Robert D Kavanagh

Aline A Vidotto

Baptiste Klein

Moira M Jardine

Jean-François Donati

Dúalta Ó Fionnagáin

Planet-induced radio emission from the coronae of M dwarfs: the case of Prox Cen and AU Mic

Keywords: stars: individual: Proxima Centauri, AU Microscopii -stars: winds, outflowsstars: mass-loss -stars: magnetic field -radio continuum: planetary systems

There have recently been detections of radio emission from low-mass stars, some of which are indicative of star-planet interactions. Motivated by these exciting new results, in this paper we present Alfvén wave-driven stellar wind models of the two active planet-hosting M dwarfs Prox Cen and AU Mic. Our models incorporate large-scale photospheric magnetic field maps reconstructed using the Zeeman-Doppler Imaging method. We obtain a mass-loss rate of 0.25 𝑀 for the wind of Prox Cen. For the young dwarf AU Mic, we explore two cases: a low and high mass-loss rate. Depending on the properties of the Alfvén waves which heat the corona in our wind models, we obtain mass-loss rates of 27 and 590 𝑀 for AU Mic. We use our stellar wind models to assess the generation of electron cyclotron maser instability emission in both systems, through a mechanism analogous to the sub-Alfvénic Jupiter-Io interaction. For Prox Cen we do not find any feasible scenario where the planet can induce radio emission in the star's corona, as the planet orbits too far from the star in the super-Alfvénic regime. However, in the case that AU Mic has a stellar wind mass-loss rate of 27 𝑀 , we find that both planets b and c in the system can induce radio emission from ∼ 10 MHz -3 GHz in the corona of the host star for the majority of their orbits, with peak flux densities of ∼ 10 mJy. Detection of such radio emission would allow us to place an upper limit on the mass-loss rate of the star.

INTRODUCTION

Many theoretical works have aimed to identify potential targets for the detection of exoplanetary radio emission (Grießmeier et al. 2007;Vidotto et al. 2015;Vidotto & Donati 2017;Kavanagh et al. 2019). A model that is commonly used to estimate the generation of radio emission from exoplanets is the radiometric Bode's law, which extrapolates the observed relation between the radio power emitted from the magnetised solar system objects and the incident solar wind power (Zarka 2007). In this model, the interaction between the planet's magnetosphere and the magnetised stellar wind of its host star leads to reconnection in the planet's magnetotail. This reconnection then accelerates electrons towards the planet's poles, creating emission through the electron cyclotron maser instability (ECMI).

In these works, hot Jupiters have often been selected as suitable

★ Contact e-mail: kavanar5@tcd.ie candidates for radio detection. As they are exposed to harsh stellar wind conditions due to their proximity to their host stars (Vidotto et al. 2015), they are expected to produce much higher radio powers than those observed in the solar system (Zarka et al. 2001). This emission can only be generated if hot Jupiters are magnetised. Given that the emission mechanism is ECMI, it occurs at the cyclotron frequency 𝜈 = 2.8 𝐵 MHz, where 𝐵 is the magnetic field strength in gauss (G). Zaghoo & Collins (2018) estimated that hot Jupiters could harbour magnetic fields with strengths of 1 -10 G. In this range, these field strengths would correspond to radio emission from 2.8 -28 MHz. Despite hot Jupiters being seemingly favourable sources of detectable low frequency radio emission, many radio surveys of these systems have found little evidence of such emission (Smith et al. 2009;Lazio et al. 2010;Lecavelier des Etangs et al. 2013;de Gasperin et al. 2020;Narang et al. 2021). Numerous explanations have been proposed for the lack of detections, such as a mismatch between the emitted and observation frequency (Bastian et al. 2000), the emission being beamed out of our line of sight (Smith et al. 2009), and absorption of the emission by the stellar wind of the host star (Vidotto & Donati 2017;Kavanagh et al. 2019;Kavanagh & Vidotto 2020). Intriguingly, Turner et al. (2021) recently detected radio emission at ∼ 30 MHz from the hot Jupiter host 𝜏 Boo. They suggest that this emission could be indicative of ECMI generated in the orbiting planet's magnetic field. Follow-up observations will be required however in order to investigate this further.

An alternative unipolar model for the generation of radio emission via ECMI in exoplanetary systems has also been proposed, analogous to the sub-Alfvénic interaction between Jupiter and its moon Io (Neubauer 1980;Zarka 1998;[START_REF] Saur | Plasma interaction of Io with its plasma torus[END_REF]Zarka 2007;Grießmeier et al. 2007). In this model, the host star and planet take the roles of Jupiter and Io respectively. If the planet orbits with a sub-Alfvénic velocity relative to the wind of its host star, it can generate Alfvén waves that travel back towards the star (Ip et al. 2004;McIvor et al. 2006;Lanza 2012;Turnpenney et al. 2018;Strugarek et al. 2019;Vedantham et al. 2020). A fraction of the wave energy produced in this interaction is expected to dissipate and produce radio emission via ECMI in the corona of the host star (Turnpenney et al. 2018).

Due to the increasing sensitivity of radio telescopes such as LOFAR, M dwarfs are beginning to light up the radio sky at low frequencies (Callingham et al., submitted). One such system is the quiescent M dwarf GJ 1151, which was recently detected to be a source of 120 -160 MHz emission by Vedantham et al. (2020). The authors illustrated that the observed emission is consistent with ECMI from the star induced by an Earth-sized planet orbiting in the sub-Alfvénic regime with a period of 1 -5 days. Prior to this detection, there had been no evidence to suggest GJ 1151 is host to a planet. There has been some discussion in the literature recently about the existence of such a planet. Mahadevan et al. (2021) have suggested that a planet orbits the star in a 2-day orbit, whereas Perger et al. (2021) have ruled this out, placing a mass upper limit of 1.2 Earth on a planet in a 5-day orbit. Follow-up observations of the system will be needed to further assess if the radio emission is of a planet-induced origin.

M dwarfs typically exhibit strong magnetic fields, with maximum large-scale strengths ranging from ∼ 100 G -1 kG (Donati et al. 2008;Morin et al. 2010;Shulyak et al. 2019). If an M dwarf is host to a planet that orbits in the sub-Alfvénic regime, the planet could induce radio emission from the star's corona via ECMI at frequencies up to 280 MHz -2.8 GHz. Planet-induced emission from M dwarfs therefore could be distinguished from emission predicted with the radiometric Bode's law, due to the different frequency ranges.

The M dwarf Proxima Centauri (Prox Cen) is our closest stellar neighbour, which also hosts a planet that is expected to be Earthsized (Anglada-Escudé et al. 2016). Naturally, being so close to us we want to understand both the habitability of the orbiting planet and the system's potential for producing planet-induced radio emission. An important step in answering this question is to assess the stellar wind environment around the planet. Prox Cen is expected to have a relatively weak stellar wind, with a mass-loss rate of 𝑀 < 0.2 𝑀 (Wood et al. 2001), where 𝑀 = 2×10 -14 𝑀 yr -1 is the solar wind mass-loss rate. The star also possesses a strong large-scale surface magnetic field of ∼ 200 G (Klein et al. 2021). Its proximity to Earth along with its potentially habitable planet make it an interesting system to search for planet-induced radio emission.

AU Microscopii (AU Mic) is a young M dwarf which also shows potential for detecting planet-induced radio emission. It lies just under 10 pc away from Earth, and has recently been discovered to host two Neptune-sized close-in planets (Plavchan et al. 2020;Martioli et al. 2020a). We list the properties of the AU Mic system along with those for Prox Cen in Table 1. While planets b and c orbiting AU Mic are not likely to be habitable, their proximity to the host star makes them ideal candidates for inducing radio emission in the corona of the host star. The stellar wind mass-loss rate of AU Mic is relatively unconstrained. Models of interactions between the stellar wind and debris disk in the system estimate a mass-loss rate from 10 𝑀 (Plavchan et al. 2009) up to 1000 𝑀 (Chiang & Fung 2017). Its large-scale surface magnetic field is also quite strong, with a strength of ∼ 500 G (Klein et al. 2020). The two close-in planets along with the star's strong magnetic field make it a very suitable candidate for planet-induced radio emission.

In this paper, we use the magnetic field maps reconstructed by Klein et al. (2021) and Klein et al. (2020) to model the stellar winds of the two M dwarfs Prox Cen and AU Mic respectively. We use our 3D Alfvén wave-driven stellar wind models to investigate if the orbiting planets can induce the generation of ECMI emission in the coronae of their host stars. We then illustrate how the detection of planet-induced radio emission can be used to constrain properties of the wind of the host star.

MODELLING THE WINDS OF M-DWARFS

To model the stellar winds of Prox Cen and AU Mic we use the Alfvén wave solar model (AWSoM, van der Holst et al. 2014) implemented in the 3D magnetohydrodynamic (MHD) code BATS-R-US (Powell et al. 1999), which is part of the Space Weather Modelling Framework (Tóth et al. 2012). In the AWSoM model, Alfvén waves are injected at the base of the chromosphere where they propagate, reflect, and dissipate. The dissipation of the Alfvén waves heats the corona, which in turn drives the stellar wind outflow. The AWSoM model has been validated against observations of the solar corona (van der Holst et al. 2014;Jin et al. 2017), and has also been applied to study the stellar wind environments of other systems (see Garraffo et al. 2017;Boro Saikia et al. 2020;Alvarado-Gómez et al. 2020;Ó Fionnagáin et al. 2021).

In our models, we set boundary conditions for the radial magnetic field, Alfvén wave flux, plasma density and temperature at the base of the chromosphere. For the magnetic field we use maximumentropy maps reconstructed from spectropolarimetric observations for the two stars by Klein et al. (2021) and Klein et al. (2020) via the Zeeman-Doppler Imaging method [START_REF] Donati | [END_REF]. These are shown in Figure 1. The input for the Alfvén wave flux in our model is given in units of flux per unit magnetic field strength 𝑆 A /𝐵. We refer to this as the 'Alfvén wave flux-to-magnetic field ratio', for which we adopt a variety of values (see Section 3). We fix the density and temperature at the base of the chromosphere as 2 × 10 10 cm -3 and 5 × 10 4 K respectively in our models, the same as chosen by Alvarado-Gómez et al. (2020). These values are also typically used for the Sun (see Sokolov et al. 2013).

The corona is heated through a turbulent energy cascade in our model, which is produced by the reflection and dissipation of Alfvén waves in the local plasma environment. The wave dissipation rate depends on the transverse correlation length 𝐿 ⊥ , which in turn is proportional to 𝐵 -1/2 (Hollweg 1986). In our model the transverse correlation length is given in terms of the proportionality constant 𝐿 ⊥ √ 𝐵. We adopt the default value of 𝐿 ⊥ √ 𝐵 = 1.5 × 10 9 cm √ G for this (van der Holst et al. 2014). We also include both collisional and collisionless heat conduction, radiative cooling, and stochastic 2020) respectively. We implement these maps at the inner boundary of our stellar wind simulations for the two stars.

heating in our models, as described in van der Holst et al. (2014).

We fix the adiabatic index to be 5/3. Our computational grid is spherical and extends from the base of the chromosphere to 100 times the stellar radius 𝑅 ★ . The maximum resolution in our grid is ∼ 5 × 10 -4 𝑅 ★ . We also adaptivelyresolve current sheet regions in our grid. The number of cells in our grid is ∼ 6 million. We iteratively solve the MHD equations until the model reaches a steady state. We take this to be when the mass-loss rate, open magnetic flux, and angular momentum-loss rate of the wind vary by less than 10 percent between iterations.

STELLAR WIND ENVIRONMENTS OF THE M-DWARFS

Prox Cen

Using an Alfvén wave flux-to-magnetic field ratio of 𝑆 A /𝐵 = 5 × 10 4 erg s -1 cm -2 G -1 , we obtain a stellar wind with a mass-loss rate of 𝑀 = 0.25 𝑀 for Prox Cen. This is in agreement with the upper limit obtained by Wood et al. (2001). The rest of the free parameters in the model are as described in Section 2. Our calculated mass-loss rate is in line with other published works on the wind of Prox Cen, which used proxies for the surface magnetic field (Garraffo et al. 2016;Alvarado-Gómez et al. 2020).

Figure 2 shows a 3D view of the simulated wind for Prox Cen. We see that the planet orbits in a super-Alfvénic regime for the entirety of its orbit. Therefore, we do not expect any unipolar planetinduced radio emission from Prox Cen during the epoch of the observations from which the magnetic field map was reconstructed (April -July 2017, Klein et al. 2021). Note that we assume that the orbital plane of the planet is perpendicular to the rotation axis of the star.

AU Mic

To investigate the two estimated mass-loss rates for AU Mic, we use two different values for the Alfvén wave flux-to-magnetic field ratio. Again, the values used for the rest of the free parameters in the model are given in Section 2. For a value of 𝑆 A /𝐵 = 1.1 × 10 5 erg s -1 cm -2 G -1 (Sokolov et al. 2013), we obtain a stellar wind mass-loss rate of 27 𝑀 for AU Mic. We refer to this as the 'Low 𝑀' model for the star. Increasing the Alfvén wave flux-to-magnetic field ratio to 𝑆 A /𝐵 = 6 × 10 6 erg s -1 cm -2 G -1 , we find that the mass-loss rate increases to a value of 590 𝑀 . We refer to this as the 'High 𝑀' model. Figure 3 shows a 3D view of both stellar wind models for AU Mic. We take both planets to orbit perpendicular to the stellar rotation axis, as both planets transit the star with orbital inclinations of ∼ 90 • (Martioli et al. 2020a), with the orbit of planet b being aligned with the rotation axis of the star (Martioli et al. 2020b).

In the case of the Low 𝑀 model (left panel of Figure 3), we find that both planets b and c orbit in the sub-Alfvénic regime for the majority of their orbit. This means that the two planets could 1). The contour in the orbital plane is coloured by the wind radial velocity (𝑢 𝑟 ).

induce the generation of radio emission along the lines connecting the planet to the star (Turnpenney et al. 2018;Vedantham et al. 2020). For the High 𝑀 model however, both planets are in the super-Alfvénic regime for their entire orbit (right panel of Figure 3), as the wind is much denser than the Low 𝑀 model. As a result, we do not expect planet-induced radio emission from AU Mic in the case that it has a high mass-loss rate.

We now estimate the maximum mass-loss rate for AU Mic at which the planets can induce radio emission. Using a least-squares method, we fit our data with a power law between the mass-loss rate and maximum size of the Alfvén surface in the orbital plane 𝑅 max A, orb . For our data, we obtain a fit of 𝑀 = 9.6 × 10 5 (𝑅 max A, orb ) -2.6 𝑀 , where 𝑅 max A, orb is in stellar radii. For both planets to be in sub-Alfvénic orbits at least partially, the maximum size of the Alfvén surface must be greater than the orbital distance of planet c. Substituting this orbital distance of 26.9 𝑅 ★ into our power law fit to our data, we find that AU Mic would need to have a mass-loss rate of 190 𝑀 in order for planet-induced radio emission to be generated in its corona. Detection of such emission would therefore allow us to place an upper limit of 190 𝑀 on the mass-loss rate of AU Mic. Note that in either case, radio emission could still be generated from the planet's own magnetosphere (see Vidotto & Donati 2017;Kavanagh et al. 2019;Kavanagh & Vidotto 2020). We explore the scenario for the generation of planet-induced radio emission in AU Mic's corona further in Section 4.1.

PLANET-INDUCED RADIO EMISSION

AU Mic

Here, we describe our model we use to estimate the radio emission induced in the corona of AU Mic by planets b and c, in the scenario where the star has a mass-loss rate of 190 𝑀 . We use our Low 𝑀 model data in our calculations. A planet is said to be in the sub-Alfvénic regime when the Alfvén Mach number is less than 1:

𝑀 A = Δ𝑢 𝑢 A < 1.
(1)

Here, Δ𝑢 is the relative velocity between the stellar wind velocity 𝑢 and planet orbital velocity 𝑢 p :

𝚫𝒖 = 𝒖 -𝒖 p .
(2)

The orbital velocity of the planet is 𝒖 p = √ 𝐺 𝑀 ★ /𝑎 φ, where 𝐺 is the gravitational constant, 𝑀 ★ is the stellar mass, 𝑎 is the orbital distance, and φ is the azimuthal angle. In equation 1, 𝑢 A is the Alfvén velocity of the stellar wind at the planet:

𝑢 A = 𝐵 √ 4𝜋𝜌 , (3)
where 𝐵 and 𝜌 are the stellar wind magnetic field strength and density respectively. The power of the Alfvén waves produced by a planet orbiting in the sub-Alfvénic regime is [START_REF] Saur | [END_REF]):

𝑃 = 𝜋 1/2 𝑅 2 𝐵𝜌 1/2 Δ𝑢 2 sin 2 𝜃, ( 4
)
where 𝑅 is the radius of the obstacle, and 𝜃 is the angle between the vectors 𝑩 and 𝚫𝒖. Assuming the planet is unmagnetised, we use the planetary radius 𝑅 p for the obstacle radius. The waves travel towards the star along the magnetic field line connecting it to the planet, and a fraction of their energy is converted into ECMI emission. The emission occurs at the local cyclotron frequency (Turnpenney et al. 2018):

𝜈 = 2.8 × 10 -3 𝐵 1 G GHz. (5)
We assume emission can be generated everywhere along the magnetic field line connecting the planet to the star, provided the cyclotron frequency exceeds the plasma frequency:

𝜈 > 𝜈 p = 9 × 10 -6
√ 𝑛 e 1 cm -3 GHz.

(6)

Here 𝑛 e is the electron number density. We trace each magnetic field line that connects to the planet's orbit back to the point where it connects to the surface of AU Mic. Both planets orbit through closed-field regions of the star's largescale magnetic field, which predominantly resembles an aligned dipole (Klein et al. 2021). This means that ECMI emission can be generated in both the Northern and Southern hemisphere of AU Mic. Checking the condition in Equation 6 is satisfied, we extract the magnetic field strength along the line and compute the cyclotron frequency with Equation 5. These regions are highlighted in blue in the left panel of Figure 3. We then compute the flux density observed at a distance 𝑑 from the system as

𝐹 𝜈 = 𝜀𝑃 Ω𝑑 2 𝜈 . ( 7
)
Here 𝜀 is the fraction of the wave energy from the interaction which is converted into ECMI emission. Observations of the Jupiter-Io interaction indicate that 𝜀 ≈ 0.01 (Turnpenney et al. 2018), which 1). The contour in the orbital plane is coloured by the wind radial velocity (𝑢 𝑟 ). The grey lines show the stellar magnetic field lines that connect to the orbit of planet b. Each of these lines is a closed loop, and connects back to the star in both the Northern and Southern hemisphere. The blue shaded region of each line illustrates where planet b can induce the generation of radio emission via ECMI (see Equation 6). Note that planet c can also induce generation, but for clarity we omit these field lines. Right: High 𝑀 model for AU Mic. Both planets b and c orbit in the super-Alfvénic regime in this scenario. Note that the magnetic field lines shown here do not connect to the orbit of either planet.

we adopt here. Ω is the solid angle of the emission beam. Again, we adopt the observed value for the Jupiter-Io interaction of Ω = 1.6 sr (Zarka et al. 2004).

We show the flux density induced by planet b in the Northern and Southern hemispheres of AU Mic as a function of its orbital phase in Figure 4. The flux density is colour-coded with the frequency of the emission. We find that planet b can induce emission from ∼ 10 MHz -3 GHz in both hemispheres, with flux densities ranging from ∼ 10 -2 -10 mJy. Higher flux densities correspond to lower frequencies (see Equation 7). Note that planet c also can induce radio emission in this frequency range, albeit with flux densities that are around an order of magnitude lower. This is due to the term 𝐵𝜌 1/2 in Equation 4 dominating at larger orbital distances, as the density and magnetic field strength drop off radially. As the planets orbit with different rotation periods however, they may produce distinct radio signals at different frequencies and periodicities that could be detected.

In Figure 4 we also highlight the emission that is generated at 140 MHz, which corresponds to the middle of the frequency range of 120 -160 MHz at which some M dwarfs have recently been detected (Vedantham et al. 2020, Callingham et al., submitted). Vedantham et al. (2020) suggested that their observations of emission from the M dwarf GJ 1151 may be generated by a planet orbiting in the sub-Alfvénic regime. At 140 MHz, our results bear a strong resemblance to the observations of GJ 1151: both have flux densities of about 1 mJy which exhibit temporal variations. It would be very useful to obtain radio and near-simultaneous spectropolarimetric observations of M dwarfs similar to these systems so that this scenario could be explored further.

Time-varying nonthermal radio emission from AU Mic at 1.5 GHz was reported by [START_REF] Cox | Radio Stars[END_REF], with flux den-sities ranging from 2 -3 mJy. Compared to our predicted fluxes induced by planet b at this frequency range, the observed flux densities are about an order of magnitude larger. If this emission was induced by either planet, it would require a much higher radio power than we predict. This could occur if the wind properties were enhanced in response to an increase in the magnetic field strength of AU Mic. Alternatively, if the radius of the obstacle were 𝑅 ≈ 3.2 𝑅 p , the radio power estimated using Equation 4 would increase by an order of magnitude. We estimate that this would be the size of the magnetopause of planet b if it had a dipolar field strength of ∼ 6 G (see Vidotto & Donati 2017;Kavanagh et al. 2019). Therefore, not only could detection of planet-induced radio emission allow us to constrain the mass-loss rate of the host star, but also the magnetic field strength of the planet itself, if it is magnetised. As there is no polarisation information reported by [START_REF] Cox | Radio Stars[END_REF] however, we unfortunately cannot assess the origin of this emission this further. Future observations with the VLA, MWA, or upcoming SKA radio telescope would certainly be beneficial to search for radio signals in the frequency range at which we predict AU Mic could emit planet-induced radio emission. A derivable quantity such as the brightness temperature of the emission could allow for detected polarised emission indicative of being planet-induced to be distinguished from flaring emission, as has been illustrated by Vedantham et al. (2020). Emission generated at 140 MHz is highlighted with a white dashed line. This is the middle frequency of the observing band at which radio emission was recently detected from the M dwarf GJ 1151 by Vedantham et al. (2020), which is suspected of being induced by an orbiting planet. The grey shaded areas illustrate the region where the orbit of planet is in the super-Alfvénic regime. No emission can generate in these regions (see Equation 6).

Prox Cen

suggested that this emission could be indicative of star-planet interactions. If this is the case, it would require the planet to be orbiting in the sub-Alfvénic regime. Our wind model of Prox Cen however, which has a mass-loss rate in agreement with the upper limit obtained by Wood et al. (2001), shows that planet b orbits in the super-Alfvénic regime for the entirety of its orbit. In order for the planet to induce radio emission from the corona of Prox Cen, the Alfvén surface would need to extend beyond the planet's orbit. This would require the mass-loss rate of the star to be significantly lower than the value we obtained of 0.25 𝑀 .

To investigate this further, we ran two additional stellar wind models for Prox Cen, with Alfvén wave flux-to-magnetic field ratios of 𝑆 A /𝐵 = 1.1 × 10 4 erg s -1 cm -2 G -1 and 1.1 × 10 5 erg s -1 cm -2 G -1 . We obtain mass-loss rates of 0.05 𝑀 and 0.57 𝑀 respectively for these two simulations. In the case of the lower mass-loss rate model, we find that the planet still orbits entirely in a super-Alfvénic regime. Naturally, this is also the case for the model 0.57 𝑀 , which has a denser wind than our model presented in Section 3.1.

We now use these results, combined with those from Section 3.1, to estimate the Alfvén wave flux-to-magnetic field ratio and mass-loss rate needed to place Prox Cen b in a sub-Alfvénic orbit. We first perform a least-squares power law fit between the Alfvén wave flux-to-magnetic field ratio 𝑆 A /𝐵 and mass-loss rate 𝑀, and find that 𝑀 = 2.6 × 10 -6 (𝑆 A /𝐵) 1.06 𝑀 for Prox Cen, where 𝑆 A /𝐵 is in erg s -1 cm -2 G -1 . The effects of changing 𝑆 A /𝐵 on the solar wind mass-loss rate were investigated by Boro Saikia et al. (2020) using the AWSoM model. For comparison, we fit the same power law to their results, and find that during solar minimum 𝑀 ∝ (𝑆 A /𝐵) 1.37 , and during solar maximum 𝑀 ∝ (𝑆 A /𝐵) 1.42 (using the 𝑙 max = 5 data in their Tables F.1 and F.2).

For Prox Cen, we compute the maximum radius of the Alfvén surface 𝑅 max A, orb in the orbital plane of the planet for each plot, and plot it with its corresponding Alfvén wave flux-to-magnetic field ratio and mass-loss rate in Figure 5. Again, performing a least-squares power law fit between 𝑅 max A, orb and 𝑆 A /𝐵, we estimate that an Alfvén wave flux-to-magnetic field ratio of < 5.3 × 10 3 erg s -1 cm -2 G -1 is required in order to place the plane in a sub-Alfvénic orbit. This corresponds to a stellar wind mass-loss rate of < 2.3 × 10 -2 𝑀 , which is an order of magnitude lower than the upper limit obtained by Wood et al. (2001).

DISCUSSION

There are some caveats to our planet-induced radio emission model. Firstly, we assume that the emission is always visible to the observer. However, it is well-established from observations of the Jupiter-Io interaction that the emission generated in this interaction is beamed in a hollow cone (see Zarka et al. 2004). As a result, if the emission can be generated it would likely be beamed in and out of the line of sight towards the system. Therefore, our results represent the best-case scenario where we always see the emission. We also do not account for the relative rotational motion between the star and planet, which would effect which stellar magnetic field line connects to the planet at any given time. Doing so would allow for us to estimate the temporal variations one would see observing the system, if this emission were detectable.

We also note that the higher radio fluxes we estimate for AU Mic in the case of a low mass-loss rate correspond to lower frequencies. However, as was shown by Kavanagh & Vidotto (2020), low frequency radio emission could suffer from significant attenuation from the stellar wind. We computed the thermal spectrum of AU Mic's low mass-loss rate wind using the model developed by the Alfvén wave flux-to-magnetic field ratio (red diamonds). The upper 𝑥-axis is scaled accordingly based on the mass-loss rate of each wind simulation. The red dashed line illustrates our least-squares power law fit to the datapoints. We extrapolate this fit to estimate the Alfvén wave fluxto-magnetic field ratio required for the Alfvén surface to exceed the orbital distance of the planet (blue horizontal line). If this condition is satisfied, the planet can induce the generation of radio emission in the corona of the Prox Cen. This region is shown in grey on the left-hand side. We find that an Alfvén wave flux-to-magnetic field ratio of 2.7 × 10 -3 erg s -1 cm -2 G -1 is needed for the planet to orbit inside the Alfvén surface. This corresponds to mass-loss rate of 1.1 × 10 -2 𝑀 for Prox Cen. Ó Fionnagáin et al. (2019a,b), and found that the wind is optically thick below ∼ 0.1 GHz. This means that a significant amount of the low frequency planet-induced emission (< 0.1 GHz) we predict could be absorbed by the stellar wind. Note that we compute a peak flux density of ∼ 0.5 𝜇Jy for the thermal wind spectrum of AU Mic, which is 2 -3 orders of magnitude smaller than the predicted planet-induced emission.

While we do not expect planet-induced radio emission from Prox Cen or AU Mic in the case that it has a high mass-loss rate, radio emission could still be generated in the magnetospheres of the orbiting planets (see Vidotto et al. 2019). For a low mass-loss rate however, AU Mic seems to be promising candidate for detectable planet-induced radio emission. Simultaneous spectropolarimetric and radio data of the system would be very complimentary to investigate this further, which would allow us to assess the stellar wind environment of AU Mic at the time of the radio observations. UV observations of the planetary transits could also be used in parallel, which have been shown to vary depending on the mass-loss rate of the host star (Carolan et al. 2020). In a broader context, the methods presented in the paper could be applied to a wide range of other suitable exoplanetary systems which may be capable of generating planet-induced radio emission.

CONCLUSIONS

In this paper, we presented Alfvén wave-driven stellar wind models for the two active M dwarfs Prox Cen and AU Mic, both of which are hosts to planets. We used our stellar wind models to investigate whether the orbiting planets could orbit sub-Alfvénically and induce radio emission in the coronae of their host stars, through a Jupiter-Io-like interaction. For Prox Cen, we constrained our stellar wind model using the upper limit for the mass-loss rate of 0.2 𝑀 obtained by Wood et al. (2001). We found that for this mass-loss rate, the planet orbits in the super-Alfvénic regime for its entire orbit. As a result, we do not expect that the planet could induce radio emission from the corona of the host star. By performing additional stellar wind simulations and extrapolating the results, we estimate that Prox Cen would need to have a mass-loss rate of 1.1 × 10 -2 𝑀 in order for the planet to orbit sub-Alfvénically. In other words, Prox Cen would need to have a mass-loss rate that is 20 times lower than the expected upper limit in order for the orbiting planet to induce radio emission in its corona.

For AU Mic, we investigated two scenarios, where the star has a low and high mass-loss rate. By adjusting the flux of Alfvén waves propagating out of the photosphere in our models, we obtained massloss rates of 27 𝑀 and 590 𝑀 respectively for the star. In the case of the low mass-loss rate (27 𝑀 ), we found that both planets b and c orbit sub-Alfvénically for the majority of their orbits. This means that both planets could induce the generation of radio emission in the corona of AU Mic. We showed that planet b could induce timevarying radio emission from ∼ 10 MHz -3 GHz, with a peak flux density of ∼ 10 mJy. The radio emission we predict also bears a striking resemblance to that reported for the M dwarf GJ 1151 by Vedantham et al. (2020), which is suspected of be induced by a terrestrial planet in a 1 -5 day orbit. Planet c could also induce radio emission in this frequency range, but with flux densities that are about an order of magnitude lower, and for a smaller fraction of its orbit. Due to the differences in orbital periods, both planets b and c could therefore produce radio emission distinct from one another. Comparing our estimated planet-induced radio emission for AU Mic to observations reported by [START_REF] Cox | Radio Stars[END_REF], we found that planet b could be magnetised, with a field strength of ∼ 6 G. However, as there is no polarisation information reported by [START_REF] Cox | Radio Stars[END_REF], we cannot assess further if this emission is indicative of being induced by the planets.

Our results illustrate that detection of planet-induced radio emission from AU Mic could allow us to constrain both the massloss rate of the star, as well as the magnetic field strength of the planet that induces the emission.

DATA AVAILABILITY

The data presented in this paper will be shared on reasonable request to the corresponding author.

Figure 1 .

 1 Figure 1. Radial photospheric magnetic field of Prox Cen (top) and AU Mic (bottom), reconstructed by Klein et al. (2021) and Klein et al. (2020) respectively. We implement these maps at the inner boundary of our stellar wind simulations for the two stars.

Figure 2 .

 2 Figure 2. Simulated stellar wind of Prox Cen. The grey lines illustrate the large-scale magnetic field of the star which is embedded in the wind. The orbit of Prox Cen b is shown as a black circle, and the white line corresponds to where the Alfvén Mach number 𝑀 A = 1 (see Equation1). The contour in the orbital plane is coloured by the wind radial velocity (𝑢 𝑟 ).

Figure 3 .

 3 Figure 3. Left: Low 𝑀 model of the stellar wind of AU Mic. The orbits of planets b and c are shown as black circles, and the white line corresponds to where the Alfvén Mach number 𝑀 A = 1 (see Equation1). The contour in the orbital plane is coloured by the wind radial velocity (𝑢 𝑟 ). The grey lines show the stellar magnetic field lines that connect to the orbit of planet b. Each of these lines is a closed loop, and connects back to the star in both the Northern and Southern hemisphere. The blue shaded region of each line illustrates where planet b can induce the generation of radio emission via ECMI (see Equation6). Note that planet c can also induce generation, but for clarity we omit these field lines. Right: High 𝑀 model for AU Mic. Both planets b and c orbit in the super-Alfvénic regime in this scenario. Note that the magnetic field lines shown here do not connect to the orbit of either planet.

  Radio emission from 1.1 -3.1 GHz was recently detected from Prox Cen by Pérez-Torres et al. (2020). These observations occurred during April 2017, at same epoch the spectropolarimetric observations began for Prox Cen (Klein et al. 2021). Pérez-Torres et al. (2020)

Figure 4 .

 4 Figure 4. Radio spectrum of AU Mic induced by planet b in the Northern and Southern hemispheres of the star's corona (left and right panels respectively).Emission generated at 140 MHz is highlighted with a white dashed line. This is the middle frequency of the observing band at which radio emission was recently detected from the M dwarf GJ 1151 byVedantham et al. (2020), which is suspected of being induced by an orbiting planet. The grey shaded areas illustrate the region where the orbit of planet is in the super-Alfvénic regime. No emission can generate in these regions (see Equation6).

Figure 5 .

 5 Figure 5. Average radius of the Alfvén surface in the orbital plane of Prox Cen b vs.the Alfvén wave flux-to-magnetic field ratio (red diamonds). The upper 𝑥-axis is scaled accordingly based on the mass-loss rate of each wind simulation. The red dashed line illustrates our least-squares power law fit to the datapoints. We extrapolate this fit to estimate the Alfvén wave fluxto-magnetic field ratio required for the Alfvén surface to exceed the orbital distance of the planet (blue horizontal line). If this condition is satisfied, the planet can induce the generation of radio emission in the corona of the Prox Cen. This region is shown in grey on the left-hand side. We find that an Alfvén wave flux-to-magnetic field ratio of 2.7 × 10 -3 erg s -1 cm -2 G -1 is needed for the planet to orbit inside the Alfvén surface. This corresponds to mass-loss rate of 1.1 × 10 -2 𝑀 for Prox Cen.

Table 1 .

 1 Stellar and planetary parameters for the Prox Cen and AU Mic systems.

	Parameter	Prox Cen	Reference	AU Mic	Reference
	Stellar mass (𝑀 ★ )	0.12 𝑀	1	0.50 𝑀	2
	Stellar radius (𝑅 ★ )	0.14 𝑅	1	0.75 𝑅	2
	Stellar wind mass-loss rate ( 𝑀 )	< 0.2 𝑀	3	10 𝑀 , 1000 𝑀	4, 5
	Average surface stellar magnetic field strength	200 G	6		500 G	7
	Spectral type	M5.5V	1		M1V	8
	Stellar rotation period	83 days	1	4.86 days	2
	Distance (𝑑)	1.30 pc	1		9.79 pc	2
		Planet b		Planet b	Planet c
	Planetary mass (𝑀 p )	> 1.27 𝑀 ⊕	1	1.00 𝑀 Nep	1.7 -27.7 𝑀 ⊕	9
	Planetary radius (𝑅 p )	-	-	1.13 𝑅 Nep	0.91 𝑅 Nep	9
	Planetary orbital distance (𝑎)	74.0 𝑅 ★	1	18.5 𝑅 ★	26.9 𝑅 ★	9
	Planetary orbital period	11.2 days	1	8.46 days	18.

MNRAS 000, 1-8 (2020)

ACKNOWLEDGEMENTS

We thank the anonymous referee for their comments and suggestions. We also thank Dr. Joe Callingham, Dr. Sebastian Pineda, Prof. Harish Vedantham, and Prof. Jackie Villadsen for their insightful discussions. RDK acknowledges funding received from the Irish Research Council (IRC) through the Government of Ireland Postgraduate Scholarship Programme. AAV acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 817540, ASTROFLOW). BK acknowledges funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement No 865624, GPRV). MMJ acknowledges support from STFC consolidated grant number ST/R000824/1. JFD and BK acknowledge the ERC for grant agreement No 740651, NewWorlds. DÓF acknowledges funding from the IRC Government of Ireland Postdoctoral Fellowship Programme. We acknowledge the Irish Centre for High-End Computing (ICHEC) for providing the computational facilities used to perform the simulations published in this work.


K Lund

M Jardine

A J B Russell

J.-F Donati

R Fares

C P Folsom

S V Jeffers

S C Marsden

J Morin

P Petit

V See

Field linkage and magnetic helicity density

Keywords: stars, magnetic field -methods, analytical

The helicity of a magnetic field is a fundamental property that is conserved in ideal MHD. It can be explored in the stellar context by mapping large-scale magnetic fields across stellar surfaces using Zeeman-Doppler imaging. A recent study of 51 stars in the mass range 0.1-1.34 M showed that the photospheric magnetic helicity density follows a single power law when plotted against the toroidal field energy, but splits into two branches when plotted against the poloidal field energy. These two branches divide stars above and below ∼ 0.5 M . We present here a novel method of visualising the helicity density in terms of the linkage of the toroidal and poloidal fields that are mapped across the stellar surface. This approach allows us to classify the field linkages that provide the helicity density for stars of different masses and rotation rates. We find that stars on the lower-mass branch tend to have toroidal fields that are non-axisymmetric and so link through regions of positive and negative poloidal field. A lower-mass star may have the same helicity density as a higher-mass star, despite having a stronger poloidal field. Lower-mass stars are therefore less efficient at generating large-scale helicity.

INTRODUCTION

Magnetic helicity is a fundamental property of magnetic fields that measures the amount of linkage and twist of field lines within a given volume. Since it is exactly conserved in ideal MHD and highly conserved for high magnetic Reynolds numbers in general (Woltjer 1958;Taylor 1974), helicity is an important factor when attempting to understand how magnetic fields are generated and evolve (e.g. Brandenburg & Subramanian 2005;[START_REF] Chatterjee | [END_REF]Pipin et al. 2019). Until recently, this could only be measured for the Sun (e.g. reviews by Démoulin 2007;Démoulin & Pariat 2009). We can, however, now map all three components of the large-scale magnetic field at the surfaces of stars using the spectropolarimetric technique of Zeeman-Doppler imaging (Semel 1989).

These magnetic field maps now exist for a large enough sample of stars that trends with stellar mass and rotation period have become apparent (Donati & Landstreet 2009). In particular, it appears that magnetic fields show different strengths and topologies in the mass ranges above and below ∼ 0.5 M , which is believed to correspond to the onset of the transition from partially to fully convective interiors. Rapidly-rotating stars in the mass range above ∼ 0.5 M tend to have fields that are predominantly toroidal (Donati et al. 2008a). The stronger the toroidal field, the more likely it is to be axisymmetric (See et al. 2015). In the mass range below ∼ 0.5 M , stars show predominantly axisymmetric poloidal fields. For the lowest masses, however, a bimodal behaviour is found, such that stars may have strong, predominantly axisymmetric poloidal fields, or much weaker, non-axisymmetric poloidal fields (Donati et al. 2008a;Morin et al. 2008b;Donati & Landstreet 2009;Morin et al. 2010).

This difference in magnetic fields in stars that are partially or fully convective is also apparent in their photospheric helicity densities. Using observations of 51 stars, Lund et al. (2020) found that the helicity density scales with the toroidal energy according to | ℎ | ∝ B tor 2 0.86 ± 0.04 . The scaling with the poloidal energy is more complex, however, revealing two groups with different behaviours. Specifically, stars less massive than ∼ 0.5 M appear to have an excess of poloidal energy when compared to more massive stars with similar helicity densities. It appears that stars with different internal structures and different total magnetic energies may nonetheless generate magnetic fields with the same helicity density at their surfaces. The aim of this paper is to explore the nature of this division and the types of flux linkage that support the measured helicity densities. In order to do that, we have developed a novel method of visualising the linkages of different field components across the surfaces of stars.

METHODS

Poloidal and toroidal magnetic field components

For the purposes of this paper the stellar magnetic fields discussed will be decomposed into their poloidal and toroidal components: 𝑩 = 𝑩 pol + 𝑩 tor . The poloidal and toroidal fields can be expressed in a general form as (see Appendix III of Chandrasekhar 1961):

𝑩 pol = ∇ × [∇ × [Φ r]],
(1)

𝑩 tor = ∇ × [Ψ r]. ( 2
)
In a spherical coordinate system1 the scalars Φ and Ψ take the form:

Φ = 𝑆(𝑟)𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 , (3)
Ψ = 𝑇 (𝑟)𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 . ( 4
)
𝑃 𝑙𝑚 ≡ 𝑃 𝑙𝑚 (cos 𝜃) is the associated Legendre polynomial of mode 𝑙 and order 𝑚 and

𝑐 𝑙𝑚 ≡ √ 2𝑙 + 1 4𝜋 (𝑙 -𝑚)! (𝑙 + 𝑚)! (5)
is a normalisation constant. 𝑆(𝑟) and 𝑇 (𝑟) are functions describing the radial behaviour of the magnetic field components. Determining the complete form of these functions from observations of stellar magnetic fields is impossible, however values can be obtained at stellar surfaces (𝑟 = 𝑅 ★ ).

The Zeeman-Doppler imaging technique (Semel 1989) describes the large-scale (low 𝑙 modes) magnetic fields at the surfaces of stars in terms of 𝛼 𝑙𝑚 , 𝛽 𝑙𝑚 , and 𝛾 𝑙𝑚 coefficients (e.g. Donati et al. 2006;Vidotto 2016):

𝑩 pol (𝜃, 𝜙) = ∑ 𝑙𝑚 𝛼 𝑙𝑚 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 r + ∑ 𝑙𝑚 𝛽 𝑙𝑚 (𝑙 + 1) 𝑐 𝑙𝑚 d𝑃 𝑙𝑚 d𝜃 𝑒 𝑖𝑚𝜙 θ + ∑ 𝑙𝑚 𝛽 𝑙𝑚 𝑖𝑚 (𝑙 + 1) sin 𝜃 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 φ, (6)
𝑩 tor (𝜃, 𝜙) = ∑ 𝑙𝑚 𝛾 𝑙𝑚 𝑖𝑚 (𝑙 + 1) sin 𝜃 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 θ - ∑ 𝑙𝑚 𝛾 𝑙𝑚 (𝑙 + 1) 𝑐 𝑙𝑚 d𝑃 𝑙𝑚 d𝜃 𝑒 𝑖𝑚𝜙 φ. ( 7
)
These expressions are consistent with the general form of the poloidal and toroidal fields evaluated at the stellar surface when

𝑆(𝑅 ★ ) = 𝛼 𝑙𝑚 𝑅 2 ★ 𝑙 (𝑙 + 1) , d𝑆(𝑟) d𝑟 𝑟 =𝑅 ★ = 𝛽 𝑙𝑚 𝑅 ★ (𝑙 + 1) (8)
and

𝑇 (𝑅 ★ ) = 𝛾 𝑙𝑚 𝑅 ★ (𝑙 + 1) . ( 9
)
Given the surface magnetic field, magnetic energies are estimated by calculating the mean squared magnetic flux density 𝐵2 . For instance, in the case of the poloidal energy 2 : 𝐵 2 pol = 1 Ω ∫ 𝑩 pol • 𝑩 pol dΩ. Accordingly, the fraction of axisymmetric poloidal magnetic field energy is given by3 : 𝐵 2 pol,m=0 / 𝐵 2 pol . The toroidal energy and axisymmetry fraction are calculated analogously.

Magnetic helicity density

Magnetic helicity can be defined as 𝐻 = ∫ 𝑨 • 𝑩 dV (Woltjer 1958), where 𝑨 is a vector potential corresponding to the magnetic field 𝑩. As magnetic helicity is a quantity measuring the linkage of fields within a volume, our surface magnetic fields limit us to evaluating the magnetic helicity density ℎ = 𝑨 • 𝑩. The separation of the magnetic field into its poloidal and toroidal components is particularly useful in this regard. It dispenses with the need to invoke a gauge (Berger & Hornig 2018) since the usual gauge field (the corresponding potential field with the same boundary flux) has zero helicity. In addition, in a spherical coordinate system, toroidal field lines lie purely on spherical surfaces while poloidal field lines pass through these surfaces. This makes visualising the linkage of field lines straightforward. Fig. 1 illustrates how poloidal fields lines (shown red) that pass through the stellar surface may link through loops of toroidal field (shown blue) that lie on the stellar surface. It is notable that in these examples, only some fraction of the poloidal field links with the toroidal field line shown. Interpreting magnetic helicity as the linking of poloidal and toroidal fields (Berger 1985;Berger & Hornig 2018) allows the helicity density to be calculated for any stellar magnetic map given only its 𝛼 𝑙𝑚 and 𝛾 𝑙𝑚 coefficients and the stellar radius 𝑅 ★ (Lund et al. 2020):

ℎ(𝜃, 𝜙) = Re ∑ 𝑙𝑚 ∑ 𝑙 𝑚 𝛼 𝑙𝑚 𝛾 𝑙 𝑚 𝑅 ★ (𝑙 + 1)𝑙 (𝑙 + 1) 𝑐 𝑙𝑚 𝑐 𝑙 𝑚 𝑒 𝑖 𝜙 (𝑚+𝑚 )
𝑃 𝑙𝑚 𝑃 𝑙 𝑚 𝑙 (𝑙 + 1) -

𝑚𝑚 sin 2 𝜃 + d𝑃 𝑙𝑚 d𝜃 d𝑃 𝑙 𝑚 d𝜃 . (10)
The magnetic helicity density, as expressed in Eq. 10, depends on the 𝛼 𝑙𝑚 and 𝛾 𝑙𝑚 coefficients, but not the 𝛽 𝑙𝑚 coefficients found in the 𝜃 and 𝜙 components of the poloidal field. This is because only the radial component of the poloidal field (𝑩 pol,r ) passes through the spherical surfaces containing the toroidal field.

When comparing different magnetic maps, e.g. for different stars, it is often useful to summarise the overall helicity with a single number. For this purpose, we consider an average value across the hemisphere facing the observer. We note that typically only one hemisphere is fully visible as part of the star never comes into view. Furthermore, we take the absolute value of the averaged helicity density as we are interested in comparing magnitudes, not signs. This absolute average helicity density (| ℎ |) will for simplicity be referred to as the "helicity density" for the remainder of this paper.

It is possible to visualise the linkage of poloidal and toroidal fields that results in helicity density at the stellar surface through maps showing the strength of 𝑩 pol,r (calculated from the 𝛼 𝑙𝑚 according to Eq. 6) with the field lines of 𝑩 tor superimposed. Expanding Eq. 2 as

𝑩 tor = ∇Ψ × r (11)
shows that the contours of Ψ correspond to the field lines of 𝑩 tor (see Fig. 2). In particular, at the stellar surface,

Ψ = 𝛾 𝑙𝑚 𝑅 ★ (𝑙 + 1) 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 , (12)
from Eqs. 4 and 9.

STELLAR SAMPLE

Our sample of stellar magnetic maps are all created using Zeeman-Doppler imaging. They describe the magnetic fields of 51 different stars, 15 of which are represented by multiple maps. The stars range in spectral type from F to M, and in mass from 0.1-1.34 M . Details of each star/map are provided in Table 1, along with the calculated helicity densities and magnetic energy components. For the sake of a fair comparison every magnetic map is evaluated to the same resolution, which means every calculation is performed up to the same 𝑙-mode (𝑙 ≤ 4) even when higher modes are available (Lund et al. 2020).

THE ROLE OF AXISYMMETRY IN HELICITY DENSITY

The very well-defined dependence of helicity density on the toroidal field of | ℎ | ∝ B tor 2 0.86 ± 0.04 was shown in Lund et al. (2020). One enduring puzzle, however, is that the dependence on the poloidal field revealed two branches, as shown in Fig. 3. The higher-mass branch (𝑀 ★ > 0.5 M ) follows | ℎ | = 𝐵 2 pol 1.04 ± 0.05 10 10.18 ± 0.13 and the lower-mass branch (𝑀 ★ ≤ 0.5 M ) follows | ℎ | = 𝐵 2 pol 0.88 ± 0.15 10 9.57 ± 0.74 . When fitting these power laws the sample is split specifically at 0.5 M because a number of magnetic properties, including helicity density, have been shown to change behaviour across this value (Donati et al. 2008b;Morin et al. 2008bMorin et al. , 2010;;See et al. 2015;Lund et al. 2020). Fig. 3 shows that the lowest mass stars have higher poloidal energies than higher-mass stars with the same helicity density. The lowest mass stars in our sample also typically have the lowest Rossby numbers, as indicated by the colours in the plot. It appears from Fig. 3 that the lower-mass fully-convective stars have excess poloidal field that does not contribute to their helicity density. In order to explore the distribution of the poloidal and toroidal fields on these two branches and to determine their contribution to the helicity density, we plot maps showing how their poloidal and toroidal fields link. As an example, Fig. 4 presents maps for GJ 182, WX UMa, GJ 49 and GJ 1245B. These stars are highlighted in Fig. 3 and represent two pairs of stars with approximately the same helicity density, and thus similar toroidal energies. Each pair consists of one star from the higher-mass branch and one star from the lowermass branch, and the two pairs are distinguished by the magnitude of their helicity densities (GJ 182 and WX UMa (top row) have higher helicity densities than GJ 49 and GJ 1245B). To illustrate field linkages at the largest scale the maps show the dipole (𝑙 = 1) mode. We note that these four stars fall into the four categories shown in Fig. 1.

The clearest trend to emerge is in the toroidal field. It is notable that the stars on the higher-mass branch, GJ 182 and GJ 49 (left column of Fig. 4), both have fairly axisymmetric toroidal fields, whilst the stars on the lower mass branch, WX UMa and GJ 1245B, have nonaxisymmetric toroidal fields. We can quantify this trend by plotting the fraction of the toroidal field energy that is held in axisymmetric modes as a function of the ratio of toroidal to poloidal energy (see Fig. 5). Lower-mass stars tend to have toroidal fields that are nonaxisymmetric and magnetic energy budgets that are dominated by the poloidal field. Lund et al. (2020) showed that stars with a similar helicity density have toroidal fields of similar strengths. As can be seen from Fig. 4, however, these toroidal fields may have different symmetries if the stars lie on two different branches. The fact that these different symmetries are able to produce the same helicity density is because the poloidal field, although typically stronger on the lower-mass branch, can also be of different symmetry to the toroidal field. In both rows of Fig. 4, the transition from the higher-mass branch to the lower-mass branch is accompanied by an increase in the strength of the poloidal field. At the same time, for the lower mass stars, toroidal field lines enclose regions of both positive and negative poloidal field such that the total linked flux is relatively small. The difference in magnetic field topology roughly cancels the difference in field strength, such that the helicity is very similar within each pair, although it is arrived at very differently.

To separate the effects of field strength and geometry Fig. 6 plots stellar mass against the "helicity energy fraction"; h ≡ | ℎ |/ 𝑅 ★ 𝐵 2 . Dividing the helicity density by the mean squared magnetic flux density and stellar radius results in a dimensionless helicity density normalised by magnetic field strength. Fig. 6 shows an even spread of low helicity energy fractions across the entire range of stellar masses, however only the higher mass stars (𝑀 ★ > 0.5 M ) exceed a fraction of 0.35. This illustrates that even though the lower-mass stars are among those whose dynamos are most efficient at injecting magnetic energy into the largest spatial scales that ZDI is able to detect and map (e.g. Morin et al. 2008b), they are apparently less efficient at generating helicity at these largest scales. This is most likely because of the inefficient linking between their poloidal and toroidal magnetic field components.

Inefficient linking can be arrived at in more than one way. For the higher-helicity pair in Fig. 4 (GJ 182, WX UMa), the lowermass star has a strongly axisymmetric poloidal field, and since the toroidal field is non-axisymmetric, this combination produces inefficient linking. In the case of the lower-helicity pair (GJ 49 and GJ 1245B), the lower-mass star's poloidal and toroidal fields are both strongly non-axisymmetric; nonetheless, they still offset each another by approximately 90 degrees, which again gives inefficient linking.

We can place these trends in a broader context by showing how the axisymmetry and helicity density varies across the stellar massrotation period plane. This is shown in Fig. 7 which also shows separately the variation of the toroidal and poloidal axisymmetry fractions. High-helicity stars exist in both mass ranges. The decline in the axisymmetry of the toroidal fields with decreasing mass is apparent, but the trends in poloidal axisymmetry are more complex. The reasons for this, and the potential role of magnetic cycles, are not yet clear. Fig. 7 also demonstrates very clearly the differences in the strength and structure of the magnetic field that is possible for stars in the bimodal regime, which includes the stars of lowest mass and shortest period. Two of our example stars (WX Uma and GJ 1245B) lie in this regime. They have similar masses (0.1 and 0.12 M respectively) and rotation periods (0.78 and 0.71 days, respectively) but their magnetic fields and the helicity densities they support are quite different.

DISCUSSION AND CONCLUSIONS

Helicity measures the linkage within a field. By studying the linkage of the poloidal and toroidal components of stellar magnetic fields, we can learn about the underlying dynamo processes generating the field, and thereby the form of the magnetic cycles that might take place and the evolution of stellar fields as stars spin down over their main sequence lifetimes. This study reveals that stars in different mass ranges, which may be either fully or partially convective, generate their helicity through different forms of toroidal/poloidal field linkage.

For partially convective stars (those that lie on the higher-mass branch in Fig. 3) the toroidal fields are mainly axisymmetric. An increase in rotation rate (or a decrease in Rossby number) generally leads to increased helicity density. For fully-convective stars (those that lie on the lower-mass branch in Fig. 3) the toroidal fields are mainly non-axisymmetric. For the lowest-mass stars, a bimodal behaviour is present in the form and strength of the magnetic field (Morin et al. 2011) which may be weak with a non-axisymmetric poloidal field, or strong with an axisymmetric poloidal field. These two types correspond to low and high helicity densities respectively. This may have implications for the possibility that this represents a bimodality in the dynamo operating in this regime (Morin et al. 2011).

Stars can evolve from one mass-branch to the other if their field structure changes, for instance as a result of their internal structure changing from mainly convective to mainly radiative at a very young age, or if they transition from one bimodal dynamo mode to the other. Furthermore, stars can evolve along each branch as their rotation rates decay with age. Their field linkages can also evolve on much shorter timescales due to magnetic cycles. It is notable in Fig. 5 that where there are multiple observations of a star, taken at different times, these typically follow the trend that an increase in the ratio of 𝐵 2 tor to 𝐵 2 pol leads to an increase in the axisymmetry of the toroidal field. A similar behaviour is seen in Fig. 3 where for each star with multiple observations, these all lie within the scatter about the best-fit line.

It is not clear what causes the non-axisymmetry of the toroidal fields in the lowest-mass stars. Their deep-seated convection may produce bipoles that emerge through the stellar surface with randomised axial tilts, leading to a lack of axisymmetry in the toroidal field. Their low surface differential rotation may also reduce the shearing of bipoles and hence the diffusive cancellation of poloidal field that results. Both of these processes, however, occur at length scales well below what can be resolved by these Zeeman-Doppler field measurements. The field characteristics that are most robustly recovered by Zeeman-Doppler imaging (field axisymmetry and the ratio of poloidal to toroidal field (Lehmann et al. 2019)) are nonetheless the very ones that underpin the helicity density.

In summary, we find that lowest-mass stars tend to be inefficient at generating helicity on the largest scales, given their magnetic energy. The helicity density at a stellar surface depends not only on the stellar radius and the strengths of the individual poloidal and toroidal field components (see Eq. 10) but also on their spatial distribution relative to each other. The fraction of the poloidal flux that links with the toroidal flux is maximised when the axes of symmetry of the two fields align perfectly. The bottom row of Figure 1 illustrates two ways in which such an alignment is possible: where both fields are axisymmetric (left) and where both are non-axisymmetric (right). Conversely, if the poloidal and toroidal axes of symmetry are orthogonal to each other there is no linkage, and consequently no helicity. Different orientations with different amounts of field linking can nonetheless result in the same helicity density due to the depen-dence on field strengths. In short, to achieve a full understanding of the source of the helicity density at a stellar surface, it is not enough simply to look at, for example, the radius and field strength, it is also necessary to produce a surface map showing the field linkage. It is only through a combination of all these components that a clear picture can be formed.

Table 1. Our stellar sample, with the four stars we focus on in this paper highlighted in bold (GJ 182, GJ 49, GJ 1245B and WX UMa). From left to right the columns show: star name, mass, radius, rotation period,Rossby number, absolute helicity density averaged across the visible hemisphere, 𝐵 2 pol , poloidal axisymmetric magnetic energy as a fraction of poloidal energy, 𝐵 2 tor , toroidal axisymmetric magnetic energy as a fraction of toroidal energy, 𝑙 max and observation epoch. The helicity density and the energies are all calculated for 𝑙 ≤ 4. References for the stellar parameters are given in the last column, where references to the papers where the magnetic maps were published are in italic. A more comprehensive table of parameters for these stars can be found in Vidotto et al. (2014)

Figure 1 .

 1 Figure 1. The cartoon shows four different combinations of symmetries (axisymmetric and non-axisymmetric relative to the rotation axis) of the poloidal (red arrows) and toroidal (blue arrows) fields. Poloidal field lines that link with the toroidal field are represented by solid lines, the ones that do not are dashed.

Figure 2 .

 2 Figure 2. (a) Contours of the function Ψ lying on the surface of the star are shown in blue. (b) This shows an enlarged version, illustrating that 𝑩 tor = ∇Ψ × r .

Figure 3 .

 3 Figure 3. Absolute helicity density averaged across a single hemisphere versus the mean squared poloidal magnetic flux density (𝑙 ≤ 4). The colour of the symbols correspond to Rossby number and the shape of the symbols splits the sample into two mass groups; circles represent 𝑀 ★ > 0.5 M and diamonds represent 𝑀 ★ ≤ 0.5 M . When there are multiple measurements for the same star these are connected by lines. The thick grey lines show the best fit of | ℎ | = 𝐵 2 pol 𝛼 10 𝛽 for stars in the two mass groups; 𝛼 = 1.04 ± 0.05, 𝛽 = 10.18 ± 0.13 for 𝑀 ★ > 0.5 M (circles) and 𝛼 = 0.88 ± 0.15, 𝛽 = 9.57 ± 0.74 for 𝑀 ★ ≤ 0.5 M (diamonds). The stars outlined are shown in Fig. 4.

Figure 4 .

 4 Figure 4. A visualisation of the linkage of the dipole (𝑙 = 1) poloidal and toroidal field components of GJ 182, WX UMa (2008), GJ 49 and GJ 1245B (2006). The colour shows the strength of the radial magnetic (poloidal) field, and the black contours represent the toroidal magnetic field lines. The heavy black contour separates regions of positive (solid) and negative (dashed) toroidal field. These examples correspond roughly to the four classes shown in Fig. 1.

Figure 5 .Figure 6 .

 56 Figure 5. Axisymmetric toroidal energy as a fraction of total toroidal energy versus the ratio of the mean squared toroidal to poloidal magnetic flux densities (𝑙 ≤ 4). The symbols are the same as in Fig. 3.

Figure 7 .

 7 Figure 7. Helicity density of the large-scale magnetic fields (𝑙 ≤ 4) of our stellar sample shown according to stellar mass and rotation period. The dashed lines split the sample at 0.5 M . The size of the symbol indicates relative strength of the helicity density. The colour corresponds to the fractional axisymmetry of the toroidal (top) and poloidal (bottom) magnetic energies. The stars outlined are shown in Fig. 4.

Table 1 .

 1 . -continued

	Star ID	𝑀 ★ (M )	𝑅 ★ (R )	𝑃 rot (d)	𝑅 𝑜	| ℎ | (Mx 2 cm -3 )	𝐵 2 pol (G 2 )	Pol Axi	𝐵 2 tor (G 2 )	Tor Axi	𝑙 max	Obs. epoch	Ref.
	Solar-like stars
	HD 3651	0.88	0.88	44.0	1.916	1.82E+11 1.49E+01	0.87	4.96E-01	0.98	10	-	1, 2, 3
	HD 9986	1.02	1.04	22.4	1.621	7.16E+09	4.71E-01	0.50	3.43E-02	0.94	10	-	1, 2, 3
	HD 10476	0.82	0.82	35.2	0.576	6.77E+09 6.23E+00	0.00	5.69E-01	0.40	10	-	1, 4, 2, 3
	HD 20630	1.03	0.95	9.00	0.593	2.09E+13 2.61E+02	0.32	4.56E+02	0.90	10	Oct 2012	1, 5, 2, 6
	HD 22049	0.86	0.77	11.76	0.366	7.19E+11 1.16E+02	0.70	3.35E+00	0.23	10	-	1, 5, 2, 7
	HD 39587	1.03	1.05	5.136	0.295	1.95E+12 1.45E+02	0.07	2.04E+02	0.84	10	-	1, 5, 2, 3
	HD 56124	1.03	1.01	20.7	1.307	2.43E+11 5.22E+00	0.90	9.32E-01	0.91	10	-	1, 2, 3
	HD 72905	1	1	5.227	0.272	1.39E+13 1.43E+02	0.15	8.70E+02	0.97	10	-	1, 5, 2, 3
	HD 73350	1.04	0.98	12.3	0.777	3.54E+11 8.99E+01	0.00	8.91E+01	0.90	10	-	1, 8, 2, 3
	HD 75332	1.21	1.24	3.870	>1.105 9.88E+11 4.58E+01	0.80	3.96E+00	0.39	15	-	1, 5, 2, 3
	HD 78366	1.34	1.03	11.4	>2.781 2.39E+12 2.11E+02	0.94	8.36E+00	0.29	10	2008	1, 2, 9
	. . .	. . .	. . .	. . .	. . .	4.27E+11 4.83E+01	0.06	7.91E+00	0.54	. . .	2010	. . .
	. . .	. . .	. . .	. . .	. . .	4.29E+11 2.18E+01	0.77	2.25E+00	0.30	. . .	2011	. . .
	HD 101501	0.85	0.9	17.04	0.663	3.02E+12 1.28E+02	0.26	5.68E+01	0.80	10	-	1, 5, 2, 3
	HD 131156A	0.90	0.80	6.25	0.256	2.11E+13 1.48E+03	0.30	4.12E+03	0.93	10	Aug	10, 2, 11
												2007
	. . .	. . .	. . .	. . .	. . .	5.24E+12 5.58E+02	0.59	3.00E+02	0.62	. . .	Feb 2008	. . .
	. . .	. . .	. . .	. . .	. . .	6.64E+12 4.91E+02	0.07	4.54E+02	0.81	. . .	Jun 2009	. . .
	. . .	. . .	. . .	. . .	. . .	4.05E+12 3.89E+02	0.08	1.91E+02	0.37	. . .	Jan 2010	. . .
	. . .	. . .	. . .	. . .	. . .	1.40E+13 2.73E+02	0.34	2.03E+02	0.95	. . .	Jun 2010	. . .
	. . .	. . .	. . .	. . .	. . .	6.60E+13 7.18E+02	0.73	1.12E+03	0.96	. . .	Aug	. . .
												2010
	. . .	. . .	. . .	. . .	. . .	2.91E+13 5.25E+02	0.27	1.77E+03	0.97	. . .	Jan 2011	. . .
	HD 131156B	0.66	0.55	11.1	0.611	5.59E+12 2.60E+02	0.25	1.25E+02	0.81	10	-	10, 2, 3
	HD 146233	0.98	1.02	22.7	1.324	8.87E+08 1.90E+00	0.09	1.48E-02	0.05	10	Aug	12, 2, 8
												2007
	HD 166435	1.04	0.99	4.2	0.259	5.12E+12 2.53E+02	0.50	1.70E+02	0.79	10	-	1, 2, 3
	HD 175726	1.06	1.06	4.0	0.272	1.71E+12 6.65E+01	0.18	2.65E+01	0.80	10	-	1, 13, 2, 3
	HD 190771	0.96	0.98	8.80	0.453	5.02E+12 8.50E+01	0.35	1.53E+02	0.98	10	2007	9, 2, 8
	HD 201091A	0.66	0.62	34.1	0.786	8.29E+10 3.02E+01	0.03	3.57E+00	0.42	10	-	1, 5, 2, 14
	HD 206860	1.1	1.04	4.6	0.388	2.77E+13 3.03E+02	0.49	3.71E+02	0.94	10	-	1, 2, 15
	Young suns
	AB Dor	0.76	1.00	0.5	0.026	2.87E+14 3.34E+04	0.14	1.21E+04	0.56	25	Dec 2001 16, 17, 2, 18
	. . .	. . .	. . .	. . .	. . .	6.80E+13 2.97E+04	0.09	5.92E+03	0.50	. . .	Dec 2002	. . . .
	BD-16351	0.9	0.88	3.21	0.14 +0.01 -0.02 5.04E+13 2.94E+03	0.04	2.45E+03	0.88	15	Sep 2012	19
	HII 296	0.9	0.93	2.61	0.13 +0.01 -0.01 4.74E+13 3.36E+03	0.38	3.96E+02	0.53	15	Oct 2009	19
	HII 739	1.15	1.07	1.58	0.25 +0.01 -0.08 1.28E+12 1.78E+02	0.28	1.09E+02	0.65	15	Oct 2009	19
	HIP 12545	0.95	1.07	4.83	0.14 +0.02 -0.02 8.28E+14 1.36E+04	0.49	7.63E+03	0.85	15	Sep 2012	19
	HIP 76768	0.80	0.85	3.70	0.09 +0.03 -0.02 4.63E+14 5.23E+03	0.75	9.10E+03	0.95	15	May	19
												2013
	TYC 0486-4943-1	0.75	0.69	3.75	0.13 +0.03 -0.03 4.42E+12 7.19E+02	0.13	2.13E+02	0.70	15	Jun 2013	19
	TYC 5164-567-1	0.90	0.89	4.68	0.19 +0.04 -0.05 3.23E+13 3.21E+03	0.59	4.54E+02	0.36	15	Jun 2013	19
	TYC 6349-0200-1	0.85	0.96	3.41	0.07 +0.01 -0.02 4.58E+13 3.27E+03	0.26	8.35E+02	0.58	15	Jun 2013	19
	TYC 6878-0195-1	1.17	1.37	5.70	0.10 +0.04 -0.03 9.02E+13 3.17E+03	0.19	1.40E+03	0.85	15	Jun 2013	19
	Hot Jupiter Hosts
	𝜏 Boo	1.34	1.42	3	>0.732 1.52E+11 1.61E+00	0.48	1.54E+00	0.86	5	Jun 2006	20, 21, 2, 22
	. . .	. . .	. . .	. . .	. . .	3.11E+11 9.42E+00	0.59	1.80E+00	0.65	8	Jun 2007	20, 21, 2, 23
	. . .	. . .	. . .	. . .	. . .	7.83E+10 3.10E+00	0.13	6.16E+00	0.91	. . .	Jan 2008	20, 2, 21
	. . .	. . .	. . .	. . .	. . .	7.76E+10 3.12E+00	0.30	5.48E-01	0.45	. . .	Jun 2008	20, 2, 21
	. . .	. . .	. . .	. . .	. . .	2.53E+10 2.31E+00	0.62	2.12E-01	0.36	. . .	Jul 2008	20, 2, 21
	. . .	. . .	. . .	. . .	. . .	1.71E+11 3.75E+00	0.56	1.18E+00	0.63	. . .	Jun 2009	20, 21, 2, 24
	HD 73256	1.05	0.89	14	0.962	3.28E+11 4.30E+01	0.03	1.18E+01	0.78	4	Jan 2008	25, 2, 24
	HD 102195	0.87	0.82	12.3	0.473	2.43E+12 7.09E+01	0.23	9.30E+01	0.88	4	Jan 2008	26, 27, 2, 24

We use a right-handed spherical coordinate system where a positive radial field component points out of the star, the 𝜃 component is positive pointing from North to South and the 𝜙 component is positive in the clockwise direction as viewed from the South pole.

When calculating the mean squared magnetic flux density we integrate over a full sphere, dividing by the solid angle of Ω = 4 𝜋.

We define axisymmetric as 𝑚 = 0.MNRAS 000,1-8 (2020)

MNRAS 000,1-8 (2020)

This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000,1-8 (2020)

ACKNOWLEDGEMENTS

The authors would like to thank the referee for thoughtful and constructive comments. MJ and KL acknowledge support from Science and Technology Facilities Council (STFC) consolidated grant number ST/R000824/1. VS acknowledges funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement no. 682393, AWESoMeStars). JFD acknowledges funding from the European Research Council (ERC) under the H2020 research and innovation programme (grant agreement 740651 NewWorlds). RF acknowledges funding from United Arab Emirates University (UAEU) startup grant number G00003269.

DATA AVAILABILITY

The data used to make Figures 3, 5 and 7 are in Table 1 and can be accessed at https://doi.org/10.17630/ 4dea9f0c-9626-4655-ba61-8af5c05fbe38. Archival data underpinning the plots is available at polarbase (http://polarbase. irap.omp.eu).