tokens: relative to the forcing from m p is Even with the uncertainties of various parameters, the inner two planets are weakly coupled to the stellar spin. The stellar obliquity (relative to ) therefore varies between |θ ⋆ p − θ 12, p | and θ ⋆ p + θ 12, p (see equations 12 – 13 ; note that since L p ≫ L 12 = L 1 + L 2 for this system, is aligned with ). The observed stellar obliquity (λ ≥ 37°) then implies θ ⋆ p + θ 12, p ≥ 37° (see Li et al. 2014 , for a previous analysis). 5 DISCUSSION The main goal of this paper is to inform the readers that there is a simple way to assess the dynamical role of external perturbers on the spin–orbit misalignments of planetary systems. The key is to evaluate the star–planet spin–orbit coupling parameter (equation 9 ) in response to the differential precession induced by the external companion. We have presented analytic results in various limiting regimes, and discussed how the general results (see Fig. 1 ) scale with different physical parameters (Section 2). Similar analysis can be done for multiplanet systems (Section 4), with the generalised spin–orbit coupling parameter given by equation ( 31 ). As alluded to in Section 1 , continuing observations on spin–orbit misalignments and external companions in HJ and other compact planetary systems may shed light on the formation mechanism of close-in planets. Several recent works of such systems have either neglected some key physical ingredients (spin–orbit coupling) or made such a relatively simple problem unnecessarily complex or obscure. We hope that the results of this paper (see also Lai 2016 ) will make future analysis simpler and more transparent. A recent analysis on spin–orbit alignments of HJs around cool stars concluded that the exterior companions are coplanar (Becker et al. 2017 ). While this may be the case of a few systems with gas giant companion at ∼1 au, similar conclusions cannot be drawn for systems with a slightly more distant companion (see Section 3 , especially Fig. 2 ) since the spin–orbit coupling parameter ε ⋆1 depends sensitively on the perturber’s semimajor axis (; see equation 24 ). Certain HJ formation mechanisms (e.g. LK migration) require an external companion with a high inclination (>40°). As a result, constraints placed on the mutual inclination via the method discussed in this paper may in principle lead to a constraint on the migration mechanism. However, we emphasize that the inclination of giant planet companion may only be constrained when it is relatively close (at ∼1 au, so that ε ⋆1 ≳ 1, see equation 24 ), such as for WASP-41 and HAT-P-13. These systems clearly could not have formed in the classic LK migration picture (with the HJ originating at ∼1 au or beyond), because the initial configuration would be unstable. LK migration requires a much more distant companion (at, say ∼5 − 10 au, so that the initial configuration is stable). Such a companion will satisfy ε ⋆1 ≪ 1, so that it is unable to affect the star-HJ
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