Head Shade Priced in Heat: A Brim-Width Rule, Its Saving in Watts and Sweat, and Its Limit at Low Sun

A hat is usually sold by appearance and bought by habit, yet its working value is measurable: it removes part of the solar load that the body would otherwise have to shed by sweating. This paper prices head shade in heat terms. Under a stated clear-sky irradiance of 1000 W/m² and a surface absorptance of 0.65, a bare head, neck, and shoulders absorb 58.5 W with the sun overhead, 63.7 W at 60 degrees elevation, and 54.9 W at 35 degrees. A cotton cap, which shades the scalp only, saves 12.7 W overhead but only 7.3 W at 35 degrees. A woven-straw hat with a 100 mm brim saves 46.8 W overhead (0.069 litres of sweat per working hour, at the stated conversion) and 26.9 W at 60 degrees, but only 8.9 W at 35 degrees, because at low sun the brim's shadow falls sideways off the neck and shoulders. The sizing rule that follows is stated against the surfaces a hat can in principle shade — scalp, face, and neck, which absorb 32.5 W at 35 degrees: the least-cost hat that removes at least half of that head load at the worst stated angle is a palm-leaf hat with a 160 mm brim at a stated price of USD 2.00, which removes 53.8% of the head load (head absorption falls to 15.0 W) and saves 17.5 W overall, or 0.026 litres of sweat per working hour. A stronger claim sometimes made for wide-brim hats — that they remove most of the total load including the shoulders — does not survive the geometry at low sun: the best hat scanned (felt, 160 mm brim) removes only 33.7% of the total head, neck, and shoulders load at 35 degrees. A hat is a powerful defence against a high sun and a partial one against a low sun, and its price in heat should be quoted at the angle at which it will actually be worn.