Here's an interesting relevant graphic: https://twitter.com/i/status/1125023185643220992 On Wed, Dec 25, 2019 at 6:03 PM Bill Gosper <billgosper@gmail.com> wrote:
HGB> What about time around an elliptical orbit? Any closed form solutions
for -- e.g., approximations to #days from solstice?
Yes! How could we possibly have gone this long without a universally recognized Kepler(a,b,t) function? —Bill
The function is ridiculously simple, but I *never* could have found it
without crucial help from Julian and NeilB. Let
lips[t_, e_] := {e + Cos[t], Sqrt[1 - e^2] Sin[t]}
be an ellipse of eccentricity e with unit semimajor axis, one focus at the origin, center at e, and other focus at 2e. If t is proportional to time, this is *not* Keplerian (= constant swept area per unit time = constant angular momentum) motion. The parameter t needs to be fudged by a pacing function, which is merely
fuj[t_,e_] := InverseFunction[e Sin[#] + # &]@t Then kepler[t_,e_]:={e + Cos[#], Sqrt[1 - e^2] Sin[#]}&@fuj[t,e] . Proof: In[40]:= Area[r kepler[t, eps], {r, 0, 1}, {t, t0, t0 + duration}]
Out[40]= 1/2 duration Sqrt[1 - eps^2]
I.e., the area of the swept elliptical sector is independent of t0. (The actual area needs to be scaled by the square of the actual semimajor axis, which we called "1". One "year" is a duration of 2π.) Area is a fancy Mathematica function that, in this case, correctly guesses that its first argument is a pair parametric rectangular coordinates.
Here are the 52 weeks of a 364 day year <http://gosper.org/keplerwks.png> (with eccentricity exaggerated to √½) (This discussion applies to hyperbolic and parabolic trajectories as well.) http://www.tweedledum.com/rwg/pizza.htm makes a fuss over numerical methods for InverseFunction[e Sin[#] + # &], but Mathematica is *so* seductive. Now I can easily realistically animate orbiting planets. Modulo Einstein. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com