# What is the background of the maximum altitude formula?

This question What determines the maximum altitude a plane can reach? has a formula at Peter Kämpf's answer :

$$\rho_{min} = \frac{2\cdot m\cdot g}{(Mach^2 \cdot c_L)_{max}\cdot a^2\cdot S}$$

Does anybody know what's the name of the formula or who has developed it? Could anyone explain why $Mach^2 \cdot c_L = 0.4$ is a good value? What determines good or bad values for airfoils or what type of airfoils is requested?

The formula can be derived equaling the weight ($W$) of an airplane with the maximum lift ($L$) generated by its wings. I'm not aware of any special name for it.

$$W = L \\ m\cdot g= \frac{\rho_{min}}{2}\cdot V^2\cdot S\cdot c_{L,max}$$

Substituting the aircraft's speed with an expression containing the Mach number ($a$ is the local speed of sound), $$V=Mach \cdot a$$ and solving for the density you get the formula in Peter's answer:

$$\rho_{min} = \frac{2\cdot m\cdot g}{(Mach^2 \cdot c_L)_{max}\cdot a^2\cdot S}$$

At high speeds the maximum lift coefficient is not constant. That's the reason why the Mach number and the maximum lift coefficient are written together in one term: $(Mach^2\cdot c_L)_{max}$. In order to fly as high as possible this term has to be maximized.

With supercritical airfoils you generate less drag and can therefore fly at higher Mach numbers, increasing this term.

I don't think the value 0.4 is a physical limit. Probably it is just an empirical value which takes many real-world constraints into account (if there is a physical limitation I'd be glad to read an answer explaining it). If you calculate $(Mach^2\cdot c_L)_{max}$ using U-2's data you get indeed a value close to it.

• I learned the formula from Mark Drela, and I also don't know a special name for it. It is easily derived, as you showed. The 0.4 is indeed empirical and what I found to be common to good designs. Maybe you get a little above that with supercritical airfoils, but not with "regular" airfoils which stay below their critical Mach number. – Peter Kämpf Apr 8 '18 at 21:31