The important atmospheric parameters are the atmospheric temperature, pressure, density, and viscosity, which depend on the distance from the earth surface, geographic location, and time. In order to describe the atmosphere in a universal way, a standard atmosphere model has been developed, where the atmospheric parameters are determined as the univariate functions of altitude from sea level. Temperature exhibits strong variations with time of year, geographic location, and altitude. And, on a daily basis, temperature depends on current weather conditions in a stochastic manner. It is impossible to develop a first‐principles model that will capture all of these parameters that influence the temperature profile; thus, the standard temperature profile is determined from an average of a large ensemble of atmospheric measurements. The variation of pressure with altitude, however, is rigorously described by some basic physical principles – we will derive these here. In fact, pressure is so intricately and reliably linked to altitude that aircraft altimeters measure pressure and convert the measurement to an indicated altitude through the definition of the standard atmosphere. Density is related to the estimated value of temperature and the derived value of pressure via the ideal gas law. Finally, we will provide a relationship that determines the viscosity of air as a function of temperature. Based on these developments, we will define a standard atmosphere that can be expressed in tabular form, or equations coded for computational analysis. This chapter will start with a physical description of the atmosphere and then present a detailed development of the standard atmosphere. Most of the development of the standard atmosphere presented in this chapter will rely on SI units, since this was the unit system used to define the standard atmosphere and the boundaries of atmospheric regions. The input and output of the standard atmosphere can be easily converted from SI units to English units as needed.
2.1 Earth's Atmosphere
Earth's atmosphere is an envelope of air surrounding the planet Earth, where dry air consists of 78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.031% carbon dioxide, and small amounts of other gases (NOAA et al. 1976). In addition, air contains a small amount of water vapor (about 1%). The entire atmosphere has an air mass of about 5.15 × 1018 kg (1.13 × 1019 lb), and three quarters of the total air mass are contained within a layer of about 11 km (∼36,000 ft) from the Earth's surface.
There is a general stratification of Earth's atmosphere, which leads to the definition of distinct regions of the atmosphere: the troposphere (0–11 km), stratosphere (11–50 km), mesosphere (50–85 km), and thermosphere (85–600 km). The atmosphere becomes thinner as the altitude increases, and there is no clear boundary between the atmosphere and outer space. However, the Kármán line has been defined at 100 km and is often used as the border between the atmosphere and outer space. Atmospheric effects become noticeable during atmospheric reentry of spacecraft at an altitude of around 120 km. Aircraft propelled by internal combustion engines and propellers are generally limited to operating in the troposphere, while jet‐propelled aircraft routinely operate in the stratosphere.
Figure 2.1 illustrates the bottom three layers of Earth's atmosphere, which is where all atmospheric flight vehicles conduct flight. The delineation between the various regions of the atmosphere is based on historical measurements of temperature profiles, which lead to distinct regions with different temperature lapse rates. In the troposphere (the layer of the atmosphere nearest the surface), the air temperature generally decreases linearly with the altitude. This temperature reduction with altitude is due to the increasing distance from Earth and a concomitant reduction in heating from Earth's surface. Weather phenomena are directly dependent on this temperature reduction with altitude, causing most storms and other weather phenomena to develop and reside within the troposphere. The dividing boundary between the troposphere and the next layer (the stratosphere) is called the tropopause, at 11 km. Within the lower portion of the stratosphere (11–20 km), the air temperature remains constant; it then increases with altitude in the upper stratosphere (20–50 km), due to absorption of the sun's ultraviolet radiation by ozone in this region of the atmosphere.
Figure 2.1 The layers of Earth's atmosphere.
In contrast with the temperature–altitude profile, the variation of pressure with altitude is highly repeatable and deterministic. Air pressure continually decreases with altitude from Earth's surface all the way to the edge of the atmosphere. The primary reason for this is the action of Earth's gravitational acceleration on air, causing a given mass of air to exert a force on the air below it. Air at a given altitude must support the weight of all of the air mass above it, and it balances this force by pressure. As altitude increases, there is less air mass above that altitude, so there is less force (weight) acting on the air at that point and the pressure decreases. Thus, pressure decreases as altitude increases. We will discuss this physical mechanism in greater detail in Section 2.2, when we derive an expression for the variation of pressure with altitude.
2.2 Standard Atmosphere Model
A standardized model of the atmosphere allows scientists, engineers, and pilots in the flight testing community to have a commonly agreed‐upon definition of the properties of the atmosphere. The definition of the standard atmosphere includes the variation of gravitational acceleration, temperature, pressure, density, and viscosity as a function of altitude. There is actually more than one definition of the standard atmosphere: the U.S. Standard Atmosphere (NOAA et al. 1976) and the International Civil Aviation Organization (ICAO) Standard Atmosphere (ICAO 1993). Thankfully, the two definitions are identical at lower altitudes where aircraft fly – the only differences are in the upper stratosphere and beyond. Our discussion here will generally follow the development of the U.S. Standard Atmosphere (NOAA et al. 1976).1
2.2.1 Hydrostatics
The development of the standard atmosphere directly results from the hydrostatic equation, which is derived here based on a control volume analysis. Figure 2.2 illustrates an arbitrary control volume, measuring dx × dy × dhG, and the forces acting upon it (here, hG is the geometric altitude, or height above mean sea level (MSL)). The forces due to pressure acting on all of the side walls balance one another out in this static equilibrium condition, and we will consider only the forces acting in the vertical direction. The force acting upward on the bottom surface of the control volume is the pressure, p, times the cross‐sectional area dx dy. Similarly, on the top surface, we have a force of (p + dp)dx dy acting downward. (Here, the differential pressure dp accounts for pressure changes in the vertical direction.) Finally, we have the weight of the air inside the control volume acting downward, W = mg, where g is the local gravitational acceleration and the mass of the air inside the control volume can be found from the product of density and the volume,
(2.1)