Keyboard shortcuts

Press ← or → to navigate between chapters

Press S or / to search in the book

Press ? to show this help

Press Esc to hide this help

Atmosphere

In short

  • What it models: the air’s temperature, pressure, density, speed of sound and viscosity by height: the 1976 U.S. Standard Atmosphere, optionally shifted to field conditions, humid air, and weather-balloon soundings or forecasts; and the pressure altitude a barometric altimeter reads.
  • Sources: the U.S. Standard Atmosphere, 1976; the WMO’s Guide to Instruments and Methods of Observation (WMO-No. 8, 2023); the CIPM-2007 moist-air density formula (Picard et al., 2008).
  • How well it is validated: every value within 0.1% of the 1976 tables at 32 altitudes from −2 to 86 km; humid density within 0.047% of CIPM-2007 over 15–27 °C. Against RocketPy, its density agrees within 3.7e-4 over the 23 heights its parachute descents sample (Recovery). Its pressure altitude matches two flight logs’ own altimeter readings of their pressure, to 0.195 m and 1.321 m (report); the air itself has no real-flight check.
  • What it leaves out: a real day’s changes aloft. A field-condition offset holds all the way up (+20 K at a 1400 m field puts density +30% off the standard’s at 30 km), so higher flights need a sounding. Viscosity ignores humidity, which lowers it 2.1% at 30 °C and saturation.

Code and sources

Code: hpr_atmos::ussa76 (the standard and its offsets), hpr_atmos::moist (humid air), hpr_atmos::profile (soundings and forecasts), and the Atmosphere trait in hpr_atmos::air.

Sources:

  • [USSA] U.S. Standard Atmosphere, 1976, NOAA-S/T 76-1562, part 1, pinned as us-std-atmosphere-1976. Equation, table and page numbers below are its own.
  • [WMO] WMO-No. 8, Guide to Instruments and Methods of Observation, Vol. I (2023), wmo-no8-vol1-2023.
  • [CIPM] A. Picard et al., “Revised formula for the density of moist air (CIPM-2007)”, Metrologia 45 (2008) 149–155, picard-2008-cipm-2007.

Height datum

Every atmosphere is queried with geometric height above mean sea level (height_msl_m). The core’s heights are ellipsoidal (Frames), so the flight engine subtracts the geoid undulation first: H_msl = h − N. Samples carry an extrapolated flag, set whenever a model answers outside the range it is defined or tabulated over.

The 1976 standard, −5 km to 86 km

Seven layers in geopotential altitude H, each with a constant gradient of the molecular-scale temperature T_M ([USSA] Table 4):

base H_b (km′)011203247517184.852 (top)
L_M,b (K/km′)−6.50+1.0+2.80−2.8−2.0
 
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))           (33a)   L_M,b ≠ 0
P   = P_b exp[−g₀′ M₀ (H − H_b) / (R* T_M,b)]           (33b)   L_M,b = 0
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)
ν   = μ / ρ                                            (52)
  • Constants:
    • R* = 8314.32 J/(kmol·K) (p. 3), not CODATA’s 8314.46. Table 2 misprints the exponent’s sign.
    • M₀ = 28.9644 kg/kmol, g₀ = g₀′ = 9.80665, r₀ = 6 356 766 m.
    • P₀ = 101325 Pa, T₀ = 288.15 K, γ = 1.40, β = 1.458e-6.
    • S = 110.4 K (p. 19). Table 2 and p. 4 print 110 K, but the tables use 110.4 K: sea-level μ is 1.7894e-5 Pa·s with it and 1.7912e-5 with 110.
  • 80 to 86 km: M/M₀ comes from Table 8, interpolated linearly in Z. The printed tables leave it out below 86 km and print T = T_M (p. 9). This model follows the equations, so its kinetic temperature there is up to 0.036%, and its viscosity up to 0.031%, below the print.
  • Outside the range: below −5 km the first layer continues, and above 86 km the atmosphere is isothermal at 186.87 K. Both are flagged. The real standard is also isothermal from 86 to 91 km, then warms, and its composition changes above 86 km. Pressure and density there are rough, but tiny.
  • Loft got this wrong (Loft lessons L2 to L4):
    • It fed geometric altitude to geopotential formulas: at 11 km it gave 216.65 K and 22 632 Pa, against 216.774 K and 22 699.96 Pa.
    • It had only four layers, so at 70 km it gave 335 K against 219.6 K.
    • Its Sutherland constants gave a sea-level viscosity 1.3% high.

Offsets and launch-site conditions

Ussa76::with_offset(ΔT, P₀) adds ΔT to T_M at every geopotential height and integrates eqs. 33a/33b from P₀, so the atmosphere stays hydrostatic. Ussa76::anchored(Z, T, P) picks ΔT and P₀ so the profile passes through a measured temperature and pressure, such as field conditions.

This is not the aviation convention (ESDU 77022), which offsets temperature at equal pressure altitude. Both are hydrostatic, but they differ aloft. At H = 3000 m′ with ΔT = +20 K, the aviation convention gives 289.95 K where this gives 288.65 K, and densities 0.38% apart (validation/oracles/atmosphere/conventions.py). This choice keeps the lapse rate attached to height, like a sounding; see the atmosphere decision, ADR-004.

An anchor’s offset holds all the way up, which a real hot or cold day doesn’t. Anchoring +20 K at a 1400 m field, at the standard’s pressure there, gives these densities against the standard (conventions.py):

height3 km20 km30 km
density−5.7%+14%+30%

Field conditions suit flights of a few kilometers; higher flights need a sounding.

Pressure altitude: what a barometric altimeter reads

A barometric altimeter measures pressure, not height. It turns the pressure into the height at which the 1976 standard has that pressure, its pressure altitude, and subtracts the pad’s. Ussa76::pressure_altitude_m(P) computes the same number, so hpr can read its own flight the way a logged flight was read (Accuracy: real flights).

It inverts eqs. 33a and 33b in the layer whose base pressures bracket P:

H = H_b + (T_M,b / L_M,b) [(P / P_b)^(−R* L_M,b / (g₀′ M₀)) − 1]      L_M,b ≠ 0
H = H_b − (R* T_M,b / (g₀′ M₀)) ln(P / P_b)                         L_M,b = 0

In the standard’s troposphere this is the altimeter formula H = 44330.8 m × [1 − (P / 101325 Pa)^0.190263]. The result is in geopotential meters, m′, as an altimeter’s is.

A worked example. An altimeter on a pad at 86000 Pa reads 1361.8 m′ there. At 58000 Pa it reads 4464.4 m′, so it logs a climb of 3102.6 m′. On a day 20 K warmer than the standard all the way up, with the same sea-level pressure, the same two pressures lie 3318.0 m′ apart: the rocket climbed 6.9% more than its altimeter says, and the altimeter reads 6.5% less than the climb. Warm air is less dense, so pressure falls more slowly with height. In the troposphere, with the sea-level pressure unchanged, the ratio is exactly (T₀ + ΔT) / T₀ = 308.15 / 288.15.

Where it is used. hpr’s flights don’t use it: they fly in the air of the day. The real-flight comparison reads hpr’s height through it, from the ERA5 pressure at the center of mass, when the log is barometric (hpr_validate::real_flight::Barometer). Two logs that record their pressure, Prometheus’s and Juno III’s, are this reading less the first row’s, to 0.195 m and 1.321 m over the rows compared (report).

Moist air

An ideal mixture of dry air (M₀, γ = 1.4) and water vapour (M_v = 18.01528 kg/kmol, [CIPM] §2.1):

e_w(t) = 6.112 exp(17.62 t / (243.12 + t)) hPa        [WMO] Annex 4.B, eq. 4.B.1, t in °C
e      = U e_w(T),   x_v = e / p
ρ      = p [(1 − x_v) M₀ + x_v M_v] / (R* T)  =  p / (R_d T_v)      [WMO] eq. 12.18
C_p    = (1 − x_v)(7/2) R* + x_v · 4 R*,   γ = C_p / (C_p − R*),   a = (γ R* T / M)^½
  • Relative humidity is taken with respect to liquid water at every temperature, as radiosondes report it ([WMO] §12.1.2). No enhancement factor is applied. Near the surface it would raise e by about 0.47%, which changes density by at most 0.013% (at 40 °C and saturation), and less in cooler or drier air.
  • Accuracy:
    • Density agrees with CIPM-2007 (a real-gas equation) to 0.047% over its range: 15–27 °C, 600–1100 hPa, dry to saturated. Ignoring humidity entirely is 0.4% off at 20 °C and 50% RH.
    • Taking water vapour’s C_p as 4R* instead of its real value near 300 K (about 1% higher) moves a by 0.009% at 30 °C and saturation.
    • Viscosity stays dry air’s Sutherland value. By Wilke’s rule with IAPWS R12-08 vapour viscosity, saturation at 30 °C lowers it 2.1%, which moves turbulent skin friction about 0.4%.
    • All these numbers come from validation/oracles/atmosphere/moist_air.py.

Sounding and forecast profiles

SoundingProfile takes the site’s latitude and levels of geometric height, temperature, optional pressure, optional relative humidity, and optional wind speed and direction. Humidity and wind are given on every level or on none. A given pressure must be below the level beneath it, which catches hPa entered as Pa.

  • Gravity: the profile works in WMO geopotential height Z(z, φ) ([WMO] eqs. 12.15–12.16), whose gravity is the normal gravity at the site’s latitude.
    • Surface gravity runs from 9.780 m/s² at the equator to 9.832 m/s² at the poles, ±0.27% around the standard’s g₀.
    • Over 2 km at 293 K the same sounding’s pressure falls 0.12% more at the pole than at the equator.
    • A latitude-free geopotential would leave errors of that size.
  • Between levels, interpolation runs in Z:
    • T and U are linear in Z.
    • Pressure uses ln P = ln P_i + ln(P_{i+1}/P_i) · ln(T/T_i)/ln(T_{i+1}/T_i), which is exact for a dry hydrostatic layer with T linear in Z and passes through both levels’ pressures.
    • Levels sampled from the standard at its geopotential levels reproduce it between them to 1e-12, at any latitude.
  • Missing pressures above the lowest level are filled hydrostatically, with virtual temperature linear in Z ([WMO] eqs. 12.17–12.18).
    • A dry fill agrees with direct integration under WGS 84 normal gravity to 2e-8. The limit is WMO’s rounded surface gravity constants, 3–5e-8 low.
    • A humid fill agrees to 1.1e-5, because vapour pressure is exponential in T and so not quite linear across the layer.
  • Beyond the levels the profile continues as the standard atmosphere anchored at the end level and evaluated at the same geopotential (so still with the local gravity), and flags the sample:
    • Below, it holds relative humidity.
    • Above, it holds the vapour mole fraction, capped at saturation, so a humid top doesn’t carry water into the cold stratosphere.
    • The continued pressure is the dry standard’s. It is hydrostatic for dry air, but in humid air it falls up to x_v(1 − M_v/M₀) faster (1.6% of the gradient at 30 °C and saturation).
  • Geopotential heights. Soundings (Wyoming’s HGHT) and forecasts (Open-Meteo’s geopotential_height) report geopotential meters above sea level. Convert them with geometric_from_wmo_geopotential_m. At 30 km that is 29.7785 km of geopotential at the equator and 29.932 km at 80° N. The standard’s latitude-free r₀ formula is only for the standard itself.
  • From an ERA5 file: hpr_io::era5 builds this profile over a launch site at launch time from the day’s reanalysis; see ERA5 weather files.
  • From Open-Meteo: hpr_net::open_meteo builds it from a forecast, or an archived one, on pressure levels; see Launch-day weather.
  • From a weather balloon: hpr_net::wyoming builds it from a radiosonde’s sounding in the University of Wyoming’s archive; see Weather-balloon soundings.
  • Loft lesson L5: “today’s conditions” kept the standard lapse from the field up, ignored humidity and never used sounding temperatures.

RocketPy 1.13.0, for the code-to-code comparison

These are findings from reading refs/rocketpy for the RocketPy comparison of the validation milestone (M2.1), not yet pinned by fixtures:

  • Standard atmosphere: ISO 2533 layers from −2 to 80 km, with R = 287.05287 (the same as R*/M₀). Temperature is linear in geometric height between converted layer boundaries. Pressure is sampled at 100 points and splined.
  • Custom profiles:
    • Every quantity, pressure included, is linear in height above sea level, with constant extrapolation.
    • Linear pressure is off hydrostatic by up to 0.06% between 1000 and 925 hPa, 1.15% between 700 and 500 hPa, and 3.4% between 50 and 30 hPa (conventions.py).
    • M2.1 comparisons need a RocketPy-compatible option or levels dense enough that this doesn’t matter.
  • Humidity is not used anywhere.
  • Wyoming heights are converted from geopotential with a radius only (no latitude).
    • The helper’s default radius, 63 781 370 m in rocketpy/tools.py:972, is ten times the Earth’s.
    • Check which callers rely on that default before the M2.1 comparisons.

Tests that pin this

  • ussa76::tests::matches_the_1976_tables_at_32_altitudes (the done when of M1.2, the atmosphere and wind milestone):
    • Covers T, T_M, H, P, ρ, a, μ and ν at 32 altitudes from −2 to 86 km.
    • Every value is within 0.1%, and within one count of its last printed digit.
    • The exceptions are those above (80–85.5 km, where the printed T equals T_M) and the 84 km density. The latter prints 9.6940E-6 where the equations give 9.69387e-6; the row’s own ρ/ρ₀ agrees with the equations.
    • The fixture was transcribed from the page images and cross-checked by validation/oracles/ussa76/tables.py against mpmath and ambiance.
  • Loft lessons:
  • Pressure altitude:
    • pressure_altitude_inverts_the_1976_tables: each printed pressure’s altitude is the printed geopotential altitude at all 32 rows, to what the prints resolve.
    • pressure_altitude_is_the_altimeter_formula_in_the_troposphere, and its refusals.
    • pressure_altitude_inverts_the_pressure: a property test over offsets and heights, the extrapolated layers included.
  • Constants and structure:
    • constants_match_the_transcription checks the constants and Tables 4 and 8.
    • Hydrostatic balance dP/dZ = −ρg is checked in every layer, and as a property test over offsets.
    • Anchors reproduce their conditions, and layers join continuously.
  • moist::tests:
    • WMO 4.B.1 against the formula evaluated separately in mpmath (WMO prints no table of it).
    • Dry air equals the standard.
    • CIPM-2007 density to 0.05% at 36 points.
  • profile::tests:
    • Standard levels are reproduced at 0°, 45° and 80°.
    • Dry fills at 0°, 33° and 90°, and a humid fill, match direct integration under WGS 84 normal gravity.
    • Pressure falls faster at the pole.
    • The continuation beyond the levels is hydrostatic when dry, with the humid shortfall pinned.
    • WMO geopotential values.
    • Errors (including rising pressures) and serde.