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tokens: reveals a clear difference in the two moon orientations. Figure 6. Exoplanet orbit (blue) with two different exomoon orbits: the red solid line is prograde and coplanar to the planet, and the black dashed line is retrograde and misaligned to the planet. As seen by the observer (right) there is a degeneracy, as both moons has the same ( x , y ) position and ( v x , y y ) velocity, despite the side view (left) betraying a clear difference between the two orbits. L denotes the angular momentum vectors. This degeneracy may be broken by nodal precession, which would not occur for the coplanar orbit but would for the misaligned orbit. Fortunately, for a moon that orbits at a fair fraction of its planet’s Hill sphere, precession will be rapid, revealing the magnitude of the misalignment in just tens of orbits of the planet. Therefore, the dynamically evolving character of TDV will betray the prograde or retrograde character of the moon. If the planetary impact parameter is low then the “TIP–TDV” may be negligible and the magnitude of non-coplanarity may not be enough to break the degeneracy. In this case, higher-order dynamical effects that differ in sign between prograde and retrograde moons may need to be taken into account, as envisioned by Lewis Fujii ( 2014 ). Two alternative methods for breaking the degeneracy, practical only with Extremely Large Telescopes, were discussed by Heller Albrecht ( 2014 ). 5.2. The Prevalence of Large TTVs for Long-period Gas Giants According to the transit times of Table S3 of Teachey Kipping ( 2018 ), the planet Kepler-1625b has a mean absolute deviation from a constant-period model, normalized by the orbital period—a “scatter”—of s O − C / T P = 2.40 × 10 −5 . The timings have a median error bar normalized by the orbital period of σ / T P = 1.55 × 10 −5 . For the TTV measurements of Holczer et al. ( 2016 ), the data are more precise than that for 40 planets with T P > 100 days. Of those 40, 15 planets have larger TTV scatter, i.e., s O − C / T P > 2.40 × 10 −5 , and all of these are deemed significant at . The large amplitude and period of these signals makes them likely due to planet–planet perturbations. We conclude that Kepler-1625b may very likely have a TTV signal due to additional planets, which may be confused for exomoons, or at least contaminate the exomoon TTV signal. A repeated photometric transit signal of the exomoon, rather than the TTV induced on the planet, is likely a more reliable signature. 5.3. Overlapping Moon and Planet Transits There are two possible scenarios for overlapping moon and planet transits. First, the moon may be entirely in front of or behind the planet, in which case the photometric signal would be identical to that of an isolated planet transit and the moon would be hidden. Such an event is not explicitly considered in our equations. We estimate it to be rare though, with a likelihood on the order of ≈ R P / a M if I M = I P = 90°, and significantly less for inclinations that