we denote PSPI, is valid for low Alfv´en Mach number MA. The underlying approximation that (1 −1/M2
A)−1/2 ≈MA
(Callingham et al. 2024) is accurate to within 20% for MA < 0.8. Pineda & Hallinan (2018) reframe this equation in cgs to show the dependence of MA on wind density:
PSPI = 1
� 4πρ . (8)
2 ¯α2R2 AWv2B sin2 θ 16 Villadsen et al.

The parameters in this model are: a dimensionless interaction strength ¯α (Section 8.2), the crosssectional radius RAW of the Alfv´en wings (Section 8.3), the stellar wind’s velocity in the planet’s rest frame at speed v at angle θ relative to ⃗B (Section 8.4), and the unperturbed stellar wind magnetic field ⃗B and mass density ρ at the planet’s location (Section 6). The unperturbed stellar wind properties are the values that would occur if the planet was not present.
8.2. Interaction strength ¯α The interaction strength parameter ¯α can vary from 0 (no interaction) to 1 (strong interaction), quantifying the satellite’s ability to halt oncoming plasma flow through electromagnetic interaction. Strong interaction (¯α ≈1) occurs in two cases: objects with strong extended magnetospheres such as Ganymede, and unmagnetized objects with an electrically-conducting ionosphere such as Io. In contrast, unmagnetized atmosphere-less objects such as Tethys and Dione can have low interaction strength (¯α ∼10−2; Saur et al. 2013). We assume ¯α = 1 in our calculations. Since most of our radio upper limits correspond to extended magnetospheres (Section 8.7), this assumption should be appropriate. However, if the resulting constraint on the Alfv´en wing radius were less than the planet size, the upper limit could reflect a low interaction strength due to the lack of an atmosphere, which is likely for our systems (Section 2).
8.3. Radius of Alfv´en wing The cross-sectional radius of the Alfv´en wing, RAW, is measured far enough from the planet that the field lines connecting the star and planet have become parallel. Here, RAW is the equivalent of R or Reff in S13, or Ro in Pineda & Hallinan (2018). Determining RAW depends on whether the planet has a closed magnetosphere that extends above its surface. Extended magnetosphere: RAW = √
2Rmag. If the planet has a strong enough magnetic field, it will form a closed magnetosphere with radius Rmag (denoted Robst in S13) in pressure balance with the stellar wind. Depending on the relative orientation ΘM of the stellar wind magnetic field and planet’s equatorial field, the Alfv´en wing may expand wider than the closed magnetosphere, or in the relatively rare case of anti-aligned fields, no stellar field lines connect to the planet and the Alfv´en wing does not exist. Thus, RAW = fgeoRmag (9) 3 depending on orientation (S13). Since R2 AW appears in SPI power, we perform a spherical average of f 2 geo across all ΘM to obtain ⟨fgeo⟩= √
where fgeo ≈ � 3 cos(ΘM/2) ranges from 0 to √ 2. This is close to the value obtained for θM = 90◦, which may be a preferred orientation in our systems, since the tidally-locked planetary rotation axis should be nearly perpendicular to the radial stellar wind field. Thus, we adopt RAW = √
2Rmag for planets with extended magnetospheres. Induced magnetosphere: RAW = Rp. If the planet is unmagnetized or its field is too weak to form an extended magnetosphere, then if it has an ionosphere it can form an induced magnetosphere. S13 adopts RAW = 1.3Rp 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).