the Mach Meter does not use a Temperature input for it's readings...I just read more carefully
....all the 'gamma' stuff s because the compression is adiabatic so the adiabatic gas laws must by used to define impact pressure...that's all a simple solution to the diff eq. obtained by direct substitution
§ 1.1 General definitions.
Calibrated airspeed means the indicated airspeed of an aircraft, corrected for position and instrument error. Calibrated airspeed is equal to true airspeed in standard atmosphere at sea level
Equivalent airspeed means the calibrated airspeed of an aircraft corrected for adiabatic compressible flow for the particular altitude. Equivalent airspeed is equal to calibrated airspeed in standard atmosphere at sea level.
True airspeed means the airspeed of an aircraft relative to undisturbed air. True airspeed is equal to equivalent airspeed multiplied by (ρ0/ρ)1/2.
Mach number means the ratio of true airspeed to the speed of sound.
No more and no less
edit: this is complex: