Answer to Question #160 in Optics for Silvia

Question #160
With the help of condensing lens one can light up a cigarette, focusing straight sun light on it. Is it possible to do the same, having an appropriate lens (for example, objective of a huge astronomical telescope), with the help of full Moon light?
Expert's answer
For simplicity let’s consider that solar radiation complies with Lambert law. Straight sun rays falling perpendicular on the earth create illuminance on it Es = πBsin²Θs, where B – surface brightness of the sun, Θs - its angular measures. From daily experience is known that such illumination is not enough for lighting up a cigarette. Average illumination of the earth will be π times less, that is Es = Bsin²Θs. If neglecting absorption and diffusion of light, then operation of lens is limited to increase of angular measures of the sun. However maximum illuminance can not exceed πB.

Since a diameter of moon orbit is negligible as compared with the distance to the sun, than average illuminance of surface of the moon will equal to the sun’s, that is Es = Bsin²Θs. If moon being a radiant complies with Lambert law, than its surface brightness will be Es = Bsin²Θs. Rays from moon falling perpendicular on the surface of the earth, create illuminance B sin²Θs sin²Θm, where Θm - angular radius on the moon. Such illuminance is smaller than Bsin²Θs. That is no matter what size of lens to take it still will not light up a cigarette.

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