Kepler & the Inverse Square Law

 We hear a lot of talk about global warming but how good a job are we doing on presenting the science of global warming? To answer this question a few posts on the nature of heat along with the history of the science might help. We start with light and the inverse square law.

Aristotle wrote in De Anima bk II, ch7 (c. 350 BC) about the nature of light. In it he notes a relationship between heat, light and color, that light is non corporeal and thus not an emission of substance, that colors require light to be revealed and that it is associated with a medium.

In 1604 in Astronomiae Pars Optica Kepler cites Aristotle and lists a number of propositions on the nature of light. He notes that light is unchanged as it moves from its origin to some distant place, that it can travel along an infinite number of lines from its source, that its path is straight and its speed is infinite. Proposition 9 deals with the quantity of light passing through the surfaces of a sphere with the quantity of light being the same for all spheres with the common center and the density varies due to different surface areas.

"Propositio IX
Sicut se habent sphæricæ superficies, quibus origo lucis pro centro est, amplior ad angustiori, ad illam in laxiori sphærica superificie, hoc est, conuersim. Nam per 67 tantundem lucis est in angustiori sphærica superficie, quantum in fusiore, tanto ergo illic stipatior & densior quam hic. Si autem radii linearis alia atque alia esset densitas, pro situ ad centrum (quod Prop. 7 negatum est) res aliter se haberet."

"Proposition 9
As they have spherical surfaces, wherein the source of light, for the center is, the larger is to the narrower, to each in lessor spherical surface, that is, interdependent. For by 6 & 7, the amount of light in a smaller spherical surface is as in the extended, so therefore as that there is more crowded and denser than that here. If however, in one way or another, as the linear radius would be, the density is as the situation to the center (as Prop. 7 is negated) would have things differently."

This is basically a statement of the inverse square law, that is, d₁:S₂::d₂:S₁ or d₂=d₁S₁/S₂, but the flow does not have to be for the entire surface. Alternatively one could consider the passage of light through a radial flux tube of solid angle dΩ and bounded at the ends by surface areas determined by the formula dS=r²dΩ. The quantity of light flowing through the tube is dQ=FdSdt which defines the flux F, a constant for steady flow. Solving for F on some surface we find that F=I/r² where I is the luminous intensity of the source in the direction of the tube. This was verified by Lambert's time.