surrounding the star that is pierced by open field lines. At the orbital distances of our targets, we assume fopen(r) ≈1, which may slightly underestimate density near GJ 367 b at 3.4R∗. The Parker solution assumes the wind is purely radial, enabling an analytic solution, whereas a numerically-integrated Weber-Davis wind (Weber & Davis 1967) allows for azimuthal wind velocity caused by stellar rotation. We compared a radial Parker wind model for YZ Cet to the Weber-Davis SPI Radio Search 13 solution of Pineda & Villadsen (2023)

for their base conditions, finding that the two are equivalent until >100R∗since the star’s slow rotation makes the azimuthal component negligible near the star (ur ≳100uϕ). Figure 2 shows our predicted stellar wind densities, which are 0.4-12×10−20 g cm−3 near the planet in our 5 target systems. Coincidentally, this mass density is comparable to that near Io (n ∼400-2300 cm−3 with 22 amu; Saur et al. 2013).
6.7. Uncertainties in Predicted Wind Conditions To interpret non-detections, we rely on the wind model for: 1) magnetic field strength near the stellar surface, which determines the maximum cyclotron frequency (Section 7.3), and 2) wind magnetic field strength, 3) wind speed, and 4) wind density near the planet, which determine the Alfv´en Mach number (Section 7.1) and the predicted SPI radio flux density (Section 8). Our predictions for all of these properties depend on stellar rotation period, which we estimate has ∼20% uncertainty due to the varying quality of rotation period estimates for our targets (see references in Table 1), but this is not the dominant source of uncertainty in the wind parameters due to large scatter in rotation-activity relationships. Throughout this work, to be conservative, we combine the orders-ofmagnitude uncertainties in the wind model multiplicatively, rather than adding in quadrature, since the latter practice assumes independent variables. 1) Magnetic field strength near stellar surface. 75% of a large sample of M dwarfs follow the observational ZB relationship of Reiners et al. (2022) to within ×2, so the uncertainty on our BZB predictions (Equation 1) is ×1.3 (from P −1.25
) ×2 (ZB scatter) ≈×2.5. In our Equation 2, using fZDI =6% and 14% from Reiners & Basri (2009) introduces difficult-to-quantify uncertainty because these scale factors come from only 6 stars, all much more active than our targets. We take the uncertainty on fZDI to be ×2 (roughly the difference between the partially and fully-convective values), so that our estimates of large-scale field have uncertainty of order ×5. 2) Magnetic field strength near planet. Based on the range of values in Vidotto et al. (2014a), we take the uncertainty in fopen,0 (in Equation 6) to be about 20%, leading to a net uncertainty of ≈×6 in the circumplanetary stellar wind field strength (Figure 2, lower left panel). Direct measurements of the stellar magnetic field are thus critical for interpreting SPI observations and identifying targets for SPI searches. While ZDI of all possible SPI targets would require prohibitive observing time, Zeeman broadening surveys could reduce the uncertainty in stellar wind magnetic field estimates to ∼×2.5 instead of ×6. 3) Wind speed near planet. We estimate that our assumed wind temperature of 2 MK for slowlyrotating M dwarfs has an uncertainty of ×2 (Section 6.3). A factor of 2 change in wind temperature has a moderate effect (∼×1.6) on the wind speed at large distances, where a hotter wind reaches a higher terminal speed. Temperature has a stronger effect in the wind acceleration region (×2 −3 at 3R∗) since a hotter wind accelerates more rapidly close to the star. Thus, X-ray observations of coronal temperature, and/or theoretical estimates of wind heating, can improve wind speed and density predictions in the near-star regions relevant for sub-Alfv´enic SPI. In an Alfv´en-wave-driven model of TRAPPIST-1’s stellar wind, R´eville et al. (2024) find that the possible range of slow and fast stellar wind speeds spans a factor of 2 depending on unknown parameters, consistent with our assumed wind speed uncertainty. 4) Wind density near planet. Our wind density predictions have two main uncertainties: wind temperature and mass loss rate. For mass loss rate, the uncertainty in rotation period and X-ray
rot 14 Villadsen et al.