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The internet tells me that the orbital period of a planet is calculated via Kepler's third law which, to my understanding, says Orbital period = semi-major axis3

I have a gas giant (radius of 1 Jupiter and mass of 1.37 Jupiter, but that's flexible) that has a habitable, tidally-locked moon where my actual worldbuilding takes place. The moon is 0.6 of Earth's mass and ideally, I want its orbital and rotational period to be larger than 6 days since at least 3 nights of "day" and "night" sound cool, but that would automatically mean it's further away from the gas giant, right?

I'd like the parent planet to be large in the moon's sky, ideally at least 15 times the size of our moon, but from my very limited understanding of the maths and playing around with Universe Sandbox, that seems impossible. I tried increasing and decreasing the speed of the moon and the mass of the parent planet (in US) since those were my best guesses. If it's straightforward [orbital period in Earth years = semi-major axis^3], then my wish for short distance but long orbital period is impossible or at least I assume so?

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    $\begingroup$ You can make the central body lighter, so it has a weaker gravitational pull, which lowers the orbital velocity at any constant distance. $\endgroup$
    – ErikHall
    Dec 22, 2023 at 15:15
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    $\begingroup$ @ErikHall So less mass but the radius stays as big as Jupiter? I'll have to read up on how much mass a gas giants needs to stay a gas giant, but if that works, that's great! Thank you $\endgroup$ Dec 22, 2023 at 16:06
  • $\begingroup$ If the mass drops, but the volume remains constant, we are talking about a change in density. You can do this easily by just putting the gas giant closer to its star. That way the planet will expand, so you remove some mass to get the original radius but at a lower mass. Tada ! Lower density. $\endgroup$
    – ErikHall
    Dec 22, 2023 at 17:30
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    $\begingroup$ That makes sense, thanks for clarifying! $\endgroup$ Dec 22, 2023 at 17:56

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Your period-radius relation is incorrect. Most crucially, you're missing a dependence on the masses of the orbiting bodies.

Wikipedia gives this formula for orbital period:

$$T = 2 \pi \sqrt{a^3 \over GM}$$

where $a$ is the semi-major axis, $M$ is the mass of the more massive body, and $G$ is Newton's gravitational constant.

With this formula and some trigonometry, we can work out exactly what the mass of your Jupiter-sized planet needs to be.

The moon's angular diameter, as seen from Earth, is about half a degree. You want your gas giant's angular size to be fifteen times larger, so that means it'll be about 8 degrees.

Now, imagine drawing a straight line from the center of your moon to the center of the gas giant. Then, turn 90 degrees and draw a line to the surface of the gas giant. Then, draw a line from there back to the moon.

What results is a right triangle, with one base equal to the radius of the gas giant (= the radius of Jupiter), the other base equal to the moon's orbital radius, and an angle at one corner of the angular radius of the gas giant of 4 degrees (half the angular diameter).

Thus, we can solve

$$\tan (4°) = {R_J \over a}$$

and, apparently, get an orbital radius of almost exactly one million kilometers.

Plugging this and your desired orbital period (6 days) into the period-radius relation above and solving for $M$, we get

$$M = {a^3 \over G\left({T \over 2\pi}\right)^2}$$

which works out to about $2.2\times10^{27}$ kg, or 1.2 times the mass of Jupiter.

Which is very convenient! Your situation is, in fact, entirely plausible in real life.

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    $\begingroup$ I appreciate the detailed and easy explanation, thank you so much! It goes to show that sometimes the first google result is, in fact, not the right one. $\endgroup$ Dec 22, 2023 at 17:58
  • $\begingroup$ Wouldn't the radius of the gas giant also depend on its mass? This seems to be missing in this answer. $\endgroup$ Dec 24, 2023 at 2:33
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    $\begingroup$ @PaŭloEbermann It would, and also on its composition and temperature. But this gas giant's density came out so close to Jupiter's that I didn't think investigating those things was very necessary. This gas giant might be a bit colder than Jupiter or a bit richer in elements denser than hydrogen. $\endgroup$ Dec 24, 2023 at 4:18

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