Geometrical optics. Photometry. Quantum nature of light > Photometry > Problem 13.4.8

Statement

An artificial satellite of the Earth with a radius of m, illuminated by the Sun, is visible from a distance of km as an ordinary star. Estimate the distance to such a star.

Solution

Let be the intensity of light from the Sun on the Earth and the satellite. Consequently, the power of light from the Sun intercepted by the cross section of the satellite is , where is the radius of the satellite. Since the light is reflected by a half of the satellite, the intensity of light from the satellite on the Earth is

where is the distance between the satellite and the Earth. Using m and km, we see that the intensity of the light on the Earth from the satellite is of that from the Sun. Since light intensity varies inversely with the square of the distance from the source, a star comparable to the Sun will give the same intensity of light on the Earth as the satellite if it is times as far from the Earth as the Sun. Because the Sun is about light-second from the Earth, the star is about light-seconds, which is roughly light-years, from the Earth. Note that the Sun as a yellow dwarf is in fact brighter than most stars which are red dwarfs, so a better estimate of the distance is less than light-year.

Answer

Less than light-years.