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Re: Astronomical Unit (AU)



It seems I need to correct an error I made in my previous post in this
thread. In describing some of the various averages that could be taken
when evaluating the average orbital distance from the Sun I wrote one of
the averages incorrectly. It was:

...
r_5 = a*(1 - (e + (1 - e^2)*arctanh(e))/(2*E(e)))
...
Average r_5 is the average distance of the orbit from the Sun where the
average is weighted by the *arc length* of the orbital path. In this
last case the arctanh(...) function is the inverse hyperbolic tangent
function and the E(...) is the complete elliptic integral of the 2nd
kind. This is the only one of these averages that differs from a by a
term of order e. All the other averages listed only differ from a by
terms of at least order e^2.

This above formula and the discussion of is is incorrect. Apparently I
made a silly error when doing a variable substitution in an integral
needed to evaluate r_5 above. It ends up that things are much simpler in
this case. The corrected arc length weighted average for the orbital
distance from the Sun is

r_5 = a

This average (as well as r_1) gives the same value as the semi-major
axis a. Thus, none of the averages considered so far deviate from a
with terms of first order in e. All such deviations are at least 2nd
order in e.

In addition, I have since considered a couple of more values from a
couple of previously unconsidered averaging processes:

r_6 = 2*a*E(e)/[pi]

r_7 = a*(1 + (e^2)/2)

Here r_6 is the radius of a circle which has the same circumference
as the orbit's perimeter. Also r_7 is the orbital distance from the
Sun weighted by the area swept by the orbit. Because of Kepler's 2nd
law (& the conservation of angular momentum) we see that r_7 *must* be
numerically equal to r_4 which is the *time-weighted* average of the
orbital distance from the Sun.

David Bowman
David_Bowman@georgetowncollege.edu