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According to my textbook, the square of orbital period $P_{orb}$ is given by $$P_{\mathrm{orb}}^{2}=\frac{4\pi^{2}a^{3}}{GM},$$ where $a$ is the semi-major axis.

My question is, does the inclination $i$ affect orbital period?

I read that the launch of a rocket often gets help from Earth's angular momentum, for example, if you launch at Earth's equator, the velocity you need to obtain is less than, say, launching near the poles. However, I don't see inclination $i$ being part of the equation for orbital period.

Does the inclination have any effect on orbital period? For example, a planet orbiting around a star at $0^{\circ}$ inclination, would its period change when the inclination rises to $60^{\circ}$?

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    $\begingroup$ The reason rockets get help if they launch from the equator is that Earth is rotating. By launching in the direction of this rotation, they can add the velocity of Earth's surface at the launch site to their velocity and, thus, they need to accelerate less to reach orbital speed. The equator moves fastest and the poles aren't moving at all. $\endgroup$
    – Jim
    Jan 25, 2022 at 13:52
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    $\begingroup$ How do you even define the inclination for an arbitrary planet orbiting an arbitrary star? We use the ecliptic as reference just because Earth is important for us. $\endgroup$
    – nasu
    Jan 25, 2022 at 16:00
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    $\begingroup$ @nasu couple of ways you might do it. 1) using the plane of rotation of the star. 2) using the average orbital plane of all the star's planets. One such that the standard deviation in the spread of inclinations is minimized. That's the one I'd probably go with. Or 3) ask any aliens that live in that system what they use $\endgroup$
    – Jim
    Jan 25, 2022 at 21:34
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    $\begingroup$ @Jim Then you recognize that it is not "the inclination" but "some inclination", an arbitary value than can be anything, depending on who you ask, and cannot affect a measurable quantity like the orbital period. $\endgroup$
    – nasu
    Jan 25, 2022 at 23:52
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    $\begingroup$ @nasu oh absolutely! Except in case 3. Whatever those aliens say, that's "the" inclination. It's their star system after all $\endgroup$
    – Jim
    Jan 26, 2022 at 13:42

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No, inclination does not affect the period (but see below).

The reason is that gravity is a central force, so for something orbiting a spherically symmetric mass there is no dependency on what reference plane or directions you use. You could always select another one and get a different inclination measure, but the period does not change if somebody tilts their coordinate system.

Around Earth a satellite is affected by the oblateness of Earth (and even more by the Moon), that does introduce a dependency on which plane it rotates in. These deviations mean that technically there is no given period since the orbit does not perfectly close (it is no longer the standard 2-body problem, and the plane precesses around Earth), although in practice this is a minor effect. One can approximate the effect, and then there is a slight effect of inclination on period.

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