Isentropic Flow Calculator (p/p₀, T/T₀, ρ/ρ₀ and A/A* from Mach number)

Understanding the isentropic flow calculator — concept and principle

This calculator gives the ratios of static to stagnation (total) pressure, temperature and density, the area ratio to the sonic throat, the dynamic pressure ratio and the Mach angle for a given Mach number and ratio of specific heats. With stagnation values entered it also returns the static pressure and temperature. It is a standard tool for nozzle, wind-tunnel and aerodynamics work.

Speeding the gas to Mach 2 trades pressure and temperature for velocity; the nozzle must widen 1.69 times past the throat.

How it is calculated

f = 1 + (γ − 1) M² ÷ 2

T/T₀ = 1 ÷ f; p/p₀ = f^(−γ/(γ−1)); ρ/ρ₀ = f^(−1/(γ−1))

A/A* = (1/M) × [(2/(γ+1)) f]^((γ+1)/(2(γ−1)))

q/p₀ = (γ/2) M² (p/p₀); Mach angle μ = arcsin(1/M) for M > 1

γ = 1.4 for air; p₀ in kPa and T₀ in K are optional.

The principle

In steady flow with no heat transfer and no friction, the total energy of the gas stays constant. As the gas speeds up, some of its thermal energy becomes kinetic energy, so its temperature drops; because the process is reversible (isentropic), pressure and density follow fixed power laws of temperature. The stagnation values are what the gas would have if brought to rest. Mass conservation through a channel gives the area ratio: subsonic flow speeds up in a narrowing duct, but supersonic flow speeds up in a widening one, which is why rocket nozzles converge and then diverge.

Worked example

Default inputs: M = 2, γ = 1.4, p₀ = 101.325 kPa, T₀ = 288.15 K.

f = 1 + 0.2 × 4 = 1.8, so T/T₀ = 0.5556.
p/p₀ = 1.8^(−3.5) = 0.1278; ρ/ρ₀ = 1.8^(−2.5) = 0.23.
A/A* = (1/2) × (1.8/1.2)³ = 1.688; q/p₀ = 0.7 × 4 × 0.1278 = 0.3579.
Static pressure = 101.325 × 0.1278 = 12.95 kPa; static temperature = 288.15 × 0.5556 = 160.08 K (−113.07 °C).
Mach angle = arcsin(1/2) = 30°; regime supersonic.

Each point is the calculator re-run with only “Mach number M” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Mach number M	Pressure ratio p/p₀
0.5	0.843
1	0.5283
1.5	0.2724
2	0.1278
2.5	0.05853
3	0.02722
4	0.006586

How to read the result

Every A/A* above 1 has two solutions, one subsonic and one supersonic; which one occurs depends on the pressure downstream. Large drops in static temperature at high Mach number can cause condensation in wind tunnels. Shock waves are not isentropic, so these ratios do not apply across them.

Common mistakes

Treating static and stagnation pressure as the same.
Using γ = 1.4 for hot combustion gases, which have lower γ.
Applying the ratios across a shock wave.
Using the formulas at very high Mach numbers, where real-gas effects grow.

Keep in mind: The relations assume a calorically perfect gas with no friction or heat transfer. Real nozzle and inlet design needs boundary-layer and shock analysis by qualified engineers.

Frequently asked questions

What is stagnation pressure?

The pressure the gas would reach if brought to rest isentropically, as measured by a pitot tube in subsonic flow.

Why does a supersonic nozzle widen?

Above Mach 1, density falls faster than velocity rises, so more area is needed to pass the same mass flow.

What is the Mach angle?

The half-angle of the cone of weak pressure waves trailing a supersonic object.

What happens at M = 1?

A/A* = 1; the flow is sonic, as at the throat of a choked nozzle.

Sources: NASA Glenn Research Center, Beginner's Guide to Aeronautics, "Isentropic Flow"

Checked 9 Nov 2026. NASA Glenn Research Center, Beginner’s Guide to Aeronautics, “Isentropic Flow” (eqs. 4–9: p/pt, T/Tt, r/rt, A/A* and q = γpM²/2). Mach angle arcsin(1/M): standard formula.

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