# Question 30c20

Apr 9, 2017

Of the order of ${10}^{16}$ photons

#### Explanation:

This is an interesting question requiring many assumptions. Also, we need to work with averages.

• Since the question asks the number of photons entering human eye per second in daylight, we need to consider the amount of solar energy reaching earth's surface during daytime.

The figure above depicts Spectrum of Solar Radiation for direct light at both the top of the Earth's atmosphere (yellow area) and at sea level (red area). As light passes through the atmosphere, some are absorbed by gasses like ozone, oxygen water vapors and carbon dioxide which have specific absorption bands. Additional light (especially blue) is redistributed by Raleigh scattering.
• Only visible wavelengths of light are considered. Ultra Violet is absorbed in the atmosphere. Infra Red even though may reach the eye would not be visible. The typical human eye will respond to wavelengths from about $390 \text{ to } 700 n m$. For the sake of calculations, we assume photons of mid values $= 550 n m$. As such, each photon is assumed of average energy of about $2.4 e V .$
This is also supported by the eye color sensitivity as shown in the figure below.

• Solar intensity variations depend on a number of cycles. These include the 11-year sunspot solar cycle, the proposed 88-year (Gleisberg cycle), 208-year (DeVries cycle) and 1,000-year (Eddy cycle). Effects of these have been not considered.
• Effect of Earth's Perihelion as Aphelion have been averaged and an intensity of $1367 W {m}^{-} 2$ is assumed to be reaching earth's surface at sea level for the purpose of calculations.
• Average Ground Reflectance Information is referenced here. It may be seen that maximum reflectance is from Light building surfaces=60%#. As such we see that reflected intensity reaching eye is $820 W {m}^{-} 2.$
• The last item I considered is the size of the pupil. The normal pupil size in adults varies from 2 to 4 mm in diameter in bright light. It may vary from 4 to 8 mm in the dark due to the accommodation of the eye. For sake of these calculations, an average of 3mm has been taken.

Ready with the assumptions and data we proceed as below:
Intensity entering one eye per sec ${E}_{\text{ave"="Average Solar intensity"xx"Area of pupil}}$

As average Energy of one photon$= 2.4 \times 1.60218 \times {10}^{-} 19 J$
$\therefore$Number of photons per second$= {E}_{\text{ave}} / \left(2.4 \times 1.60218 \times {10}^{-} 19\right)$
$= 820 \times \pi {\left(\frac{3}{2} \times \frac{1}{10} ^ 3\right)}^{2} \times \frac{1}{2.4 \times 1.60218 \times {10}^{-} 19}$
$\approx {10}^{16}$