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-   -   [Space] Triple Full Moons (http://forums.sjgames.com/showthread.php?t=152334)

Jinumon 10-18-2017 01:01 AM

[Space] Triple Full Moons
 
Hey All,

I'm (finally) about to get my Fantasy game off the ground and one of my players is playing a werewolf. Trouble is, my Fantasy world has 3 moons (technically moonlets), and I've decided (as to not make the game too disruptive) to only force a transformation on a simultaneous triple full moon. This shouldn't happen too infrequently as each moon goes through a full cycle every 2 - 6 hours (depending on the moon), but every calculation I come up with seems way to frequent to be right. I've found online calculations for planets with 2 moons, but none with 3, and I don't know how it's further complicating things. Any and all Mathematics (Applied) (IQ/H) or Astronomy (IQ/H) assistance would be greatly appreciated. Here are the figures I'm currently working with:
  • 2 Suns, orbiting each other at 0.4AU every ~64 days with 0 Eccentricity, Fantasy planet orbits the pair.
  • The Fantasy Planet, called Lerrom. Earthlike, orbiting the Suns every ~510 24-hour days at 1.6 AU (from system center).
  • 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)

I understand that the Suns (or their combined center of mass) and all three moons must be approximately aligned to all show full, but for the life of me can't figure out how to calculate it. Ya'll are awesome. 'Preciate ya!
Jinumon

Daigoro 10-18-2017 03:22 AM

Re: [Space] Triple Full Moons
 
A full moon is when you have the Sun, Moon and Earth in linear conjunction. If you have 2 suns, then each moon will be in conjunction at 2 points in its orbit. You'll have to decide if moons are "full" when they're in conjunction with one or other of the suns or at the midpoint between these, which would be the brightest, assuming equal intensity of the suns.

However, when Lerrom is at the point in its orbit where it's perpendicular to the axis between the 2 suns, they'll be about 15 degrees apart in the sky. This will happen twice in their 64 day orbit, plus an offset due to the planet's movement around its orbit, so a bit more than every 32 days.

As the moons are orbiting the planet 4 to 12 times a day, just go for the solar cycle as being the werewolf's cycle, about every 40 days or so.

When the suns rotate their perpendicular away from the planet, their relative brightness will drop, until the lowest point when the planet and 2 suns are in conjunction.

malloyd 10-18-2017 03:36 AM

Re: [Space] Triple Full Moons
 
Quote:

Originally Posted by Jinumon (Post 2129144)

I understand that the Suns (or their combined center of mass) and all three moons must be approximately aligned to all show full, but for the life of me can't figure out how to calculate it. Ya'll are awesome. 'Preciate ya!
Jinumon

Actually it's worse than that. The suns don't need to be aligned, and in fact the less aligned they are the longer the "full" moons last - if they are at maximum separation there is a 14 degree slice of sky the moons can be in in which they will be fully lit viewed from the planet.

With only one sun I suppose you need to calculate the solar (rather than sidereal) periods of the moons, and find the integer solutions to
p1 x - p2 y = 0
p2 y - p3 z = 0
p3 z - p1 x = 0

These kinds of problems are called Diophantine equations, and are not simple to solve.

Daigoro 10-18-2017 04:36 AM

Re: [Space] Triple Full Moons
 
Quote:

Originally Posted by malloyd (Post 2129153)
the integer solutions to

Speaking of which, moons often fall into harmonic orbits of simple integral ratios. You could set them to a 2:5:8 resonance or something, meaning it'd be easy to calculate how often they're all full.

a humble lich 10-18-2017 06:54 AM

Re: [Space] Triple Full Moons
 
Quote:

Originally Posted by malloyd (Post 2129153)
Actually it's worse than that. The suns don't need to be aligned, and in fact the less aligned they are the longer the "full" moons last - if they are at maximum separation there is a 14 degree slice of sky the moons can be in in which they will be fully lit viewed from the planet.

With only one sun I suppose you need to calculate the solar (rather than sidereal) periods of the moons, and find the integer solutions to
p1 x - p2 y = 0
p2 y - p3 z = 0
p3 z - p1 x = 0

These kinds of problems are called Diophantine equations, and are not simple to solve.


I don't think it is quite as bad as this. It doesn't make sense to make it a Diophantine equation because the period of the moons are so short. The period of one of the moons is only about 2 hours. With those sorts of periods each full moon is only going to last a few minutes, depending on what is considered "full."

Also, because the period of the moons is much shorter than the orbital period, the difference between the solar and sidereal periods should be negligible.

Looking at it numerically, depending on what you mean by a full moon, it looks like to me you will have a full moon about every 33.2 hours, and each full moon lasts about 20 minutes.

For details....

This ignores the difference between the solar and sidereal periods because they should be small. Also, it ignores any effect of two suns because from the point of view of the planet the two suns remain fairly close together. To first approximation, the motion of the suns shouldn't effect the frequency of the full moon too much. 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.

When we graph that function, the result is a messy squiggle (technical term) but it gets close to 3 (i.e. all three moon are aligned) about once every 33 hours. Sometimes the alignment is close enough so that you'll get a second or even third full moon with only a few (about 4) hours difference.

Apollonian 10-18-2017 11:52 AM

Re: [Space] Triple Full Moons
 
Personally, I'd duck the question and have only one moon matter for... reasons. (Maybe each sort of lycanthrope is the result of the mating of one sun god and one moon god, so they're only forced to change when the gods "mate" again.)

Jinumon 10-18-2017 01:31 PM

Re: [Space] Triple Full Moons
 
Quote:

Originally Posted by a humble lich (Post 2129172)
Looking at it numerically, depending on what you mean by a full moon, it looks like to me you will have a full moon about every 33.2 hours, and each full moon lasts about 20 minutes.

Thanks a ton, lich, and everyone else here. Considering the moons are visible for an average of 10.25 hours each 24 hour day, I'll multiply by 24/10.25 to find the frequency of a triple full moon visible at night. Gives me once every 3.24 days. He was wanting something more frequent than once every month anyway. Not every night, but pretty frequently.

As a follow-up question, assuming his transformation lasts from the triple full moon and for the remainder of the night, what would the frequency classification be for a Disadvantageous Alternate Form? A regular full moon is Rare (from GURPS Horror).

Jinumon

malloyd 10-18-2017 02:52 PM

Re: [Space] Triple Full Moons
 
Quote:

Originally Posted by Jinumon (Post 2129250)
Thanks a ton, lich, and everyone else here. Considering the moons are visible for an average of 10.25 hours each 24 hour day, I'll multiply by 24/10.25 to find the frequency of a triple full moon visible at night. Gives me once every 3.24 days. He was wanting something more frequent than once every month anyway. Not every night, but pretty frequently.

All full moons happen at night. By definition the moon is on the opposite side of the planet as the sun, it's night anywhere you can see it. Note that with this short period, the phases of the moon go through there entire cycle several times over the course of any given night - moon A is full *six times* every night, moon C is twice. The real frequency question is how much of the time they count as full so those might happen to overlap.

On the other hand, I'm not sure the phases are *visible*. Assuming Lerrom is earth-like, all these moons appear to be inside the Roche limit, so they're necessarily tiny. None of them are likely to show a visible disk, they're just points of light that vary in brightness.

whswhs 10-18-2017 03:06 PM

Re: [Space] Triple Full Moons
 
Quote:

Originally Posted by malloyd (Post 2129259)
All full moons happen at night. By definition the moon is on the opposite side of the planet as the sun, it's night anywhere you can see it. Note that with this short period, the phases of the moon go through there entire cycle several times over the course of any given night - moon A is full *six times* every night, moon C is twice. The real frequency question is how much of the time they count as full so those might happen to overlap.

I think we generally consider our moon to be "full" about one day a month. As an approximation, that says "no more than 6 of arc away from opposition." You could say that each of the three moons spends 1/30 of each day being full. That's 48 minutes per day, split up into some number of shorter intervals.

malloyd 10-18-2017 03:34 PM

Re: [Space] Triple Full Moons
 
Quote:

Originally Posted by whswhs (Post 2129262)
I think we generally consider our moon to be "full" about one day a month. As an approximation, that says "no more than 6 of arc away from opposition." You could say that each of the three moons spends 1/30 of each day being full. That's 48 minutes per day, split up into some number of shorter intervals.

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.


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