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Math question-any experts?
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Feb 20, 2014 11:45AM
I'm writing a novel in which twin planets of almost equal size (approx Earth diam) orbit each other. The orbit can be circular. I wanted the two planets to keep the same face to each other but with an orbital period of ideally about 24 hours. According to my math, this is impossible; the core of each planet would have to be about 111.5 km from the other, meaning that both would disintegrate. The formula I used was P-squared (period)=(4pi-squared/G(M1+M2))*a-cubed. G is the gravitational constant, M is the mass of the planet, a is the distance from core to core. I rearranged the equation to find a. Any math experts in the group? And can you figure orbital mechanics? My calcs say this scenario is impossible.
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I'm not an expert by any means but I don't understand why you're getting a result which isn't in the same ballpark as geostationary orbits. Did you mess up the units (by using hours instead of seconds or something)?
Outis wrote: "I'm not an expert by any means but I don't understand why you're getting a result which isn't in the same ballpark as geostationary orbits. Did you mess up the units (by using hours instead of seco..."
The equation I found used "year" as a unit of measure for "period." I calculated using 1/365 squared. I'm wondering myself if I used a wrong unit, but I don't think so. A geostationary orbit around a baricenter, a point equidistant from both planets, would likely take more than 24 hours, and that's probably why I can't get it to work using the 24-hour constant. Keep in mind that our own moon takes a month to complete an orbit.
The equation I found used "year" as a unit of measure for "period." I calculated using 1/365 squared. I'm wondering myself if I used a wrong unit, but I don't think so. A geostationary orbit around a baricenter, a point equidistant from both planets, would likely take more than 24 hours, and that's probably why I can't get it to work using the 24-hour constant. Keep in mind that our own moon takes a month to complete an orbit.
You normally use seconds in these types of equations. The regular gravitational constant is used to calculate accelerations in meter/second^2 units. Not meter/year^2.Try it and see if it works. Sure, the distance won't be the same as with geostationary satellites but the order of magnitude should be right. 115 km is way, way too small.
Substituting seconds instead of year gives a realistic figure of about 626,356 km, but the equation did specify year, and that's what I'm not sure about. They would orbit a barycenter of about half that distance, but could they do it in 24 hours? THIS IS WRONG; THE EQUATION ALSO SPECIFIED THE DISTANCE IN TERMS OF AUs.
The units in any equation have to check out.Unless the equation includes a constant designed to correct for mismatched units (like /60 to match minutes with seconds), if you have a unit like seconds on the one side, you must have the same unit on the other.
626,356 km seems too large. But that seems plausible. Maybe my guess was off. I can't be bothered to check it.
I think what you have to make sure you get right if you don't want the setting to appear obviously wrong to readers who don't bother to check calculations is tides, seasons and stuff.
Ideal circular orbits with no tilts would make all that simple but would that seem plausible as a natural occurrence?
If I don't mention it as an ideal circular orbit, and the figures are in the ball park, I don't think anyone would complain, even if they took the trouble to calculate it. As long as it isn't grossly and ridiculously wrong, I think it would be okay. But I'm not a math wizard, though I do understand algebra, so I don't really trust my own calcs in something as complex as this appears to be. Failing this, I will probably just have the "twin planets" orbiting the star instead of each other, adjacent to each other, and both within the zone that allows life to flourish.
I'm not sure what you mean by "adjacent" but there may be more potential for obvious wrongness there. Over an astronomical timescale, there are signficant gravitational interactions even between the distant bodies of the solar system.
Outis wrote: "I'm not sure what you mean by "adjacent" but there may be more potential for obvious wrongness there. Over an astronomical timescale, there are signficant gravitational interactions even between th..."
Exactly, and close enough to affect each other. Planet A (outer) would lag slightly behind Planet B (inner) over time, perhaps pull Planet B into a higher orbit as B pulls A into a lower orbit, and they can swap orbits from time to time. Possible? Or really out there?
Exactly, and close enough to affect each other. Planet A (outer) would lag slightly behind Planet B (inner) over time, perhaps pull Planet B into a higher orbit as B pulls A into a lower orbit, and they can swap orbits from time to time. Possible? Or really out there?
Found the solution, and you can access any two-body calculation you want on this link: http://www.1728.org/kepler3a.htm
For my problem I solved for orbital radius, where the sum of the two masses is 12.2064 X 10exp24 kg, and period is 86,400 seconds, which is 24 hours. It gives the answer in 5 different units, the smallest of which is meters. The answer?--about a 1/2 meter for the orbital radius, and clearly impossible. You cannot have two bodies the approximate size of Earth orbiting each other in 24 hours. THIS IS WRONG; I MISPLACED THE DECIMAL.
For my problem I solved for orbital radius, where the sum of the two masses is 12.2064 X 10exp24 kg, and period is 86,400 seconds, which is 24 hours. It gives the answer in 5 different units, the smallest of which is meters. The answer?--about a 1/2 meter for the orbital radius, and clearly impossible. You cannot have two bodies the approximate size of Earth orbiting each other in 24 hours. THIS IS WRONG; I MISPLACED THE DECIMAL.
With such an equation, doubling one of the inputs is obviously not going to affect to affect the output so much!This calculator outputs 53'000 km. I didn't check the calculation but that was never the tricky part anyway.
The planets which swap orbits sounds like it'd be highly unstable on large timescales due to the likelyhood of collision or ejection into different orbits.
And if something is that unstable, it's not plausible as a natural occurence in a mature system.
Outis wrote: "With such an equation, doubling one of the inputs is obviously not going to affect to affect the output so much!
This calculator outputs 53'000 km. I didn't check the calculation but that was never..."
I didn't recheck, but I think swapping orbits is a common occurrence among some of Jupiter's or Saturn's satellites.
This calculator outputs 53'000 km. I didn't check the calculation but that was never..."
I didn't recheck, but I think swapping orbits is a common occurrence among some of Jupiter's or Saturn's satellites.
Outis wrote: "With such an equation, doubling one of the inputs is obviously not going to affect to affect the output so much!
This calculator outputs 53'000 km. I didn't check the calculation but that was never..."
I agree with that calculation, and with both being outside the point at which they would interact enough for tidal forces to cause them to break up (Roche limit). Maybe it would work after all, but I'm still calculating. Next problem is to concoct a scenario for how they got that way...
This calculator outputs 53'000 km. I didn't check the calculation but that was never..."
I agree with that calculation, and with both being outside the point at which they would interact enough for tidal forces to cause them to break up (Roche limit). Maybe it would work after all, but I'm still calculating. Next problem is to concoct a scenario for how they got that way...
All,I see nothing wrong with the formula. It can be simply derived for circular orbits because the gravitational force has to equal the centrifugal force. The m1 + m2 appears, and not the usual product m1*m2, as in the gravitational force, because the reduced mass must be used in the centrifugal force term to reduce the two-body problem to an equivalent point-mass problem. For the geosat problem, the mass of the satellite is ignorable, so m1 + m2 is just the mass of the Earth, but the formula is simple then: P = 2*pi*sqrt(a/g), where g is the "local acceleration of gravity."
Many binary star systems have this kind of dynamics.
Everything in the equation has units, and they have to agree. The units for G are always such that Gm1m2/R**2 is force, dynes or newtons.
Wow! I haven't done this stuff for a while. Nostalgia trip! Thanks for the memories.
r/Steve
