Question

The brightest star in the sky is the star sirius. It is 2.64 pc distant, and has an apparent visual magnitude of mv= -1.4. If the pupil of the human eye has a diameter of 5mm, how many photons from sirius enter the eye per second?

Answer 1

First we find flux from Sirius. Using definition of apparent magnitude and knowing it for the Sun we can find

$F_{sirius}=F_{sun}\cdot 2.512^{m_{sun}-m_{sirius}}=1361\,\frac{kWt}{m^{2}}\cdot 2.512^{-26.7-1.4}\approx 1.03\cdot 10^{-4}\,\frac{kWt}{m^{2}}$

We also need temperature of surface of Sirius. From Wikipedia we take it $T=9940K$. Wien’s displacement law tells us that most intense line will be

$\lambda_{max}T=b,\qquad\lambda_{max}=b/T$

where $b=2897768.6nm\cdot K$, we find that $\lambda_{max}=291.5nm$ We suppose that 40 percent of light are emitted nearly at this wavelength, with energy

$E=\hbar\nu=\hbar\frac{\lambda_{max}}{c}=6.81456\cdot 10^{-19}J$

Now we can find number of photon per human eye per second

$N=\frac{1.03\cdot 10^{-4}}{0.4\cdot 6.81456\cdot 10^{-19}}\cdot 3.14\cdot(2.5\cdot 10^{-3})^{2}\approx 0.19\cdot 10^{10}\,\mbox{photons}$

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