Finding Source of Heating Rate Equation

I am currently working on creating a tool to calculate the maximum heating rate on a re-entry vehicle given starting conditions and coefficients. I am using Re-Entry Aerodynamics by Wilbur L. Hankey as my current source. On page 30, he introduces the following equation as a way to calculate the heating rate:

The problem is that I do not understand the equation. The way I interpret it is:

What doesn't make sense to me is the ρ^m in the equation. If you attempt to solve for the units of ρ^m, you get it as K s^3/m^4, which makes no sense to me.

My researching myself has been unable to lead me to an origin or explanation for the equation, and neither my coworkers or my student friends have ever seen it before. If someone is able to give me some idea what I'm looking at, that would be great.

$m$ is an empirical curve-fitted constant, depending on the composition of the atmosphere, not a mass. For earth and mars, $m = 0.5$.

Actually the "3" is also a curve-fitted constant. Your reference to Hankey is behind a paywall so I don't know why he fixed that at 3 but left $m$ variable.

• "3" is a curve fitted constant - is it set at three as power is v cubed ? at least that's my guess. Thinking of the equation for the power produced by a wind turbine... – Solar Mike May 5 '17 at 6:05