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Re: ellipse, mean distance from point on perimeter to a focus ?
Posted:
Feb 12, 2010 5:56 PM


In article <20100212.111907@whim.org>, Rob Johnson <rob@trash.whim.org> wrote: >In article <ut8bn5hhqceuqnjnnmk261gcrc14setco2@4ax.com>, >Cary <cary@domain.invalid> wrote: >>On Fri, 12 Feb 2010 10:22:33 +0000 (UTC), rob@trash.whim.org (Rob >>Johnson) wrote: >> >>>In article <20100211111323.200$wv@newsreader.com>, >>>David W. Cantrell <DWCantrell@sigmaxi.net> wrote: >>>>rob@trash.whim.org (Rob Johnson) wrote: >>>>> In article >>>>> <a36f3aaf77d5499bbd30f603613dcc8d@j6g2000vbd.googlegroups.com>, G >>>>> Patel <gaya.patel@gmail.com> wrote: >>>>> >Is the mean distance from a point on perimeter to a focus equal to the >>>>> >semi major axis length? >>>>> >>>>> Yes. >>[snip] >> >>>>But I initially (before seeing Achava's post) calculated the mean distance >>>>with respect to theta, thinking of the ellipse in polar coordinates as >>>>given by >>>> >>>>r = a (1  e^2) / (1  e cos(theta)) >>>> >>>>Doing that, we find instead >>>> >>>>mean distance = a sqrt(1  e^2) = b, the semiminor axis length. >>> >>>I get that as well. Furthermore, averaging with respect to time, >>>using equal area in equal time, I get a mean of a(1+e^2/2). >> >> >>This paper may be of interest to those following this thread. >><http://faculty.matcmadison.edu/alehnen/kepler/kepler.htm> > >Thanks for the reference; that paper goes the distance. It covers >the means that David Cantrell (True Anomaly) and I (Time) discussed. >I note that they also discuss average over arclength, and since arc >length is symmetric with respect to the foci, that average is a. > >It may not be obvious at first, but eccentric anomaly is also >symmetric with respect to the foci. The eccentric anomaly is the >angle from the center of the ellipse after the ellipse has been >scaled in the direction of either the major or the minor axis to a >circle. Thus, the mean with respect to the eccentric anomaly is >also a.
Another mean with possible physical meaning is the mean distance weighted by the amount of energy received from the star. The intensity of the energy is k/r^2 while the time spent receiving that energy is 1/2 r^2 d(theta). Thus, the average distance weighted by energy would be identical to the average distance with respect to the True Anomaly, which, as David has pointed out, is the semiminor axis.
Rob Johnson <rob@trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font



