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If you can determine the capacitance, you can determine capacitive reactance (Xc in Ω) where C=capacitance in farads, ƒ=frequency in Hertz, ∏=3.1416, as:

Xc=1/(2∏ƒC)

… but that will only help. I haven’t run across a magic bullet engineering formula for determining efficiency of hat materials, arm lengths, radius, or shape. All I know for sure is that Xc is reduced by C.

It makes loading more effective, meaning you could reduce the number of turns, and therefore loss, in a loading coil. So make your antenna, a LC with half the number of turns for a given frequency, add your hat arms longer than you think they should be, and then start clipping the ends off a little at a time until it resonates.