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Old 10-18-2017, 03:38 PM   #11
a humble lich
 
Join Date: Jun 2017
Default Re: [Space] Triple Full Moons

Putting some numbers in, the moons are so close, I think they could be visible. You're right that they are inside the Roche limit, although for the farther moons they are close to the limit, so they are probably small captured asteroids. If the closest one is the size of Phobos (i.e. about 20 km) and Lerrom is about earth-sized, then Moon A will subtend an arc of about 0.18 degrees. For comparison, our moon subtends 0.5 degrees, so Moon A would be a little smaller than half the size of our moon.

The Roche limit also depends on the density of the satellite, so if these moons are captured asteroids with a high iron content, they could be bigger.

I'm wondering what tides are like. If the moons are tiny there won't be much tides, I calculate for the case of Phobos being Moon A the tidal acceleration from Moon A would be about 1% of what Earth gets from our moon. However, if the moons are larger and denser, that would go up.
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Old 10-18-2017, 03:55 PM   #12
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Default Re: [Space] Triple Full Moons

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Originally Posted by a humble lich View Post
I'm wondering what tides are like. If the moons are tiny there won't be much tides, I calculate for the case of Phobos being Moon A the tidal acceleration from Moon A would be about 1% of what Earth gets from our moon. However, if the moons are larger and denser, that would go up.
You still have a solar tide. That's about 1/3 the lunar one here on Earth. For the moons, it's likely totally negligible. Not only are they small, but most of the effect of tides here on Earth are a result of oscillations *pumped* by the tidal forces, not directly responding to them at that moment. But these things are so close in the speed the input changes faster than anything it was driving can propagate across any particular tide basin - the speed of sound in water isn't high enough to outrace the moving moon...
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Old 10-18-2017, 03:56 PM   #13
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Default Re: [Space] Triple Full Moons

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Originally Posted by malloyd View Post
If you adopt that definition, then for all of them to be full the sum of three cosines function would seem to need to be greater than 2.983 (technically some fraction of the time it was might still not qualify, but probably close enough).

Of course if the effect lasts until the end of the night once it happens, instead of never more than 3.9 minutes - the time it takes for Moon A to move through 1/30th of its phase cycle, that's even more complication. Something closer to 1/4 of the time than the 1/27000th you'd get for just the period of phase match.
I was using a much more generous definition of what counts as full (I have always considered the full moon to last three days). By using a threshold of 2.983, that puts the time between all three moons being full to about once per 200 hours. In addition, there is only a period of about 10 earth years where you get all three moons being full. After that, there is a stretch of about 14 years where it never happens.
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Old 10-18-2017, 04:09 PM   #14
malloyd
 
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Default Re: [Space] Triple Full Moons

One side issue maybe worth thinking about is that if it's a fantasy planet, there's no particular reason for the phases of the moon(s) or the tides to be related to their positions at all. They could both perfectly well be driven by magical or divine forces rather than celestial geometry.
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Old 10-18-2017, 07:23 PM   #15
Jinumon
 
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Default Re: [Space] Triple Full Moons

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Originally Posted by a humble lich View Post
The perpendicular component of each moon's displacement is given by x= A cos(2 pi t/T +phi). We could solve for where those all intersect, but that will get messy because you have to define how close they need to get to count as an intersection. Simpler, just look at the function x=cos(2 pi t/T_1) + cos(2 pi t/T_2) + cos(2 pi t/T_3). When this function approaches 3, then all three moons are aligned behind the planet. We will ignore the phases because they hold the information of when the next full moon will occur and we are only interested in the frequency of full moons.
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.

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Old 10-18-2017, 10:10 PM   #16
a humble lich
 
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Default Re: [Space] Triple Full Moons

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Originally Posted by Jinumon View Post
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
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Old 10-19-2017, 12:11 AM   #17
Jinumon
 
Join Date: Mar 2013
Default Re: [Space] Triple Full Moons

Thank you SO MUCH lich.

I was able to graph the function using some more exact figures and intersected it with a flat line indicating about 99% combined luminosity, resulting in about 1 in every 3 nights. Fun part is that it seems to be largely at random, so I think I'm gonna do it every 1d nights, or on a roll of 5 or 6 on 1d. Should make for a very interesting game.

Jinumon

PS: I went as far as College Algebra and dabbled with some Trig in High School, but it's been a fair number of years.
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Old 10-19-2017, 01:43 PM   #18
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Default Re: [Space] Triple Full Moons

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Originally Posted by Jinumon View Post
  • Moon A, orbiting Lerrom every ~0.08138 days at 1.25 Earth Radii (neg mass)
  • Moon B, orbiting Lerrom every ~0.19845 days at 2.25 Earth Radii (neg mass)
  • Moon C, orbiting Lerrom every ~0.23243 days at 2.5 Earth Radii (neg mass)
Quote:
Originally Posted by a humble lich View Post
Putting some numbers in, the moons are so close, I think they could be visible.
I haven't done the trig, but if they're that close, wouldn't they be in the planet's umbra most of the time? They might need a steep inclination to still give full moons. However, having two suns illuminating them helps here, reducing the region of shadow somewhat.
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Old 10-19-2017, 05:06 PM   #19
a humble lich
 
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Default Re: [Space] Triple Full Moons

You are right, with the moons that close there is a good chance that every full moon is actually a lunar eclipse. I don't know what that means for werewolves. It could be that a werewolf changes not because the moons are full, but because all the moons turn to blood; i.e., when you have a triple lunar eclipse.
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Old 10-19-2017, 05:50 PM   #20
Anthony
 
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Default Re: [Space] Triple Full Moons

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Originally Posted by a humble lich View Post
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.
While this is correct, why on earth are you bothering? You don't need to know x and y -- the moon is full when its angle is opposite the sun. If we define towards the sun as an angle of zero, that means the decimal portion of ( t/T + phi/2pi ) is, say, between 0.45 and 0.55 (equivalent to a 3 day full moon).

We can even account for solar motion. The inner moon undergoes 4488.2 sidereal months per year; it will undergo either 4489.2 or 4487.2 solar months per year, depending on the direction of its orbit (4489.2 would be considered 'standard') and thus its solar month will be 0.08136 or 0.08140 days. Same math applies to the other moons.
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