Endurance⚠ unverified

Understanding the Parameters

• No speed term — the defining difference from range. Endurance measures time aloft, and time does not care how fast you cover ground. Dropping the $V$ that appears in Range Breguet leaves $E = (1/c_t)(L/D)\ln$. Consequently, maximum endurance and maximum range occur at different speeds: range wants max $L/D$; jet endurance wants minimum thrust (min drag, same as max $L/D$ for a jet), while propeller endurance wants minimum power — a slower speed still.

• $1/c_t$ — the engine's efficiency, identical in role to its appearance in range: lower fuel consumption means longer loiter. This is why long-endurance drones (Global Hawk, Reaper) pair very efficient engines with very high $L/D$ airframes.

• $L/D$ — the airframe efficiency. High-aspect-ratio wings (gliders, surveillance UAVs, the U-2) maximise it.

• The weight-ratio logarithm — again the fuel fraction enters as $\ln(W_i/W_f)$, with the same diminishing returns: the second half of the fuel buys less loiter than the first half.

• Jet vs propeller optimum — for a jet, best endurance is at the max-$L/D$ speed (minimum drag → minimum thrust → minimum fuel flow). For a propeller aircraft, best endurance is at the minimum-power speed, which is slower (about $0.76\times$ the min-drag speed), because a propeller's fuel flow tracks power, not thrust.

Derivation (Approaching a Proof)

Start from the same fuel-burn bookkeeping as Range Breguet, but integrate in time rather than distance. For a jet, fuel weight burned per second is $c_t\,T$, so

$$\frac{dW}{dt} = -c_t\,T.$$

In steady level flight $T = D = W/(L/D)$ (since $L = W$), giving

$$\frac{dW}{dt} = -c_t\,\frac{W}{L/D} \quad\Longrightarrow\quad dt = -\frac{1}{c_t}\,\frac{L}{D}\,\frac{dW}{W}.$$

This is exactly the range integrand without the factor $V$ (range had $dR = V\,dt$; here we keep $dt$). Integrating with $L/D$ and $c_t$ constant:

$$E = -\frac{1}{c_t}\,\frac{L}{D}\int_{W_i}^{W_f}\frac{dW}{W} = \frac{1}{c_t}\,\frac{L}{D}\,\ln\frac{W_i}{W_f}. \qquad\blacksquare$$

Why the propeller form is genuinely different. For a propeller aircraft, fuel flow tracks shaft power, $\dot W = c_p\,P/\eta_p = c_p\,DV/\eta_p$. The drag at the endurance condition is written with the aerodynamic coefficients, and eliminating $V$ through $L = W = \tfrac12\rho V^2 S C_L$ yields the standard propeller endurance:

$$E_{\text{prop}} = \frac{\eta_p}{c_p}\,\frac{C_L^{3/2}}{C_D}\,\sqrt{2\rho S}\left(\frac{1}{\sqrt{W_f}} - \frac{1}{\sqrt{W_i}}\right).$$

This depends on air density (so it is altitude-sensitive) and is maximised by the $C_L^{3/2}/C_D$-optimal speed (minimum power), not the $L/D$-optimal speed. The registry's shown formula is not this — it is the jet form, despite the "propeller" label. That is the flag.

Dimensional check (jet form). $$\frac{1}{c_t}\,\frac{L}{D}\,\ln\frac{W_i}{W_f} = \frac{1}{1/\text{s}}\times(\text{–})\times(\text{–}) = \text{s}. \checkmark$$

History and Development

• Shared Breguet lineage. Endurance and range emerged together from Louis Breguet's and contemporaries' 1920s analyses of long-distance flight; they are the two integrals of one fuel-burn equation.

• Endurance records. Loiter time, not distance, drove a distinct thread of aviation: the 1929 Question Mark aerial-refuelling endurance flights, the mid-century era of piston patrol aircraft, and the 1986 Rutan Voyager round-the-world flight (which was really a range record enabled by extreme fuel fraction — the logarithm pushed to its limit).

• The UAV era. Endurance became a headline metric with unmanned surveillance: aircraft like the U-2, the RQ-4 Global Hawk (30+ h), and solar aircraft (Airbus Zephyr, months aloft) are endurance-optimised — high $L/D$, low $c_t$, and, for solar craft, an effectively infinite weight ratio because the "fuel" is sunlight. These push exactly the levers this equation identifies.

Related Concepts: Range Breguet, Specific Range, Lift-to-Drag Ratio, Best Climb Speed, Drag Force, Lift Force, Thrust

Notes: Formula shown is the jet endurance form despite the "propeller-aircraft" description — the true propeller form is $\tfrac{\eta_p}{c_p}\tfrac{C_L^{3/2}}{C_D}\sqrt{2\rho S}(W_f^{-1/2}-W_i^{-1/2})$ and is density-dependent. Differs from Range Breguet only by the absent factor $V$ — which is why best-endurance and best-range speeds differ. Jet endurance optimum = max $L/D$ (min thrust); prop endurance optimum = min power (slower). Defaults $1.0$ ⇒ zero endurance.