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Topic: ellipse, mean distance from point on perimeter to a focus ?
Replies: 8   Last Post: Feb 12, 2010 5:56 PM

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Rob Johnson

Posts: 1,771
Registered: 12/6/04
Re: ellipse, mean distance from point on perimeter to a focus ?
Posted: Feb 12, 2010 5:56 PM
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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
>>>>> <a36f3aaf-77d5-499b-bd30-f603613dcc8d@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 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 <rob@trash.whim.org>
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