I have been formulating the motions of two bodies that travel in elliptical orbits due to the effects of each other's gravity and have come up with some interesting results. I originally posted this in The Grand Puzzle but got no response about this particular part of it so I decided to start a whole new thread. I am wondering if these observations are already known or if I may have discovered something new. I have looked through some websites to find out but have only found close approximations to the actual values. Some sites even say that they are not completely accurate.
Kepler's third law of motion states 4pi2a3=GMT2 for a body in orbit around another, so if M is the mass of the sun and (a) is the semi-major axis of a planet's orbit, then the period of its revolution (T) can be found with this formula. In this case, since GM/4pi2 is a constant for the sun, then a3/T2 is a constant for all planets that orbit it. Now, it is indeed already known that this law isn't quite accurate and a large error is found for larger planets, but I have yet to see the accurate version of this law. Another version says that for larger planets, one can use the sum of the masses in place of the mass of the sun to gain a more accurate result. This is as far as I have seen it taken. The actual formulas are so simple that I would have trouble believing that I am the first to come up with them. But if this is the case, why haven't I seen it and why are only the approximations available? Could the reason noone has responded to the original post be that nobody has found anything either?
The precise formulas which I have found that directly relate to Kepler's third law of motion are simply 4pi2(a1+a2)3=G(M1+M2)T2, pi2(Df+Dc)3=2G(M1+M2)T2, 4pi2a13=GM23T2/(M1+M2)2, and 4pi2a23=GM13T2/(M1+M2)2. More can be found in the original post.
If I get no responses here, should I post a fuller version in another part of the forum (perhaps astronomy)?Originally Posted by The Grand Puzzle
I am still working on the binary problem and I am close to the results, but I need other decaying binary systems to compare it to. I have looked for some but have not found any with results quite like the one you gave me. It would have be be two bodies only that orbit in a single plane with the semi-axis and rate of decay of the semi-axis over time which are both expressed in units of distance. Can you recommend any specific sites?