A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Robin Stuart
Date: 2021 Nov 5, 07:29 -0700
I haven't seen any responses to Frank's puzzle from July http://fer3.com/arc/m2.aspx/Average-Sun-SD-FrankReed-jul-2021-g50933
It would be a pity if this interesting question, which has quite a simple closed form solution, were to fade away and be forgotten.
a semi-major axis of the orbit
e orbital eccentricity
s Sun’s radius
Then in the small angle approximation sin-1 s/a = s/a the average of the maximum and minimum semidiameters is s / ( a ( 1 - e2 ) ). The time averaged value is smaller being just s / a.
Frank asked "For how many days out of the year is the Sun larger than the simple average".
This is equivalent to asking what proportion of the orbital period does the the Earth spend at a distance less than a and can be shown to be 1/2 - e/π which is a nice simple result. My derivation is attached for anyone who may be interested,