8 Villadsen et al.
The possible causes of non-detections (Section 7) depend on the magnetic field and density of the stellar wind, but our targets do not have observational constraints on their wind properties. In this section, we first predict the stellar wind base conditions: stellar magnetic field, mass loss rate, and temperature. We then use a radial wind model to predict circumplanetary wind density and field strength, shown in Figure 2. We conclude this section with a rough estimate of the uncertainties on our predictions.
6.1. Wind base conditions: Large-scale photospheric magnetic field Our target planets orbit at 3−8R∗, where the effects of small-scale surface fields are negligible and large-scale field dominates the wind properties (Lang et al. 2014), so we seek to predict the large-scale surface field that would be detectable by Zeeman Doppler Imaging (ZDI), BZDI. While rotation-BZDI correlations exist for more massive stars (Vidotto et al. 2014b), slowly-rotating M dwarfs exceed these correlations by an order of magnitude (Lehmann et al. 2024; See et al. 2025), but new rotation-BZDI correlations are not yet available for this mass/rotation parameter space as ZDI characterization of these targets is new and ongoing (Donati et al. 2023). Instead, to estimate BZDI, we first use rotation to predict the average surface field detectable by Zeeman Broadening (ZB), which includes small-scale features. We apply the observational ZB relationship of Reiners et al. (2022) [R22] (their Eq. 2). All of our targets are slow rotators with Ro > 0.13 (Table 3), so the relationship predicts:
BZB = (8570 G)R−2
∗P −1.25 rot
, (1) where R∗is in solar units and Prot in days. This formula predicts fields of 130-600 G for our targets (Table 3) and 600-2000 G for our comparison sample since they are later spectral type. We then translate from the predicted ZB field, to estimate the magnitude of the large-scale field BZDI observable by ZDI: BZDI = fZDI BZB (2) using the typical ratios measured by Reiners & Basri (2009) of fZDI = |BV /BI| ∼6% for partially convective stars and 14% for fully convective stars. We use a partially/fully-convective boundary of M∗= 0.35M⊙. This calculation predicts large-scale surface fields of 8-80 G. We assume that the ratio BZDI/BZB is the same as BV /BI, which may underestimate BZDI since BV is reduced by projection effects. However, the error introduced is small compared to other uncertainties, and Wulff et al. (2026) find that the dipole field of slowly-rotating M dwarfs (See et al. 2025; Lehmann et al. 2024) is typically about 10% of BZB, matching our 6-14% ratio. Based on the few ZDI observations of slowly-rotating M dwarfs, the large-scale field is probably almost entirely dipolar (Lehmann et al. 2024) [hereafter L24]. While none of our 5 targets have ZB or ZDI measurements, some of the comparison sample do, but we use the rotation-based magnetic field predictions throughout this work for simplicity.
6.2. Wind base conditions: Mass loss rate None of our targets have wind measurements. To estimate their mass loss rates, we employ the power-law fit of Wood et al. (2021) for a population of main-sequence stars including about half SPI Radio Search 9 [Table:

Table 3. Predicted stellar wind parameters]