Why must an atmospheric calculation stay inside its stated altitude and temperature range?

The boundaries built into the model

Crabwise's altitude calculations use the ICAO standard atmosphere in the troposphere. The pressure ratio is δ = (1 − 6.8755856e-6 × PA)^5.2558797, the temperature ratio is θ = (OAT + 273.15) ÷ 288.15, and density altitude comes from σ = δ ÷ θ. The Library worksheet enforces the same domain explicitly: pressure altitude from −2000 ft to 36089 ft, the top of the troposphere, and a temperature that must stay above −273.15°C.

Inside those limits the formula always returns a number. That number describes a dry standard-atmosphere model, not the weather at an airfield. Humidity, inversions and the actual pressure field are not inputs, and density altitude is not a performance figure.

Check an input against the domain

1. Decide whether your number is pressure altitude or a height read from a map; only pressure altitude belongs in the field.
2. Confirm it lies between −2000 and 36089 ft.
3. Enter the outside-air temperature in °C and check it is a physical value above absolute zero.
4. If a value falls outside, leave the result unavailable instead of clamping the input to the nearest edge.

Results at both ends of the range

At the upper edge, 36089 ft with an OAT of −56.5°C, the model's ISA temperature is 15 − 1.98 × 36.089 ≈ −56.45622°C, so the deviation is only −0.04378°C. The pressure ratio is about 0.223363 and density altitude comes out at about 36088.943839 ft, almost the same as the input because the day is close to standard.

At the lower edge, −2000 ft with 25°C, the ISA temperature is 18.96°C, the deviation is 6.04°C and density altitude is about −1292.944861 ft. Both are mathematically valid, yet neither says anything about a real aircraft or airport.

ISA temperature at 36089 ft (°C)

−56.45622

Density altitude at 36089 ft, −56.5°C (ft)

36088.943839

Density altitude at −2000 ft, 25°C (ft)

−1292.944861

Common mistakes

• Typing a stratospheric cruise level such as 39000 ft and expecting the troposphere formula to extend.
• Entering temperature in °F; 30°F and 30°C give very different density ratios.
• Treating a finite density altitude as proof that a take-off or climb is possible.

Scope

The domain check marks where this implementation is defined, not where flight is realistic. The NASA isentropic-flow pages and the FAA handbook listed below describe the underlying relations; performance always comes from the aircraft's own charts.

Primary sources and related work

• NASA: isentropic flow relationships
• FAA Pilot’s Handbook of Aeronautical Knowledge