# Exoplanet-induced radio emission from M-dwarfs

Sam Turnpenney Department of Physics and Astronomy
University of Leicester
Leicester, LE1 7RH, UK  Jonathan D. Nichols Department of Physics and
Astronomy
University of Leicester
Leicester, LE1 7RH, UK  Graham A. Wynn  Department of Physics and Astronomy
University of Leicester
Leicester, LE1 7RH, UK  Matthew R. Burleigh  Department of Physics and
Astronomy
University of Leicester
Leicester, LE1 7RH, UK

(Received July 1, 2016; Revised September 27, 2016)

###### Abstract

We consider the magnetic interaction of exoplanets orbiting M-dwarfs,
calculating the expected Poynting flux carried upstream along Alfvén wings to
the central star. A region of emission analogous to the Io footprint observed
in Jupiter’s aurora is produced, and we calculate the radio flux density
generated near the surface of the star via the electron-cyclotron maser
instability. We apply the model to produce individual case studies for the
TRAPPIST-1, Proxima Centauri, and the dwarf NGTS-1 systems. We predict steady-
state flux densities of up to $\sim$ 10 $\upmu$Jy and sporadic bursts of
emission of up to $\sim$ 1 mJy from each case study, suggesting these systems
may be detectable with the Very Large Array (VLA) and the Giant Metrewave
Radio Telescope (GMRT), and in future with the Square Kilometre Array (SKA).
Finally, we present a survey of 85 exoplanets orbiting M-dwarfs, identifying
11 such objects capable of generating radio emission above 10 $\upmu$Jy.

planets and satellites: magnetic fields — plasmas — radio continuum: planetary
systems — stars: late-type

††journal: ApJ

##  1 Introduction

Recently, seven terrestrial planets have been discovered orbiting the nearby
ultracool dwarf star TRAPPIST-1 (Gillon et al., 2016, 2017), and an Earth-
sized planet, Proxima b, has been observed orbiting our nearest stellar
neighbour, Proxima Centauri (Anglada-Escudé et al., 2016). These and other
observations of planets orbiting M-dwarfs present an opportunity to study the
magnetic interaction between extrasolar planets and their host stars. Proxima
b and at least three of the planets at TRAPPIST-1 orbit in the nominal
‘habitable zone’, defined as the region around a star in which liquid water
may exist at the surface of a planet. Many factors influence the potential
habitability of a planet, and the existence of an intrinsic magnetic field may
possibly play a vital role by protecting the atmospheres of exoplanets from
erosive stellar radiation and winds (Khodachenko et al., 2007; Lammer et al.,
2009; Seager, 2013; Vidotto et al., 2013). This is especially significant for
M-dwarfs, since their habitable zones lie close to the star and planets would
be subject to high XUV flux and stellar wind dynamic pressure (e.g. Cohen et
al., 2015; Garraffo et al., 2017; Wheatley et al., 2017).

Coherent, nonthermal radio emission attributed the electron-cyclotron maser
instability (ECMI) is observed at magnetised planets in the solar system and
is associated with auroral activity (Wu & Lee, 1979; Zarka, 1992; Ergun et
al., 2000; Treumann, 2006; Imai et al., 2008). Although auroral emission from
exoplanets has yet to be detected (e.g. Lazio et al., 2009; Luger et al.,
2017b), the expected radio emission from them has been the subject of a number
of studies. The ‘Radiometric Bode’s Law’ (RBL) is an empirical scaling
relation based on observations of radio emission from magnetised Solar System
planets that is often extrapolated to estimate the radio power expected from
exoplanets (e.g. Lazio et al., 2004; Zarka, 2007). The RBL relates the output
radio power from a planetary body to the Poynting or kinetic energy flux
convected onto the obstacle, and has been used to demonstrate that hot
Jupiters may be detectable with the next generation of radio telescopes
(Farrell et al., 2004; Lazio et al., 2004; Grießmeier et al., 2007; Zarka,
2007). However, the RBL is empirical in nature and emission driven by
ionospheric flow resulting from the reconnection of interplanetary and
exoplanetary magnetic fields, analogous to the process generating Earth’s
auroral radio emission, has also been studied (Nichols & Milan, 2016), and is
predicted to be detectable with the next generation of radio telescopes.
Nichols (2011) also examined the detectability of auroral radio emission from
Jupiter-like exoplanets driven by currents generated by magnetosphere-
ionosphere coupling, and identified 91 potential targets for radio detection
within 25 pc of the Solar System (Nichols, 2012). Saur et al. (2013)
considered the case of close-orbiting exoplanets in the sub-Alfvénic regime,
calculating the magnetic energy flux communicated away from the planet back
towards the central star, and found that the total Poynting flux can reach
values of $>$ 1019 W in certain cases. Since habitable zone exoplanets around
M-dwarfs orbit very close to the star, it is expected that the star-planet
interactions there will be largely sub-Alfvénic. In light of the TRAPPIST-1
and Proxima Centauri b discoveries we compute radio emission powers due to
M-dwarf-exoplanet interactions in the framework of Saur et al. (2013) and
estimate the radio emission generated at the star arising from the upstream
transport of Poynting flux owing to the existence of a planetary obstacle to
the stellar wind flow. Earth-sized exoplanets with plausible magnetic field
strengths would produce radio emission directly from the planet that is below
the ionospheric cutoff frequency of $\sim$10 MHz, therefore making it
undetectable from Earth, regardless of the incident flux density (e.g.
Burkhart & Loeb, 2017). If, however, the relative velocity of a planetary body
through a magnetised plasma is sub-Alfvénic, then energy can be transported
upstream of the flow along Alfvén wings. Jupiter’s interaction with its
Galilean satellites is a well known example of a sub-Alfvénic interaction,
producing powerful radio emissions and auroral footprints (Clarke et al.,
1996; Zarka, 1998; Saur et al., 2004; Jones & Su, 2008; Bonfond et al., 2009;
Wannawichian et al., 2010). The interaction between Jupiter and its satellites
has been discussed in depth by Kivelson et al. (2004) and Saur et al. (2004).
Similar interactions do not occur between solar system planets and the Sun,
since the orbital distances are large enough that the solar wind is super-
Alfvénic in the planets’ reference frame, prohibiting significant upstream-
flow of energy. The interaction of exoplanets in close orbit with their host
star, however, may be sub-Alfvenic, in which case emission analogous to the Io
footprint and Io decametric (DAM) radio bursts may be produced near the
surface of the star and modulated at the orbital period of the planet.
Considerable attention has been given to the sub-Alfvénic interaction of
Jupiter with the Galilean satellites, serving as a useful benchmark for this
work, due to the TRAPPIST-1 planetary system being comparable in scale to the
Jovian satellite system.

In this paper we examine the detectability of radio emission resulting from
sub-Alfvénic star-planet interaction at M-dwarf exoplanet systems. We apply
Saur et al.’s (2013) analytic formulation for the magnetic energy communicated
from an exoplanet to the central star via Alfvén wings to three case studies,
each providing a distinct motivation for examination. Firstly, to TRAPPIST-1,
a star hosting a system of multiple terrestrial exoplanets planets; secondly,
to Proxima Centauri, the closest star to the Solar System, and host to a
terrestrial exoplanet; and finally to NGTS-1, a more distant M-dwarf (224 pc)
hosting a recently discovered hot Jupiter. In each case we estimate the radio
flux generated by the ECMI from the surface of the star. We also apply the
same method to study a wider sample of exoplanet-hosting M-dwarfs, identifying
those systems which may produce radio emission via the star-planet interaction
that is detectable with current or next generation radio telescopes.

##  2 Theoretical background

###  2.1 Sub-Alfvénic interaction

A planetary body moving relative to an external magnetised plasma creates an
obstacle to the plasma flow, thereby interacting with and modifying the
surrounding environment (see e.g. Saur et al., 2013, and references therein).
Of particular importance is the generation of Alfvén waves in the wake of the
flow, which are able to carry energy and momentum via the magnetic field. If
the velocity $v_{0}$ of the plasma relative to the planetary body is less than
the Alfvén speed $v_{\mathrm{A}}$, i.e. if

$M_{\mathrm{A}}=\frac{v_{0}}{v_{\mathrm{A}}}<1,$ (1)

where $M_{\mathrm{A}}$ is the Alfvén Mach number, then the interaction
generates two standing Alfvén waves, making up Alfvén wings (Neubauer, 1980),
which can propagate upstream of the flow, transporting energy and momentum in
that direction (see Figure 1). The condition for this case of the Alfvén mode
is satisfied in the solar system in the interaction of satellites with
magnetised planets, and the sub-Alfvénic interaction between Jupiter and its
moons, particularly Io, has been widely studied (Prangé et al., 1996; Clarke
et al., 1996; Saur et al., 2004; Jones & Su, 2008; Wannawichian et al., 2010).
Alfvén waves propagate along the magnetic field, and one of the wings, which
are draped with respect to the field, is oriented back towards the central
star provided the radial stellar wind velocity $v_{\mathrm{sw}}$ is less than
the radial Alfvén speed, i.e. if

$v_{\mathrm{sw}}<\frac{B_{r}}{\left(\upmu_{0}\rho_{\mathrm{sw}}\right)^{1/2}},$
(2)

where $B_{r}$ is the radial component of the interplanetary magnetic field
(IMF), and $\rho_{\mathrm{sw}}$ is the mass density of the stellar wind.
Equations (1) and (2) define the necessary conditions that must be met in
order for the Poynting flux to communicate energy from the star-planet
interaction back to central star.

###  2.2 Cases of plasma flow-obstacle interaction

There are at least four evident cases of plasma flow interaction with a
planetary body. These are (i) a magnetised planet with an atmosphere (e.g.
Earth, Jupiter, Saturn), (ii) a magnetised planet with no atmosphere (e.g.
Mercury), (iii) an unmagnetised planet possessing an atmosphere (e.g. Venus),
and (iv) an unmagnetised, non-conducting body without an atmosphere (e.g. the
Moon). In the first case of a magnetised planet with an atmosphere, currents
due to the interaction with the stellar wind flow in the magnetosphere and
Pedersen layer of the atmosphere, coupling the stellar wind to the planetary
magnetosphere. In the case of an unmagnetised planet with an atmosphere the
magnetopause is replaced by a magnetic pile-up boundary and a well defined
plasma boundary. The ionosphere replaces the magnetosphere acting as the
boundary with which the IMF interacts in this case. The case of a magnetised
but airless planet still produces an interaction with the stellar wind, as is
observed in the Sun-Mercury interaction, and in these cases the currents flow
through the conducting region inside the planet (e.g. the iron core at
Mercury). The final case of an unmagnetised, airless, non-conducting planetary
body is expected to produce only a weak perturbation of the IMF in the wake of
the body. In this paper we consider cases (i) and (iii), and the conclusions
we reach apply to both magnetised and unmagnetised planets which posses
atmospheres. We suggest the second case for future work.

Figure 1: Schematic showing the geometry of the star-planet interaction. The
two Alfvén wings generated by the interaction are indicated by the dotted
lines. The solid spiral emanating from the central star is the Parker spiral
magnetic field. Highlighted are the stellar wind velocity $v_{\mathrm{sw}}$;
the magnetic field of the stellar wind $B_{\mathrm{sw}}$; the orbital velocity
$v_{\mathrm{orb}}$ of the exoplanet; the resultant velocity of the impinging
stellar wind plasma $v_{0}$; and the angle $\theta$ between $v_{0}$ and
$B_{\mathrm{sw}}$.

###  2.3 Poynting flux within Alfvén wings

The morphology envisaged for exoplanets in close orbit around M-dwarfs differs
slightly from the Io-Jupiter interaction, in that the plasma flow is
predominantly radial due to the outflow of stellar wind, rather than azimuthal
as in the case of Io, and the expected geometry is shown in Figure 1. The
total Poynting flux carried within Alfvén wings from an exoplanetary obstacle
was calculated by Saur et al. (2013) who also showed that for small Alfvén
Mach number the Poynting flux is approximated by

$S_{\mathrm{total}}=2\uppi
R^{2}\bar{\upalpha}^{2}\frac{E_{\mathrm{sw}}B_{\perp}}{\upmu_{0}}M_{\mathrm{A}},$
(3)

where $R$ is the radius of the obstacle to the plasma flow, $E_{\mathrm{sw}}$
is the magnitude of the motional electric field (i.e.
$\boldsymbol{E}_{\mathrm{sw}}=-\boldsymbol{v}_{\mathrm{sw}}\times\boldsymbol{B}_{\mathrm{sw}}$)
seen in the rest frame of the planetary body, $\bar{\upalpha}$ denotes the
strength of the sub-Alfvénic interaction, and $B_{\perp}$ is the component of
the stellar magnetic field perpendicular to the impinging plasma velocity at
the orbital distance of the planet, given by

$B_{\perp}=B_{\mathrm{sw}}\sin\theta,$ (4)

where $\theta$ is the angle between the IMF $B_{\mathrm{sw}}$ and the incident
stellar wind velocity, as shown in Figure 1. The factor $\bar{\upalpha}$
represents the degree by which the motional electric field and plasma flow
velocity are reduced by the interaction. For the cases considered in this
paper, $\bar{\upalpha}=1$ is a reasonable approximation, the justification for
which is contained in Appendix A.

Note that we do not consider the exact nature of the flows within a
magnetospheric or ionospheric obstacle. The formulation developed by Saur et
al. (2013) employs the radius $R$ of the obstacle presented to the plasma flow
in order to calculate the Poynting flux radiating away from the planet along
interplanetary magnetic field lines due to the star-planet interaction. For an
unmagnetised planet this is the radius of the ionosphere, which in this work
we approximate as the planetary radius. For a magnetised planet we consider
the radius of the magnetosphere, providing stellar wind and IMF conditions
allow this to form.

The total Poynting flux, as expressed in equation (3), and hence the spectral
flux density, is a function of the magnetic and motional electric fields.
Values of these parameters are determined by the form of the stellar wind and
magnetic field. The model of the stellar wind and magnetic field we employ is
described in Appendix B.

###  2.4 Exoplanetary magnetic field strengths

To estimate unknown exoplanetary magnetic field strengths we employ the
magnetic field scaling law proposed by Sano (1993), whereby the planetary
dipole moment $\mathcal{M}$ is estimated by

$\mathcal{M}\,\,\propto\,\,\rho_{c}^{1/2}\Omega\ r_{c}^{7/2},$ (5)

where $\Omega$ is the rotational velocity of the planet, $\rho_{c}$ is the
mass density in the dynamo region, for which we adopt the terrestrial value,
and $r_{c}$ is the core radius of the planet, which can be determined from the
mass of the planet by the empirical scaling law (Curtis & Ness, 1986)

$r_{c}\,\,\propto\,\,M^{0.44}.$ (6)

The surface magnetic field strength of the planet can then be obtained by

$B_{p}=\frac{\mathcal{M}}{R_{p}^{3}}.$ (7)

In the absence of any data on the rotation rates of exoplanets, we assume
throughout this work that the planets are tidally locked, and thus that the
rotational velocity is equal to the orbital velocity, justified by the extreme
close orbits occupied by the majority of planets around M-dwarfs. This
assumption represents a lower limit on the planetary magnetic field, and a
larger magnetic field would be calculated from this method if the planet has
not yet become tidally locked. Other scaling laws exist (e.g. Busse, 1976;
Stevenson, 1983; Mizutani et al., 1992) which provide similar planetary
magnetic moment estimates as Sano (1993).

###  2.5 Radio power from near the stellar surface

Providing the conditions of equations (1) and (2) are satisfied, then the
total Poynting flux calculated from equation (3) radiating away from the
planet, is carried by one of the two Alfvén wings back towards the central
star (see Figure 1). Shear Alfvén waves carry magnetic field-aligned currents,
which are associated with the ECMI, responsible for producing radio emission
at Jupiter and Saturn. We consider the radio emission generated near the
surface of the star via the ECMI mechanism, at a frequency corresponding to
the local gyrofrequency of the emission region (Treumann, 2006). The total
radio power emitted from one hemisphere of the star is given by

$P_{r}=\epsilon S_{\mathrm{total}},$ (8)

where $\epsilon$ represents the radio efficiency factor, i.e. the fraction of
total Poynting flux converted to radio emission at the stellar surface. To
estimate $\epsilon$ we compare with Io-induced DAM radio emission at Jupiter.
Saur et al. (2013) calculated that for Io the total Poynting flux from the
magnetic interaction is $\sim$ 1011 \- $10^{12}$ W, while the observed Io DAM
reaches 109 \- 1010 W (Clarke et al., 2004), thus giving an overall radio
efficiency of $\epsilon=0.01$. This represents a combination of the ECMI
efficiency in addition to wave energy lost from the Alfvén wing by reflection
and conversion to other forms of energy, as is observed in sub-Alfvénic
interaction at Jupiter and Saturn (Wright & Schwartz, 1989; Jacobsen et al.,
2007; Hess et al., 2011). This is also consistent with in situ measurements of
the ECMI growth rate at Saturn (Lamy et al., 2011), and for these reasons we
employ this value of $\epsilon$ throughout our analysis.

Spectral flux density from the star can be obtained by

$F_{r}=\frac{P_{r}}{\Omega\,s^{2}\Delta\upnu},$ (9)

where $s$ is the distance from the star to the Earth, $\Delta\upnu$ is the
bandwidth of the radio emission, and $\Omega$ is the solid angle into which
the ECMI emission is beamed. No reliable estimates exist for the beaming
angles of ECMI emission from low-mass stars. Hence, we assume throughout this
work a reasonable value of $\Omega=1.6$ sr, in conformity with observations of
Jupiter’s HOM and DAM emission (Zarka et al., 2004), and also for the Io-
related DAM emission (Ray & Hess, 2008). The bandwidth is assumed to be equal
to the gyrofrequency at the surface of the star in the equatorial region, an
approximation consistent with solar system planet observations (Zarka, 1998),
and is hence given by

$\Delta\upnu=\frac{eB_{\star}}{2\uppi m_{e}},$ (10)

where $B_{\star}$ is the average surface magnetic field strength of the star,
and $e$ and $m_{e}$ are the charge and mass of the electron respectively. In
this work we explore M-dwarf magnetic field strengths in the range 0.05 - 0.15
T, based on typical values measured at low-mass stars (Reiners & Basri, 2007,
2010; Lynch et al., 2015). From equation (10) this translates to an ECMI
cutoff frequency range of $\sim$ 1.4 - 4.2 GHz. However, we expect radio
emission will be generated below and up to the cutoff frequency, and assume
that the emission spectrum will be flat. A number of radio telescopes are
therefore be capable of testing the predictions proposed in this paper. The
Murchison Widefield Array (MWA) has achieved sensitivities of $\sim$ 10 mJy at
150 MHz (Tingay et al., 2013), while the Giant Metrewave Radio Telescope
(GMRT) is currently undergoing a major upgrade, which is expected to enable
sensitivities of 190 $\upmu$Jy at 120 - 250 MHz, and 45 $\upmu$Jy at 1050 -
1450 MHz (Gupta et al., 2017). The Low Frequency Array (LOFAR) has achieved a
flux density sensitivity of 10 mJy at 60 MHz (e.g. Van Weeren et al., 2014),
while in the high-frequency regime the Very Large Array has obtained
sensitivities of the order 10 $\upmu$Jy at 3 GHz (e.g. Hallinan et al., 2017).
Finally, the Square Kilometre Array (SKA) is a next generation radio
telescope, due to begin observations in 2020, with an anticipated flux density
sensitivity of $\sim$ 10 $\upmu$Jy (Zarka et al., 2015). All of these
telescopes will be considered when discussing the predictions of our model.

In the following section we apply the formulations described to three
exoplanetary case studies, along with a wider survey of M-dwarf-exoplanet
systems.

##  3 Results

###  3.1 TRAPPIST-1

We consider first the radio power expected from sub-Alfvénic interactions
between TRAPPIST-1 and its seven known exoplanets, orbiting between 0.011 and
0.063 AU with periods ranging between 1.51 and 18.77 days (Gillon et al.,
2016, 2017; Luger et al., 2017a). The key stellar parameters are the rotation
period, mass and radius of the star, 3.3 days, 0.08 $M_{\odot}$ and 0.117
$R_{\odot}$ respectively, in addition to the magnetic field strength and mass
outflow rate of the star. The average surface magnetic field strength at
TRAPPIST-1 was measured by Reiners & Basri (2010) as $B_{\star}$ = 0.06 T
(where 1 Tesla = 104 Gauss). We employ a stellar mass outflow rate of
$\dot{M_{\star}}=1\dot{M_{\odot}}=3\times 10^{-14}M_{\odot}\mathrm{yr}^{-1}$
(Garraffo et al., 2017).

Figure 2: Plots of derived parameters for the TRAPPIST-1 exoplanetary system
versus orbital distance $d$ in units of stellar radii $R_{\star}$ and AU for
intrinsic planetary magnetic field strengths of 0.1 $B_{\oplus}$ (blue lines),
1 $B_{\oplus}$ (green lines) and 10 $B_{\oplus}$ (red lines). Solar values of
the XUV luminosity $L_{\mathrm{XUV}}$ and stellar wind mass outflow rate
$\dot{M}$ were used, along with a planetary radius of 1 $R_{\oplus}$ for all
seven planets. Panel (a) shows the magnetopause standoff distance
$R_{\mathrm{mp}}$ in units of planetary radii. Panel (b) shows the radio power
$P_{\mathrm{r}}$ from the star-planet interaction (left axis) and the
corresponding total Poynting flux (right axis) for the idealised Parker spiral
(solid lines), and the maximum power of radio bursts (dashed lines), with the
black lines representing the case of an unmagnetised planet. Finally, panel
(c) shows the radio flux density $F_{\mathrm{r}}$ from the surface of the star
in units of mJy, assuming a distance from Earth of 12.1 pc. The horizontal
lines indicate sensitivity thresholds of the respective radio telescopes.

We now derive parameters characterising the interaction of the exoplanets with
the stellar wind conditions. For the purpose of plotting, we have assumed that
each of the exoplanets around TRAPPIST-1 has a radius of 1 R⊕, while Table. 1
shows the derived parameters for the individual planets using their measured
masses and radii. Firstly, in Figure 2(a) we show the size of the magnetopause
standoff distance, calculated using equation (B8), for a range of three
plausible planetary magnetic field strengths bracketing the terrestrial value.
The results show that for an intrinsic planetary field strength of 10 B⊕,
small magnetospheres ranging in magnetopause standoff distance of $\sim$1.1 -
1.4 RP form at all seven planets. For a terrestrial field strength only the
outer five planets form magnetospheres, while a planetary field strength of
0.1 B⊕ precludes the formation of a magnetosphere at six of the seven
exoplanets.

Turning now to the ECMI-induced radio power from the surface of the star
calculated using equations (3) and (8), where the additional black lines in
Figure 2(b) show the power resulting from unmagnetised exoplanets, i.e. where
the radius $R$ of the obstacle is simply the radius of the planet. The solid
lines represent the lower limit of the power in the idealised case of a
perfect Parker spiral, i.e. where there is no $B_{z}$ component of the
magnetic field, only radial $B_{r}$ and azimuthal $B_{\phi}$ components.
However, this is unrealistic, since a number of factors, such as a magnetic
field tilted with respect to the planetary orbit, the occurrence of transient
stellar activity such as coronal mass ejections, or a tilting of the current
sheet, would result in deviation from the idealised Parker spiral. Hence, the
‘notch’ in the vicinity $35R_{\star}$, which occurs when the incident plasma
velocity is parallel to the IMF, would not actually drop to zero, and even if
the above mentioned factors were not present, we would still expect
interaction due to viscous forces which would slow the stellar wind and
produce Alfvén wings. Therefore, we have also shown with the dashed coloured
lines in Figure 2(b) the upper limit of the radio power, which is calculated
using the total IMF strength $B_{\mathrm{sw}}$ in equation (3) rather than the
component perpendicular to the incident stellar wind velocity. The magnetic
field actually experienced will lie somewhere between the two. The dashed
lines therefore represent the upper limit for bursts of radio power resulting
from intervals of high magnetic field strength above the steady state value
represented by the solid lines. We observe that the steady state radio power
(solid lines) induced by sub-Alfvénic interaction peaks at the orbits of the
innermost and outmost planets around TRAPPIST-1, at $\sim$ 1013 W, although
sporadic bursts may reach radio powers of up to two orders of magnitude
greater. By comparison, the radio power from the Io DAM emission at Jupiter is
$\sim$ 109 \- 1010 W (Prangé et al., 1996; Clarke et al., 1996; Gérard et al.,
2006). Figure 2(c) shows the spectral flux density from TRAPPIST-1 given by
equation (9). The horizontal lines running across the plot indicate the
sensitivities of the radio telescopes, as discussed in Section 2.5. With
steady state flux densities of up to $\sim$ 10 $\upmu$Jy, and bursts producing
flux densities of almost 1 mJy, our results show that we do not expect steady
state emission to be presently detectable, assuming plausible planetary
magnetic fields. However, planets with field strengths greater than 1
$\mathrm{B_{\oplus}}$ may induce sporadic radio bursts which are detectable
with the VLA and in the future with the SKA. The GMRT may also be capable of
detecting sporadic bursts if the planetary field strength is $\sim$ 10
$\mathrm{B_{\oplus}}$. We note, however, that the planetary magnetic field
strengths shown in Table 1, which were calculated using Sano’s (1993) scaling
law, preclude the formation of magnetospheres at any of the seven planets.
Hence, in this case the flux density would be that indicated by the black line
in Figure 2(c), implying that radio bursts from the four innermost planets
only would be detectable with the VLA or SKA. Garraffo et al. (2017) estimate
magnetopause standoff distances of up to $\sim$ 2.7 $R_{p}$ at the outermost
exoplanets, since they adopt Earth-like planetary magnetic field strengths. In
contrast, Sano’s (1993) law yields the smaller planetary magnetic field
strength found in this work due to the smaller masses and longer rotation
periods of the planets compared with Earth.

We note that the coronal temperature of 2 MK employed in our model is an
estimate of a weakly constrained quantity which we have adopted following the
work of Vidotto et al. (2011, 2013). However, coronal temperatures at M-dwarfs
may reach 10 MK (Schmitt et al., 1990; Giampapa et al., 1996), therefore we
also examined the effects of a 10 MK corona, corresponding to a stellar wind
sound speed of 379 km s-1. A temperature of 10 MK is also the value which
corresponds to stellar wind sound speeds consistent with those of Garraffo et
al. (2017). We find that the resulting radio powers and flux densities are
reduced by less than one order of magnitude for this higher coronal
temperature and sound speed.

In Figure 3 we show a map of radio power $P_{r}$ versus orbital distance $d$
and radius of the obstacle $R$. Apparent is the general trend that radio power
increases with obstacle radius, and decreases with orbital distance. This plot
has been produced for an idealised Parker spiral IMF, and thus the ‘notch’
evident in Figure 2 is also a feature here at $\sim$ 0.02 AU. Such color plots
may be used in conjunction with future radio detections to infer the radii of
detected obstacles, assuming the planetary orbital radius and relevant stellar
parameters are known. Significant persistent deviations from the points
plotted (corresponding to known radii) will indicate the existence and
strength of planetary magnetic fields. Multiple planetary systems such as
TRAPPIST-1 will provide the tightest constraints, and similar plots can in
principle be produced for many stars.

Figure 3: A color plot for TRAPPIST-1 of radio power $P_{r}$ versus distance
from the central star $d$ in units of AU, and the radius of the obstacle to
the plasma flow $R$ in units of Earth radii. Markers indicate the positions of
the exoplanets, where the measured radii, as in Table 1, have been used. Table
1: Properties of the planets orbiting TRAPPIST-1

Planet |  $d$a, b | Orbital perioda, b |  $n_{\mathrm{sw}}$c |  $v_{\mathrm{sw}}$d |  $B_{\mathrm{sw}}$e |  $M_{p}$f |  $R_{p}$a |  $B_{p}$g |  $R_{\mathrm{mp}}$ / $R_{p}$h |  $P_{r}$i |  $F_{r}$j
---|---|---|---|---|---|---|---|---|---|---|---
| (AU) | (days) | (cm-3) | (km-1) | (nT) | ($M_{\oplus}$) | ($R_{\oplus}$) | ($B_{\oplus}$) |  | (W) | ($\upmu$Jy)
b | 0.0111 | 1.511 | 38519 | 492 | 137169 | 0.79 | 1.086 | 0.353 | 1 | 5.92 $\times$ 1012 | 1.58
c | 0.0152 | 2.422 | 19131 | 528 | 73276 | 1.63 | 1.056 | 0.731 | 1 | 3.71 $\times$ 1011 | 0.099
d | 0.0215 | 4.050 | 8923 | 566 | 36746 | 0.33 | 0.772 | 0.096 | 1 | 3.56 $\times$ 1010 | 0.00949
e | 0.0282 | 6.100 | 4942 | 594 | 21451 | 0.24 | 0.918 | 0.023 | 1 | 2.26 $\times$ 1011 | 0.0603
f | 0.0371 | 9.206 | 2730 | 621 | 12479 | 0.36 | 1.045 | 0.019 | 1 | 4.22 $\times$ 1011 | 0.113
g | 0.0451 | 12.353 | 1793 | 640 | 8506 | 0.566 | 1.127 | 0.023 | 1 | 5.14 $\times$ 1011 | 0.137
h | 0.0596 | 18.766 | 987 | 666 | 4947 | 0.086 | 0.715 | 0.0033 | 1 | 1.91 $\times$ 1011 | 0.0510

Note. — a From Gillon et al. (2017); b From Luger et al. (2017a); c From
equation (B11); d From equation (B1); e From equation (B6); f From Wang et al.
(2017); g From equation (7); h From equation (B8) but equal to unity if
$R_{\mathrm{mp}}<R_{\mathrm{p}}$; i From equation (8); j From equation (9).

###  3.2 Proxima Centauri

As a second case study we also examine the radio emission from Proxima
Centauri due to sub-Alfvénic interaction with its exoplanet, Proxima b, which
orbits at a distance of 0.04 AU. The planet has a minimum measured mass of 1.3
$\mathrm{M_{\earth}}$, implying a radius of 1.1 $\mathrm{R_{\earth}}$ assuming
a density equal to that of Earth. A number of crucial factors differentiate
the results for Proxima Centauri compared with TRAPPIST-1, namely: the much
longer rotation period of Proxima Centauri, 82.6 days; the larger mass and
radius of Proxima Centauri, 0.122 $M_{\odot}$ and 0.154 $R_{\odot}$
respectively; and the distance of Proxima Centauri to the Earth, just 1.3 pc
compared with 12.1 pc for TRAPPIST-1 (Anglada-Escudé et al., 2016). The
measured average surface magnetic field strength of Proxima Centauri is 0.06 T
(Reiners & Basri, 2008), and the mass outflow rate from is estimated to be
similar to the solar value, therefore we again use $\dot{M_{\star}}=3\times
10^{-14}M_{\odot}\mathrm{yr}^{-1}$ (Garraffo et al., 2016).

Figure 4 shows the derived parameters taking a planetary magnetic field
strength of $0.1\;\mathrm{B_{\oplus}}$, close to the exact value of
$0.0952\;\mathrm{B_{\oplus}}$ calculated from equations (5) - (7), and
bracketing this with values an order of magnitude either side. We see from
Figure 4(a) that a magnetic field strength greater than the value of
$0.1\mathrm{B_{\oplus}}$ given by Sano’s (1993) scaling law is required in
order to form a magnetosphere at the planet. The radio power from Proxima
Centauri shown in Figure 4(b) has a value of $\sim 10^{12}$ W assuming the
terrestrial value for the field strength, and $\sim 10^{11}$ W if the
planetary field strength is low enough to prohibit the formation of a
magnetosphere, while radio bursts of up to $\sim 10^{14}$ W are plausible.
Despite lower radio power, the flux density from Proxima Centauri (Figure
4(c)) is slightly larger than that from TRAPPIST-1 owing to the system’s much
closer proximity to Earth, yielding a value of $\sim\;10\;\upmu$Jy for a
terrestrial magnetic field strength. Radio bursts producing flux densities of
up to $\sim$ 1 mJy are also possible. Hence, coherent radio emission from
sporadic bursts from Proxima Centauri may be detectable by the GMRT and SKA.
We note that with a declination of approximately -60∘, Proxima Centauri is not
visible at the latitude of the VLA. Bell et al. (2016) observed Proxima
Centauri with the Murchison Widefield Array, resulting in a non-detection.
They place an upper limit on radio emission from the system at 42.3 mJy/beam
(3-sigma) at 200 MHz, a limit at least two orders of magnitude greater than
the flux density predicted by our model.

Figure 5 shows a color plot of the radio power $P_{r}$ from the star-planet
interaction. A noticeable difference from the equivalent plot for TRAPPIST-1
(Figure 3) is the lack of the ‘notch’ in the results. This is due to the much
slower rotation of Proxima Centauri, meaning that the incident plasma velocity
does not become parallel to the IMF until a greater orbital distance, beyond
the range of the plot. As with TRAPPIST-1 this power map may be used to
determine the obstacle radius and thus the nature of any planetary magnetic
field at Proxima-b, by comparison with any future radio emission detected.

Figure 4: As for Figure 2 but for Proxima Centauri b. Again solar values of
$L_{\mathrm{XUV}}$ and $\dot{M}$ were used, along with a planetary radius of
1.1 $R_{\oplus}$ Figure 5: As for Figure 3, but with Proxima Centauri stellar
parameter values.

###  3.3 NGTS-1

The previous two case studies, Proxima Centauri and TRAPPIST-1, are systems
containing approximately Earth-sized planets. For a final case study we
examine a system comprising a hot Jupiter in orbit around an M-dwarf, situated
224 pc from Earth. NGTS-1b is a planet of mass 0.812 $\mathrm{M_{Jup}}$ and
radius 1.33 $\mathrm{R_{Jup}}$, orbiting an M0.5 dwarf of mass 0.617
$\mathrm{M}_{\odot}$ and radius 0.573 $\mathrm{R}_{\odot}$ in a P = 2.647 d
orbit (Bayliss et al., 2017). The magnetic field strength of NGTS-1 is
unknown, and therefore we have assumed a value of 0.06 T, consistent with the
two previous case studies. A stellar rotation period of 43 days is estimated
from photometric analysis of spot coverage (Bayliss et al., 2017). In common
with our studies of TRAPPIST-1 and Proxima Centauri, we have assumed
$L_{\mathrm{XUV}}=L_{\mathrm{XUV\odot}}$ at NGTS-1.

Figure 6: As for Figures 2 and 4 but for NGTS-1 and with the following
difference: Planetary magnetic field strengths of 0.3 $B_{J}$ (blue lines), 3
$B_{J}$ (green lines) and 30 $B_{J}$ (red lines) are plotted. Figure 7: As for
Figures 3 and 5, but with NGTS-1 stellar parameter values. Here we now
consider an obstacle radius $R$ range of 10 - 100 $\mathrm{R_{\oplus}}$ to
reflect the fact that the exoplanet in this system is a hot Jupiter of radius
1.33 $\mathrm{R_{Jup}}$.

The derived planetary parameters are shown in Figure 6. For this particular
case study of a hot Jupiter, we use a scaling law appropriate for giant
planets in the form given by Reiners et al. (2009) to estimate the strength of
the magnetic field of NGTS-1b:

$B_{\mathrm{dyn}}=4.8\times\left(\frac{ML^{2}}{R^{7}}\right)^{1/6}[\mathrm{kG}],$
(11)

where $B_{\mathrm{dyn}}$ is the mean strength of the magnetic field at the
surface of the dynamo, and $M$, $L$, and $R$ are the mass, luminosity, and
radius of the planet respectively, all normalized with solar values. The
equatorial dipole field strength is related to the field strength at the
surface of the dynamo by (Reiners & Christensen, 2010)

$B_{p}=\frac{B_{\mathrm{dyn}}}{2\sqrt{2}}\left(1-\frac{0.17}{M/M_{J}}\right)^{3}.$
(12)

We use in equation (11) a typical luminosity value of $L=10^{-7}L_{\odot}$
taken from the evolutions tracks calculated by Burrows et al. (1997). For
NGTS-1b, equations (11) and (12) give an equatorial field strength of $B\simeq
1.5$ mT, i.e. approximately three times greater than the surface field
strength of Jupiter, and we also examine field strengths an order of magnitude
either side of this value. Figure 6(a) shows that a magnetopause may form with
a stand-off distance $\leq 4\;\mathrm{R_{p}}$ at the orbital radius. Figure
6(b) shows that the radio power carried in the Alfvén wing is $\sim
10^{16}-10^{18}$ W, which translates to a flux density from the surface of the
M-dwarf (Figure 6(c)) of $\sim$ 0.01 mJy - 1 mJy. These results suggest that a
planetary field strength at the upper end of the range considered may produce
emission that is detectable by the VLA, GMRT, and SKA. A color plot of radio
power is shown in Figure 7, which may again enable determination of planetary
magnetic field strength from future radio observations, via the method
described in Section 3.1. We note that planets such as NGTS-1 which
potentially possess magnetic fields comparable to Jupiter in strength are
important, since they may in principle be also detectable from the ground via
direct planetary radio emissions. Comparison of stellar and planetary
emissions will yield vital information regarding the nature of the interaction
over and above the scope of the present model.

###  3.4 M-dwarf exoplanets

We now present analysis of a survey of M-dwarf-exoplanet systems using data
listed in the https://exoplanetarchive.ipac.caltech.edu and
http://exoplanet.eu catalogues. Selecting only those exoplanets orbiting
within 1 AU of the host star, we identify a total of 85 exoplanets for
inclusion in our study, and calculate the expected radio flux for each via
sub-Alfvénic magnetic interaction. As discussed above, the properties required
to calculate the flux density are: the incident stellar wind velocity $v_{0}$;
stellar magnetic field components $B_{r}$ and $B_{\phi}$; the density
$\rho_{\mathrm{sw}}$ of the stellar wind; and the radius of the planetary
obstacle $R$. None of these quantities are directly measurable at present.
Therefore, just as with our case study results, we estimate the required
quantities as described in Appendix B.

Unlike the magnetic field strengths of TRAPPIST-1 and Proxima Centauri which
have reasonably well constrained estimates, the magnetic field strengths of
the other M-dwarfs in our study are, for the most part, unknown. Typical low-
mass star magnetic field strengths are commonly of the order 0.1 T (e.g.
Reiners & Basri, 2007; Lynch et al., 2015). Therefore, in the absence of any
data, we have applied our model for three magnetic field strengths of
$B_{\star}=0.05,\,0.1,\,0.15$ T for all stars in the survey. To estimate the
effective radius of the obstacle, knowledge of the intrinsic planetary
magnetic field is required. In the case studies we examined a range of three
separate values for the magnetic field strength. However, in this survey we
desire a single flux density value for each system, and we therefore again
employ the scaling law of Sano (1993) to arrive at a magnetic field strength
value given by equations (5) - (7). As with the case studies of TRAPPIST-1 and
Proxima b, we assume that all exoplanets in this study are tidally locked; an
assumption likely to be valid in most cases due to the close orbits and ages
of the majority of systems.

In the cases where either database is incomplete, we estimate the values of
any missing properties as follows: If the radius of the star $R_{\star}$ is
missing, we estimate the value based on typical values for the spectral type
of the star, as given in Appendix G of Carroll & Ostlie (2007). If either the
radius $R_{\mathrm{p}}$ or mass $M_{\mathrm{p}}$ of the exoplanet is unknown,
we assume a terrestrial density to estimate the missing quantity. In case the
orbital period $P_{\mathrm{orb}}$ is unmeasured, we use Kepler’s third law to
calculate the missing value. Since data on rotation rates of M-dwarfs is
sparse, we have assumed a period of 19 days for all dwarfs based on results
from a survey of M-dwarf rotation periods by McQuillan et al. (2013).
Additionally, we have assumed throughout a stellar mass loss rate of
$\dot{M_{\star}}=\dot{M_{\odot}}$, based on the mass loss rates of a similar
magnitude used in our earlier case studies.

We calculate the radio power and flux density for the selected targets using
the same method as in the earlier case studies. Of the 85 M-dwarf-orbiting
exoplanets, 11 are estimated to generate radio flux above 10 $\upmu$Jy, and
the results for these targets are shown in Table 2. Only one of the exoplanets
generating potentially detectable emission is estimated to possess a
magnetosphere, and thus in the majority of cases the effective radius of the
planetary obstacle to the stellar wind is simply the planetary radius. We note
that the results show that flux density is independent of stellar field
strength $B_{\star}$ for cases where no magnetosphere forms, i.e. where
$R_{\mathrm{mp}}/R_{p}=1$. Closer inspection of equation (3) reveals that
although $S_{\mathrm{total}}\propto B_{\star}$, this is cancelled by a
1/$B_{\star}$ term in the flux density calculation arising from our assumption
that the bandwidth of the emission is the cyclotron frequency in the
generation region at the surface of the star.

Table 2: Flux density estimates for exoplanets orbiting M-dwarfs Exoplanet | Spectral | $d$ | $M_{\star}$ | $R_{\star}$ | $M_{\mathrm{p}}$ | $B_{\mathrm{p}}$ | $s$ | $R_{\mathrm{mp}}/R_{\mathrm{p}}$ | $P_{r}[B_{\star}=0.05,0.1,0.15\mathrm{T}]$ | $F_{r}$
---|---|---|---|---|---|---|---|---|---|---
| type | (AU) | ($M_{\odot}$) | ($R_{\odot}$) | ($M_{\mathrm{Jup}}$) | (nT) | (pc) |  | (W) | ($\upmu$Jy)
GJ 876 d | M4 | 0.021 | 0.33 | 0.35 | 0.021 | 4.38 $\times 10^{4}$ | 4.7 | 1 | 1.44 $\times 10^{14}$, 2.89 $\times 10^{14}$, 4.33 $\times 10^{14}$ | 306.6
GJ 436 b | M2.5 V | 0.029 | 0.47 | 0.46 | 0.07 | 2.00 $\times 10^{4}$ | 10.23 | 1 | 4.02 $\times 10^{14}$, 8.04 $\times 10^{14}$, 1.21 $\times 10^{15}$ | 180.1
GJ 674 b | M2.5 | 0.039 | 0.35 | 0.48 | 0.035 | 2.38 $\times 10^{4}$ | 4.54 | 1 | 2.09 $\times 10^{13}$, 4.18 $\times 10^{13}$, 6.26 $\times 10^{13}$ | 47.51
GJ 3138 b | M0 | 0.0197 | 0.68 | 0.5 | 0.0056 | 3.41 $\times 10^{4}$ | 28.48 | 1 | 4.03 $\times 10^{14}$, 8.07 $\times 10^{14}$, 1.21 $\times 10^{15}$ | 23.33
GJ 3470 b | M1.5 | 0.031 | 0.51 | 0.48 | 0.043 | 9.31 $\times 10^{3}$ | 25.2 | 1 | 3.14 $\times 10^{14}$, 6.29 $\times 10^{14}$, 9.43 $\times 10^{14}$ | 23.31
GJ 649 c | M1.5 | 0.043 | 0.54 | 0.56 | 0.03 | 2.30 $\times 10^{4}$ | 10.34 | 1 | 2.94 $\times 10^{13}$, 5.87 $\times 10^{13}$, 8.81 $\times 10^{13}$ | 12.88
GJ 876 b | M4 | 0.20832 | 0.33 | 0.35 | 2.28 | 1.75 $\times 10^{4}$ | 4.7 | $1.90\begin{subarray}{c}+0.49\\\ -0.24\end{subarray}$ | 8.35 $\times 10^{12}$, 1.06 $\times 10^{13}$, 1.21 $\times 10^{13}$ | $11.21\begin{subarray}{c}+6.52\\\ -2.64\end{subarray}$
Kepler-42 c | M5 V | 0.006 | 0.13 | 0.17 | 0.0011 | 3.89 $\times 10^{4}$ | 38.7 | 1 | 3.49 $\times 10^{14}$, 6.98 $\times 10^{14}$, 1.05 $\times 10^{15}$ | 10.92
GJ 581 e | M3 V | 0.02815 | 0.31 | 0.29 | 0.0053 | 1.28 $\times 10^{4}$ | 6.27 | 1 | 9.06 $\times 10^{12}$, 1.81 $\times 10^{13}$, 2.72 $\times 10{13}$ | 10.81
GJ 1132 b | M3.5 | 0.0154 | 0.181 | 0.207 | 0.0051 | 2.71 $\times 10^{4}$ | 12.04 | 1 | 3.28 $\times 10^{13}$, 6.56 $\times 10^{13}$, 9.84 $\times 10^{13}$ | 10.61
GJ 3634 b | M2.5 | 0.0287 | 0.45 | 0.43 | 0.026 | 3.60 $\times 10^{4}$ | 19.8 | 1 | 8.65 $\times 10^{13}$, 1.73 $\times 10^{14}$, 2.59 $\times 10^{14}$ | 10.34

##  4 Discussion and Summary

Our results have yielded estimates of the steady state radio flux expected
from the exoplanets around TRAPPIST-1, Proxima Centauri, and NGTS-1, as well
as placing upper limits on the bursts of radio flux which may detected due to
extreme conditions in the stellar wind, caused by the factors discussed in
Section 3.1. The prediction from our results that exoplanet-induced radio
emission should be detectable from M-dwarfs with the VLA, GMRT, and SKA, must
take into account whether a signal can be extracted from any background noise
in radio observations. A recent radio survey of M-dwarfs found that for the
majority of the sample the flux density was $<$ 100 $\upmu$Jy (McLean et al.,
2012). This background emission is approximately one order of magnitude larger
than the flux density that our model predicts from exoplanets. However, the
known orbital periods of the exoplanets is advantageous when attempting to
isolate the signal from the background, as the exoplanet-induced emission will
be modulated at the same period, and therefore the light curves from
observations can be folded at the orbital period, facilitating detection of
the signal.

Although magnetic fields have not been observed at exoplanets, their existence
is generally expected (Christensen et al., 2009), and a number studies have
considered potential methods for detecting their presence, as discussed above.
For instance, Scharf (2010) proposed that X-ray emissions could probe
exoplanets internally through the dynamo-generated magnetic field. Additional,
Vidotto et al. (2011) studied asymmetries in observations of transiting
exoplanets, which may be an indication of bow shocks, which could, in turn, be
used to infer the presence of intrinsic magnetic fields. In studies
considering ECMI emission directly from the planet, the magnetic field
strength may be constrained by the frequency of the radio emission (Hess &
Zarka, 2011). This is not possible for terrestrial planets with weaker
magnetic fields, owing to the ionospheric cutoff frequency as discussed
earlier, thus hindering the probing of such planetary fields, and
strengthening the case for future space-based observatories to access low-
frequencies (Burkhart & Loeb, 2017). Multi-planet systems, such as TRAPPIST-1
provide an important opportunity to identify the presence of exoplanetary
magnetic fields. The poorly constrained nature of a number of parameters, such
as the stellar magnetic field strength and mass outflow rate, means that a
flux density measurement of a single exoplanet-interaction cannot determine
whether that planet possesses an intrinsic magnetic field. Multiple sources of
radio emission, however, from the same system, where factors such as stellar
magnetic field, XUV luminosity and stellar wind should vary in a well
understood manner, could be compared with the profiles we predict from Figure
2 to constrain the relative differences in the magnetic field strength of the
planets. This factor distinguishes TRAPPIST-1 from the other systems examined
in this study, and for this reason we suggest it as a priority focus for
future radio observations.

Although the flux density predictions of our results for NGTS-1 are similar in
magnitude to the predictions for TRAPPIST-1 and Proxima Centauri, we note that
NGTS-1 is a much more distant star than the other two case studies (224 pc
compared with 12.1 and 1.3 pc of TRAPPIST-1 and Proxima Centauri
respectively). Discovery of a similar system to NGTS-1, comprising a hot
Jupiter in close orbit around an M-dwarf, but closer to the Solar System,
would have a greater probability of radio detection, due to a higher flux
density.

It is generally a necessary condition that there is a small separation
distance between star and exoplanet (typically $\leq 0.2$ AU) for this method
to bear results, for a number of reasons. Most crucially, the sub-Alfvenic
condition is typically only satisfied for close orbits. Secondly, equation (3)
shows that $S_{\mathrm{tot}}\propto v_{0}^{2}\propto v_{\mathrm{orb}}^{2}$,
and hence the total Poynting flux is very sensitive to orbital distance, as
illustrated in Table 2 where the majority of semi-major axes are $<$ 0.1 AU.
Thirdly, as shown in Figure 1, one of the Alfvén wings points back towards the
star. However, if the azimuthal component of the stellar wind is sufficiently
large, as occurs at greater orbital distance, then both Alfvén wings may be
produced at angles directed away from the star.

The results of our study investigating the coherent radio emission generated
from sub-Alfvénic interaction between exoplanets and M-dwarfs have shown that
such emission should be widely detectable currently with the VLA, with the
GMRT following its imminent upgrade, and in the future with SKA. This work
focused on radio emission originating from near the surface of the star, to
overcome the issue of ionospheric blocking of the low frequency signals
expected directly from planets, despite the fact that radio flux density of
exoplanet origin would be greater than flux from the stellar surface due to
the comparatively weak planetary magnetic fields. In order to access the low
frequency radio emission from terrestrial exoplanets, observations must be
taken from above the ionosphere. To this end, Zarka et al. (2012) examined the
prospects of future radio arrays on the lunar surface. Alternatively, (Rajan
et al., 2016) has proposed a space-based array of CubeSats for low-frequency
radio observations. Future measurements of properties such as the magnetic
field strength, rotation period, XUV luminosity and mass outflow rate of
M-dwarfs would greatly improve the accuracy of the flux density predictions in
this study, and we also suggest that this work is extended in future beyond
M-dwarfs to other main sequence stars hosting exoplanets.

ST was supported by an STFC Quota Studentship. JDN was supported by STFC grant
ST/K001000/1.

## References

  * Anglada-Escudé et al. (2016) Anglada-Escudé, G., Amado, P. J., Barnes, J., et al. 2016, Nature, 536, 437
  * Bayliss et al. (2017) Bayliss, D., Gillen, E., Eigmuller, P., et al. 2017, arXiv:1710.11099, in pres
  * Bell et al. (2016) Bell, M., Lynch, C., Kaplan, D., et al. 2016, The Astronomer’s Telegram, 9465
  * Bonfond et al. (2009) Bonfond, B., Grodent, D., Gérard, J.-C., et al. 2009, Journal of Geophysical Research: Space Physics, 114, A7
  * Burkhart & Loeb (2017) Burkhart, B., & Loeb, A. 2017, arXiv:1706.07038, in pres
  * Burrows et al. (1997) Burrows, A., Marley, M., Hubbard, W. B., et al. 1997, The Astrophysical Journal, 491, 856
  * Busse (1976) Busse, F. 1976, Physics of the Earth and Planetary Interiors, 12, 350
  * Carroll & Ostlie (2007) Carroll, B. W., & Ostlie, D. A. 2007, An introduction to modern astrophysics (Addison-Wesley, San Francisco)
  * Christensen et al. (2009) Christensen, U. R., Holzwarth, V., & Reiners, A. 2009, Nature, 457, 167
  * Clarke et al. (2004) Clarke, J. T., Grodent, D., Cowley, S. W., et al. 2004, Jupiter. The planet, satellites and magnetosphere, 1, 639
  * Clarke et al. (1996) Clarke, J. T., Ballester, G. E., Trauger, J., et al. 1996, Science, 274, 404
  * Cohen et al. (2015) Cohen, O., Ma, Y., Drake, J., et al. 2015, The Astrophysical Journal, 806, 41
  * Cranmer (2004) Cranmer, S. R. 2004, American Journal of Physics, 72, 1397
  * Curtis & Ness (1986) Curtis, S., & Ness, N. 1986, Journal of Geophysical Research: Space Physics, 91, 11003
  * Ergun et al. (2000) Ergun, R., Carlson, C., McFadden, J., et al. 2000, The Astrophysical Journal, 538, 456
  * Farrell et al. (2004) Farrell, W., Lazio, T., Zarka, P., et al. 2004, Planetary and Space Science, 52, 1469
  * Garraffo et al. (2016) Garraffo, C., Drake, J. J., & Cohen, O. 2016, The Astrophysical Journal Letters, 833, L4
  * Garraffo et al. (2017) Garraffo, C., Drake, J. J., Cohen, O., Alvarado-Gómez, J. D., & Moschou, S. P. 2017, The Astrophysical Journal Letters, 843, L33
  * Gérard et al. (2006) Gérard, J.-C., Saglam, A., Grodent, D., & Clarke, J. T. 2006, Journal of Geophysical Research: Space Physics, 111, A4
  * Giampapa et al. (1996) Giampapa, M. S., Rosner, R., Kashyap, V., et al. 1996, The Astrophysical Journal, 463, 707
  * Gillon et al. (2016) Gillon, M., Jehin, E., Lederer, S. M., et al. 2016, Nature, 533, 221
  * Gillon et al. (2017) Gillon, M., Triaud, A. H., Demory, B.-O., et al. 2017, Nature, 542, 456
  * Grießmeier et al. (2007) Grießmeier, J.-M., Zarka, P., & Spreeuw, H. 2007, Astronomy & Astrophysics, 475, 359
  * Gupta et al. (2017) Gupta, Y., Ajithkumar, B., Kale, H., et al. 2017, Current Science, 113, 707
  * Hallinan et al. (2017) Hallinan, G., Corsi, A., Mooley, K., et al. 2017, Science, eaap9855
  * Hess et al. (2011) Hess, S., Delamere, P., Dols, V., & Ray, L. 2011, Journal of Geophysical Research: Space Physics, 116
  * Hess & Zarka (2011) Hess, S., & Zarka, P. 2011, Astronomy & Astrophysics, 531, A29
  * Imai et al. (2008) Imai, M., Imai, K., Higgins, C. A., & Thieman, J. R. 2008, Geophysical Research Letters, 35, L17103
  * Jacobsen et al. (2007) Jacobsen, S., Neubauer, F., Saur, J., & Schilling, N. 2007, Geophysical research letters, 34, L10202
  * Jones & Su (2008) Jones, S., & Su, Y.-J. 2008, Journal of Geophysical Research: Space Physics, 113
  * Khodachenko et al. (2007) Khodachenko, M. L., Ribas, I., Lammer, H., et al. 2007, Astrobiology, 7, 167
  * Kivelson et al. (2004) Kivelson, M. G., Bagenal, F., Kurth, W. S., et al. 2004, Jupiter: The planet, satellites and magnetosphere, 513
  * Lammer et al. (2009) Lammer, H., Bredehöft, J., Coustenis, A., et al. 2009, Astronomy and Astrophysics Review, 17, 181
  * Lamy et al. (2011) Lamy, L., Cecconi, B., Zarka, P., et al. 2011, Journal of Geophysical Research: Space Physics, 116, 2156
  * Lazio et al. (2004) Lazio, T., Farrell, W., Dietrick, J., et al. 2004, The Astrophysical Journal, 612, 511
  * Lazio et al. (2009) Lazio, T. J. W., Carmichael, S., Clark, J., et al. 2009, The Astronomical Journal, 139, 96
  * Luger et al. (2017a) Luger, R., Lustig-Yaeger, J., Fleming, D. P., et al. 2017a, The Astrophysical Journal, 837, 63
  * Luger et al. (2017b) Luger, R., Sestovic, M., Kruse, E., et al. 2017b, Nature Astronomy, 1, 0129
  * Lynch et al. (2015) Lynch, C., Mutel, R., & Güdel, M. 2015, The Astrophysical Journal, 802, 106
  * McLean et al. (2012) McLean, M., Berger, E., & Reiners, A. 2012, The Astrophysical Journal, 746, 23
  * McQuillan et al. (2013) McQuillan, A., Aigrain, S., & Mazeh, T. 2013, Monthly Notices of the Royal Astronomical Society, 432, 1203
  * Mead & Beard (1964) Mead, G. D., & Beard, D. B. 1964, Journal of Geophysical Research, 69, 1169
  * Mizutani et al. (1992) Mizutani, H., Yamamoto, T., & Fujimura, A. 1992, Advances in Space Research, 12, 265
  * Neubauer (1980) Neubauer, F. 1980, Journal of Geophysical Research: Space Physics, 85, 1171
  * Neubauer (1998) Neubauer, F. M. 1998, Journal of Geophysical Research: Planets, 103, 19843
  * Nichols (2012) Nichols, J. 2012, Monthly Notices of the Royal Astronomical Society: Letters, 427, L75
  * Nichols & Milan (2016) Nichols, J., & Milan, S. 2016, Monthly Notices of the Royal Astronomical Society, 461, 2353
  * Nichols (2011) Nichols, J. D. 2011, Monthly Notices of the Royal Astronomical Society, 414, 2125
  * Parker (1958) Parker, E. N. 1958, The Astrophysical Journal, 128, 664
  * Prangé et al. (1996) Prangé, R., Rego, D., Southwood, D., et al. 1996, Nature, 379, 323
  * Rajan et al. (2016) Rajan, R. T., Boonstra, A.-J., Bentum, M., et al. 2016, Experimental Astronomy, 41, 271
  * Ray & Hess (2008) Ray, L., & Hess, S. 2008, Journal of Geophysical Research: Space Physics, 113
  * Reiners & Basri (2007) Reiners, A., & Basri, G. 2007, The Astrophysical Journal, 656, 1121
  * Reiners & Basri (2008) —. 2008, Astronomy & Astrophysics, 489, L45
  * Reiners & Basri (2010) —. 2010, The Astrophysical Journal, 710, 924
  * Reiners et al. (2009) Reiners, A., Basri, G., & Christensen, U. R. 2009, The Astrophysical Journal, 697, 373
  * Reiners & Christensen (2010) Reiners, A., & Christensen, U. R. 2010, Astronomy & Astrophysics, 522, A13
  * Sano (1993) Sano, Y. 1993, Journal of geomagnetism and geoelectricity, 45, 65
  * Saur et al. (2013) Saur, J., Grambusch, T., Duling, S., Neubauer, F., & Simon, S. 2013, Astronomy & Astrophysics, 552, A119
  * Saur et al. (2004) Saur, J., Neubauer, F. M., Connerney, J., Zarka, P., & Kivelson, M. G. 2004, Jupiter: The Planet, Satellites and Magnetosphere, 1, 537
  * Saur et al. (1999) Saur, J., Neubauer, F. M., Strobel, D. F., & Summers, M. E. 1999, Journal of Geophysical Research: Space Physics, 104, 25105
  * Scharf (2010) Scharf, C. A. 2010, The Astrophysical Journal, 722, 1547
  * Schmitt et al. (1990) Schmitt, J., Collura, A., Sciortino, S., et al. 1990, The Astrophysical Journal, 365, 704
  * Seager (2013) Seager, S. 2013, Science, 340, 577
  * Stevenson (1983) Stevenson, D. 1983, Reports on Progress in Physics, 46, 555
  * Tingay et al. (2013) Tingay, S., Goeke, R., Bowman, J. D., et al. 2013, Publications of the Astronomical Society of Australia, 30
  * Treumann (2006) Treumann, R. A. 2006, The Astronomy and Astrophysics Review, 13, 229
  * Van Weeren et al. (2014) Van Weeren, R. J., Williams, W. L., Tasse, C., et al. 2014, The Astrophysical Journal, 793, 82
  * Vidotto et al. (2011) Vidotto, A., Jardine, M., & Helling, C. 2011, Monthly Notices of the Royal Astronomical Society: Letters, 411, L46
  * Vidotto et al. (2013) Vidotto, A., Jardine, M., Morin, J., et al. 2013, Astronomy & Astrophysics, 557, A67
  * Wang et al. (2017) Wang, S., Wu, D.-H., Barclay, T., & Laughlin, G. P. 2017, arXiv preprint arXiv:1704.04290
  * Wannawichian et al. (2010) Wannawichian, S., Clarke, J., & Nichols, J. 2010, Journal of Geophysical Research: Space Physics, 115
  * Wheatley et al. (2017) Wheatley, P. J., Louden, T., Bourrier, V., Ehrenreich, D., & Gillon, M. 2017, Monthly Notices of the Royal Astronomical Society: Letters, 465, L74
  * Wright & Schwartz (1989) Wright, A. N., & Schwartz, S. J. 1989, Journal of Geophysical Research: Space Physics, 94, 3749
  * Wu & Lee (1979) Wu, C., & Lee, L. 1979, The Astrophysical Journal, 230, 621
  * Zarka (1992) Zarka, P. 1992, Advances in Space Research, 12, 99
  * Zarka (1998) —. 1998, Journal of Geophysical Research: Planets, 103, 20159
  * Zarka (2007) —. 2007, Planetary and Space Science, 55, 598
  * Zarka et al. (2004) Zarka, P., Cecconi, B., & Kurth, W. S. 2004, Journal of Geophysical Research: Space Physics, 109, A9
  * Zarka et al. (2015) Zarka, P., Lazio, J., & Hallinan, G. 2015, Advancing Astrophysics with the Square Kilometre Array (AASKA14)
  * Zarka et al. (2012) Zarka, P., Bougeret, J.-L., Briand, C., et al. 2012, Planetary and Space Science, 74, 156

##  Appendix A Calculating interaction strength factor $\bar{\upalpha}$

The plasma flow-obstacle interaction strength factor $\bar{\upalpha}$ can be
determined by considering the ionospheric Pedersen conductance in the case of
either a magnetised or unmagnetised planet. Neubauer (1998) and Saur et al.
(1999) showed that $\bar{\upalpha}$ can be approximated by

$\bar{\upalpha}=\frac{\Sigma_{\mathrm{P}}}{\Sigma_{\mathrm{P}}+2\Sigma_{\mathrm{A}}},$
(A1)

where $\Sigma_{\mathrm{P}}$ is the ionospheric Pedersen conductance, and
$\Sigma_{\mathrm{A}}$ is the Alfvén conductance, given by

$\Sigma_{\mathrm{A}}=\frac{1}{\upmu_{0}v_{\mathrm{A}}},$ (A2)

where the Alfvén speed $v_{\mathrm{A}}$ in a stellar wind of mass density
$\rho_{\mathrm{sw}}$ is

$v_{\mathrm{A}}=\frac{B_{\mathrm{sw}}}{\left(\upmu_{0}\rho_{\mathrm{sw}}\right)^{1/2}}.$
(A3)

Here, the ionospheric Pedersen conductivity is estimated using the empirical
power law of Nichols & Milan (2016), i.e.

$\Sigma_{\mathrm{P}}=\upkappa\left(\frac{d}{1\mathrm{AU}}\right)^{\uplambda}\left(\frac{B_{J}}{B_{p}}\right)\left(\frac{L_{\mathrm{XUV}}}{L_{\mathrm{XUV\odot}}}\right)^{\upmu}\mathrm{mho},$
(A4)

where $d$ is the orbital distance of the planet, $B_{p}$ is the equatorial
exoplanetary magnetic field strength, $B_{J}$ is the surface field strength at
Jupiter, $L_{\mathrm{XUV}}$ is the stellar XUV luminosity, and
$L_{\mathrm{XUV\odot}}$ is the solar value. The constants in equation (A4)
take the values $\upkappa$ = 15.475, $\uplambda$ = -2.082, and $\upmu$ = 0.5.

For M-dwarfs we assume a value of $L_{\mathrm{XUV}}=L_{\mathrm{XUV\odot}}$,
consistent with X-ray observations of TRAPPIST-1 (Wheatley et al., 2017). In
all the cases we consider of close-orbiting exoplanets,
$\Sigma_{\mathrm{P}}\gg\Sigma_{\mathrm{A}}$, therefore $\bar{\upalpha}\simeq
1$. We note that for an unmagnetised or weakly magnetised planet this
approximation is also valid, since in this case $\Sigma_{\mathrm{P}}$
approaches infinity.

##  Appendix B Model of the stellar wind and magnetic field

We use an isothermal stellar wind (Parker, 1958) fully parameterised by the
sound speed $c_{s}$ for which we assume a value of 170 km s-1, corresponding
to a cornal temperature of $\sim$ 2 $\times$ 106 K, consistent with the
temperature adopted by Vidotto et al. (2013) in their study of M-dwarf stellar
winds. Specifically, we employ Cranmer’s (2004) closed-form analytic solution
of the isothermal wind equation, given by

$v_{\mathrm{sw}}^{2}=\begin{cases}-v_{c}^{2}W_{0}[-D(d)]&\text{if}d\leq
d_{c},\\\ -v_{c}^{2}W_{-1}[-D(d)]&\text{if}d\geq d_{c},\end{cases}$ (B1)

where $W_{0}$ and $W_{-1}$ are branches of the Lambert $W$ function, and
$D(d)$ is given by

$D(d)=\left(\frac{d}{d_{c}}\right)^{-4}\exp\left[4\left(1-\frac{d_{c}}{d}\right)-1\right],$
(B2)

where $d_{c}$ is the critical distance at which the stellar wind speed
$v_{\mathrm{sw}}$ passes through the sound speed $c_{s}$, given by

$d_{c}=\frac{GM_{\star}}{2c_{s}^{2}}$ (B3)

for a star of mass $M_{\star}$.

The magnetic field components of the Parker Spiral are given by

$B_{\mathrm{r}}=B_{\star}\left(\frac{d_{0}}{d}\right)^{2},$ (B4)

and

$B_{\phi}=B_{\mathrm{r}}\frac{\Omega_{\star}d}{v_{\mathrm{sw}}},$ (B5)

where $\Omega_{\star}$ is the stellar rotational velocity, and here we take
$d_{0}=R_{\star}$, i.e. we assume that the field is radial at the stellar
surface. The resultant IMF magnitude is given by

$B_{\mathrm{sw}}=\sqrt{\left(B_{r}^{2}+B_{\phi}^{2}\right)}.$ (B6)

The angle $\theta$ between the stellar wind magnetic field and the impinging
plasma velocity can hence be defined by

$\theta=\arctan\left(\frac{B_{\phi}}{B_{r}}\right)-\arctan\left(\frac{v_{\mathrm{orb}}}{v_{\mathrm{sw}}}\right).$
(B7)

Interaction of the stellar wind with an intrinsic planetary magnetic field
governs the location of the substellar magnetopause standoff distance
$R_{\mathrm{mp}}$, which can be calculated via a consideration of pressure
balance, and is given by

$\left(\frac{R_{\mathrm{mp}}}{R_{p}}\right)=\left(\frac{k_{m}^{2}B^{2}_{\mathrm{p}}}{2\upmu_{0}(p_{\mathrm{dyn\,sw}}+p_{\mathrm{th\,sw}}+B^{2}_{\mathrm{sw}}/2\upmu_{0})}\right)^{1/6},$
(B8)

where $k_{m}$ = 2.44 represents the factor by which the magnetopause currents
enhance the magnetospheric magnetic field at the magnetopause for a realistic
boundary shape (Mead & Beard, 1964), $p_{\mathrm{th\,sw}}$ is the thermal
pressure of the stellar wind, and $p_{\mathrm{dyn\,sw}}$ is the stellar wind
dynamic pressure given by

$p_{\mathrm{dyn\,sw}}=\rho_{\mathrm{sw}}v_{0}^{2},$ (B9)

where $\rho_{\mathrm{sw}}$ is the density of the stellar wind, which, for a
stellar mass loss rate of $\dot{M_{\star}}$ is determined by

$\rho_{\mathrm{sw}}=\frac{\dot{M_{\star}}}{4\uppi d^{2}v_{\mathrm{sw}}},$
(B10)

and the corresponding plasma number density is given by

$n_{\mathrm{sw}}=\rho_{\mathrm{sw}}/m_{\mathrm{av}},$ (B11)

where we take a Sun-like value of the average particle mass in the stellar
wind $m_{\mathrm{av}}$ of 1.92 $\times$ 10-27 kg.

Figures 8, 9, and 10 show profiles of various stellar wind parameters for our
case studies of TRAPPIST-1, Proxima Centauri, and NGTS-1 respectively.

Figure 8: Stellar wind parameters for TRAPPIST-1 versus orbital distance $d$
in units of stellar radii $R_{\star}$ and AU. Panel (a) shows the stellar wind
speed $v_{\mathrm{sw}}$ (dashed line), the Keplerian speed (close-dotted
line), and the resultant incident stellar wind speed $v_{0}$ (solid line).
Also shown are the constant sound speed $c_{\mathrm{s}}$ (loose-dotted line),
and the Alfvén speed $v_{\mathrm{A}}$ (dot-dashed line). Panel (b) shows the
stellar magnetic field radial component $B_{\mathrm{r}}$ (dotted line),
azimuthal component $B_{\phi}$(dot-dashed line), resultant magnetic field
strength $B$ (solid line), and the component perpendicular to the impinging
stellar wind $B_{\perp}$ (dashed line). Panel (c) shows the motional electric
field or $E_{\mathrm{sw}}$. Panel (d) shows the Alfvén Mach number
$M_{\mathrm{A}}$. Panel (e) shows the stellar magnetic pressure (dot-dashed
line), the dynamic pressure (solid line) and the thermal pressure (loose-
dotted line). The vertical dotted lines running through all panels denote the
orbital distances of the planets TRAPPIST-1(b) - (h). Figure 9: As for Figure
8 but for Proxima Centauri Figure 10: As for Figures 8 and 9 but for NGTS-1.