Wikipedia:Reference desk/Archives/Mathematics/2015 August 11

= August 11 =

Parabolic or elliptical orbit
The article Lexell's Comet reads: "A number of orbital calculations were made, some indicating a perihelion date [...] of August 9–10, and some a date of 13–14, depending on whether the orbital solutions were parabolic or elliptical.". But that seems strange, given that a parabola is just the borderline case of an ellipse. Wouldn't there be elliptical solutions all over the place, including for August 11-12? &mdash; Sebastian 19:14, 11 August 2015 (UTC)
 * The probability of an actual parabolic orbit is 0, leaving possible trajectories as elliptical or hyperbolic. But assuming a parabola is useful in some situations since its the limiting case of thin ellipses. The most familiar example is that we assume that an object thrown near the ground follows a parabolic arc, when in reality it follows an elliptical orbit. The difference between the ellipse in this case and a parabola is negligible and the math is a lot easier for a parabola. For a comet, you assume it started some very large distance from the sun and follows an orbit so long and thin that for some computations you can take it be be a parabola. If there is enough data that it can't be made to fit a parabola then an ellipse can be computed, which I assume is how they get the orbital period mentioned in the article. --RDBury (talk) 15:14, 12 August 2015 (UTC)


 * In orbital mechanics, a parabolic trajectory specifically means an escape trajectory - one that does not form a closed orbit. While it's true that in the mathematical sense, the parabola is the extreme case of an ellipse, in the physical sense, it models a totally different situation. If the comet was on a parabolic orbit, its origin was outside the solar system - if it was on an elliptical orbit, its origin was within the Solar System. People back then mostly assumed that comets came from outside the solar system (since it didn't make sense that the solar system, which was so neat and tidy, with all the planets on nearly-circular orbits in the same direction on the same orbital plane, also had big chunks of rock flying in wildly eccentric orbits just willy-nilly), and did all their sums assuming a parabolic trajectory. When they found that these didn't give the right answer, they then tried elliptical orbits, fitting the parameters to the comet's observed trajectory, and also took into account a previous encounter with Jupiter. See A History of Physical Theories of Comets, From Aristotle to Whipple. Smurrayinchester 08:01, 13 August 2015 (UTC)
 * Is your source on that reliable? An open orbit which starts outside the solar system (at what can be considered infinity) can be either a parabola (if it started with velocity 0) or a hyperbola (starting with any nonzero velocity). An exact parabola is of course much rarer, and I highly doubt that people would say "parabolic trajectory" when they really mean "parabolic or hyperbolic trajectory", or alternatively, that they would just assume the object's (relative) velocity was exactly 0 when it was outside the solar system. If people tried to fit a hyperbolic model, they would immediately figure out that it's actually an ellipse. -- Meni Rosenfeld (talk) 08:04, 14 August 2015 (UTC)
 * I was wondering about that myself. According to the book in the link above, a comet on a hyperbolic trajectory is moving so fast that by the time you've made enough observations to differentiate it from a parabola the comet is no longer visible. Which might have seemed likely enough in Lexell's time but I think nowadays the consensus is that comets start out in the far distant Oort cloud and are knocked toward the sun by some passing star. In the Oort cloud theory a near parabolic orbit would be more likely than not. --RDBury (talk) 10:32, 14 August 2015 (UTC)