2. Many animals see well at night. (Actually, humans are pretty good at this, be

Question

2. Many animals see well at night. (Actually, humans are pretty good at this, being only a factor of two less sensitive than owls.) Let's consider the minimal light level you can see in and how it depends on the parameters of the eyeball. Suppose that there is some minimal image brightness on the retina that is needed in order to see an object, and that this minimal brightness level is the same for all animals. (Brightness is essentially equal to the number of photons per square meter per second striking a surface. The brightness s directly proportional to but it also depends on the focal ength f of the eye, and the diameter of the eye D. A How would you expect the number of photons collected to depend on D? Give the functional form of the dependence. B The brightness of the image on the retina also depends on the focal length f. Give the functional form of the dependence

Explanation / Answer

Given the eye with focal length, f and diameter, D.

Let an object of height, h, be at a distance of L.

From the Telescope equations, we know that the magnification of the image is given by

M = f/L = hi/h

The area, dSi of the image perceived by the eye is proportional to the square of the magnification.

dSi = M²dS = (f² / L²)*dS -----------------(1)

where dS is the object area

Now, amount of light (# of photons) received on the eye is

dE = theta² dET

where dET is the total light emitted (in case of object being a light source) or diffused by the object. And theta is the angle of the object subtended from the eye at a distance L. Since the object is far away from the eye,

theta = D/L

dE = D²/L² dET -------------------(2)

(A) Number of photons collected is square dependent on D

Now for brightness being # of photons per area, take the dE per area, dSi from (1)

Therefore from (1) and (2), we have

B = dE/dSi = D²/L² dET *1/((f² / L²)*dS)

B = D²/f² * dET / dS

(B) Brightness depends on the inverse square of focal length, f

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