We see right away that z varies greatly from latitude to latitude. Only near the equator does sunlight reach the earth at close to perpendicular (i.e., z = 0). At other latitudes z is much greater. Because the majority of the earth's surface receives solar radiation at large zenith angles, it is worthwhile to consider the effect of the zenith angle on the intensity of sunlight. The decrease of sunlight intensity with increasing latitude is reflected in studies of the incidence of skin cancer relative to geographical location [2, pp. 121-123].

The zenith angle has a very significant effect on the intensity of the radiation we receive from the sun. The fact that the more northern and southern latitudes of the earth have clearly distinguishable seasons is primarily due to the angle of the radiation striking a particular latitude. It is not due to distance changes between the earth and the sun. In the northern hemisphere, for example, the zenith angle is least during the summer months. Temperatures are higher because the sun's rays are striking the earth more directly at these latitudes even though the earth is actually farther away from the sun during the months of June, July, and August.

Without much difficulty the equation for I(a) can be modified to take into account the angle of the path of light. Consider a path of light 1 m wide. The view from the earth (see the following figure) shows that the light falls on the horizontal surface of the earth with a length of x > 1.

So how many photons are absorbed on this angular path from the sun to the surface of the earth? Express x in terms of the zenith angle z.

Now we need to calculate the amount of O₂ that the light travels through as it moves from the sun to the surface of the earth. We begin by considering the amount of light that passes through a typical cross sectional area along the path 1 m wide at zenith angle z. For a given change in altitude,  , the cross sectional figure (see the following diagram) is a parallelogram with height  and base x.

Express the area of this figure in terms of  and the zenith angle z.

To determine the fraction of photon loss that occurs as the light travels through this particular segment, we multiply the cross-sectional area by the concentration of O₂ present at that particular height and the ability of O₂ to absorb light.

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Use your computer algebra system to graph I(a) for three different zenith angles, z = 30^o, z = 45^o, and z = 60^o. What do your graphs tell you about the intensity of the sunlight as the zenith angle decreases? Explain your observation in terms of the physical difference in the path the sunlight takes for the different angles.