1. 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

2. 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]

3. 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

4. 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

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

6. 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
= lb/ft2
Use equation of state for density

7. 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

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

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

10. 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

11. Reading the Atmosphere Tables

12. 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 °R
other properties calculated as before

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

14. 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 = [ft/s2] or 3.72 [m/s2]
T0 = “sea-level” temperature = [°R] or 228 [K]
P0 = “sea-level” pressure = [lb/ft2] or 774 [Pa]
r0 = “sea-level” density = 3.43×10-5 [slug/ft3] or [kg/m3]
dT/dh = lapse rate = [°R/ft] or [K/m]