Old Smokey,
Just being pedantic for a sec.... it's not exponential, but a simple square law.
But yes, it's non-linear, which is the main point.
For me as an engineer, it's still counter-intuitive, that, whether you barrel along at 2000 km/h TAS at sealevel or at 2000 km/h TAS at 60,000 ft, the temperature rise is the same, notwithstanding the far lower air density.
I'm not querying the formulae... worked with them long enough. It's just that it doesn't 'feel' right.
I expect the answer is elementary...
Temperature is a property of each individual molecule.
In a total temperature probe each indivdual molecule is brought to rest, so it experiences the same temperature rise, while the number of molecules does not enter into the equation.
Sounds right....
As a boffin, I now ask myself the question: to what extent does the air density affect the heat transfer, be it to the total temperature sensor, or to the structure.
To take an extreme case, think of the space station, where the total air temp works out at something like 30,000°C, but there are simply not enough air molecules to result in any significant kinetic heating.
CJ