for the difference.

For example, in Page 2, line no. 71-72: at what scales is this averaging done? Is that scale expected to be the same in the Sun-Earth system and the exoplanet-host star system? Please go through the whole text and add relevant strengths and scales to the text.

We add comments in the text! We note here that the above scheme of aw-dynamos considers magnetic field averaged over convective pulsations only. Of course, the magnetic field known from observations contains fluctuations as well.

2. In detail discussion of the existing observations?: While working on the above point no. 1, it would also be very important to describe the observational probes which were used to infer those magnetic field strengths and scales. There is some discussion about the observations of magnetic fields in the Sun-Earth system and very little about the exoplanet-host star system but some more would be helpful. I suggest having another section “Observational efforts” after the introduction where such things can be discussed. I also suggest to try and explain those observations based on the theory and symmetry arguments (present Sec. 2., 4., and 5.).

For example, the results from the reference 7 (Cauley et al., Nature Astronomy, 2019) can be used to make a more direct comparison with the theoretical arguments. By the way, this reference also does a detailed study of magnetic field strengths in hot Jupiters.

We have introduced in the text the new section 6 “The structure of astrosphere as the possible indication to the dynamo symmetries” with the discussion of available experimental estimates of magnetic fields.

“Now let us consider some aspects of observations of exoplanet/host star magnetic fields and their extension in the surrounding space. While the existence of magnetized stars is known for a quite long time, there are little data about exoplanetary magnetic fields. They were discovered and estimated recently by indirect measurements based particularly on the observations of anomalous enhanced emissions (in Radios, UV, or X-ray) that likely indicate the existence of the magnetic field of exoplanets \cite{2018SantosASSP, 2016StrugarekApJ}, see also \cite{lazio2016planetary}. 

The irradiation of the planet outer layers by the star’s wind can lead to a substantial atmospheric outflow  and therefore give the observable signatures for a distant observer (e.g., \cite{2013TremblinMNRAS, 2015KhodachenkoApJ}). This phenomenon can not be directly related to planet-host star magnetic interactions, but magnetic fields can influence and accelerate the outflows of ionized gas \cite{2011AdamsApJ}. The exact method was recently proposed \cite{Oklop_i__2020} for direct detecting of the presence of magnetic fields in the atmospheres of transiting exoplanets. This method likes one that has been initially introduced in the solar physics by \cite{2002TrujilloBuenoNatur} based on detecting radiation polarization in the helium line triplet at $1083 nm$ during transits of close-in exoplanets with extended or escaping atmospheres. This method is defined by authors as exact and extremely sensitive to the presence of magnetic fields in exoplanet atmospheres. Therefore one can expect further rapid development of investigations in this area.

The magnetic fields of extrasolar planets are of great interest because they can play the substantial role in the shielding of planetary atmospheres from cosmic and galaxy rays and in the regulation of atmospheric losses, which, in turn, are important for the estimates of their bowel and atmospheric contents and, finally, their habitability. Also magnetic fields of exoplanets can be influenced strongly by mutual exoplanets - host star interactions. As a result the investigations of impact of host star magnetic fields and corresponding environment are also the object of intensive researches. In this context hot Jupiters (exoplanets with masses M greater than 13 mass of Jupiter ($M>13{{M}_{J}}$) orbiting close to their host stars at distances less than 10 star’s radii $d>10 R_{*}$, as well as brown dwarfs (a separate group of celestial bodies with $12<M<80 M_J$ moving alone or with their satellites) are the important objects for observations of magnetic activities. First observations of their magnetic fields were done in \cite{2017BerdyuginaApJ}. Recently the auroral emission for four dwarfs was detected that corresponded to magnetic fields from about 3.2 to 6.2 kG \cite{2018KaoApJS}. It was concluded that rapid dwarf rotation may be important for producing strong dipole fields in convective dynamos and it possibly contribute for driving the current systems powering auroral radio emission. 

Observations of Ly$\alpha$ line (that is a signature of neutral H atoms moving with high velocities toward and away from the star) during exoplanet HD 209458b transit in front of its host star revealed the motion of stellar wind with a velocity  $v_{sw} \approx 400 km/s$, plasma density ${{n}_{sw}}\approx 5\cdot {{10}^{2}}{{m}^{-3}}$  and finally allowed estimating the planetary magnetic moment as ${{D}_{p}}\approx 1.6\text{ }\times \text{ }{{10}^{26}}\,A/{{m}^{2}}$, that is approximately ten percent 

from the Jupiterian one \cite{2014KislyakovaSci}. 

The star–planet interactions in the form of planet-modulated chromospheric emission of the star were investigated for four hot Jupiters \cite{cauley2019magnetic}. Magnetic star–planet interactions involving the release of energy of both stellar and planetary magnetic fields allowed estimating exoplanetary magnetic fields. It was found that the value of the surface magnetic field for hot Jupiters is in range from 20 G to 120 G. That is approximately one--two order larger than it is predicted by dynamo scaling laws, but is in agreement with scaling laws relating the internal heat flux and planetary magnetic field. 

We should mention here that these first results of experimental researches of exoplanetary magnetic fields are very important, on the one side, but, on the other side, they are sometimes quite difficult for interpretation because of observational complexities of such distant planetary systems and the limited set of methods generally providing indirect data.  For comparison with abovementioned results, the dipole magnetic moment of the Earth is about $8\cdot {{10}^{22}}A\cdot {{m}^{2}}$ while the Jupiterian one is $2\cdot {10^4}$ times larger \cite{Bagenal2013}. The values of magnetic induction at the surfaces are equal, correspondingly, $(0.3-0.6) G$ at the Earth and about $5 G$ at Jupiter. We see that magnetic fields of the planets of the solar system and extrasolar 

planets can be substantially different and the later ones sometimes could not agree with the theory. 

It depends on many factors that may be missing in our knowledge about extrasolar planets and their interactions with paternal stars. The fact that exoplanets have a much stronger magnetic field than Jupiter makes them suitable objects for further research. As a result the known methods of the Sun’s observations principally can 

be applied for investigations of their magnetic fields and interactions with their host stars. 

Concerning the consequences of dynamo symmetry for the formation of the astrosphere structure (by analogy with the heliosphere we name such region around the star where both magnetic field and plasma environment of the star dominate), one can highlight two important questions: 1) the shapes and characteristics of star’s magnetic fields as the source of a large-scale current system in astrosphere; 2) the host star-exoplanet interaction depending from magneto-plasma structures in astrosphere. The large magneto-plasma structures in the surrounding of the star depend generally on is magnetic field shape, plasma density, temperature, rotation speed, tilt of the magnetic axis to the rotation and other factors. The star’s dynamo processes can indirectly lead formation of a system of currents sheets with characteristic scales comparable with the whole astrosphere. This system of currents is important for the transport of ionized matter of the star and for the interaction of the stellar environment with exoplanets. Because interior dynamo processes in the bowels of stars are practically inaccessible for observations we might suppose that these large-scale current systems in astrospheres, if they can be observed and estimated, can be the good indicators of the stars magnetic configurations.

Large-scale currents in the astrosphere, as in the heliosphere, are supposed to be the extension of neutral lines in the star’s corona separating regions with oppositely directed magnetic fluxes \cite{svalgaard1974model, svalgaard1975long, balogh1998magnetic}. In the solar system during the period of minimum solar activity (i.e. the predominance of the dipole heliomagnetic field), a disk-like heliospheric current sheet is formed in the near-equatorial region of the heliosphere. During the periods of maximum of solar activity when the quadrupole field dominates two quasi-stationary conical current sheets can be formed at high latitudes as the continuation of neutral lines of quadrupole \cite{reville2015effect, maiewski2020magnetohydrodynamic}. Figure 2 illustrates the structure of possible large-scale currents in the astrosphere for different stars, including the Sun, with the dipole magnetic configuration (figure 2a) and axisymmetric quadrupole field (figure 2b). Figure 2c shows the corresponding configuration of currents in a mixed dipole-quadrupole configuration, which is characterized by the asymmetry of the magnetic field and current sheets localizations in the Northern and Southern hemispheres (see also Figure 1c with the basic magnetic configuration).

Here one can mention the remote measurements in the heliosphere of the Lyman-alpha radiation  scattered on interstellar hydrogen atoms that can be the useful tool to investigate both hydrogen atom concentration and the mass flux of the solar wind. In \cite{Petrukovich:2020} the observed by SOHO spacecraft celestial maps of the intensity of scattered Lyman-alpha radiation allowed reconstruct the dependence of the solar wind mass flux on heliolatitude. It is shown that during the periods of solar minima the dependence of solar wind concentration on heliolatitude has maximum in the Sun's equatorial region and minima at the poles. However, concentration maxima are clearly seen at mid-latitudes of ±(30 – 50)º during periods of weaker solar activity. These observational data are in good qualitative agreement with the results of MHD modeling \cite{maiewski2020magnetohydrodynamic, kislov2019quasi} indicating the formation of two quasi-stationary large-scale conic current and plasma sheets during solar maximum. Figures 2a-c illustrate our  view that two current density maxima at high heliolatitudes indicate the dominant quadrupole structure of the Sun’s magnetic field while one current sheet at equatorial region is characteristic for the dipole field. 

Let us continue our analogy between the characteristic properties of the solar system that can be useful in investigations of exoplanetary systems. It is known that large-scale current sheets as well as heliospheric current sheet can be formed at the distances beyond the Sun’s Alfven surface (the surface around the Sun where the velocity of accelerating solar wind becomes equal to the Alfven velocity), i.e. they are observed at the distances approximately larger than approximately ten of Sun’s radii. If we use the analogy of the Sun and host star with closely moving hot Jupiters (their usual separation from the host star is less than about 10 star’s radii \cite {cauley2019magnetic}), one can expect that hot Jupiters should move in the regions of subsonic plasma flows and do not cross surfaces of large-scale current sheet. By other words, the shape of the average star’s magnetic field seen from the hot Jupiters should be dipole or quadrupole-like as it is presented in figure 1a-c. At the same time hot Jupiter as well as Mercury might strongly depend on magneto-plasma structures in the star’s corona and non-stationary processes in them.

During their motion the exoplanets located at larger distances (e.g. the most common type of multiple planets system in the Galaxy having masses between 1 and 20 Earth masses are orbiting within 0.5–1 AU [Santos 2018]) from their host stars like solar system should probably cross the current sheets of the astrosphere. Therefore during their rotation in the gravitational field of the star their magnetospheres can be perturbed due to influence of both plasma and magnetic environment of the star. In \cite{Khodachenko_2011, alvarado2020earth} the characteristic shape of exoplanetary magnetosphere streamlined by star wind flow are considered with magnetopause, magnetosheath and magnetotail. In such conditions we expect the similar processes of magnetospheric perturbation as magnetic storms and substorms (e.g., \cite{zelenyi2019current}). Like the Earth the processes of atmospheric escape from exoplanets should be considered taking into account the form of planetary magnetic fields. The differences of atmospheric escape in the quadrupole and dipole magnetic fields are considered, particularly, in \cite{tsareva2020atmospheric}.”

3. Analytical estimates?: The paper is based on very sound theoretical arguments but it would be good to add some equations and estimate the importance of various terms. This would help us physically understand the importance of those terms in various systems and also more quantitatively compare the Sun-Earth system with the exoplanet-host star systems.

We agree that the basic equations and estimates will be useful in the frame of the work and can help with understanding, so we add the governing equation for the averaged magnetic field in the text, add explain main terms, give their estimates and add short comments by blue text in the second paragraph. 

For example, in Page 4, line no. 132-135: estimate the Dynamo number for a typical star hosting exoplanet and compare the number with that of the Sun. Another example, in Page 2, line no. 71-73: estimate properties of small-scale magnetic fluctuations from the magneto convection theory and provide the length scale over which the mean-field dynamo theory is applicable and how that scale depends on the properties of the host star? Please go through the whole text and add relevant equations and estimates as and when necessary.

Note that abilities of contemporary science to estimate how large is dynamo number $D$ in particular celestial bodies are very limited. The point is that direct observations of mirror asymmetry of convection in stellar or planetary interior remain unrealistic. Scattering in theoretical estimates may be as large as order of magnitudes. The viewpoint of \cite{KN17} that solar dynamo is just above the excitation threshold is close to our understanding of the situation however it is very far to be unanimously accepted by the expect community. We add some comments in the text!

4. More details of the small-scale fluctuations?: The paper, in Page 2, line no. 71-73, mentions that the observations also contains ‘fluctuations’. Some more description about the reason for these fluctuations and how does this makes the comparison with observations difficult would be a useful addition to the paper.

We are sorry: of course, the small-scale fluctuations are very interesting in the frame of astrophysics magnetism, however we removed a sentence concerning the role of magnetic fluctuations just because it would be irresponsible to discuss this topic in context of contemporary knowledge about physics of exoplanets.

5. What is not the same?: Is there anything which will be different in the Sun-Earth system and the exoplanet-host star system, for example, due to the higher temperature of host stars? or due to the higher rotation rate of the host star?

Of course, some very qualitative scalings for $D$ are possible. Say, rapid rotation is helpful for dynamo action however the gap between such general ideas and a quantitative estimate of dynamo number in a particular celestial body is quite wide (see, e.g., \cite{BNR} which is often cited in this context). We add an explanation in the text.

6. Coherence in writing: The paper has very good ideas and will certainly be very useful to the community, especially to observers, for their research. Since the paper would attract readers from a variety of fields (dynamo theory, MHD simulations, observations, etc. ), I would suggest rephrasing parts of the paper to ensure better coherence. Also, make sure the sentences are complete and grammatically correct (some examples in minor suggestions). This makes it easier for the reader to appreciate the text.

Thank you very much for all your remarks! We checked the entire text once again and tried to make it more clearer and more grammatically correct for the readers  from a variety of fields.

Minor suggestions

Thanks a lot! We check minor suggestions though the text and write it in the correct form.

1. Page 1, line no. 4: “The main our results . . . ” ! “The main part of our results . . . ” or “Our main result . . . ”
2. Page 1, line no. 7: “. . . abovementioned exoplanet systems . . . ” !“. . . above mentioned exoplanethost star systems . . . ” (Fig. 2)
3. Page 1, line no. 12: “ . . . is far to be straightforward . . . ” !“ . . . is far from being straightforward . . . ”
4. Page 1, line no. 17: “Then it becomes possible to use for exoplanet studies the available bulk of available knowledge . . . ” ! “Then it becomes possible to use for the exoplanet studies, the available bulk of available knowledge . . . ”
5. Page 1, line no. 25: “Theoretical abilities here are however limited because . . . ” ! “Theoretical abilities, here, are however limited because . . . ”
6. Page 2, line no. 35: “magnetoshperic studies.” !“magnetospheric studies for exoplanet-host star system”
7. Page 2, line no. 42: “is known now as dynamo . . . ” ! “is known now as a dynamo”
8. Page 2, line no. 46-47: “The point however is that according to so-called Lenz rule electromagnetic induction acting in an electrical circuit do suppress rather enhance a seed magnetic field.” ! “The point is that according to the so-called Lenz rule, electromagnetic induction acting in an electrical circuit suppresses a seed magnetic field.”
9. Page 2, line no. 65-67: “Dynamo model based in joint action of differential rotation with angular velocity ! and _-effect known as the spherical _!-dynamos are extensively investigated numerically nowadays.” ! “Dynamo models based on the joint action of the differential rotation with angular velocity ! and the _-effect known as the spherical _!-dynamos are extensively investigated numerically nowadays.” 
10. Page 2, line no. 75-77: “Geomagnetic field of the Earth is a poloidal magnetic field closed to magnetic field of a magnetic dipole almost parallel to the rotation axis. Deviation of magnetic dipole from the rotation axis is considered as a large-scale fluctuation.” ! “Geomagnetic field of the Earth is a poloidal magnetic field close to the magnetic field of a magnetic dipole, which is almost parallel to the rotation axis. Deviation of the magnetic dipole from the rotation axis is considered as a large-scale fluctuation.”
11. Page 2, line no. 78: What kind of observations?

We mean here that the toroidal component of the geomagnetic field is presumed to be located somewhere deep inside the Earth being inaccessible for contemporary observations because the Earth's mantle prevents its propagation to the surface. Concerning first ideas for the Earth's toroidal magnetic field observations see, e.g., Levy, E.H.; Pearce, S.J.  Steady state toroidal magnetic field at Earth’s core-mantle boundary.1991,29396, 3935–3942. We add some comments in the text.

12. Page 3, line no. 81: “deep in solar interior however is traced at the solar surface due to sunspots.” ! “deep in the solar interior, however, it is traced at the solar surface due to sunspots.”
13. Page 3, line no. 85: “. . . sunspots are rare just in the equator vicinity”!“. . . sunspots are rare just in the vicinity of the equator”
14. Page 3, line no. 108: “. . . are not compatible with symmetry . . . ” ! “. . . are not compatible with the symmetry . . . ”
15. Page 4, line no. 123-124: “One have to accept, that during that times solar only one current sheet was available in solar vicinity however the sheet was located in middle latitudes rather . . . ” ! “One have to accept, that during that times, only one current sheet was available in the solar vicinity, however, the sheet was located in the middle latitudes rather than . . . ”
16. Page 4, line no. 128-129: “The point is that the both dynamo drivers, namely differential rotation and _-effect, can be positive as well as negative.” ! “The point is that both the dynamo drivers, namely the differential rotation and _-effect, can be positive as well as negative.”
17. Page 4, line no. 134-135: “For the sake of definiteness we accept here that D < 0 in solar case.” ! “For the sake of definiteness, we accept here that D < 0 in solar case.”
18. Page 4, line no. 152: “. . . otation low of a spherical body may be not . . . ” !“. . . low rotation of a spherical body may be not . . . ”
19. Page 5, line no. 170: “ Thus both exoplanet and central star . . . ” ! “ Thus, both the exoplanet and the host star . . . ”

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Dear authors,

Thank you very much for your new version of the manuscript. In my opinion, you have improved it notably after an effort for clarifying it and adding a new interesting section. I send you some comments in what follows. I also send you a .pdf with some highlighted words to be checked, some commas added or some minor corrections.

1) Major comments

Page 3:

-New 1st paragraph, let's say from line 93 to line 110. I'm just only missing the definition of "v" that you mention in L102, 103.

- From L117-120, is it possible to provide some numbers on this spatial scales, on the radius of the sphere? Perhaps some reference? I'm just asking.

2) Minor comments or corrections.

I have marked through the .pdf file some corrections, that I can summarize here:

- I have underlined/highlighted with yellow, some 'errata'

- Check the use of 'modelling' or 'modeling', I mean, both of them appear through the text. I think, I underline all of them but you must check them.

- Check "in respect of", I'm not sure if it is "with respecto to"

L4: miss "their" --> and their corresponding host stars

L13: because of the novelty of the topic: miss "the"

L21: System: capital S

L28: rather than that... : miss "than"

L38: basing on --> based on

L42-43: larger --> higher

L44-45: more deep --> deeper

L50: rather that --> rather than

L53: system --> systems

L54: we give... --> we give their...

L60: miss a "," at the end of the line?

L63: for... --> as an...

L87: under surface --> under the surface

L111: dynamo considered --> dynamo is considered

L113: the effects --> these effects (?)

L118-120: starting from "i.e. averaging..." do you mean something like "i.e. averaging over the convective vortexes inside the spherical shell, the scale of the dynamo driven magnetic field is..." ?

L180: has desired --> has the desired

L181: oparities --> parities (?)

L185: rather what --> rather than what...

L200: several percent --> few percent (?)

L264: Radios --> radio

L270: likes --> is similar to..

I don't know if the word "bowel" could be replaced by "interior", it seems more suitable.

L287: that --> and

L297: allowed estimating --> allow to estimate

L310: paternal --> parent

L324: lead to formation --> lead to the formation

L343: check Lyman-$\alpha$

L359, 360: Alfven --> Alfvèn

L364: By other words --> In other words

L376: Incorrect form to include the reference

In summary, after this revision, I could recommend the publication of the manuscript. Thank you very much for your effort. It is a qualitative paper but it provides some interesting hints on exoplanetary magnetism.

Comments for author File: Comments.pdf

Author Response

Dear Reviewer, thank you again for your comments! We try to correct the text and give point-to-point answers below; in the text we mark corrections and explanations by bold type

1) Major comments

Page 3:

-New 1st paragraph, let's say from line 93 to line 110. I'm just only missing the definition of "v" that you mention in L102, 103.

In fact the quantity v has been introduced slightly earlier however we agree that additional clarification might be useful and introduced additionally in the place mentioned {\bf the convective velocity}

- From L117-120, is it possible to provide some numbers on this spatial scales, on the radius of the sphere? Perhaps some reference? I'm just asking.

We add some discussion and references in the text. {\bf which thickness is few time smaller rather than the radius of exoplanet or hosting star respectively (approximately on the order of one tenth of the planetary or stellar radii, see, e.g., \cite {1990Choudhuri, 2015Labrosse, 2017Schaeffer, 2020Gastine})}

2) Minor comments or corrections.

I have marked through the .pdf file some corrections, that I can summarize here:

- I have underlined/highlighted with yellow, some 'errata'

Thanks! We go through the  text, correct all errata, and mark corrections in the text by bold type.

- Check the use of 'modelling' or 'modeling', I mean, both of them appear through the text. I think, I underline all of them but you must check them.

Thank you! Indeed, we mixed the both spelling. We correct the text and replace 'modelling' by 'modeling'.

- Check "in respect of", I'm not sure if it is "with respecto to"

Here we also replace "in respect of" by "with respect to"

L4: miss "their" --> and their corresponding host stars

L13: because of the novelty of the topic: miss "the"

L21: System: capital S

L28: rather than that... : miss "than"

L38: basing on --> based on

L42-43: larger --> higher

L44-45: more deep --> deeper

L50: rather that --> rather than

L53: system --> systems

L54: we give... --> we give their...

L60: miss a "," at the end of the line?

L63: for... --> as an… 

L87: under surface --> under the surface

L111: dynamo considered --> dynamo is considered

L113: the effects --> these effects (?)

L118-120: starting from "i.e. averaging..." do you mean something like "i.e. averaging over the convective vortices inside the spherical shell, the scale of the dynamo driven magnetic field is..." ?

Yes, thank you!

L180: has desired --> has the desired

L181: oparities --> parities (?)

L185: rather what --> rather than what...

L200: several percent --> few percent (?)

L264: Radios --> radio

L270: likes --> is similar to..

I don't know if the word "bowel" could be replaced by "interior", it seems more suitable.

Yes, you are right. It is more suitable!

L287: that --> and

L297: allowed estimating --> allow to estimate

L310: paternal --> parent

L324: lead to formation --> lead to the formation

L343: check Lyman-$\alpha$

L359, 360: Alfven --> Alfvèn

L364: By other words --> In other words

L376: Incorrect form to include the reference

Thank you! We add all the corrections to the text!

Author Response File: Author Response.pdf

Reviewer 2 Report

The new version is much longer but does not improve upon the original one. The extra parts are trivial statements about planetary magnetic fields in the solar system and the symmetries of planetary dynamos. There are still no practical applications for exoplanets. On top of that, the language of the new version is unreadable. Unfortunately, I do not see any reason for me to change my original evaluation of this paper.

Author Response

We fully appreciate this criticism and avoid writing such kind of papers systematically. This case is however an exception determined by the general situation in the topic under discussion.

Author Response File: Author Response.pdf