How can TAS be calculated from indicated temperature and Mach number?

Question

Situation

I am working on programming my own E6B while I learn more about aviation, but I have run into some confusion. Two pairs of the equations are very similar except for using true or indicated temps based on which equation. If anyone has a Sporty's E6B, it is listed as Planned Mach # and Actual Mach # as well as Planned TAS and Actual TAS.

Important Info

The inputs for the Mach # formulas are as follows: Temp, Mach #

While the output is: TAS

The inputs for the TAS formulas are as follows: Pressure Altitude, Temp, CAS

While the outputs are: TAS, Mach #, Density Altitude

"Planned" versions take "True Temp" according to the manual, while the "Actual" versions use "Indicated Temp". From what I can tell from research, True temp is known as SAT (Static) and Indicated would be RAT (Ram). I have also been told that there usually isn't a difference between the 2 unless you are over certain speeds... but I am thinking that maybe there is some form of temperature conversion that I need to do with the RAT/Indicated input. As you can see below, using the same inputs will provide slightly different outputs.

Examples

Planned Mach # with Indicated Temp of 22C and Mach # of 0.33 provides a True Airspeed output of 221

Actual Mach # with Indicated Temp of 22C and Mach # of 0.33 provides a True Airspeed output of 218.6

Where I'm at and have tried

I am able to calculate the correct answers when using True Temp but not Indicated. Currently, I've been trying to convert RAT to SAT with some math I found using a "recovery factor". I get quite close to the result that my Digital E6B but it is not the same result. I have used many different formulas from an AI I won't name... and overall I am just stumped. After a week of trying, I am going to ask this forum, which has helped me many times before.

Does anyone have some insight on how to handle this and use the Indicated Temperature in these equations?

Answer 1

TAS is Mach number ($M_a$) multiplied by the speed of sound ($a$).

$$\textrm{TAS}=M_a\cdot a$$

The speed of sound depends on the temperature of the (static) air $T_s$.

$$ a = \sqrt{\gamma RT_s}$$

The relation between the measured total air temperature (or ram 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.

Putting it all together yields:

$$\textrm{TAS} = M_a \sqrt{\frac{\gamma R T_{\textrm{total}}}{1+\frac{\gamma -1}{2}eM_a^2}}$$

• The ratio of specific heats $\gamma$ is 1.4
• R, the specific gas constant is 287.052874 J K/kg

The formula is for SI units, so temperature should be in Kelvin [K], TAS will be given in meters per second [m/s]