# Orbital resonance and the precession of the periapsis

I was reading up on orbital resonance and came across something on Wikipedia stating that the 1:2 orbital resonance of Io-Europa is "imperfect". I mean that when you simply take the period of one and multiply it by the ratio, you won't get the correct answer; they won't be equal.

That you have to take into account the precession of the perijove. Which, when taken into account, makes it a perfect 1:2.

How does this work exactly? I would think that the mean motion ratio would be the same, regardless of the perijove as I thought that that was the fundamental definition of orbital resonance.

Let's give an example planet, so you can help explain it to me. Let’s say a regular 1 mass Earth, named Kana, with two 0.5 lunar mass moons. Let’s just call them Kana A and B, in a 1:2 resonance. I would think this would mean that they have a period of 27 days and 54 days respectively (which should be 380,944 km for A; and 604,711 km for B, in terms of semi-major axis, according to my previous understanding of orbital resonance), with mean motions in an exact 1:2 ratio. But it appears this is not the case.

Is it due to differing eccentricities? I don't think it would be, as eccentricity doesn't really play a part in orbital period equations. Only relevant body masses and the semi-major axis do.

So how exactly does the precession of the perikana allow for the mean motions to not be perfectly aligned, whilst still having the correct resonance? I don't really understand it at the moment and the page on it explains it very poorly. I want both the physics behind it, and the mathematics behind it, please.