International Standard Atmosphere (ISA)

The International Standard Atmosphere is a static atmospheric model that defines how pressure, temperature, density, and viscosity change with altitude. It provides a common reference for aircraft performance calculations, engine design, ballistic trajectories, and virtually every aerospace engineering discipline.

Why Standardize?

Real atmospheric conditions vary with latitude, season, weather, and solar activity. Without a standard model:

• Aircraft performance data from one test site would be incomparable to another
• Engine manufacturers could not guarantee thrust specifications
• Altimeter calibrations would be ambiguous
• Re-entry trajectory predictions would have unacceptable uncertainty

The ISA model, codified in ISO 2533:1975, defines a “typical” mid-latitude atmosphere based on average sea-level conditions at 45 N latitude.

ISA Model Definition

Sea-Level Reference Conditions

Fundamental Equations

The ISA model is built on three physical laws applied layer by layer:

1. Hydrostatic Equilibrium — the pressure gradient balances gravity:

2. Ideal Gas Law — relates pressure, density, and temperature:

3. Temperature Profile — each layer has a defined lapse rate $\lambda = dT/dh$:

Where $T_b$ and $h_b$ are the temperature and altitude at the base of the current layer.

Pressure and Density Integration

Combining hydrostatic equilibrium with the ideal gas law:

For layers with non-zero lapse rate ($\lambda \neq 0$):

For isothermal layers ($\lambda = 0$, $T = T_b = \text{constant}$):

Speed of Sound

The speed of sound in an ideal gas depends only on temperature:

At sea level: $a_0 = \sqrt{1.4 \times 287.05 \times 288.15} = 340.29$ m/s.

Geopotential vs Geometric Altitude

The ISA model uses geopotential altitude $h$ rather than geometric altitude $z$. This accounts for the decrease in gravitational acceleration with height. The relationship is:

where $r_e = 6,356,766$ m is Earth’s effective radius. For the altitudes covered by the ISA (0–105 km), the difference is small — at 105 km geometric altitude, the geopotential altitude is approximately 103.3 km. The hydrostatic equation uses geopotential altitude so that $g_0$ remains constant.

Scale Height

The atmospheric scale height $H$ is the altitude over which pressure decreases by a factor of $e$ in an isothermal layer:

At sea level ($T = 288.15$ K): $H = 287.05 \times 288.15 / 9.80665 \approx 8,434$ m.

In the tropopause ($T = 216.65$ K): $H \approx 6,342$ m — pressure drops faster because the air is denser and colder.

The scale height concept appears when rearranging the isothermal pressure equation:

Every increase of one scale height reduces pressure to $1/e \approx 36.8\%$ of its initial value. After 5 scale heights, less than 1% of the original pressure remains.

Sutherland’s Law for Viscosity

Dynamic viscosity varies with temperature according to Sutherland’s law:

where $S = 110.4$ K is Sutherland’s constant for air, $\mu_0 = 1.716 \times 10^{-5}$ Pa·s at $T_0 = 273.15$ K. At 105 km altitude ($T \approx 208$ K), viscosity drops to approximately $1.4 \times 10^{-5}$ Pa·s.

Atmospheric Layers

The ISA divides the atmosphere into distinct layers, each with a characteristic temperature gradient:

Why Does Temperature Vary This Way?

The temperature profile is driven by the absorption of solar radiation at different altitudes:

• Troposphere (0–11 km): Heated from below by Earth’s surface. Temperature decreases because the surface is the primary heat source. The lapse rate of −6.5 K/km is close to the moist adiabatic lapse rate, reflecting the role of water vapor convection.

• Stratosphere (20–47 km): The ozone layer absorbs UV radiation (200–300 nm), converting it to thermal energy. The heating is strongest around 50 km where ozone concentration peaks, producing the stratopause temperature maximum of ~270 K (−2.5 °C).

• Mesosphere (51–85 km): Above the ozone layer, there are few molecules to absorb solar radiation, and CO₂ radiates heat to space. Temperature drops to the coldest point in the atmosphere — ~187 K (−86 °C) at the mesopause.

• Thermosphere (85+ km): Direct absorption of extreme UV and X-ray radiation by O₂ and N₂ molecules. Temperature rises rapidly, reaching hundreds of kelvin by 200 km. However, the air is so thin that a thermometer would read far lower due to the low heat capacity.

Mach Number and True Airspeed

The Mach number $M$ depends on the local speed of sound:

Since temperature decreases with altitude in the troposphere, the same true airspeed produces a higher Mach number at altitude. For example, an aircraft flying at 250 m/s:

• Sea level: $M = 250 / 340.3 = 0.73$
• 11 km: $M = 250 / 295.1 = 0.85$

This is why aircraft cruising at high altitude approach their Mach limits even at moderate true airspeeds. The relationship between true airspeed (TAS), calibrated airspeed (CAS), and density is:

At 10 km, density is ~34% of sea level, so $V_{TAS} \approx 1.71 \times V_{CAS}$ — the aircraft is moving 71% faster than the airspeed indicator suggests.

Non-Standard Day Corrections

Real atmospheres deviate from ISA. Temperature deviations are expressed as $\Delta T = T_{actual} - T_{ISA}$. For a +15 °C deviation at sea level, the density altitude (altitude at which ISA density equals actual density) can be approximated:

A +15 °C hot day at sea level produces a density altitude of approximately $15 / 0.0065 \approx 2,300$ ft — equivalent to a 700 m altitude increase for aircraft performance purposes.

Pressure deviations create pressure altitude corrections. When barometric pressure differs from ISA, altimeter readings must be corrected:

ISA Properties Calculator

Enter an altitude below and click Calculate to compute temperature, pressure, density, and speed of sound using the empirical ISA model.

Engineering Applications

Aircraft Performance

The ISA model is fundamental to aircraft performance calculations:

• Altimeter calibration: Pressure altitude is derived from $P(h)$
• True airspeed: $V_{TAS} = V_{CAS} \sqrt{\rho_0 / \rho(h)}$
• Engine power: Reciprocating engine power decreases as $\rho/\rho_0$ with altitude
• Density altitude: $h_d = h + (T - T_{ISA}) \times 120$ ft (approximate)

Re-entry Trajectories

Atmospheric density drives the deceleration profile during ballistic re-entry:

Gas Turbine Design

Off-standard day corrections use:

Corrected parameters: $N_{corr} = N/\sqrt{\theta}$, $\dot{m}_{corr} = \dot{m}\sqrt{\theta}/\delta$.

References

1. ISO 2533:1975 — Standard Atmosphere
2. U.S. Standard Atmosphere, 1976 — NOAA/NASA/USAF
3. Anderson, J.D. — Introduction to Flight, 8th Edition, McGraw-Hill, 2016
4. ESDU 77022 — Equations for Calculation of International Standard Atmosphere
5. ICAO Document 7488/3 — Manual of the ICAO Standard Atmosphere

All calculations use the ISO 2533:1975 / ICAO Standard Atmosphere model, valid from 0 to 105 km geopotential altitude.