# Is the Mach number shown by an Air Data Computer considered a “true” Mach number?

I am trying to find more information on "true" vs. "indicated" Mach number (not airspeed). I've found older fighter aircraft manuals online that have a conversion chart from one to the other, but the modern jets I've flown don't have such charts.

During a recent flight at 41000 feet, at a Mach number of 0.81, a true airspeed of 476 knots was displayed on the EFIS. In the standard atmosphere, 81% of the speed of sound is 464 knots at that altitude. Sure enough, the actual temperature was some 13 degrees C warmer than the "standard" -56.5 C, hence the true airspeed of 476 knots instead of 464.

But that leads me to believe that the Mach number I am flying is not really "true".

The relevant flight manuals do not use the words "indicated" or "true" to qualify the Mach number.

For subsonic aircraft the Mach number is normally measured from dynamic pressure and static pressure, very similar to the way calibrated airspeed is measured.

$$M_a=\sqrt{\frac{2}{\gamma-1}\left[\left(\frac{q_c}{p}+1\right)^\frac{\gamma-1}{\gamma}-1\right]}$$

with:

• $M_a$, the Mach number
• $q_c$, the dynamic pressure
• $p$, the static pressure
• $\gamma$, the ratio of specific heats (1.4)

If there is an error in the measurement of either the dynamic or the static pressure both the indicated airspeed and Mach number will be affected.

Note that the Mach number has no direct dependency on temperature.

The true airspeed is calculated by the ADC from the Mach number and the outside air temperature.

$$V_{TAS} = M_a \cdot \sqrt{\gamma R T_S}$$

with

• $R$, the specific gas constant (287.05 for dry air)
• $T_S$, the static air temperature

The indicated true airspeed has a direct dependency on the static air temperature ($T_S$,SAT) also known as the outside air temperature (OAT). The static temperature cannot be measured directly since the air will heat up on impact with the temperature sensor due to adiabatic compression. Instead the total air temperature ($T_\mathrm{total}$, TAT) will be measured.

The relation between the measured total air temperature and the static temperature is given by:

$$\frac{T_\mathrm{total}}{T_{s}}={1+\frac{\gamma -1}{2}M_a^2}$$

Since the measurement probe may not recover all the energy from the adiabatic heating a correction factor is introduced:

$$\frac{T_\mathrm{total}}{T_{s}}={1+\frac{\gamma -1}{2}eM_a^2}$$

This 'recovery factor' $e$ is determined empirically for the temperature sensor used.

The display on the EFIS will show the OAT, which may be slightly off if $e$ is not entirely correct. This will also affect the "true" airspeed indication.

To conclude; anything indicated in the cockpit can contain measurement errors, regardless whether it is called 'indicated' or 'true'.

With the introduction of digital flight data systems the error correction tables have been moved from the manuals to the computer, which automatically corrects for them.

• With all that MathJax, I thought this was a Peter Kämpf answer, and I mean that as high praise! – FreeMan May 29 '15 at 16:31
• The point is that there is indicated speed and true speed and they represent different physical quantities (dynamic pressure and speed), but there is no such pair for Mach number. – Jan Hudec May 29 '15 at 17:21
• I was chasing this because the North Atlantic procedures manual specifies that when assigned a Mach number to fly, this should be the "true Mach number". The emphasis made me curious. – Ugo May 29 '15 at 18:30
• @Ugo the North Atlantic Operations & Airspace Manual does indeed distinguish between indicated and true Mach number. It says: For many aircraft types the cockpit instrument displays the True Mach being flown. However, for some types the AFM notes a correction that must be made to the Indicated Mach to provide the True Mach. I don't know your aircraft type, but if there is nothing in the AFM then you can assume the Mach number displayed is the True Mach number. In that case the air data computer will apply the corrections before data is send to the display. – DeltaLima May 29 '15 at 18:44
• @FreeMan Thanks, I take that as a compliment! I am not exactly shy of using MathJax either, this site was two days into private beta when I first needed it :-) – DeltaLima May 29 '15 at 22:57

Mach is a number in relation to the speed of sound. The speed of sound is solely dependent on the outside air temperature. Hence unlike airspeed there is no such thing as 'indicated' mach.

$$M=\frac{TAS}{a}$$

$$a= \sqrt{1.4*288.15*T}$$

Where $a$ is the speed in m/s of sound and $T$ is the temperature in Kelvin.

The old jets needed conversion charts since the true airspeed was not computed by computer and machmeters used a best estimate value limited by the error in pressure. The following is from a USAF handbook on how the Machmeter works:

This document by Airbus also suggests you are looking the ratio of True Airspeed.

• The eyes of all fighter pilots I've met would glaze over looking at that. They just want to look at a gauge or readout :) – Simon May 29 '15 at 14:42

Indicated Mach is what your Mach indicator reads during flight. True Mach is basically indicated Mach that has been corrected for pitot-static system position error and instrument error. (Same as the difference between indicated airspeed and calibrated airspeed).

Position error is caused by the location of the pitot tubes and static ports on the aircraft. This is due to the shape of the aircraft and the way the air flows around it. Engineers try to locate them in areas where the error is at a minimum, but there's always some error. The amount of this error is determined during flight testing and charted in the flight manuals. Some more modern aircraft have software that will automatically compute their calibrated airspeed and true Mach number. Most aircraft, however, still use indicated airspeed and Mach number.

Instrument error is the difference between a calibrated input to the Mach indicator and the actual reading it gives. The Mach indicators I've worked with usually have a tolerance of +/- 0.03 Mach. So, for example, if a technician inputs a calibrated value of 0.70 Mach and the indicator reads 0.72, it's within limits.

Temperature has nothing to do with the way Mach number is calculated in aircraft flight instruments; however, temperature does affect true airspeed.

In your example, 41,000 feet, Mach 0.81 and temperature 13 degrees above standard (that means -43.5 degrees C static air temperature), I calculate the true airspeed to 478 knots. Your total air temperature was probably reading about -13 degrees C, right?

• Thx for the clarification. Since it has been a while, I'm afraid I can't say anymore what the TAT was. – Ugo Oct 30 '19 at 22:58

Equivalent airspeed is not speed at all. It is dynamic pressure, indicated as speed at which that dynamic pressure would occur at sea-level standard conditions. And indicated airspeed is equivalent airspeed plus measurement errors (calibrated airspeed is cleared of those errors, that can be effectively estimated).

So there is an indicated/equivalent airspeed and there is true airspeed and they are different physical quantities.

There is no quantity that would be expressed as Mach number, but would actually have different physical significance. So “Mach number” only has one meaning. How it is measured and what are the errors is already well covered by the other answers.

It is not correct that on most aircraft including the newest Boeing, Dassault, bombardier products only display indicated Mach. You do need to check your manuals. Only the Gulfsteam aircraft display true Mach.

• Welcome to Aviation.se! Can you provide a link to support your statement? – Ralph J Apr 12 '16 at 4:11