.
We also show how the sensitivity to Alfv´en wing size varies with orbital distance, for a radio sensitivity
of 100 µJy on the reference system. Closer-in orbits offer sensitivity to smaller Alfv´en wings, and thus to
weaker exoplanet magnetic fields. The error bar at left shows the wind model uncertainty for all systems
that enters when converting flux density to Alfv´en wing radius, which is not included in the 3σ upper limits.
(Bottom right) Exoplanet magnetic dipole field strength consistent with observed flux densities or upper

[Figure: Figure 3.]

SPI Radio Search 19 We converted observational radio upper limits and detected radio flux densities for some comparison systems (Table 1) to spectrally-integrated flux (Figure 3 upper right) using the common assumption for radio SPI observations that ∆ν = νcen, where νcen is the center frequency of the observed band. The observational constraints lie above the predictions for all systems except GJ 376, where the upper limit is close to the observed value. This difference implies that most systems require an extended magnetosphere to be detectable in our observations.
8.7. Upper limits on magnetospheric radius
We can then solve for Alfv´en wing radius:
� observed flux predicted flux for RAW = Rp (12)
RAW
=
Rp
The results are shown in Figure 3, where our 5 radio upper limits correspond to Alfv´en wing sizes of 1.3Rp (GJ 367) up to 12.9Rp (LHS 1678). In the comparison sample, GJ 486’s radio upper limit leads to RAW < 2.4Rp, whereas the three systems with radio burst detections require RAW ∼8 −40Rp to explain the range of observed flux densities. In the same figure, we also show theoretical predictions of the Alfv´en wing radius (see next section) for an Earth-like planetary field of 0.5 G, which range from 1-2.4 Rp for our 5 targets and 1.7-4.1 Rp for the comparison sample since its more distant planets experience lower wind pressures. In our sample, a prediction of 1Rp occurs when the stellar wind field is stronger than 0.5 G (LHS 3844 and GJ 367). For comparison, Earth’s magnetosphere has a dayside extent of 6-10RE (Pulkkinen 2007), larger than expected for our targets due to the lower wind pressures at Earth. Next, we use RAW = √
2Rmag (Section 8.3) to estimate the magnetospheric sizes of our targets, obtaining upper limits of Rmag <1-9Rp. If RAW < √
2Rp, we impose Rmag < 1Rp.
8.8. Upper limits on exoplanet magnetic field We follow S13 in assuming that at the magnetospheric boundary, the exoplanet’s magnetic pressure balances the stellar wind pressure. We apply our stellar wind model (Section 6) to include magnetic, dynamic, and thermal pressure; Figure 3 (lower right) shows the resulting upper limits on a dipolar Bp, which range from 0.8 G (GJ 367) to 320 G (LHS 1678). While our calculations include all 3 types of stellar wind pressure, we note that thermal pressure is negligible for all targets, and dynamic pressure nearly so, since Pdyn = 2M 2 APmag and all of our targets are sub-Alfv´enic. Thus, our results effectively simplify to the planetary and stellar magnetic pressure being equal at the magnetospheric boundary. We assume the exoplanet field Bp is dipolar so: Bwind ≈Bp (Rmag/Rp)−3.
(13) GJ 367 highlights an interesting limitation on sub-Alfv´enic SPI measurements of planetary magnetic field for close-in planets. If the stellar wind field is stronger than the planet’s field, then the planet has no extended magnetosphere. Thus, since our radio upper limits for GJ 367 were consistent with Rmag = Rp, our upper limit on GJ 367 b’s planetary magnetic field is essentially the predicted strength of the stellar wind field at that orbital distance (also seen in Figure 2, lower left). This situation is also reflected in the curve showing the reference system’s hypothetical 100-µJy upper limit on Bp as a function of orbital distance, which bends upwards left of <∼7R∗to follow Bp <= Bwind.