Standard Atmosphere and Airspeeds: Method

Standard Atmosphere and Airspeeds

What the tool does

The tool gives the state of the air at a geometric altitude from the U.S. Standard Atmosphere, 1976 (temperature, pressure, density, speed of sound, dynamic and kinematic viscosity, geopotential altitude), accepts a temperature offset from the standard day and reports the pressure and density altitudes, and converts one airspeed quantity into every other: indicated, calibrated, equivalent and true airspeed, Mach number, impact and dynamic pressure, plus the Reynolds number per unit length. The atmosphere model and the airspeed relations are public documents: the 1976 standard (NOAA, NASA and the USAF, published as NASA TM X-74335) and NACA Report 837 (Aiken 1946), which fixed the nomenclature still used in flight manuals and in the FAA Pilot's Handbook.

The 1976 standard atmosphere

The standard idealises the dry atmosphere as a perfect gas in hydrostatic equilibrium with a prescribed temperature profile. Its sea-level constants are T₀ = 288.15 K (15 °C), p₀ = 101,325 Pa and, through the gas constant R = R*/M₀ = 8.31432 × 10³ / 28.9644 = 287.053 J/(kg·K), the density ρ₀ = p₀/(R T₀) = 1.2250 kg/m³ (0.0023769 slug/ft³). The acceleration of gravity is the constant g₀ = 9.80665 m/s² when the altitude is measured as geopotential altitude H, which is related to the geometric altitude Z by H = r₀ Z / (r₀ + Z) with the effective Earth radius r₀ = 6,356,766 m; the difference is 59 m at 20 km and 1,148 m at 86 km. The temperature is piecewise linear in H with seven layers:

0 to 11 km: −6.5 K/km; 11 to 20 km: 0 (216.65 K); 20 to 32 km: +1.0; 32 to 47 km: +2.8; 47 to 51 km: 0 (270.65 K); 51 to 71 km: −2.8; 71 to 84.852 km (86 km geometric): −2.0 K/km.

In a gradient layer T = T_b + L_b(H − H_b) and the hydrostatic equation integrates to p = p_b (T_b/T)^{g₀/(R L_b)}; in an isothermal layer p = p_b exp[−g₀(H − H_b)/(R T_b)]. The exponent in the troposphere is g₀/(R L) = 5.2559, so the pressure at the tropopause (11 km geopotential, 36,089 ft) is 22,632 Pa, 22.3 % of sea level, where the density is 29.5 %. Density is p/(R T), the speed of sound a = √(γ R T) with γ = 1.4 (340.29 m/s = 661.5 kt at sea level), the dynamic viscosity follows Sutherland's law μ = β T^3/2/(T + S) with β = 1.458 × 10⁻⁶ kg/(s·m·K^½) and S = 110.4 K (1.7894 × 10⁻⁵ Pa·s at sea level), and ν = μ/ρ. The tool's implementation reproduces every digit of the standard's Tables I, III and IV at the rows read from the document (see the test script), and the tool itself lists the standard at the usual altitudes.

The standard is a mean mid-latitude year-round atmosphere: a real day differs in temperature by tens of degrees, in surface pressure by a few percent, and the humidity it leaves out lowers the density by up to 1 % in hot, wet air. Performance work handles the temperature with the offset below and the surface pressure through the altimeter setting (enter the pressure altitude as the altitude).

ISA + ΔT, pressure altitude and density altitude

Flight manuals quote performance against pressure altitude and a temperature deviation from the standard, ISA + ΔT. The tool follows that convention: the pressure at the entered altitude is the standard pressure (so the entered altitude is the pressure altitude, what the altimeter shows at 29.92 inHg / 1013.25 hPa), the temperature is the standard value plus ΔT, and the density follows from the gas law. The density altitude is then the standard-day altitude having the same density; a warm day raises it (ISA + 20 °C at 35,000 ft gives a density altitude of 37,050 ft), and it is the altitude that governs lift, engine power and true airspeed. The offset is a temperature difference, so 36 °F equals 20 °C.

The airspeed quantities

Report 837 defines the quantities the airspeed system can give. The pitot-static system measures the impact pressure q_c = p_t − p, the excess of the total (pitot) pressure over the static pressure. For a perfect gas brought to rest isentropically,

q_c = p [ (1 + (γ − 1)/2 · M²)^γ/(γ−1) − 1 ] = p [ (1 + 0.2 M²)^3.5 − 1 ]   (eq. 1, M ≤ 1),

and above the speed of sound a normal shock stands ahead of the probe, so the Rayleigh pitot formula applies: q_c/p = [ (γ+1)² M² / (4γ M² − 2(γ−1)) ]^γ/(γ−1) · (1 − γ + 2γ M²)/(γ + 1) − 1, which for γ = 1.4 is 166.92 M⁷/(7M² − 1)^2.5 − 1. The two agree at M = 1.

Indicated airspeed (IAS) is the reading of the instrument, with its own mechanical error and the installation's position error (the static port does not sense the free-stream pressure exactly). Calibrated airspeed (CAS) is IAS corrected for both: CAS = IAS + ΔV_instrument + ΔV_position, the corrections coming from the instrument's card and the flight manual's airspeed calibration chart. Since the 1920s airspeed indicators have been calibrated to equation (1) at standard sea-level conditions, q_c = p₀[(1 + 0.2 (V_c/a₀)²)^3.5 − 1] (eq. 2): the calibrated airspeed is therefore the impact pressure expressed as the speed that would produce it at sea level, and it equals the true airspeed only there. Equivalent airspeed (EAS) is the speed that gives the actual dynamic pressure at sea-level density, V_e = √(2q/ρ₀) (eq. 13), where q = ½ ρ V² is the dynamic pressure of the actual flight (eq. 9); structural load specifications are written in EAS because every aerodynamic load is proportional to q. True airspeed (TAS) is the speed relative to the air, V = M a (eq. 6), and the Mach number follows from q_c/p at the altitude by inverting equation (1).

The chain is therefore: CAS → q_c by the sea-level calibration; q_c/p at the altitude → M; TAS = M a; q = 0.7 p M² (eq. 12, the same as ½ ρ V²); EAS = √(2q/ρ₀) = TAS √(ρ/ρ₀) = a₀ M √(p/p₀) (eq. 14). The tool runs the chain from whichever quantity is entered. Two corrections separate the speeds: the compressibility correction EAS − CAS, zero at sea level and negative at altitude (−12 kt at 35,000 ft and 250 KCAS), arises because the same impact pressure corresponds to a higher Mach number in the thinner air, where more of q_c is compressibility rise rather than dynamic pressure (Report 837 writes it as the factor f = √(q/q_c)); the density correction TAS = EAS/√σ is the familiar growth of true airspeed with altitude at a given indicated speed.

Reynolds number and total conditions

The Reynolds number per unit length is Re/L = V/ν with the true airspeed and the kinematic viscosity at the altitude (Report 837 eq. 21); the tool gives it per foot or metre and at an entered reference length (a chord or a body length). The total pressure p_t = p + q_c and the total temperature T_t = T (1 + 0.2 M²) are listed for probe and engine-inlet work; a real temperature probe recovers somewhat less than the full total temperature.

Reading the results

The first section prints the layer the altitude falls in, its base values and the layer equations with the numbers substituted, then the state of the air in the units of the selected system (both knots and ft/s or m/s for the speed of sound). The airspeed section writes each step of the chain with its Report 837 equation number and tabulates IAS, CAS, EAS, TAS and the speed of sound in knots, mph, km/h, m/s and ft/s. The table of the atmosphere lists the standard day at the usual altitudes; Figure 2 draws the three ratios against altitude over the whole 86 km with this case marked, and Figure 3 holds the entered quantity constant and shows the other airspeeds against altitude to 65,000 ft (20 km).

Limits

The altitude range is −5,000 m to 86,000 m geometric (−16,404 to 282,152 ft), the range of the standard's lapse-rate layers; above 86 km the standard's Part 2 describes a diffusively separating atmosphere that the tool does not model. The atmosphere is dry; the gas constant and γ are those of sea-level air throughout (the standard holds the molecular weight constant to 86 km). Supersonic impact pressures use the Rayleigh pitot formula, which assumes a normal shock ahead of a probe aligned with the flow; the hypersonic region beyond about M = 5 is outside the report's tables. The instrument and position corrections are taken as entered; the tool does not carry a calibration curve, and a position correction that varies with flap setting or angle of attack must be read from the airplane's chart for the condition in hand.