Overview
Small sounding rockets are far more sensitive to the atmosphere than orbital launchers: they spend their whole flight in the lower atmosphere, so the launch-day temperature and the height of the troposphere change drag, Mach number and apogee. The memo therefore keeps the constants of the U.S. Standard Atmosphere but builds a local temperature profile:
- the tropospheric temperature line passes through the measured launch-site temperature T^* at site altitude h^*;
- it ends at a latitude-dependent tropopause h_T(\varphi);
- the stratosphere above is isothermal.
Geodesy and gravity are kept simple: a sphere tangent to the WGS-84 ellipsoid at the launch latitude, Potsdam surface gravity, and an inverse-square altitude correction.
Atmosphere model
Temperature
In the troposphere convective mixing gives a nearly constant lapse rate. With the lapse rate \alpha=-0.00357448 °R/ft from the 1976 Standard Atmosphere,
Pressure
Combining the perfect-gas law p=\rho RT with hydrostatic equilibrium dp/dh=-\rho g gives dp/p=-g\,dh/RT. With T linear in h this integrates to a power law. Writing the sea-level temperature of the local profile as T_{SL}=T^*-\alpha h^*, so that T(h)=T_{SL}+\alpha h:
with p_{SL}=2116.217 lb/ft² on the geoid. At the tropopause p_T=p(h_T). In the isothermal stratosphere the integral is exponential:
Note on the memo’s exponent. The memo prints the tropospheric law as p_{SL}\big(1/(1+\alpha h/(T^*-\alpha h^*))\big)^{-g/R\alpha}. With the reciprocal inside the bracket the exponent must be +g/R\alpha (equivalently, -g/R\alpha without the reciprocal) — otherwise pressure would rise with altitude. The calculator uses the correct sign.
Density and speed of sound
with R=1716 ft·lb/(slug·°R) (mean molecular weight 28.9644) and \gamma=1.4. In the calculations g is the surface gravity at the launch latitude.
Tropopause
The tropopause is much lower over the poles than over the equator. The memo models it as an ellipse in latitude:
with a_T=52{,}500 ft (equatorial) and b_T=27{,}900 ft (polar). A lower tropopause means a warmer stratosphere, so the polar stratosphere is warmer than the tropical one — a counter-intuitive but well-documented result.
Geodesy
Latitude is geodetic: the angle between the equatorial plane and the local vertical (normal to the ellipsoid), as used on maps and in most trajectory codes. The Earth is modelled locally as the sphere tangent to the ellipsoid at that latitude, with radius
where a=20{,}925{,}597.9 ft and b=20{,}855{,}437.3 ft are the WGS-84 equatorial and polar radii. The Earth rotates at \Omega=2\pi/86164.09 rad/s (the sidereal day).
Gravity
Centripetal relief and the changing distance to the Earth’s centre both make surface gravity vary with latitude. The Potsdam result is
The “standard g” of 32.174 ft/s² is simply this formula evaluated at mid-latitude. For altitude, the approximate inverse-square correction is
Flight condition: Mach, dynamic pressure, Reynolds number
For a flight speed V at altitude h, the calculator gives the quantities the drag, stability and loads calculators need:
The viscosity uses Sutherland’s law with the constants of the U.S. Standard Atmosphere, 1976 (its NASA copy misprints them; the erratum bound into it gives S = 110.4 K), \beta = 1.458 × 10⁻⁶ kg/(s·m·K1/2) and S = 110.4 K, which gives 1.7894 × 10⁻⁵ Pa·s at 288.15 K. Use the body diameter as L for the nose and fins, or the overall length for the skin friction of the whole rocket.
Assumptions and limits
- Two layers only. Above about 66,000 ft (20 km) the real stratosphere warms with height; the model keeps it isothermal.
- Dry air and a constant lapse rate. Inversions near the ground, weather and humidity are not modelled; humid air is slightly less dense than dry air at the same temperature and pressure.
- Pressure uses the surface gravity at the launch latitude (no altitude variation inside the pressure integral).
- The memo’s lapse rate (−0.00357448 °R/ft) is 0.23 % steeper than the 1976 value (−0.0035662 °R/ft). For a standard sea-level day the model matches the 1976 table to about 0.02 % in temperature and 0.03 % in pressure at 10,000 ft.
Worked example
Launch site at h^*=4000 ft, T^*=550 °R (90 °F), latitude 34°, evaluated at 40,000 ft:
- Tropopause: h_T=39{,}190.87 ft; T_T=550-0.00357448(39190.87-4000)=424.21 °R.
- Surface gravity g_0=32.13922 ft/s²; local radius R_E=20{,}903{,}785 ft.
- Tropopause pressure p_T=474.487 lb/ft².
- 40,000 ft is above the tropopause, so T=424.21 °R and p=474.487\,e^{-32.139(809.13)/(1716\cdot424.21)}=457.835 lb/ft².
- \rho=6.2894\times10^{-4} slug/ft³ and a=1009.52 ft/s.
Equations ↔ code
| Quantity | Code (js/calc/geophysics.js) |
|---|---|
| Tropopause h_T | tropopause() |
| Surface gravity g_0 | surfaceGravity() |
| Earth radius R_E | earthRadius() |
| T_T,\ p_T | atmosphere().TT, .pT |
| T,\ p,\ \rho,\ a | atmosphere() |
| \Omega | atmosphere().omega |
| Constants | C |
Verification: tests/geophysics.test.js reproduces the reference example to 10⁻¹², checks the memo form against the 1976 standard atmosphere, and checks that memo-form pressure is continuous across the tropopause.
Notes on the method
- Tropospheric pressure datum. The tropospheric law uses altitude h above the geoid, not height h-h^* above the launch site. Sea-level pressure therefore sits on the geoid, and pressure is continuous across the tropopause.
- Gravity at altitude. g(h) from the memo’s inverse-square correction is reported alongside the surface value g_0.
References
- C. P. Hoult, “Sounding Rocket Geophysical Environment,” RS&T memo, 10 February 2010. View in library
- NOAA, NASA, USAF, U.S. Standard Atmosphere, 1976, NASA-TM-X-74335, 1976 (with the erratum bound into the NASA copy). NTRS
- Air Force Geophysics Laboratory, Handbook of Geophysics and the Space Environment, 3rd ed., 1985.
- Source file: GEOPHYSICS.xls (rsandt.com).