[math-fun] (Corollary of) Kepler's 3rd Law:
The orbital period of a planet depends only on its major axis. gosper.org/4planets.gif . Does anybody feel like proving that with this phasing, these planets are collinear only twice per period? —rwg
Exercise: explain why right-hand planet reverses at perihelion, rather than proceeding to terminal collision with the sun. WFL On 12/26/19, Bill Gosper <billgosper@gmail.com> wrote:
The orbital period of a planet depends only on its major axis. gosper.org/4planets.gif . Does anybody feel like proving that with this phasing, these planets are collinear only twice per period? —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Here are 9 co-periodic planets <http://gosper.org/9planets.gif>. Shades of Shoemaker Levy 9. The string appears to periodically develop an inflection. —rwg I always thought that orbits with unit eccentricity were unbounded parabolas. Counterexample(?): Radial trajectories. On Thu, Dec 26, 2019 at 3:03 AM Bill Gosper <billgosper@gmail.com> wrote:
The orbital period of a planet depends only on its major axis. gosper.org/4planets.gif . Does anybody feel like proving that with this phasing, these planets are collinear only twice per period? —rwg
Well, your "bouncing" planet only occurrs w/ point-like planets and sun which can pass through each other, so it's understandable that that is not normally considered. Any eccentricity zero orbit w/ non zero perihelion we'll be an escaping parabolic path. On Thu, Dec 26, 2019, 13:21 Bill Gosper <billgosper@gmail.com> wrote:
Here are 9 co-periodic planets <http://gosper.org/9planets.gif>. Shades of Shoemaker Levy 9. The string appears to periodically develop an inflection. —rwg I always thought that orbits with unit eccentricity were unbounded parabolas. Counterexample(?): Radial trajectories.
On Thu, Dec 26, 2019 at 3:03 AM Bill Gosper <billgosper@gmail.com> wrote:
The orbital period of a planet depends only on its major axis. gosper.org/4planets.gif . Does anybody feel like proving that with this phasing, these planets are collinear only twice per period? —rwg
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Speaking of unit eccentricity, consider the idle question: A sine wave rotated +π/4 <http://gosper.org/tiltedsine.png> is the graph of what function? Evidently, the pacefudging function that describes Kepler motion along radial trajectories. The idle question leads to impacting comets. But notice my sinewaves are slightly asymmetrical. I don't know why. I wish I could blame Einstein. —rwg On Thu, Dec 26, 2019 at 11:20 AM Bill Gosper <billgosper@gmail.com> wrote:
Here are 9 co-periodic planets <http://gosper.org/9planets.gif>. Shades of Shoemaker Levy 9. The string appears to periodically develop an inflection. —rwg I always thought that orbits with unit eccentricity were unbounded parabolas. Counterexample(?): Radial trajectories.
On Thu, Dec 26, 2019 at 3:03 AM Bill Gosper <billgosper@gmail.com> wrote:
The orbital period of a planet depends only on its major axis. gosper.org/4planets.gif . Does anybody feel like proving that with this phasing, these planets are collinear only twice per period? —rwg
On Fri, Dec 27, 2019 at 11:25 PM Bill Gosper <billgosper@gmail.com> wrote:
Speaking of unit eccentricity, consider the idle question: A sine wave rotated +π/4 <http://gosper.org/tiltedsine.png>
BULL! A tilted sine wave looks like the 2nd plot in tilted sin <http://gosper.org/tiltedbs.pdf> . I have no idea what the upper plot is. —rwg
is the graph of what function? Evidently, the pacefudging function that describes Kepler motion along radial trajectories. The idle question leads to impacting comets. But notice my sinewaves are slightly asymmetrical. I don't know why. I wish I could blame Einstein. —rwg
On Thu, Dec 26, 2019 at 11:20 AM Bill Gosper <billgosper@gmail.com> wrote:
Here are 9 co-periodic planets <http://gosper.org/9planets.gif>. Shades of Shoemaker Levy 9. The string appears to periodically develop an inflection. —rwg I always thought that orbits with unit eccentricity were unbounded parabolas. Counterexample(?): Radial trajectories.
On Thu, Dec 26, 2019 at 3:03 AM Bill Gosper <billgosper@gmail.com> wrote:
The orbital period of a planet depends only on its major axis. gosper.org/4planets.gif . Does anybody feel like proving that with this phasing, these planets are collinear only twice per period? —rwg
participants (3)
-
Bill Gosper -
Fred Lunnon -
William R Somsky