[Space] Triple Full Moons

[a humble lich]

Originally Posted by Jinumon

It's been a long time since I've had to graph trigonometric functions, and I'm a little unfamiliar with your notation. I'd greatly appreciate a bit more in-depth explanation of your variables. I'd love to be able to calculate this kind of thing myself in the future.

Jinumon

Sure, it is always hard to know what sort of background your audience online has---it can range from high school freshmen to PhD in physics. (Not that there is anything wrong with high school students.) I was writing in a kind of shorthand, I'll try to go in more detail. (And if anyone finds a mistake please let me know, I have been known to make those.)

The motion of something in a circular orbit is given by
x = A sin( 2 pi t/ T + phi)
y = A cos( 2 pi t/ T + phi)
where x and y are the coordinates, A is the amplitude or radius of the orbit, t is time, T is the period of the orbit, phi is the phase and pi is 3.14159.... For this purpose the phase (phi) can be ignored---the phase will tell us where in the orbit it starts but that information is not useful to us. It is like if you are asking how long the Earth's year is, you don't need to know if you are starting in winter or summer. Note that the time (t) needs to be in the same units as the period (T).

For each individual moon, it will be full when it is directly behind the planet. If we say the y coordinate is the direction of the planet's orbit, and the the x coordinate is the direction perpendicular to the planet's orbit, then the moon will be full when y is at its maximum and x=0. Because we are interested in when the moon is full and not the distance, we can neglect the amplitude and look for when the function cos(2 pi t/T) is a maximum.

For one moon, that means we will have one maximum once per period, as we expect. For a bunch of moons, each moon has a separate period. We define a function f= cos(2 pi t/T_1) + cos(2 pi t/T_2) + cos(2 pi t/T_3).* Now T_1 is the period of moon 1, T_2 is the period of moon 2, etc. Each term of this tell us how full each moon is; i.e., when cos(2 pi t/T_3) is one, then moon three is full. By adding all three, we know all three moons are full when the sum is close to 3. (Also if you'd ever care all three moons would be close to new when the sum is close to -3).

Now we need to find the time between times when all three are close to full. If the ratios of the periods T_1, T_2, and T_3 are not rational numbers, then the function f is know a quasi-periodic, which you can think of as a step between periodic and chaotic.** A quasi-periodic function will never repeat, but we can look at some overall properties of it. If we plot the function f*** we get a squiggly line that goes between 3 and -3. We can look at when the function is close to 3 and see how long it will be in the area close to 3 and also how long it will be between times close to three****. These times will generally be approximate, in this case it is generally about 35 hours between periods, but sometimes there are two close ones separated by only about 4.5 hours

Let me know if that still doesn't make sense or if you have more questions.

* I think before I might have named this function x, but I've already used that letter, so it is called f now. Also T_1 should be T subscript 1.

** In this case, technically T_1/T_2 is a rational number because the values for the periods given were given to a finite number of decimal places.(and the same for T_2/T_3 and T_1/T_3) However, I assume that is because for some reason you didn't feel like tying an infinite number of digits. In any case, the function as given will be close to quasi-periodic.

*** As far as graphing goes, it will depend of the software you use. I use Octave or Matlab, something like Excel should work too. For this problem I graphed t from 0 to 500 hours so you can see a lot of periods. It you graph for much longer (say to 500,000 hours assuming your program can handle it) you can see some long time scale features too. I would not recommend trying to graph this by hand :-)

**** When I was eyeballing it I felt "close to three" was about 2.5. You can use whatever threshold of closeness you like