Isentropic Flow Ratios
T/T₀, P/P₀, ρ/ρ₀, and A/A* from Mach number — the compressible-flow table, minus the table.
Your recent runs (stored only in your browser)
No calculations yet — results land here so you can compare runs.
The engineering
Slow a compressible stream to rest without losses and you recover the stagnation state — these ratios say how far the static values sit below it at each Mach number. At M 2 in air the static pressure is only 12.8% of total: most of what a pitot probe feels at speed is recovery, not weather.
A/A* is the nozzle designer's row: it says what duct area passes a given Mach relative to the sonic throat. It's double-valued — 1.6875 at M 2 is the same area ratio as some subsonic Mach — which is why converging-diverging nozzles need back-pressure, not just geometry, to go supersonic.
Where this math comes from
Euler wrote the equations of compressible flow in 1757, but for a century they were mathematics without machinery. The machinery arrived when Gustaf de Laval, chasing steam-turbine speed in 1888, flared his nozzle *outward* past the throat and got supersonic steam — a result so counterintuitive that Aurel Stodola had to instrument the nozzle (circa 1903) to convince the profession the expansion was real.
Rocketry inherited the same ratios unchanged: every engine bell from Goddard's to Raptor's is the A/A* row of this card, solved for a design altitude. NACA Report 1135 (1953) tabulated the whole family so engineers could stop re-deriving it — this card is that table with the interpolation done.
- 1757Leonhard EulerEquations of compressible fluid motion.
- 1888Gustaf de LavalThe converging–diverging nozzle goes supersonic.
- 1903Aurel StodolaMeasures pressure along a Laval nozzle — theory confirmed (circa).
- 1953NACA (Ames)Report 1135 tabulates the isentropic-flow relations.
See the full timeline of the math behind every calculator →
Runs entirely in your browser — nothing you enter leaves this page. Your recent runs are stored only on your device.