Isentropic Relations Calculator

Temperature, pressure, density ratios and area ratio A/A* for isentropic compressible flow. Enter Mach number and γ; optionally supply T₀ and P₀ to get static values.

Compressible Flow · Isentropic

Gas presets — click to fill γ

Inputs

Mach number M (dimensionless)

Specific heat ratio γ (dimensionless)

c_p/c_v — use gas presets above

Stagnation conditions — optional, needed to compute static T and P

Total (stagnation) temperature T₀

Total temperature T₀ (K)

Total (stagnation) pressure P₀

Total pressure P₀ (kPa)

Theory

Isentropic flow relations:

T/T₀ = [1 + (γ−1)/2 × M²]⁻¹

P/P₀ = (T/T₀)^γ/(γ−1)

ρ/ρ₀ = (T/T₀)^1/(γ−1)

A/A* = (1/M) × [(2/(γ+1)) × (1 + (γ−1)/2 × M²)]^{(γ+1)/(2(γ−1))}

Critical conditions at M = 1 (throat):

T*/T₀ = 2/(γ+1) [= 0.8333 for air]

P*/P₀ = (2/(γ+1))^γ/(γ−1) [= 0.5283 for air]

ρ*/ρ₀ = (2/(γ+1))^1/(γ−1) [= 0.6339 for air]

Key values for air (γ = 1.4) at common Mach numbers:

M	T/T₀	P/P₀	ρ/ρ₀	A/A*
0.0	1.0000	1.0000	1.0000	∞
0.5	0.9524	0.8430	0.8852	1.3398
0.8	0.8865	0.6560	0.7400	1.0382
1.0	0.8333	0.5283	0.6339	1.0000
1.5	0.6897	0.2724	0.3950	1.1762
2.0	0.5556	0.1278	0.2300	1.6875
3.0	0.3571	0.0272	0.0762	4.2346
5.0	0.1667	0.0019	0.0113	25.000

Isentropic relations apply to adiabatic, reversible (frictionless) flows — convergent nozzles, divergent diffusers, and shock-free streamtubes. The area ratio A/A* determines the required throat-to-exit area ratio for a nozzle operating at a given design Mach number; each A/A* > 1 has one subsonic and one supersonic solution.