Answer to Question #160 in Optics for Silvia

Question

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?

Answer 1

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.