Can we detect aurora in exoplanets orbiting M dwarfs?

TL;DR

This paper estimates the potential for detecting auroral radio emissions from exoplanets orbiting M dwarfs, highlighting promising candidates and emphasizing the importance of stellar wind properties.

Contribution

It introduces a method to predict exoplanet radio emissions based on stellar wind models and identifies key systems with detectable signals under certain wind conditions.

Findings

GJ 674 b and Proxima b are promising for radio detection.

Detection likelihood depends on stellar wind mass-loss rates and velocities.

Current instruments may not detect emissions from GJ 436 b and Proxima b without episodic events.

Abstract

New instruments and telescopes, such as SPIRou, CARMENES and TESS, will increase manyfold the number of known planets orbiting M dwarfs. To guide future radio observations, we estimate radio emission from known M-dwarf planets using the empirical radiometric prescription derived in the solar system, in which radio emission is powered by the wind of the host star. Using solar-like wind models, we find that the most promising exoplanets for radio detections are GJ 674 b and Proxima b, followed by YZ Cet b, GJ 1214 b, GJ 436 b. These are the systems that are the closest to us (<10 pc). However, we also show that our radio fluxes are very sensitive to the unknown properties of winds of M dwarfs. So, which types of winds would generate detectable radio emission? In a "reverse engineering" calculation, we show that winds with mass-loss rates dot{M} > kappa_sw /u_sw^3 would drive planetary radio emission detectable with present-day instruments, where u_{sw} is the local stellar wind velocity and kappa_sw is a constant that depends on the size of the planet, distance and orbital radius. Using observationally-constrained properties of the quiescent winds of GJ 436 and Proxima Cen, we conclude that it is unlikely that GJ 436 b and Proxima b would be detectable with present-day radio instruments, unless the host stars generate episodic coronal mass ejections. GJ 674 b, GJ 876 b and YZ Cet b could present good prospects for radio detection, provided that their host-stars' winds have dot{M} u_sw^3 > 1.8e-4 Msun/yr (km/s)^3.

Tables

Table 1. Table 1: Sample of exoplanets studied in this work. Data are from the NASA Exoplanet Archive for most of the properties. Planets with estimated radii assuming Fe composition are highlighted with an asterisk. See Section 2 for a detailed explanation. The last column is the wind constant κ sw subscript 𝜅 sw \kappa_{\rm sw} that obeys the relation M ˙ ≳ κ sw / u sw 3 greater-than-or-equivalent-to ˙ 𝑀 subscript 𝜅 sw superscript subscript 𝑢 sw 3 \dot{M}\gtrsim\kappa_{\rm sw}/u_{\rm sw}^{3} , with mass-loss rate M ˙ ˙ 𝑀 \dot{M} given in M ⊙ ​ yr − 1 subscript M direct-product superscript yr 1 {\rm M}_{\odot}~{}{\rm yr}^{-1} and local stellar wind velocity u sw subscript 𝑢 sw u_{\rm sw} in km/s. This equation provides the minimum mass-loss rates required for a stellar wind to produce detectable radio emission with present-day instruments ( > 1 absent 1 >1  mJy, cf. Section 4.2 ).

Planet	$M_{\star }$	$R_{\star }$	$d$	$M_p$	$R_p$	$⟨{\rho }_p⟩$	$P_{\mathrm{orb}}$	sma	sma	${\kappa }_{\mathrm{sw}}$
name	($M_{\odot }$)	($R_{\odot }$)	(pc)	($M_{\oplus }$)	($R_{\oplus }$)	(g cm-3)	(d)	(au)	($R_{\star }$)	($M_{\odot }/\mathrm{yr}{\left(\mathrm{km}/s\right)}^3$)
Proxima b∗	$0.12$	$0.14$	$1.29$	$1.3$	$1.3$	$3.1$	$11$	$0.049$	$74.5$	$1.56\times {10}^{-5}$
GJ 876 d∗	$0.32$	$0.30$	$4.68$	$6.7$	$2.1$	$4.1$	$1.9$	$0.021$	$14.9$	$7.92\times {10}^{-5}$
GJ 674 b∗	$0.35$	$0.43$	$4.55$	$11$	$2.3$	$4.7$	$4.7$	$0.039$	$19.5$	$1.50\times {10}^{-4}$
GJ 436 b	$0.47$	$0.46$	$9.76$	$22$	$4.2$	$1.7$	$2.6$	$0.029$	$13.6$	$1.78\times {10}^{-4}$
YZ Cet b∗	$0.13$	$0.17$	$3.60$	$0.75$	$1.1$	$3.0$	$2.0$	$0.016$	$19.7$	$1.82\times {10}^{-4}$
GJ 411 b∗	$0.39$	$0.39$	$2.55$	$3.0$	$1.7$	$3.5$	$13$	$0.079$	$43.3$	$2.34\times {10}^{-4}$
YZ Cet c∗	$0.13$	$0.17$	$3.60$	$0.98$	$1.2$	$3.0$	$3.1$	$0.021$	$26.4$	$2.39\times {10}^{-4}$
YZ Cet d∗	$0.13$	$0.17$	$3.60$	$1.1$	$1.3$	$3.1$	$4.7$	$0.028$	$34.9$	$3.48\times {10}^{-4}$
GJ 581 b∗	$0.31$	$0.29$	$6.30$	$16$	$2.5$	$5.2$	$5.4$	$0.041$	$30.1$	$4.31\times {10}^{-4}$
GJ 1214 b	$0.15$	$0.22$	$12.9$	$6.3$	$2.8$	$1.5$	$1.6$	$0.014$	$13.8$	$5.96\times {10}^{-4}$
Ross 128 b∗	$0.17$	$0.20$	$3.38$	$1.4$	$1.3$	$3.2$	$9.9$	$0.050$	$53.3$	$6.86\times {10}^{-4}$
Wolf 1061 b∗	$0.29$	$0.31$	$4.31$	$1.9$	$1.5$	$3.3$	$4.9$	$0.037$	$26.0$	$7.23\times {10}^{-4}$
GJ 273 c∗	$0.29$	$0.29$	$3.80$	$1.2$	$1.3$	$3.1$	$4.7$	$0.036$	$27.0$	$7.24\times {10}^{-4}$
GJ 3323 b∗	$0.16$	$0.12$	$5.32$	$2.0$	$1.5$	$3.3$	$5.4$	$0.033$	$58.8$	$1.21\times {10}^{-3}$
GJ 273 b∗	$0.29$	$0.29$	$3.80$	$2.9$	$1.7$	$3.5$	$19$	$0.091$	$67.5$	$1.62\times {10}^{-3}$
GJ 687 b∗	$0.45$	$0.43$	$4.55$	$19$	$2.7$	$5.6$	$38$	$0.170$	$85.0$	$1.74\times {10}^{-3}$
Wolf 1061 c∗	$0.29$	$0.31$	$4.31$	$3.4$	$1.7$	$3.6$	$18$	$0.089$	$61.7$	$2.13\times {10}^{-3}$
GJ 581 e∗	$0.31$	$0.29$	$6.30$	$1.7$	$1.4$	$3.2$	$3.1$	$0.028$	$20.9$	$2.15\times {10}^{-3}$
GJ 667 C b∗	$0.33$	$0.42$	$6.80$	$5.7$	$2.0$	$4.0$	$7.2$	$0.051$	$25.8$	$2.44\times {10}^{-3}$
GJ 1265 b∗	$0.18$	$0.19$	$10.3$	$7.4$	$2.1$	$4.2$	$3.7$	$0.026$	$29.4$	$2.57\times {10}^{-3}$
GJ 581 c∗	$0.31$	$0.29$	$6.30$	$5.5$	$2.0$	$4.0$	$13$	$0.072$	$53.4$	$3.82\times {10}^{-3}$
LHS 3844 b	$0.15$	$0.19$	$14.9$	$-$	$1.3$	$-$	$0.46$	$0.006$	$7.04$	$4.67\times {10}^{-3}$
GJ 3779 b∗	$0.27$	$0.28$	$13.7$	$8.0$	$2.2$	$4.3$	$3.0$	$0.026$	$20.0$	$7.67\times {10}^{-3}$
GJ 832 c∗	$0.45$	$0.48$	$4.97$	$5.4$	$2.0$	$4.0$	$36$	$0.163$	$73.0$	$7.69\times {10}^{-3}$
Kapteyn c∗	$0.28$	$0.29$	$3.93$	$7.0$	$2.1$	$4.2$	$120$	$0.311$	$231$	$8.39\times {10}^{-3}$
GJ 1132 b∗	$0.18$	$0.21$	$12.0$	$1.7$	$1.4$	$3.2$	$1.6$	$0.015$	$15.7$	$8.62\times {10}^{-3}$
GJ 876 e∗	$0.32$	$0.30$	$4.68$	$15$	$2.5$	$5.1$	$120$	$0.334$	$240$	$9.54\times {10}^{-3}$
GJ 625 b∗	$0.30$	$0.31$	$6.49$	$2.8$	$1.6$	$3.5$	$15$	$0.078$	$54.3$	$1.05\times {10}^{-2}$
HD 285968 b∗	$0.45$	$0.45$	$9.47$	$8.3$	$2.2$	$4.4$	$8.8$	$0.066$	$31.5$	$1.08\times {10}^{-2}$
Gl 686 b∗	$0.42$	$0.42$	$8.16$	$7.1$	$2.1$	$4.2$	$16$	$0.091$	$46.6$	$1.31\times {10}^{-2}$
GJ 3323 c∗	$0.16$	$0.12$	$5.32$	$2.3$	$1.6$	$3.4$	$41$	$0.126$	$226$	$1.54\times {10}^{-2}$
GJ 667 C c∗	$0.33$	$0.42$	$6.80$	$3.8$	$1.8$	$3.7$	$28$	$0.125$	$64.0$	$2.30\times {10}^{-2}$
Wolf 1061 d∗	$0.29$	$0.31$	$4.31$	$7.7$	$2.1$	$4.3$	$220$	$0.470$	$326$	$2.51\times {10}^{-2}$
GJ 3634 b∗	$0.45$	$0.43$	$19.8$	$8.3$	$2.2$	$4.4$	$2.6$	$0.029$	$14.3$	$3.89\times {10}^{-2}$
LHS 1140 c	$0.18$	$0.21$	$12.5$	$1.8$	$1.3$	$4.8$	$3.8$	$0.027$	$27.4$	$4.54\times {10}^{-2}$
GJ 163 b∗	$0.40$	$0.44$	$15.1$	$11$	$2.3$	$4.7$	$8.6$	$0.061$	$29.7$	$4.66\times {10}^{-2}$
GJ 1132 c∗	$0.18$	$0.21$	$12.0$	$2.6$	$1.6$	$3.5$	$8.9$	$0.048$	$48.7$	$4.92\times {10}^{-2}$
GJ 667 C f∗	$0.33$	$0.42$	$6.80$	$2.5$	$1.6$	$3.4$	$39$	$0.156$	$79.8$	$5.61\times {10}^{-2}$
GJ 3998 b∗	$0.50$	$0.49$	$17.8$	$2.5$	$1.6$	$3.4$	$2.6$	$0.029$	$12.7$	$9.38\times {10}^{-2}$
GJ 667 C e∗	$0.33$	$0.42$	$6.80$	$2.5$	$1.6$	$3.4$	$62$	$0.213$	$109$	$1.05\times {10}^{-1}$
LHS 1140 b	$0.18$	$0.21$	$12.5$	$7.0$	$1.7$	$7.5$	$25$	$0.094$	$95.8$	$1.67\times {10}^{-1}$
K2-25 b	$0.29$	$0.29$	$45.0$	$-$	$3.4$	$-$	$3.5$	$0.030$	$22.1$	$1.84\times {10}^{-1}$
GJ 4276 b∗	$0.41$	$0.41$	$21.3$	$17$	$2.6$	$5.3$	$13$	$0.082$	$43.0$	$2.22\times {10}^{-1}$
GJ 163 c∗	$0.40$	$0.44$	$15.1$	$6.8$	$2.1$	$4.2$	$26$	$0.125$	$61.3$	$3.09\times {10}^{-1}$
GJ 3998 c∗	$0.50$	$0.49$	$17.8$	$6.3$	$2.0$	$4.1$	$14$	$0.089$	$39.0$	$3.23\times {10}^{-1}$
GJ 667 C g∗	$0.33$	$0.42$	$6.80$	$4.5$	$1.9$	$3.8$	$260$	$0.549$	$281$	$3.76\times {10}^{-1}$
GJ 3341 b∗	$0.47$	$0.44$	$23.2$	$6.6$	$2.1$	$4.1$	$14$	$0.089$	$43.5$	$8.77\times {10}^{-1}$
HIP 57274 b∗	$0.29$	$0.78$	$25.9$	$6.4$	$2.0$	$4.1$	$8.1$	$0.070$	$19.3$	$8.81\times {10}^{-1}$
GJ 3293 e∗	$0.42$	$0.40$	$20.2$	$3.3$	$1.7$	$3.6$	$13$	$0.082$	$44.1$	$9.11\times {10}^{-1}$
HD 85512 b∗	$0.43$	$0.71$	$11.3$	$3.2$	$1.7$	$3.6$	$58$	$0.260$	$78.7$	$9.20\times {10}^{-1}$
K2-18 c∗	$0.36$	$0.41$	$34.0$	$7.5$	$2.1$	$4.3$	$9.0$	$0.060$	$31.5$	$1.63\times {10}^0$
HD 125595 b∗	$0.29$	$0.74$	$28.2$	$6.4$	$2.0$	$4.1$	$9.7$	$0.080$	$23.2$	$1.63\times {10}^0$
K2-28 b	$0.26$	$0.29$	$63.1$	$-$	$2.3$	$-$	$2.3$	$0.021$	$15.9$	$1.75\times {10}^0$
Kepler-42 c	$0.13$	$0.17$	$38.7$	$-$	$0.73$	$-$	$0.45$	$0.006$	$7.59$	$2.00\times {10}^0$
GJ 3293 d∗	$0.42$	$0.40$	$20.2$	$7.6$	$2.1$	$4.3$	$48$	$0.194$	$104$	$2.09\times {10}^0$
BD-08 2823 b∗	$0.50$	$0.71$	$41.4$	$13$	$2.4$	$4.9$	$5.6$	$0.060$	$18.2$	$2.14\times {10}^0$
GJ 3293 c∗	$0.42$	$0.40$	$20.2$	$21$	$2.7$	$5.8$	$120$	$0.362$	$194$	$2.81\times {10}^0$
Kepler-42 b	$0.13$	$0.17$	$38.7$	$-$	$0.78$	$-$	$1.2$	$0.012$	$14.7$	$5.57\times {10}^0$

Equations

$$\frac{r_M}{R_p}=\xi {\left(\frac{(B_p/2{)}^2}{8\pi {\rho }_{\mathrm{sw}}{u}^2}\right)}^{\frac{1}{6}},$$

$$P_{\mathrm{sw}}={\rho }_{\mathrm{sw}}{u}^3\pi r_M^2.$$

$$P_{\mathrm{radio}}={\eta }_K{\rho }_{\mathrm{sw}}{u}^3\pi r_M^2.$$

$${\alpha }_0=\arcsin \left[{\left(\frac{R_p}{r_M}\right)}^{1/2}\right]$$

$$B({\alpha }_0)=\frac{1}{2}{B}_p[1+3{\cos }^2({\alpha }_0){]}^{1/2}.$$

$${\Omega }_{\mathrm{cyc}}=\frac{eB}{2\pi m_{e}c},$$

$$\Delta f=2.8\left(\frac{B({\alpha }_0)}{1\mathrm{G}}\right)MHz.$$

$${\varphi }_{\mathrm{radio}}=\frac{P_{\mathrm{radio}}}{d^2\omega \Delta f},$$

$$\omega =2{\int }_{{\alpha }_0-\delta \alpha /2}^{{\alpha }_0+\delta \alpha /2}\sin \alpha \mathrm{d}\alpha \mathrm{d}\phi =4\pi [\cos ({\alpha }_0-\frac{\delta \alpha }{2})-\cos ({\alpha }_0+\frac{\delta \alpha }{2})],$$

$$\rho u\frac{du}{dR}=-\frac{G{M}_{\star }}{R^2}-\frac{dp}{dR}$$

$${\varphi }_{\mathrm{radio}}={\eta }_K^{\prime }\frac{R_p^2}{d^2}{\rho }_{\mathrm{sw}}^{1/2}{u}^{2}f({\alpha }_0),$$

$$f({\alpha }_0)=\frac{{\sin }^2{\alpha }_0}{[\cos ({\alpha }_0-\delta \alpha /2)-\cos ({\alpha }_0+\delta \alpha /2)](1+3{\cos }^2{\alpha }_0{)}^{1/2}},$$

$${\rho }_{\mathrm{sw}}^{1/2}{u}^2=\frac{{\varphi }_{\mathrm{radio}}}{{\eta }_K^{\prime }}\frac{d^2}{R_p^2}\frac{1}{f({\alpha }_0)}.$$

$${\left\{{\left(\frac{n_{p,\mathrm{sw}}}{{1\mathrm{cm}}^{-3}}\right)}^{1/2}{\left(\frac{u}{1km/s}\right)}^2\right\}}_{\min }\simeq \frac{{\varphi }_{\mathrm{radio}}/(1\mathrm{mJy})}{7.6\times 10^{11}}{\left(\frac{d}{R_p}\right)}^2,$$

$${\left\{{\left(\frac{\dot{M}}{{\mathrm{M}}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{1/2}{\left(\frac{u}{1km/s}\right)}^{3/2}\right\}}_{\min }\simeq \frac{{\varphi }_{\mathrm{radio}}/(1\mathrm{mJy})}{2.8\times 10^{20}}{\left(\frac{d}{R_p}\right)}^2\frac{a}{1\mathrm{au}},$$

$$\dot{M}\gtrsim {\kappa }_{\mathrm{sw}}/u_{\mathrm{sw}}^3,$$

$${\kappa }_{\mathrm{sw}}={\left[\frac{{\varphi }_{\mathrm{radio}}/(1\mathrm{mJy})}{2.8\times 10^{20}}{\left(\frac{d}{R_p}\right)}^2\frac{a}{1\mathrm{au}}\right]}^2$$

$$\dot{M}\gtrsim 1.6\times 10^{-5}/u_{\mathrm{sw}}^3\mathrm{for}\mathrm{Proxima}b.$$

$$\dot{M}\gtrsim 1.8\times 10^{-4}/u_{\mathrm{sw}}^3\mathrm{for}\mathrm{GJ}436\mathrm{b},$$

Full text

††pubyear: ††pagerange: Can we detect aurora in exoplanets orbiting M dwarfs?–2

1 Introduction

The search for and characterisation of exoplanets is central to a number of current astrophysical research questions and space exploration. There has been rapid progress in exoplanet detection in recent years, with now over 3,800 confirmed exoplanets on record. The most successful detection methods so far have been the transit and the radial velocity methods. Both of these methods are indirect methods of exoplanet detection, which infer the existence of an exoplanet from the effect it has on the host star.

Exoplanets can be directly detected through imaging. A limitation of such method is the high contrast ratio between the intensity of electromagnetic radiation of the planet and its host star in the visible and infrared ranges; approximately 109 in the visible and 106 in the infrared (Zarka, 2007). However, there may exist a direct detection method in the low frequency radio range, between ten and a few hundred MHz (Grießmeier et al., 2011), given that some planets in the solar system are known to emit radiation within this range. This radiation arises via the interaction between planetary magnetic fields and the solar wind (Desch & Kaiser, 1984) and is known as auroral radio emission. For all magnetised solar system planets, this radiation is only 1 to 2 orders of magnitude less intense than the radiation produced by the Sun in the low frequency radio range (Zarka, 2007), thus making it favourable for direct detection. A nearly linear relation between the emitted radio power of these planets and the dissipated kinetic power of the incident solar wind has been observed (Desch & Kaiser, 1984) and is known as the ‘radiometric Bode’s law’ .

It has been theorised that magnetised exoplanets may emit at radio frequencies, analogously to the solar system planets (e.g., Farrell et al., 1999; Lazio et al., 2004). If the radiometric Bode’s law holds true for exoplanetary systems in a similar way as for the solar system planets, one can use stellar wind models to estimate radio emission from exoplanets. It is expected that hot Jupiters produce radio emission that are many orders of magnitude larger than Jupiter (Grießmeier et al., 2005, 2007; Vidotto et al., 2010a; Vidotto et al., 2012, but see also Weber et al. 2017; Kavanagh et al. 2019), the strongest emitter in the solar system at radio wavelength. This is because hot Jupiters orbit at regions where the stellar wind has a large ram pressure, which can power stronger planetary radio emissions.

The detection of exoplanetary radio emissions would not only be a revolutionary method of exoplanet detection, but also an indicative of the presence of an intrinsic planetary magnetic field. Although some studies have proposed the presence of exoplanetary magnetism to interpret spectroscopic transit observations (e.g., Vidotto et al., 2010b; Llama et al., 2011; Kislyakova et al., 2014), the interpretations are not unique (Vidotto et al., 2015a) and there is still no conclusive detection of an exoplanet magnetic field. Planetary magnetic fields are believed to be one of the key ingredients for determining planetary habitability (e.g. Lammer et al., 2007; McIntyre et al., 2019).

In that regard, M dwarf stars have been the prime targets for detecting terrestrial planets in the habitable zone, which is the region around a star in which a planet could host liquid water on its surface (Kasting et al., 1993). An important issue affecting habitability of M dwarf planets is that their host stars remain active for a long part of their lives (West et al., 2008; Irwin et al., 2011) and have high flare and coronal mass ejections (CME) rates (Davenport et al., 2014; Vida et al., 2016; Vida et al., 2017; Kay et al., 2016). This intense activity could be dangerous for a habitable zone planet, as these planets would receive high dosages of high-energy radiation and intense stellar wind/CME, which could strip away their atmospheres (Lammer et al., 2007; Khodachenko et al., 2007; Scalo et al., 2007; Vidotto et al., 2013). Although intense winds could be dangerous for the survival of planetary atmospheres, intense winds and CMEs would power stronger exoplanetary radio emission, thus allowing one to probe magnetism in exoplanets.

Recently, Burkhart & Loeb (2017) predicted the radio emission of the closest exoplanet to us – Proxima b, which orbits an M dwarf star. They estimate Proxima b has radio emission as high as 1 Jy, in a frequency of 0.02 MHz. Due to its close distance to us, Proxima b is one of the most likely candidates for producing radio emissions that could be detectable from Earth (although, with such predicted frequencies, this would not be observed with ground-based instrumentation). In this work, we estimate the radio emission of exoplanets orbiting M-dwarf stars (for predictions relating to hot Jupiters around solar-type stars see, for example, Grießmeier et al. 2007, 2011). At the time of writing, there are nearly 200 planets orbiting stars in the mass range between 0.1 and 0.5 $M_{\odot}$. The number of known exoplanets orbiting M dwarfs, including exoplanets in their habitable zones, will increase manyfold, with surveys conducted by, e.g., SPIRou (Cloutier et al., 2018), CARMENES (Quirrenbach et al., 2014) and TESS (Sullivan et al., 2015). Due to detection bias, most of the known M-dwarf planets have small semi-major axis, making them favourable candidates for detecting exoplanetary radio emissions.

This paper is divided as follows. Section 2 presents our candidate selection and Section 3 shows the radio emission model, which is based in the work of Vidotto & Donati (2017). Section 4 shows our results, where we estimate the radio emission of exoplanets using two different approaches. In the first approach (Section 4.1), we use simple stellar wind models to derive radio fluxes, and we demonstrate the large influence that stellar wind models have on the predicted radio emission from exoplanets. Since stellar wind properties of M dwarfs are uncertain, in Section 4.2 we present a second approach based on a ‘reverse engineering’ investigation, in which we derive what are the stellar wind properties that would power detectable radio emission with present-day instrumentation. This is then followed by our discussions (Section 5) and conclusions (Section 6).

2 Candidate selection

We make use of the NASA Exoplanet Archive111https://exoplanetarchive.ipac.caltech.edu/ to select 120 exoplanets (Table 2) that fulfil the following selection criteria.

We select stars in the mass range between 0.1 and 0.5$M_{\odot}$. Some of the selected stars do not have quoted errors in their masses (namely, KOI-55, Wolf 1061, GJ 3323, GJ 3341, KOI-55, HD 285968, GJ 674, Ross 458, GJ 273, GJ 3293, GJ 3323). For all the others, the errors quoted in stellar masses are within $25\%$, with the exception of the following stars: K2-72, Kepler-42, K2-9, BD-08 2823, HD 125595, HIP 57274, HD 99492. For the last three stars, masses are quite uncertain ($\sim$ a factor of 2). 2. 2.

Kepler’s third law was used to convert from orbital period to semi-major axis in the case of planets without defined semi-major axis values on the catalogue. 3. 3.

Given that the planetary radius is necessary for the computation of planetary radio power and flux, we use the analytical expression from Seager et al. (2007, see their equation (23)) to derive the radii of planets without quoted radii (mostly planets that are not transiting). These planets are highlighted in Table 2 with an asterisk. This analytical expression offers an estimate of planetary radii and is valid for planets with masses below $\sim 23~{}M_{\oplus}$. We use the formalism for planets with Fe composition, which gives a lower limit on the size of rocky planets (Seager et al., 2007). Since the square of the planet radius $R_{p}$ enters in the radio power computation, a lower limit on $R_{p}$ produces a conservative estimate for planetary radio power and flux. Assuming a silicate composition (MgSiO3), for example, would increase the quoted radii of these planets by a factor of $1.29$ to $1.35$, which would increase our estimated radio power by a factor of up to $1.8$. Planets with masses above the $\sim 23~{}M_{\oplus}$ limit or without quoted masses and radii were not included in our sample. 4. 4.

Finally, some stars did not have quoted radii in the catalogue. For the following stars, we used the radii quoted in Pasinetti Fracassini et al. (2001): GJ 667 C, GJ 163, GJ 674 and GJ 832.