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Consider a system in which a central star is orbited by a planet with liquid water oceans, which is itself orbited by a moon.

Given the masses and distances between these three objects, is there some formula that outputs the minimum and maximum tide heights the planet's oceans cycle through for every orbit of the moon?

For simplicity, the effects of local topography on the tides are being ignored.

Also, if there is such a formula, could it be applied to solar systems in which there is more than one central star and/or more than one moon orbiting the planet?

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2 Answers 2

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Wikipeida (quoting Icarus ) gives

$$\frac{15}{8}\frac{mA^4}{Mr^3}$$

Where $m$ and $M$ are the masses of the moon and planet, respectively; $r$ is the orbit radius of the moon and $A$ is the radius of the planet. For Earth this is a little less than a metre.

Systems with multiple moons (or moon and sun, like on Earth) will have multiple bulges which can add up.

This tells you very little about the sizes of actual tides, which are strongly magnified by topography. It predicts tides of 0.7 m, whereas real tides have ranges of between 0 and 16 m.

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    $\begingroup$ Calculating tides without considering topography is like working out the sound of a violin only considering the strings and ignoring the hollow body. Nearly all the sound of a violin comes not from the strings but from resonance in the body. Likewise, real tides are a resonance effect driven by the moon, not a tidal bulge. $\endgroup$
    – James K
    Mar 30, 2021 at 9:35
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You could figure this out with a GIS system like QGIS by creating a representation of your tides in a raster dataset, or other grid dataset, where the value represents Max tide in metres. Then once you worked out all the maths to get your tidal force, as noted in the previous answer, you could then apply that value to the raster which will give you tide heights with your new gravitational masses. I had a similar question wondering how tides would change if Earth had a near miss from a rouge planet that passed between the Earth and the Moon. Basically, I calculated the tidal force of the moon and the rouge planet, in my theoretical example the rouge planet was a 93% increase in tidal effect. Then I added 93% to the tidal range to see how it would effect an area. Some issues, I couldn't find a tide range dataset for all of Earth, and if I did it would likely be of a very low resolution. A high resolution dataset would require quite the PC to create. To find out what portion of land would be submerged you would need to do some additional analysis. In your case, since your planet is fictional, you would have to make the high tide raster; which means you would have to have some idea of the topology of your world and how that topology effects tides. Would make a very interesting activity that would even surpass Frank Herbert's world building in terms of sheer detail. Hope that helps.

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