I am working on my homebrewed world and I am going to have planetary alignments affect magic.
Two objects are traveling in a perfect circular orbit, at the same speed, one is inside the other traveling on a loop that is 1/4 the total distance of the other. If the total time to complete one circuit of the outer loop is 364 days, and the inner loop is 91 days, then how often will they be within five degrees of each other? (I know this is not exactly how our planets are.)
What if there was a path 12 times longer than the 364 days? How long would they be within five degrees?
Please show me the math involved in answering these questions.
Thanks for your help in advance.
I'm afraid that such calculations are beyond my skill set, so I apologize that I have nothing of value to add for you. However, can I just ask, and please don't take offense, are you posting your math homework in the guise of a gaming question? Because if you are, I owe you a cold one, my friend. Heck, I'll give you a cold one anyway, as anyone willing to contemplate such complex mathematical relations for the purpose of assigning mechanical effects in D&D is truly a gamer worthy of refreshment.
C'mon, you 20th level Mathemagicians, helps this guy out!
I am working on my homebrewed world and I am going to have planetary alignments affect magic.
Two objects are traveling in a perfect circular orbit, at the same speed, one is inside the other traveling on a loop that is 1/4 the total distance of the other. If the total time to complete one circuit of the outer loop is 364 days, and the inner loop is 91 days, then how often will they be within five degrees of each other? (I know this is not exactly how our planets are.)
What if there was a path 12 times longer than the 364 days? How long would they be within five degrees?
Please show me the math involved in answering these questions.
Thanks for your help in advance.
AAAAAAAAAH!!!!!!!!! MATH???!!! Here????!!!! AAAAAAAAAAAAAAAAH!!!!!!!
AAAAAAAAAH!!!!!!!!! MATH???!!! Here????!!!! AAAAAAAAAAAAAAAAH!!!!!!!
I've got news for ya, buddy, underneath all that cool sword swinging and spell slinging? Its all math. Welcome to the Matrix. Bring a slide rule.
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For the 91/364 combo, they would be within 5 degrees of each other about every 121 days, for about 3 days each occurrence.
Assume they both start at degree 0. When the 91 day orbit is complete, the 364 day orbit will be 1/4 complete (at degree 90). Knowing the 91 day orbit is 4 times faster (degree-wise), we get 0 + 4t = 90 + t, where 0 + 4t represents the position of the faster planet, and 90 + t represents the position of the slower one. They are equal at t = 30, which is 30 degrees, or in the case of the 364-day orbit, about 30 days as well (1:1 degree/day ratio, or very close). Adding this to the 91 days it took for the fast planet to complete the first revolution, we get 121. This lining up continues to occur every 120-121 days, approximately, and since the 91-day orbit consumes 4 degrees per day (360/91), I guesstimated 3 days of being withing 5 degrees of each other (day 1 -5, day 2 0, day 3 +5). If you want decimal hours, someone else can volunteer :)
The 364/12x combo occurs every 392-393 days, for about 10 days each (t - 360 = 1/12 t, the 364 day planet travels 1 degree per day, -5 to +5 is 10 days).
AAAAAAAAAH!!!!!!!!! MATH???!!! Here????!!!! AAAAAAAAAAAAAAAAH!!!!!!!I've got news for ya, buddy, underneath all that cool sword swinging and spell slinging? Its all math. Welcome to the Matrix. Bring a slide rule.
Duh, everyone knows that. Does that mean I gotta like it?
Duh, everyone knows that. Does that mean I gotta like it?
In the same sense that you have to like dungeon crawls, Vancian magic, and d12s. So, I suppose not. Still, bring a slide rule just in case. Its useful for protecting your dice from unlucky hands.
Taking that, how would you then position ancient monolithic circles based on this knowledge?
May I suggest darts and a globe? Oh, you want something GOOD, I see how it is. Well, my ancient monoliths may not be accurately placed, or monolithic, or ancient, but at least something of value was destroyed in the process! Take THAT, geography!
Seriously though, I'm dieing to see what you do with all of this. Please tell me you'll be sharing with the class?
Duh, everyone knows that. Does that mean I gotta like it?In the same sense that you have to like dungeon crawls, Vancian magic, and d12s. So, I suppose not. Still, bring a slide rule just in case. Its useful for protecting your dice from unlucky hands.
Thanks for the advice.
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For the 91/364 combo, they would be within 5 degrees of each other about every 121 days, for about 3 days each occurrence.
Assume they both start at degree 0. When the 91 day orbit is complete, the 364 day orbit will be 1/4 complete (at degree 90). Knowing the 91 day orbit is 4 times faster (degree-wise), we get 0 + 4t = 90 + t, where 0 + 4t represents the position of the faster planet, and 90 + t represents the position of the slower one. They are equal at t = 30, which is 30 degrees, or in the case of the 364-day orbit, about 30 days as well (1:1 degree/day ratio, or very close). Adding this to the 91 days it took for the fast planet to complete the first revolution, we get 121. This lining up continues to occur every 120-121 days, approximately, and since the 91-day orbit consumes 4 degrees per day (360/91), I guesstimated 3 days of being withing 5 degrees of each other (day 1 -5, day 2 0, day 3 +5). If you want decimal hours, someone else can volunteer :)
The 364/12x combo occurs every 392-393 days, for about 10 days each (t - 360 = 1/12 t, the 364 day planet travels 1 degree per day, -5 to +5 is 10 days).
ok Pygon - this one's for all the marbles - when do all 3 line up?
For the 91/364 combo, they would be within 5 degrees of each other about every 121 days, for about 3 days each occurrence.
Assume they both start at degree 0. When the 91 day orbit is complete, the 364 day orbit will be 1/4 complete (at degree 90). Knowing the 91 day orbit is 4 times faster (degree-wise), we get 0 + 4t = 90 + t, where 0 + 4t represents the position of the faster planet, and 90 + t represents the position of the slower one. They are equal at t = 30, which is 30 degrees, or in the case of the 364-day orbit, about 30 days as well (1:1 degree/day ratio, or very close). Adding this to the 91 days it took for the fast planet to complete the first revolution, we get 121. This lining up continues to occur every 120-121 days, approximately, and since the 91-day orbit consumes 4 degrees per day (360/91), I guesstimated 3 days of being withing 5 degrees of each other (day 1 -5, day 2 0, day 3 +5). If you want decimal hours, someone else can volunteer :)
The 364/12x combo occurs every 392-393 days, for about 10 days each (t - 360 = 1/12 t, the 364 day planet travels 1 degree per day, -5 to +5 is 10 days).
I am a bit surprised that more information has not been given. When IQ is posted there are many listed above 150 and quite a few personal characters have int of 17+. I guess not too many people go to the off topic discussion area. Regardless I have been doing my own work, mostly guestimating and will enter it later today, thanks.
ok Pygon - this one's for all the marbles - when do all 3 line up?
Alas, there are no circular orbits, only nearly-circular ones getting less so as you get further away from the focus, so your further object presents a bit of a pickle. Kepler's 1st law of planetary motion: objects orbit in elliptical paths, with the central point of the orbit at one focus of the ellipse. Also, the rate of movement should change around the focus ("slingshot effect"), rather than bodies orbiting at a constant speed, per Kepler's 2nd law.
Kepler's 3rd law dictates the time of one orbit, given how long the other one takes and the difference in orbital distances, so you can't even just pick those at random if you want the math to work. (Period squared is proportional to the cube of half the length of the major axis of the ellipse.)
Real-world physics makes for annoying game mechanics. The level of effort involved in making the math work out isn't worth it in this case (in my opinion), considering that the players will probably never know the difference unless they're astrophysicists. So you can probably just rule a frequency by divine fiat and be none the worse off for it.
RPG Superstar 2010 Top 16
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Alas, there are no circular orbits, only nearly-circular ones getting less so as you get further away from the focus, so your further object presents a bit of a pickle.
Actually, Kirth, the original post is okay. Circles, in this context, are a special case of ellipses (where the two foci coincide). In that case, the object they're in motion around is at that focus, and the speed around the object is stable.
As Pygon pointed out, the two inner bodies will line up again when the fastest body "laps" the middle one, going 4/3 of the way around to the middle body's 1/3. That happens every 4/3 of the faster body's year
Likewise, the outer two bodies line up when the middle body "laps" the outermost one, going 12/11 of the way around to the outer body's 1/11. That happens every 12/11 * middle body's year.
I go to the trouble of rehashing someone else's perfectly correct mathematics, because I wanted to illustrate a potentially useful generalization. Two orbiting planets, one of which has a year r times the others, will synch up every r/(r-1) times the faster planet's year. And r doesn't have to be an integer, or even positive, for this to be true.
Now, for the three-body problem, I cheated. (grin) I ran through an Excel spreadsheet, checking how many orbits each of the planets had made after a certain amount of time. For example, after 136.5 days, the fastest body had made 136.5/91 = 1.500 orbits; the middle body, 0.375 orbits; and the slowest body, 0.031. When two planets line up, they have the same fractional amounts. For example, after 464.68 days, the bodies had orbitted:
- fastest: 5.10638 orbits
- middle: 1.2765 orbits
- slowest: 0.10638 orbits
Notice that the inner and outer bodies are in synch, but the middle one is not.
So, I just looked for when all three bodies had the same remainder.
It didn't happen --they didn't even come close-- until the outer body had completed 1.00 revolution; the middle body, 12.00 revolutions; and the inner body, 48.00.
Notably, whenever they all line up, they are always occluding the same stars.
Alas, there are no circular orbits, only nearly-circular ones getting less so as you get further away from the focus, so your further object presents a bit of a pickle.Actually, Kirth, the original post is okay. Circles, in this context, are a special case of ellipses (where the two foci coincide). In that case, the object they're in motion around is at that focus, and the speed around the object is stable.
Mathematically, you're 100% correct--a circle is indeed an ellipse. I'd point out, though, that in our solar system, nearer objects to the sun have, in general, more nearly circular orbits than the further ones. We can observe this phenomenon with respect to Jupiter's moons as well (look at Io's orbit). Certainly, we could rule by divine fiat that all orbits in this campaign world are perfect circles, but that would fail to mimic observed "real world" conditions. Which is fine--no harm at all in it--but to me it then begs the question of why we want the math to work out in the first place. See where I'm coming from?
Edit: Various references on orbital mechanics, e.g., http://www.braeunig.us/space/orbmech.htm#types can give formulae to work out the eccentricity of elliptical orbits, not including attraction of larger (Jovian) planets, etc. Eccentricity is directly proportional to the periapsis distance.
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RPG Superstar 2010 Top 16
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Certainly, we could rule by divine fiat that all orbits in this campaign world are perfect circles, but that would fail to mimic observed "real world" conditions. Which is fine--no harm at all in it--but to me it then begs the question of why we want the math to work out in the first place. See where I'm coming from?
We don't need to rule that all orbits are circles, just these three. And, granting such things as active Lawful Neutral deities, and planets and moons serving as the means through which the gods influence the world, I can see three very stable, regular, lawful planets. (The axiom that they orbit in integer-multiples of each other is perhaps noteworthy here.)
And the gods of chaos have weird eccentrically-orbitting planets with invisible crystal spheres that induce all sorts of wacky tilt-a-whirl retrograde motion.
In any case, our fantasy worlds can have a different set-up, and magic may suspend a particular physical law or two, but they might still obey all the other laws of science. Elves don't really exist, but half-elves are the product of hybrid fertilization. Spellbooks aren't real, but they catch on fire just as you would expect of a mundane book. Dead wyvren fall to the ground. There's no such thing as a medusa, but you can still resist its power the same way you'd avoid a very bright light in the real world, by closing your eyes.
And planets in circular orbits obey celestial mechanics unless the gods say otherwise.
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Mathematically, you're 100% correct--a circle is indeed an ellipse. I'd point out, though, that in our solar system
Don't even need to go this far.
If you have two objects of differing mass, the "orbit" will never be circular, if you have more than two objects, the orbit will never be circular.....This is becuase object 1 does not orbit object 2 (or vice versa), both objects orbit the combined center of mass.
If one object is significantly larger than the other there may be the "appearence" of one orbiting the other, but it is only that, an appearence....
And, granting such things as active Lawful Neutral deities, and planets and moons serving as the means through which the gods influence the world, I can see three very stable, regular, lawful planets. (The axiom that they orbit in integer-multiples of each other is perhaps noteworthy here.)
And the gods of chaos have weird eccentrically-orbitting planets with invisible crystal spheres that induce all sorts of wacky tilt-a-whirl retrograde motion.
Ooh! I like that! Planets and moons would all be associated with the various deities, then. This gives astrology a whole new significance... imagine if one orbit gradually becomes less circular: "the gods of chaos are waxing stonger."
We're still ruling by divine fiat, rather than physics (there are reasons in physics why eccentric ellipses--rather than circles--predominate, particularly for further satellites), but at least it's divine fiat with in-game logic and internal consistency. Consider me on board!
If one object is significantly larger than the other there may be the "appearence" of one orbiting the other, but it is only that, an appearence....
Right indeed! But see above. Christ has given divine fiat new palatability, and for game purposes, that trumps realism any day.
Christ has given divine fiat new palatability
Chris, that is. Talk about typos! Must be all this mentioning of the divine around the holidays.
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Right indeed! But see above. Christ has given divine fiat new palatability, and for game purposes, that trumps realism any day.
Oh, hell yeah, I was just trying to say that there isn't a lot of point trying to calculate the numbers. If you have players that will go to the trouble of checking your numbers you *already have* players who will says what I posted above.....
Keep the numbers simple/significant to you, and tell them that is the way it is...
I just want to point out that Pygon is a genius.
Oh, hell yeah, I was just trying to say that there isn't a lot of point trying to calculate the numbers. If you have players that will go to the trouble of checking your numbers you *already have* players who will says what I posted above.....
Keep the numbers simple/significant to you, and tell them that is the way it is...
Yeah, that was kind of my point as well. But the thread opened up some cool ideas, so even if you and I might disagree with the others on the value of doing some of the math correctly, we all come away with a net profit for our time spent. Thanks, everyone!