Illuminated fraction of the Moon

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SUMMARY

The discussion focuses on deriving the expression for the ratio of the illuminated disk of the Moon to the whole disk, specifically using the formula f_i = 0.5 * (1 + cos E_s), where E_s is the phase angle. The phase angle E_s is calculated as (t/T) * 360°, with t being the elapsed time since the last full moon and T being approximately 29.5 days. The conversation clarifies that the formula for E_s can also be expressed in radians by substituting 360° with 2π. The illuminated fraction is derived from the ratio of the illuminated portion to the entire disk, leading to the conclusion that the illuminated fraction is (1 + cos E_s) / 2.

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JeffOCA
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Dear all,

It's late in the night and I have some trouble in deriving the expression of the ratio of the illuminated disk to the whole disk. Is it a formula "by definition" ?

See http://docs.google.com/viewer?a=v&q...vuuO5&sig=AHIEtbQ4C_gHumCJFdfEylxojg0t1MD6Vw" at page 16, figures 6 and 7.
It's written "Thus the ratio (...) can be expressed as f_i=\frac{1}{2}(1+\cos E_s)" where Es is the phase angle.

Thanks for helping...

Jeff
 
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The phase angle Es is \frac{t}{T} \cdot 360^o, where t is the elapsed time since the last full moon and T is the period (elapsed time between full moons, about 29.5 days). For example, Es is 0 at full moon, 90o at first quarter, 180o at new moon, etc.

If you want to work in radians rather than degrees (for example, using Excel or google for the calculation), then replace 360o with 2π in the formula for Es.

Hope that helps.

EDIT: my expression for Es is an approximation, assuming a circular orbit for the moon.
 
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I understand your approximation for Es. What I don't understand is the expression of fi given in my first post. Why it is 0.5 * (1 + cos Es) ?

Thanks
Jeff
 
Anyone ?
 
Sorry for the delay. This is best explained with a figure, and it was not until just now that I had time to make a decent one.

MoonFraction.gif

r is the radius of the moon, so of course the entire disk has a diameter of 2r. And, as the figure shows, the illuminated portion viewed from Earth is r + r \cos {E_s} = r(1 + \cos{E_s}). So the illuminated fraction is their ratio,

\frac{r(1 + \cos{E_s})}{2r} = \frac{(1 + \cos{E_s})}{2}
 
It's very clear ! Thanks a lot Redbelly98 !

Best regards