The threshold intensity for a human eye to sense the presence of light is 50 pho

Question

The threshold intensity for a human eye to sense the presence of light is 50 photons falling on the retina within a period of 0.5 s. Assume a star radiates with an output of 8 1026 W in the visible spectrum, with a peak of ?=450 nm. (This star would be similar to the Sun, which radiates with 2.0 1026 W). Estimate how far away the star can be away and still be visible to the unaided eye on a dark night. Assume the pupil of the observer's eye has a diameter of 1.0 cm. Give your answer in light years (do not insert a unit).

Explanation / Answer

eye:

E = nhf = nh(c/) = 50*(6.626e-34)*(3e8/550e-9) = 1.80709091e-17 J

P = E/t = 1.80709091e-17/0.1 = 1.80709091e-16 W

A = r2 = 3.14*(1.0e-2/2)*(1.0e-2/2) = 0.0000785 m2

I = P/A = 1.80709091e-16/0.0000785 = 2.30202663578e-12 W/m2

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star:

I = P/A = P/(4R2)

>>> R = (P/(4I))

         = (5e26/(4*3.14*2.30202663578e-12))

   = 4.1584837647939e18 m

         = 4.1584837647939e18 * 1.05702341e-16 lightyears

         = 439.56 lightyears

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