Where does the gravity equation come from?

From Thomas Chaney, in the latest JPE:

The gravity equation in international trade states bilateral exports are proportional to economic size, and inversely proportional to geographic distance. While the role of size is well understood, that of distance remains mysterious. I offer an explanation for the role of distance: If (i) the distribution of firm sizes is Pareto, (ii) the average squared distance of a firm’s exports is an increasing power function of its size, and (iii) a parameter restriction holds, then the distance elasticity of trade is constant for long distances. When the firm size distribution follows Zipf’s law, trade is inversely proportional to distance.

Here are earlier, ungated versions.