in that case for unmagnetized solar system moons such as Io. However, for the gravity of a terrestrial planet rather than a moon, the ionosphere may be thinner than Io’s, so we adopt the simple assumption that RAW = Rp for weakly magnetized planets.
8.4. Stellar wind direction: simplifying v sin θ The SPI power equation (Equation 8) depends on the angle θ between the stellar wind magnetic field ⃗B and its flow direction ⃗v in the planet’s rest frame. We assume the planets are in circular SPI Radio Search 17 orbits (Section 2).

For our radial wind model (Section 6), the wind velocity and stellar magnetic field are both radial in the star’s rest frame. In the planet frame, the magnetic field remains radial away from the star, but the stellar wind speed ⃗v has an added azimuthal component of the orbital velocity. Taking v sin θ gives only the component of ⃗v perpendicular to the radial ⃗B, so only the azimuthal component, so v sin θ ≈vorb. To refine this assumption, in Appendix C, we show that for a Weber-Davis wind appropriate for a rotating star, if stellar rotation and orbital axes are aligned, the expression instead simplifies to v sin θ = vorb(Br/B)(1 −Porb/Prot) ≈vorb(1 −Porb/Prot) for our slowly-rotating sub-Alfv´enic systems. We implement this latter approximation in our SPI power predictions in Figure 3.
8.5. Predicted SPI power for unmagnetized planet The top left panel in Figure 3 shows our predictions for the SPI power transferred from near the planet back towards the star, for the unmagnetized case where the Alfv´en wing is the width of the planet. The radial dependence of this power with orbital distance a is:
�2 1 a2
� 1
PSPI ∝v2 orbB√ρ ∝ �1 √a
∝ 1 a4√vwind
. (10)
a2vwind
Accounting for wind acceleration, PSPI ∝a−5 to a−4 at 3-100R∗. This strong dependence is somewhat reduced for magnetized planets. For a given planetary field strength, more distant planets will carve out a larger magnetosphere in balance with the decreased stellar wind magnetic pressure, reflected in Rmag ∝a2/3 (Section 8.8), so PSPI ∝a−3.7 to a−2.7. Close-in planets dramatically improve the prospects for detecting SPI, but better stellar conditions (late type stars close to Earth with strong stellar fields and mass loss flux) can outweigh this advantage.
8.6. Predicted SPI radio flux for unmagnetized planet For the case of an unmagnetized planet, we next translate our predictions of SPI power into predictions of radio flux. We predict the spectrally-integrated radio flux FR since it is independent of the unknown bandwidth ∆ν, which means these predictions can be easily adapted to the observing frequencies of various radio telescopes. The predicted radio flux is:
FR = Fν∆ν = ηPSPI
Ωd2 (11) where η is the efficiency of converting SPI power into ECM radio luminosity, Ωis the solid angle of the beamed ECM radiation, and d is the distance to Earth. We adopt the Solar System-grounded assumptions used in e.g., Pineda & Villadsen (2023) of η = 0.01 and Ω= 0.16 ster. Figure 3 (bottom left) shows our predicted fluxes for the unmagnetized planet case. These range from 4-440 µJy·GHz, where GJ 367’s prediction is the highest due to strong stellar wind conditions at a/R∗∼3. In Table 1, we give flux density upper limits for an integration time of 5 minutes. For comparison system GJ 486, we scaled from the dynamic-spectrum Stokes V sensitivity in Pe˜na-Mo˜nino et al. (2025) up to an integration time of 5 minutes and the full bandwidth. While M dwarf coherent bursts show a wide range of durations in the literature, we chose a 5-minute duration to be order-ofmagnitude consistent with the time for a beam thickness of a few degrees (beam width plus planet magnetosphere’s angular extent) to sweep across the line of sight during a ∼1-day orbit.