International Standard Atmosphere and altitude definitions

The ISA model of temperature, pressure and density with height, and the pressure, density and geopotential altitude definitions used in performance work.

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Why it matters

Every performance number an aircraft manufacturer publishes — stall speed, takeoff distance, rate of climb, range — is quoted for a standard atmosphere, because the real atmosphere changes every hour. The International Standard Atmosphere (ISA) gives one agreed variation of temperature, pressure and density with height, so that tests flown on a hot day in Chennai and a cold day in Leh can be reduced to the same baseline and compared. Almost every numerical in aircraft performance starts with "find ρ at this altitude".

Key ideas

What the ISA is. The ISA is a model atmosphere: dry air treated as a perfect gas, in hydrostatic equilibrium, with a prescribed temperature profile. Only the temperature profile is assumed; pressure and density then follow from two laws:

  • the hydrostatic equation, dp = −ρ·g₀·dh (pressure falls because each layer supports the weight of the air above it);
  • the equation of state, p = ρ·R·T.

Sea-level values (given data, used throughout this subject): T₀ = 288.15 K (15 °C), p₀ = 101 325 Pa, ρ₀ = 1.225 kg/m³, g₀ = 9.806 65 m/s², R = 287.05 J/(kg·K) for air (many textbooks round to 287), a₀ ≈ 340.3 m/s.

Layers used in performance work.

  • Troposphere (0–11 km, gradient layer): temperature falls linearly with lapse rate L = 6.5 K/km = 0.0065 K/m. At 11 km, T = 216.65 K.
  • Lower stratosphere (11–20 km, isothermal layer): T stays at 216.65 K; pressure and density decay exponentially. At 11 km, p ≈ 22 632 Pa and ρ ≈ 0.3639 kg/m³. Above 20 km the temperature rises again; transport-aircraft performance rarely needs that layer.

Altitudes — six words that mean different things.

  • Geometric altitude h_G: true height above mean sea level, measured with a tape.
  • Geopotential altitude h: height in a field of constant g₀ that gives the same potential energy. h = R_E·h_G / (R_E + h_G); below 20 km the two differ by less than 0.4 %, so performance problems normally ignore the difference. The ISA tables are in geopotential altitude.
  • Absolute altitude: height above the ground directly below.
  • Pressure altitude: the ISA altitude at which the standard pressure equals the measured pressure. An altimeter set to 1013.25 hPa reads pressure altitude. Flight levels are pressure altitudes in hundreds of feet (FL350 = 35 000 ft pressure altitude).
  • Temperature altitude: ISA altitude with the measured temperature.
  • Density altitude: the ISA altitude at which the standard density equals the actual density. Aerodynamic forces and engine output depend on density, so this is the altitude the aircraft "feels".

On a standard day all of these coincide. On a hot day the air is less dense than ISA at that pressure, so density altitude is higher than pressure altitude — the classic "hot and high" penalty on takeoff and climb.

ISA deviation. Real days are described as "ISA + ΔT": the measured temperature minus the ISA temperature at that pressure altitude. ISA + 20 at a 2 km airfield means the outside air is 20 K warmer than 275.15 K.

Limits. The ISA ignores humidity (moist air is slightly less dense), weather systems and latitude. It is a reference, not a forecast.

Formulas

T = T₀ − L·h (troposphere, 0 ≤ h ≤ 11 000 m)

p / p₀ = (T / T₀)^(g₀ / (L·R)), with g₀/(L·R) = 5.256

ρ / ρ₀ = (T / T₀)^(g₀ / (L·R) − 1), with exponent 4.256

p = p₁₁ · exp[−g₀·(h − 11 000) / (R·T₁₁)] and ρ = ρ₁₁ · exp[−g₀·(h − 11 000) / (R·T₁₁)] (isothermal layer, 11–20 km)

p = ρ·R·T (equation of state)

σ = ρ / ρ₀, δ = p / p₀, θ = T / T₀, so σ = δ / θ

a = √(γ·R·T) (speed of sound, γ = 1.4)

Symbols: T temperature (K); p pressure (Pa); ρ density (kg/m³); h geopotential altitude (m); L lapse rate (K/m); g₀ standard gravity (m/s²); R gas constant of air (J/(kg·K)); σ, δ, θ density, pressure and temperature ratios (dimensionless); subscript 0 = ISA sea level, 11 = tropopause. Use the power laws only inside the troposphere and the exponential only in the isothermal layer; for a point above 11 km, first get p₁₁, ρ₁₁ from the troposphere law.

Worked examples

Example 1 (standard): conditions at 5000 m in ISA. Given: h = 5000 m; ISA sea-level data above.

  1. Temperature: T = T₀ − L·h = 288.15 − 0.0065 × 5000 = 255.65 K.
  2. Pressure: p = p₀·(T/T₀)^5.256 = 101 325 × (255.65/288.15)^5.256 = 101 325 × 0.5331 = 54 020 Pa.
  3. Density: ρ = p/(R·T) = 54 020 / (287.05 × 255.65) = 0.736 kg/m³ (σ = 0.601).

Example 2 (GATE level): density altitude of a hot, high airfield. Given: an airfield where the station pressure is 79 500 Pa and the outside air temperature is 20 °C (293.15 K). Find the pressure altitude, the ISA deviation and the density altitude.

  1. Pressure altitude from p/p₀ = (1 − L·h/T₀)^5.256: h_p = (T₀/L)·[1 − (p/p₀)^(1/5.256)] = 44 331 × [1 − (0.78460)^0.19026] = 44 331 × 0.04510 ≈ 2000 m.
  2. ISA temperature at 2000 m = 288.15 − 13.0 = 275.15 K, so ISA deviation = 293.15 − 275.15 = ISA + 18 K.
  3. Actual density: ρ = p/(R·T) = 79 500 / (287.05 × 293.15) = 0.9448 kg/m³, so σ = 0.9448/1.225 = 0.7712.
  4. Density altitude from σ = (1 − L·h/T₀)^4.256: h_d = 44 331 × [1 − 0.7712^(1/4.256)] = 44 331 × 0.05921 ≈ 2625 m. The aircraft performs as if it were 625 m higher than its pressure altitude.

Common mistakes

  • Using the troposphere power law above 11 km. Between 11 and 20 km the temperature is constant and the variation is exponential.
  • Using °C instead of K in p = ρRT or in T/T₀.
  • Using exponent 5.256 for density. Density uses 4.256 (one less), because ρ ∝ p/T.
  • Mixing up pressure altitude and density altitude: an altimeter at 1013.25 hPa gives pressure altitude; performance charts need density altitude, which needs temperature as well.
  • Writing the exponent as g·M/R·L on a calculator without brackets; it is g₀/(L·R) = 9.80665/(0.0065 × 287.05).
  • Assuming hot air means higher pressure. At a given pressure, hotter air is less dense.

For GATE AE

Expect direct numericals: T, p or ρ at a given altitude in the troposphere or the isothermal layer, the density ratio σ needed for a later performance step, and the altitude at which a given density or pressure occurs. Short conceptual questions test the definitions of pressure, density and geopotential altitude and which way they move on a hot day. Practise the inverse problem (altitude from a given ratio), and practise carrying a value across the 11 km tropopause.

Quick check

  1. What is the ISA temperature at 8 km?
  2. At a given pressure altitude, the day is hotter than ISA. Is the density altitude higher or lower than the pressure altitude?
  3. What is the exponent in ρ/ρ₀ = (T/T₀)^n in the troposphere?
  4. How does density vary with height between 11 km and 20 km in ISA?

Answers: 1. 236.15 K (−37 °C). 2. Higher. 3. n = g₀/(L·R) − 1 ≈ 4.256. 4. Exponentially, ρ = ρ₁₁·exp[−g₀(h − 11 000)/(R·216.65)].

Interview questions

All Aircraft Performance interview questions

Try answering each one aloud before you open it.

  1. 1.What is the International Standard Atmosphere (ISA)?Concept

    The International Standard Atmosphere (ISA) is a model used to represent the average atmospheric conditions at various altitudes. It provides standard values for temperature, pressure, and density at different altitudes, which are used for calibrating instruments and designing aircraft. The ISA assumes a sea-level temperature of 15°C and a pressure of 1013.25 hPa, with a temperature lapse rate of 6.5°C per kilometer up to 11 km.

  2. 2.Explain the different altitude definitions used in aviation and why each is needed.Concept

    Geometric altitude is the true height above mean sea level; absolute altitude is the height above the ground below. Pressure altitude is the ISA altitude whose standard pressure equals the measured pressure, which is what an altimeter set to 1013.25 hPa reads and what flight levels use for traffic separation. Density altitude is the ISA altitude with the same density as the actual air; it governs lift, drag and engine output, so performance charts use it. Geopotential altitude is the height in a constant-g field with the same potential energy and is the variable the ISA tables are written in.

  3. 3.What is pressure altitude and how is it measured?Concept

    Pressure altitude is the altitude above the standard datum plane where atmospheric pressure is 1013.25 hPa. It is measured using a barometric altimeter, which is set to the standard pressure setting of 1013.25 hPa. Pressure altitude is used for flight planning and performance calculations, especially when flying above the transition altitude where standard pressure settings are used.

  4. 4.Why is the ISA model used in aircraft performance calculations?Application

    The ISA model is used in aircraft performance calculations because it provides a standardized reference for atmospheric conditions. This allows for consistent and comparable performance data across different aircraft and flight conditions. By using ISA, engineers and pilots can predict how an aircraft will perform under average conditions, which aids in design, testing, and operational planning.

  5. 5.What happens to aircraft performance if the actual atmospheric conditions deviate from the ISA?Application

    If actual atmospheric conditions deviate from the ISA, aircraft performance can be affected. For example, higher temperatures than ISA can lead to reduced engine performance and lift, requiring longer takeoff distances. Conversely, lower temperatures can improve performance. Pilots must adjust their calculations and flight plans to account for these deviations to ensure safety and efficiency.

  6. 6.How does density altitude affect aircraft takeoff performance?Application

    Density altitude is a measure of air density, which affects aircraft performance. Higher density altitudes, often caused by high temperatures, high humidity, or high elevation, result in lower air density. This reduces engine power, lift, and propeller efficiency, leading to longer takeoff distances and reduced climb rates. Pilots must consider density altitude when planning takeoffs, especially from high-altitude airports.

  7. 7.An airfield is at sea level and the local QNH is 30.12 inHg. Estimate its pressure altitude.Numerical

    Near sea level the pressure falls by roughly 1 inHg per 1000 ft, so pressure altitude ≈ field elevation + (29.92 − QNH) × 1000 ft. Here that is 0 + (29.92 − 30.12) × 1000 = −200 ft. The negative value means the local pressure is higher than standard, so the field sits below the 29.92 inHg standard datum. This is a rule of thumb; the exact value comes from the ISA pressure law.

  8. 8.Explain how temperature affects the calculation of density altitude.Application

    Temperature affects density altitude because it influences air density. Higher temperatures cause air to expand, reducing its density and increasing the density altitude. Conversely, lower temperatures increase air density, decreasing the density altitude. Pilots use temperature data to adjust their calculations of density altitude, which is critical for assessing aircraft performance during takeoff and climb.

  9. 9.What is the lapse rate in the ISA model, and why is it important?Concept

    The lapse rate in the ISA model is the rate at which temperature decreases with an increase in altitude. In the troposphere, the standard lapse rate is 6.5°C per kilometer. This rate is important because it helps in predicting temperature changes with altitude, which is essential for performance calculations, weather forecasting, and understanding atmospheric stability.

  10. 10.An aircraft is at a pressure altitude of 10,000 ft with an outside air temperature of −5 °C. What is the ISA deviation?Numerical

    ISA temperature at 10,000 ft (3.048 km) is 15 − 6.5 × 3.048 = −4.8 °C. ISA deviation = actual − ISA = −5 − (−4.8) = −0.2 °C, so the day is essentially standard (ISA − 0.2). The deviation is always referred to pressure altitude, because that is the ISA altitude for the measured pressure. Pilots often use the rule of thumb 2 °C per 1000 ft, which gives ISA −5 °C and a deviation of 0.

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