Relation between IAS and Mach number

[Rivet gun]

Lets get rid of these darn moles! The universal gas constant is 8.314 J/mol/K. If we multiply this by the average moles per kilogram for dry air (34.53) we get the specific gas constant for air which is 287.05 J/Kg/K. Call this R

We can now express the ideal gas law as:

P = R rho T

where P is pressure (Pa), rho is density (Kg/ m^3) and T is temperature (K). Humidity complicates matters because it changes R, so lets us stick to considering dry air.

Mad (Flt) scientist: the ideal gas law certainly DOES apply across air masses provided we are considering dry air. Furthermore aircraft can and do maintain constant FL across air mass boundries, they do not maintain geometric altitude (ATC would complain otherwise). The reason they can do so is because the temperature change is gradual, not a step change. For this discussion we need only consider the end states, i.e at one time the aircraft is flying in air at temperature T1, at a later time it is flying at the same FL and Mach number in air at temperature T2.

Now to deal with the speed of sound (SoS). This depends on the square root of the ratio P/rho. But from the gas law P/rho = R T. The exact equation also includes the ratio of specific heats (gamma) which for dry air is 1.4. So:

SoS = sqrt(1.4 R T)
= 20.05 sqrt(T) m/s
=39 sqrt (T) knots.

Now back to the original question.

At constant FL pressure is constant, so from the gas law rho T is constant. This is the important point.

If the aircraft flies from a cold air mass (T1) to a warm air mass (T2) and mach number is held constant the TAS increases by the ratio sqrt(T1/T2). But we have established that rho T is constant, so if T increases rho decreases in exact proportion.

The ratio EAS/TAS depends on sqrt(rho1/rho2). Therefore the increase in TAS is exactly cancelled by the decrease in EAS/TAS ratio and EAS remains constant.

To go from EAS to CAS we adjust for compressibility. However compressibility is a function of Mach number and FL, both of which are constant. Therefore compressibility error is constant and CAS is constant.

To go from CAS to IAS we adjust for position error. This is empirical and aircraft specific, however modern flight instrument systems have built in position error correction and for practical purposes we may assume CAS = IAS. Besides at constant FL and Mach number position error would be constant anyway.

Therefore IAS remains constant

QED