The Atmosphere Calculating atmospheric properties Governing gas equations Solutions for troposphere and stratosphere The US standard atmosphere Reading atmosphere tables Non-standard atmosphere Altitude definitions

Governing Gas Equations Equation of state P = rRT True for a “perfect gas” Here, P = pressure [lb/ft2]  = density [slug/ft3] R = gas constant = 1716 [ft lb/slug °R] T = temperature (absolute) [°R]

Governing Gas Equations Hydrostatic equation dP = -r g dh Ordinary differential equation Geopotential altitude, h Gravitational acceleration, g varies below 50,000 ft, differences are small assume geometric altitude = geopotential altitude

Calculating Atmospheric Properties Mathematical relationships for properties in the atmosphere Based on governing equations and temperature model Needed for predicting aircraft / rocket performance at different altitudes Solve the hydrostatic equation (an ordinary differential equation) Temperature used as boundary conditions Two regions of interest require two solutions

Troposphere Equations Temperature gradient Ts = sea-level temperature hs = sea-level altitude = 0 ft dT/dh = lapse rate = -0.00356 °R/ft Solve hydrostatic equation for pressure Ps = sea-level pressure = 2116.2 lb/ft2 Use equation of state for density

Stratosphere Equations Isothermal (to 82,000 ft) T36,152 = temperature at 36,152 ft Solve hydrostatic equation for pressure P36,152 = pressure at 36,152 ft = 472.7 lb/ft2 Use equation of state for density

Calculating Atmospheric Properties 1) Start with sea-level conditions 2) Check altitude 3) If in troposphere (h  36,152 ft) a) find T at h b) find P at h 4) If in stratosphere (36,152 ft < h < 82,000 ft) a) find T at 36,152 ft b) find P at 36,152 ft c) set T = T36,152 d) use T36,152 and P36,152 to find P at h 5) Find  from equation of state

Calculating Atmospheric Properties Find temperature, pressure and density at 12,000 ft 1) Start with sea-level conditions Ts = 518.69°R (about 59°F) Ps = 2116 lb/ft2 rs = 0.002377 sl/ft3 2) Check altitude (which atmospheric region?) 3) a) Find T at h

Calculating Atmospheric Properties 3) b) Find P at h 4) Calculate r

The US Standard Atmosphere Based on mean annual US temperature Uses empirical temperature data Reference of atmospheric conditions Appendix B, Introduction to Aeronautics Table B.1 English (USCS) units, Table B.2 SI units Includes speed of sound, a and viscosity, m

Reading the Atmosphere Tables

Non-Standard Atmosphere Design to extreme (hot) conditions Phoenix in summer Denver in summer Navy hot day Generally T e.g., T = +50°R changes Ts from standard 518.69°R other properties calculated as before

Definitions of Altitude From US Std. Atmosphere flight data: P = 786.3 lb/ft2 r = 0.001104 sl/ft3 T = 436.8 °R h = 26,000 ft Geometric altitude Density altitude Pressure altitude Temperature altitude

Martian Atmosphere Mostly carbon dioxide – close to perfect gas Hydrostatic equation and equation of state still valid Different constants and “sea-level” conditions R = gas constant = 1148 [ft lb/slug °R] or 192 [J/kg K] g = gravitational constant = 12.21 [ft/s2] or 3.72 [m/s2] T0 = “sea-level” temperature = 410.4 [°R] or 228 [K] P0 = “sea-level” pressure = 16.165 [lb/ft2] or 774 [Pa] r0 = “sea-level” density = 3.43×10-5 [slug/ft3] or 0.01768 [kg/m3] dT/dh = lapse rate = -0.0016 [°R/ft] or -0.0030 [K/m]