The Sun: Advanced math isn't needed to find the solar constant, i.e. the mean intensity of solar radiation received on Earth per unit area per unit time).
For many stars, the solar constant S can be computed if its angular diameter is known. If the angular radius of a star is: a =R/r ( a is measured in radians) with r the distance to Earth and R the star’s linear radius then:
p F = S (r/R)2
If the Sun’s angular radius is 959.63 arcsec then the solar constant S can be obtained and without the use of calculus. The method which follows shows how this is done.
We re-arrange the earlier equation to find:
S = π F/ (r/R)2
Where we already know the solar flux (e.g. from an astrophysical data source, e.g. Astrophysical Concepts by Harwit) is:
π F = 6.3 x 10 7 Jm-2 s-1.
Now, let a = 959.63 "
but this must be in radians before one can use the equation.
One radian = 57.3 degrees
Change to seconds ("):
= 57.3 deg/rad x (3600"/ deg)= 206 280 "
a = (R/r) = 959.63"/ 206 280"/rad = 0.00465 rad
So: (r/R) = 1/0.00465 rad = 215 rad-1
S = [6.3 x 10 7 Jm-2 s-1 ]/ [215 rad-1]2 = 1360 W/m2