Tides on a double planet

Question

I am making a world to be the setting of my future fantasy works, which is to say I really have no story planned but I want to have a world for it when I do and I want said world to be likely give me story ideas.

I want to make things look different but be functionally similar to earth's own circumstances for the purposes that I like science and just because magic is a thing doesn't mean that it is a thing I want to use to hand wave physics. As such I have done a lot of math to determine how this system would work.

As of now the system is a binary star system made up of a main sequence star with a mass of about .5 solar masses and a radius of .8 that of the sun's, and a white dwarf with the mass of .5 solar masses and a radius of 10,000 km. The surface temperatures of both is such that their total luminosity is equal to that of the sun's.

The planet on which my stories will take place is on a double planet, with two planets slightly larger than that of Earth that orbit around the stars while orbiting around a central point between the two of them. This is where my question comes in: how big would the tides be if the distance between them is about 3 times that of the distance between the earth and the moon so about 981,540 km apart from each other.

I am having a hard time of finding any math about that so I could just calculate it myself. Also these planets are not old enough to be tidally locked yet, and the orbit around their common center of gravity is about 28 days (cause even numbers are my friend).

To restate my question: How large would the difference between low tide and high tide on this planet be?

Answer 1

The scale of the tides exerted on one body by another can be calculated rather easily. (This is from Stephen Gillett's excellent book World-Building.) The formula is

$$T = {M \over R³}$$

where $T$ is the tidal acceleration, $M$ is the mass of the body that exerts the tide, and $R$ is the distance between the two bodies. It is assumed that $R$ is much greater than the size of the bodies themselves.

This works if you use relative terms. So if your planets are the size of Earth (about 81 times the mass of the Moon each) and they orbit one another at three times the Earth-Moon distance, you can plug the numbers in and get:

$$T = {81 \over 3³} = {81 \over 27} = 3$$

That is, the tides produced in each of your planets by the other planet will be roughly three times as strong as those exerted on Earth by the Moon. Incidentally, if they're not tidally locked yet, this would be massaging their interiors quite forcefully.

To these mutual planetary tides you should add the solar tides. The Sun produces tides on Earth which are about 45% as strong as those produced by the Moon.

P.S. You actually asked about the difference between low and high tide. That's a tricky one, since even on Earth there are extremely different ranges. Tidal acceleration is not that difficult to calculate but how that plays out on on actual physical body depends on the overall shape and distribution of the continents and seas, the winds, the ocean currents, and even the shape of the coasts. I suppose you can hand-wave most of the fine detail here and go for whatever is useful for your story, within reason.