Module ussa76 

The U.S. Standard Atmosphere, 1976, from −5 km to 86 km geometric altitude, with an optional temperature offset and sea-level pressure.

Source: U.S. Standard Atmosphere, 1976, NOAA-S/T 76-1562 (NOAA, NASA and USAF, Washington, 1976), part 1, pinned as us-std-atmosphere-1976. Equation, table and page numbers below are that document’s. docs/physics/atmosphere.md has the details and the tests that pin them.

Model. Below 86 km the atmosphere is a sequence of layers in geopotential altitude H, each with a constant gradient L_M,b of the molecular-scale temperature T_M (Table 4):

H   = r₀ Z / (r₀ + Z)                                            (18)
T_M = T_M,b + L_M,b (H − H_b)                                    (23)
P   = P_b [T_M,b / T_M]^(g₀′ M₀ / (R* L_M,b))        L_M,b ≠ 0     (33a)
P   = P_b exp[−g₀′ M₀ (H − H_b) / (R* T_M,b)]        L_M,b = 0     (33b)
ρ   = P M₀ / (R* T_M)                                            (42)
T   = T_M M / M₀                                                 (22)
a   = (γ R* T_M / M₀)^½                                          (50)
μ   = β T^(3/2) / (T + S)                                        (51)

Z is geometric altitude, T the kinetic temperature, and M/M₀ the molecular-weight ratio, 1 below 80 km and tabulated from 80 to 86 km (Table 8). The printed tables leave M/M₀ out below 86 km and print T = T_M there (p. 9); this module follows the equations, so from 80 to 85.5 km its kinetic temperature is up to 0.036% and its viscosity up to 0.031% below the printed values.

Offsets. Ussa76::with_offset adds a constant ΔT to T_M at every geopotential height and integrates the hydrostatic equation from a chosen sea-level pressure, so pressure and density stay consistent with the warmer or colder temperature. With ΔT = 0 and P₀ = 101325 Pa it is the standard itself. Ussa76::anchored picks ΔT and P₀ to pass through a measured temperature and pressure at one height, such as the launch site.

Outside −5 km to 86 km the model extends the lowest layer downward and continues isothermally above 86 km, and flags every such sample as extrapolated. The real standard is also isothermal (186.87 K) from 86 to 91 km and warms above that, and its composition changes above 86 km, so pressure and density there are rough.

Structs

Ussa76

The U.S. Standard Atmosphere, 1976, optionally offset in temperature and sea-level pressure.

Constants

DRY_AIR_GAS_CONSTANT_J_PER_KG_K

Specific gas constant of dry air, R* / M₀, J/(kg·K).

EARTH_RADIUS_M

Effective Earth radius r₀ for geopotential altitude, m (pp. 4 and 8).

GAS_CONSTANT_J_PER_KMOL_K

Universal gas constant R* as the 1976 standard adopts it, J/(kmol·K) (p. 3; Table 2 on p. 2 misprints the exponent’s sign). It is not the current CODATA value (8.314462…e3); the standard’s tables are computed with this one.

MAX_HEIGHT_M

Highest geometric altitude of the model, m. Above this the standard uses a different formulation, which this crate does not implement.

MIN_HEIGHT_M

Lowest geometric altitude the standard defines, m.

RATIO_OF_SPECIFIC_HEATS

Ratio of specific heats of air γ (Table 2).

SEA_LEVEL_MOLECULAR_WEIGHT_KG_PER_KMOL

Sea-level mean molecular weight of air M₀, kg/kmol (p. 9, eq. 21).

SEA_LEVEL_PRESSURE_PA

Sea-level pressure P₀, Pa (Table 2).

SEA_LEVEL_TEMPERATURE_K

Sea-level temperature T₀, K (Table 2).

SUTHERLAND_BETA

Sutherland’s constant β for the viscosity of air, kg/(s·m·K^½) (Table 2 and p. 19).

SUTHERLAND_S_K

Sutherland’s constant S for the viscosity of air, K (p. 19). Table 2 and p. 4 print 110 K, but the tables are computed with 110.4 K: sea-level viscosity is 1.7894e-5 Pa·s with it and 1.7912e-5 with 110.

Functions

geometric_from_geopotential_m

Geometric altitude Z (m) from geopotential altitude H (m′), the inverse of eq. 18: Z = r₀ H / (r₀ − H). H must be below r₀, the geopotential altitude of infinity.

geopotential_from_geometric_m

Geopotential altitude H (m′) from geometric altitude Z (m), eq. 18, with the standard’s constant g₀ and effective radius r₀. Heights at or below −r₀ have no geopotential.

sutherland_viscosity_pa_s

Dynamic viscosity of air by Sutherland’s law with the 1976 standard’s constants, eq. 51: μ = β T^(3/2) / (T + S), Pa·s, for kinetic temperature T in K.