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The plot shows that: when I fly with fixed MACH, the true air speed decreases when the altitude increase; when I fly with fixed CAS, the true airspeed increases when altitude increase. So how could I compute compute how much TAS change I will get based on the altitude change when I fly with fixed MACH or CAS? enter image description here

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  • $\begingroup$ What properties of physics have you thought about and What equations have you found? $\endgroup$
    – Jim
    Commented Jul 21, 2022 at 23:29
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    $\begingroup$ Can you please credit the image source? I am pretty sure I have seen the exact same diagramm before $\endgroup$
    – DeltaLima
    Commented Aug 3, 2022 at 7:12

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At the most basic level the Mach formula is: Mach = TAS/Speed of Sound

The speed of sound is a slightly non-linear function of temperature: 38.967854*SQRT(OAT+273.15) where OAT is expressed in degrees Celsius.

So:

  1. TAS = Mach * 38.967854*sqrt(OAT+273.15)
  2. Mach = TAS / 38.967854*sqrt(OAT+273.15)

PART II - CAS/TAS

This is about to get real... Mach <-> TAS is a bit easier because I’m mostly just accommodating temperature’s (OAT’s) affect on the speed of sound. You can get there directly by entering an observed OAT at your present altitude, or you can estimate what your OAT will be at various altitudes by using a surface observation and using the standard atmosphere (ISA) lapse rate which is -0.0019812°C per foot, or -1.9812°C per thousand feet. (Caution-For FAA written test purposes, you’re expected to use the approximation of -2°C per thousand feet.)

Converting CAS <-> TAS requires us to look at air density, which is a function of both the pressure and temperature of the air. We’ll ignore humidity to keep things “less complicated.” We’re either going to have to calculate rho (air density) or calculate density altitude. We already use density altitude in GA, and it’s a less complicated formula. So now we need:

  1. Your pressure altitude in feet (what height above MSL your altimeter says you are, assuming the correct altimeter setting.) We’ll call that PA.
  2. We’ll need to use the standard temperature lapse rate that I mentioned above, which is -0.0019812°C per foot. We don’t need to convert this to Kelvin since it’s a rate of change. We’ll call this LR.
  3. We need to know the ISA Standard temperature for that PA. In physics, all temperature comparisons are done in Kelvin, so we have to express this in °K, which is °C + 273.15. This is going to be sea level standard temperature with the lapse rate applied: 15°C – (0.0019812 * PA) +273.15. We’ll call this STK, for Std Temp Kelvin.
  4. We also need to convert your observed OAT from °C to °K, again by adding 273.15. We’ll call this OATK.
  5. Density Altitude will be: DA = PA + (STK/LR)*(1-(STK/OATK)^0.2349690)

Only part way there. Now we’ll take our DA and plug it into this:

TAS = CAS / ( 1 - ( .0000068755856 * DA ) )^2.127940

Or

CAS = TAS * ( 1 - ( .0000068755856 * DA ) )^2.127940

If you put the DA formula and the Kelvin conversion all into one you end up with:

TAS = CAS / ( 1 - ( .0000068755856 * (PA + (  (15-(0.0019812 * PA) + 273.15)  / 0.0019812 )*( 1-( ( 15-(0.0019812 * PA) + 273.15)/( OAT + 273.15) )^0.2349690)) ) )^2.127940

Sample problem: PA = 10,000’; OAT = 3.5°C; CAS = 150 Kts. Should get a TAS of 177. I typed it up in Excel format so you should be able to paste this into Excel and it should work:

=150 / ( 1 - ( .0000068755856 * (10000 + (  (15-(0.0019812 * 10000) + 273.15)  / 0.0019812 )*( 1-( ( 15-(0.0019812 * 10000) + 273.15)/( 3.5 + 273.15) )^0.2349690)) ) )^2.127940

It turn out that TAS goes up by about 1.015x per thousand feet assuming the standard atmospheric lapse rate, accurate to less than 1Kt.

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  • $\begingroup$ Thanks Max, from your formula, I could compute fly fix Mach, how much TAS changes based on the altitude change(due to OAT change), but how about CAS? What is CAS' relationship with TAS and OAT? $\endgroup$
    – VvV
    Commented Aug 5, 2022 at 3:20
  • $\begingroup$ @VvV Have you seen this answer with conversion formulas? $\endgroup$
    – Bianfable
    Commented Aug 5, 2022 at 8:30
  • $\begingroup$ @VvV I edited the answer and added the whole new section about CAS <-> TAS. Let me know if you need any clarification. After you see the math you'll realize why Dalton invented the E6B! $\endgroup$
    – Max R
    Commented Aug 6, 2022 at 4:47
  • $\begingroup$ These formulas are near impossible to read. Please use MathJax (here is a tutorial and quick reference). $\endgroup$
    – Bianfable
    Commented Aug 6, 2022 at 9:58
  • $\begingroup$ @Bianfable, I've used MathJax in the past when the question is being asked by someone who I believe has math/engineering background and is looking for an equation in general form. For those who I believe are looking to create their own "solver" to actually play around with in order to develop their own intuition, I post them in as near an "excel pastable format" as I can. My assumption is quite often that questions here are being asked by pilots who are "math curious," as opposed to mathematicians that are "aviation curious." $\endgroup$
    – Max R
    Commented Aug 6, 2022 at 18:10

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