# Density Altitude > Use when you must compute the density altitude from the pressure altitude and the outside temperature: the ISA deviation, the density ratio sigma from the ISA pressure ratio and the temperature ratio, and the density altitude via the troposphere closed-form inverse with the stratosphere branch. Produces the density altitude in meters and feet for hot-day and cold-day takeoff and performance checks. Trigger: density altitude, pressure altitude, outside air temperature, non-standard day, ISA deviation, density ratio, hot day, cold day, takeoff performance, performance reduction. - Skill: `ashfordeou/density-altitude` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/density-altitude` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/density-altitude/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/density-altitude --- # Density Altitude (cross-cutting/units-atmos/density-altitude) Use when the task is the density altitude on a non-standard day: the pressure altitude and the outside air temperature (OAT) collapse to the altitude the air would occupy on an ISA day, which gates takeoff distance, climb rate and performance reduction. This leaf owns the inverse-density step: the ISA deviation, the density ratio sigma = delta(hp) * T0 / T_oat, and the density altitude from the troposphere closed-form inverse with the stratosphere branch, in pure Python stdlib only. It pairs with cross-cutting/units-atmos/ isa-atmosphere, the forward ISA model this leaf inverts, and serves the domain consumers that previously duplicated this function. ## Domain quick reference - Constants: T0 = 288.15 K, L = 0.0065 K/m, g = 9.80665 m/s2, R = 287.0 J/(kg K), p0 = 101325 Pa, tropopause 11000 m, isothermal stratosphere 216.65 K. - ISA temperature at pressure altitude hp: T0 - L * hp in the troposphere, 216.65 K above it (isa_temperature_k). - ISA pressure ratio: delta(hp) = (1 - L hp / T0)^(g/(R L)) in the troposphere, exponential decay from the tropopause state above (isa_pressure_ratio). - ISA deviation: DeltaT = OAT - T_ISA(hp) (isa_deviation_k). - Density ratio: sigma = delta(hp) * T0 / T_oat. At fixed pressure altitude a warm day gives sigma below the ISA value and raises the density altitude; a cold day lowers it. - Troposphere density altitude, closed form: h_rho = (T0 / L) * (1 - sigma^e) with e = 1 / (g/(R L) - 1), the exact inverse of the density power law (about 0.2349). Note: the exponent e equals R L / (g - R L) with the module constants; the classical R L / g inverts the pressure law, not the density law, and would break the ISA identity, so the exact density inverse is used here. - Stratosphere branch (sigma below the tropopause density ratio): h_rho = 11000 - (R * 216.65 / g) * ln(sigma / sigma_trop). The two branches meet exactly at the tropopause. - Units: hp and altitude in meters (ft wrapper accepts feet), OAT in kelvin for the meter forms and degrees C for the ft wrapper. ## Workflow 1. Fix the pressure altitude and the OAT at the test point. 2. Read the ISA temperature there with isa_temperature_k. 3. Get the day's non-standard character with isa_deviation_k. 4. Read the ISA pressure ratio with isa_pressure_ratio, then the density ratio sigma with density_ratio_from_pressure_temperature or the direct product delta * T0 / oat_k. 5. Compute the density altitude with density_altitude_m (meters, kelvin) or density_altitude_ft (feet, degrees C). 6. Collect the full set with density_altitude_summary. 7. Confirm the deterministic checks with the contract test. ## Worked example Module outputs on the spec anchors (real values, R = 287.0): - ISA identity: density_altitude_m(0, 288.15) = 0.00 m and density_altitude_m(3048, isa_temperature_k(3048)) = 3048.00 m (h_rho = h_p on the standard day, exact at 10000 ft); the identity also holds at the tropopause and in the stratosphere branch. - Hot day, sea level: OAT 30 C, 15 K above the standard day: density_altitude_m(0, 303.15) = 525.34 m, density_altitude_ft(0, 30) = 1723.55 ft (about 525.46 m = 1723.94 ft by the independent check). - Warm day at 10000 ft pressure altitude, OAT = ISA + 10 K (5.19 C): density_altitude_ft(10000, 5.188) = 11159.18 ft, above the pressure altitude (about 11159.44 ft by the independent check). - Cold day at 10000 ft, OAT = ISA - 10 K (-14.81 C): density_altitude_ft(10000, -14.812) = 8786.21 ft, below the pressure altitude (about 8785.93 ft by the independent check). - Domain cross-check: density_altitude_ft(10000, 15) = 12247.62 ft, reproducing the climb-performance-flight-test bisection result of 12248.13 ft at OAT 15 C absolute on a 10000 ft pressure altitude. ## Pitfalls - Feeding OAT in degrees C to the meter form (or kelvin to the ft wrapper): density_altitude_m takes kelvin and density_altitude_ft takes degrees C; mixing the scales shifts the result by 273.15 K. - Confusing density altitude with pressure altitude: the two are equal only on the standard day; an OAT deviation moves them apart (ISA +10 K at 10000 ft reads about 11159 ft, ISA -10 K about 8786 ft). - Using the ISA temperature instead of the day's measured OAT at the test point: the whole quantity exists to capture the non-standard deviation, so the standard-day temperature gives back the pressure altitude by construction. - Passing OAT at or below 0 K or a pressure altitude below -1000 m: the module raises ValueError rather than extrapolating, though small negative geopotential altitudes are allowed down to -1000 m. - Expecting one lapse behavior across the whole atmosphere: the ISA temperature and pressure ratio branches switch at the tropopause, and the closed-form inversion matches the bisection only within the correct branch. ## Verification - ISA identity: density altitude equals pressure altitude at 0 m, 3048 m and the tropopause on the standard day, to 0.2 m. - Worked magnitudes: sea level OAT 30 C about 525.4 m / 1723.6 ft; 10000 ft ISA +10 K about 11159 ft; ISA -10 K about 8786 ft. - Monotonicity: at fixed pressure altitude a warmer OAT raises the density altitude and a colder one lowers it; at fixed OAT a higher pressure altitude raises it. - Closed form versus 200-iteration bisection of the forward ISA density law agrees to 1e-6 on both the troposphere and the stratosphere branches; the branch switch is continuous. - Determinism: identical outputs run to run, no RNG, offline. - ValueError rejection: pressure altitude below -1000 m (small negative geopotential is allowed) and OAT at or below 0 K. - Run the contract test offline: python3 scripts/test_density_altitude.py (35 tests, deterministic). ## Related leaves - cross-cutting/units-atmos/isa-atmosphere: the forward ISA state model this leaf inverts. - cross-cutting/units-atmos/airspeed-conversion: takes pressure altitude as input and pairs with the density altitude for air-data reduction. - flight-test-operations/performance/climb-performance-flight-test: the domain consumer that previously duplicated this function. ## Behavior contract (gate 3) Run the deterministic contract test (stdlib unittest, offline): python3 scripts/test_density_altitude.py The test covers the module constants and sea-level anchors, the ISA temperature and pressure ratio branches, the ISA deviation, the density ratio from pressure and temperature, the ISA identity at 0 m, 3048 m, the tropopause and in the stratosphere, the worked magnitude anchors (sea level OAT 30 C, 10000 ft ISA +10 K and ISA -10 K), the domain-precedent cross-check at 10000 ft OAT 15 C, monotonicity in temperature and pressure altitude, closed-form agreement with an independent 200-iteration bisection on both branches, continuity across the tropopause branch switch, ValueError rejection of non-physical inputs including the -1000 m boundary, the ft wrapper against the meter form, the exact summary dict key set and its consistency with the functions, and run-to-run determinism. ## Compliance - Standards referenced, not reproduced: ECSS frames the atmosphere reference context; the ISA model is common reference data and the relations above are standard engineering methodology, summary-only per standards-map.yaml. - compliance: STANDARDS-REF, gated: false.