In article <email@example.com>, Rob Johnson <firstname.lastname@example.org> wrote: >In article <email@example.com>, >Cary <firstname.lastname@example.org> wrote: >>On Fri, 12 Feb 2010 10:22:33 +0000 (UTC), email@example.com (Rob >>Johnson) wrote: >> >>>In article <firstname.lastname@example.org>, >>>David W. Cantrell <DWCantrell@sigmaxi.net> wrote: >>>>email@example.com (Rob Johnson) wrote: >>>>> In article >>>>> <firstname.lastname@example.org>, G >>>>> Patel <email@example.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 semi-minor 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 arc-length, 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 semi-minor axis.
Rob Johnson <firstname.lastname@example.org> take out the trash before replying to view any ASCII art, display article in a monospaced font