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(edited to fix obvious confusion, as detected by AlexP)

The idea is that the plane of the moon orbit is aproximately at a 90 degree angle to the plane of the planet orbit. Would that be stable, or even possible?

Supposing size, mass, distance, axial tilt, rotational period, similar to the Earth-Moon system, would this peculiarity have any noticeable effects (or requirements) on tides, geomagnetism, evolution, history of astronomy, superstitions, culture in general?

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  • $\begingroup$ Since the example of Uranus (below) proves that it is possible, perhaps your question might be better posed as whether a planet can have (large) moons in orbits that are nearly polar. $\endgroup$ – jamesqf Apr 30 '17 at 4:55
  • $\begingroup$ It's not stable. You might care whether it persists over geologic time scales — billions of years after forming, is it still at a steep angle to the equator? $\endgroup$ – JDługosz Apr 30 '17 at 6:24
  • $\begingroup$ There is already a question specifically on tides for such a system. Check that for general remarks too! $\endgroup$ – JDługosz Apr 30 '17 at 6:25
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Although I won't presume to understand the mathematics, there's a problem with the stability of highly-inclined orbits if there are other bodies close enough or massive enough to perturb the system, namely the Kozai mechanism.

In celestial mechanics, the Kozai mechanism, or the Lidov–Kozai mechanism, is a perturbation of the orbit of a satellite by the gravity of another body orbiting farther out, causing libration (oscillation about a constant value) of the orbit's argument of pericenter. As the orbit librates, there is a periodic exchange between its inclination and its eccentricity. (...)

The Lidov-Kozai mechanism places restrictions on the orbits possible within a system (...) if the orbit of a planet's moon is highly inclined to the planet's orbit, the eccentricity of the moon's orbit will increase until, at closest approach, the moon is destroyed by tidal forces.

Of course this inclination is relative to the equator of the planet, not to the plane of its orbit around its star. But since you specified that the planet's axial tilt is a moderate one, I'm assuming "inclined with respect to the equator" implies "inclined with respect to the orbital plane". If the axial tilt is zero, then that would be a polar orbit.

I have no idea how much time it would take for the Kozai mechanism to destabilize a highly-inclined orbit.

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I assume that the question is correct in the title, with the text "along a meridian" being an unfortunate misthought.

  1. Moon orbit perpendicular to planet orbit - possible?

    Yes. The regular satellites of Uranus orbit more or less in the plane of the equator or Uranus, which is almost perpendicular to its orbit.

  2. Would this peculiarity have any noticeable effects (or requirements) on tides, geomagnetism, evolution, history of astronomy, superstitions, culture in general?

    • Tides: yes.

    • Geomagnetism: no.

    • Evolution: no.

    • History of astronomy: maybe.

    • Superstitions: maybe, most likely yes.

    • Culture in general: no.

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  • $\begingroup$ Yup, it was a misthought. I assumed a planet with a moderate axial tilt. Of course in a planet with a 90 degree tilt, a perpendicular orbit would be an equatorial orbit. $\endgroup$ – Luís Henrique Apr 30 '17 at 0:04

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