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# 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?
1
2010-05-21T04:29:58-0400
For simplicity let&rsquo;s consider that solar radiation complies with Lambert law. Straight sun rays falling perpendicular on the earth create illuminance on it Es = &pi;Bsin&sup2;&Theta;s, where B &ndash; surface brightness of the sun, &Theta;s - 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 &pi; times less, that is Es = Bsin&sup2;&Theta;s. 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 &pi;B.

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&rsquo;s, that is Es = Bsin&sup2;&Theta;s. If moon being a radiant complies with Lambert law, than its surface brightness will be Es = Bsin&sup2;&Theta;s. Rays from moon falling perpendicular on the surface of the earth, create illuminance B sin&sup2;&Theta;s sin&sup2;&Theta;m, where &Theta;m - angular radius on the moon. Such illuminance is smaller than Bsin&sup2;&Theta;s. That is no matter what size of lens to take it still will not light up a cigarette.

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