13.4.6. Estimate how many times the illumination of the same surface on a lunar night during a full moon is less than on a sunny day? The height of the Moon and Sun above the horizon is the same. Assume that parts of the moon scatter all the light falling on them in all directions evenly. Take the distance from the Moon to the Earth as $400000$ km, and the Moon's radius as $2000$ km.
Solution
Let $I$ be the intensity of light from the Sun on the Earth. As the distance between the Moon and the Earth is much less than that between the Sun and the Earth, we may assume that the intensity of light from the Sun on the Moon is $I$ as well. Consequently, the power of light from the Sun intercepted by the cross section of the Moon is $\pi r^2I$, where $r$ is the radius of the moon. Since light from the Sun is reflected by one side of the Moon, the intensity of light from the Moon on Earth is
where $d$ is the distance between the Moon and the Earth. Using $r=2000$ km and $d=400000$ km, we see that the intensity of light on the Earth from the Moon is $1/80000$ of that from the Sun.