Starred ratios and the length to sonic
For a perfect gas, T/T* = (γ+1) / (2 + (γ−1) M²), P/P* = (1/M) √((γ+1) / (2 + (γ−1) M²)), and the stagnation-pressure ratio P0/P0* follows the usual isentropic factor on those static values. The friction length from Mach M to the sonic state is 4fL* / D = (1−M²)/(γ M²) + (γ+1)/(2γ) ln{ (γ+1) M² / (2+(γ−1)M²) }. At Mach 1 every starred ratio is 1 and the remaining length is zero. Friction drives a subsonic Fanno flow toward Mach 1 and a supersonic Fanno flow toward Mach 1; a long enough duct chokes.
This is the closed-form table, not a numerical march of f along a pipe with a varying Reynolds number. For isentropic area change see isentropic flow; for heat addition in a constant-area duct, see Rayleigh flow; for shocks, see normal shock.
Sources
- John and Keith, Gas Dynamics; Zucker and Biblarz, Fundamentals of Gas Dynamics.
- Anderson, Modern Compressible Flow.
- Standard γ = 1.4 air tables (4fL* / D ≈ 1.069 at M = 0.50).
Frequently asked questions
What is the starred state?
The sonic (M = 1) reference on the same Fanno line — the state friction would produce if the duct were just long enough to choke.
Can I enter a length and solve for Mach?
Not on this page. It tabulates the starred ratios and 4fL* / D from a Mach number. Inverting 4fL* / D for M is a root find you can do against the same formula.
Does this include heat transfer?
No. Heat addition in a constant-area duct is Rayleigh flow, on a separate page.