Laplace's longitudes

Here we go: for generalised Laplace longitude λ, resonant-angle φ₁₂ = qλ₁ + pλ₂ + (p - q)ϖ_1,2. p and q here are the p:q integers although, yeah, first-order such that p = q+1.

We know Europa/Ganymede (like Mimas/Tethys &c.) is just φ₂₃ = λ₂ - 2λ₃ - ϖ_2,3. After some definitionals and mathematics which the arXiv did for us already, the angles on our station:

φ2s = 3λ2 - 4λs       + ϖs
φs3 =       2λs - 3λ3 + ϖs

The latter is like φ_bc for TOI-1136 although, of course, φ_cd was 2:1 over there. Subtract 'em out and: φ_2s3 = 3λ₂ - 6λ_s + 3λ₃. So, when λ₂=λ₃ at conjunction which we'll define as zero, λ_s = -180°/6 = -30° so, behind them in their common rotation. I'll leave to the nerds how to transform the vector.

For Europa/Ganymede (far more so than Ariel/Umbriel) we must always remember that at 0° conjunction, Io's λ₁ = 180°. Really -180°, pulling our station back. Simultaneously Europa and Ganymede are all pulling our station forward; mighty Ganymede upward, at that. Hence libration. Which we're not competent to do here; especially considering Jupiter's own tide.