Chapter 3: The Standard Atmosphere – Notes

The Standard Atmosphere (Chapter 3) – Comprehensive Notes

  • Overview and purpose

    • Aerospace vehicles operate in two basic categories: atmospheric vehicles (airplanes/helicopters) and space vehicles (satellites, lunar/Mars probes, deep-space probes).
    • Space vehicles may briefly interact with atmospheres during launch, reentry, or planetary flybys; thus atmospheric properties must be considered in design and performance.
    • The Earth's atmosphere is a dynamically changing system. Properties like pressure, temperature, and density depend on altitude, location, time of day, season, and solar activity.
    • A standard atmosphere is defined to relate flight tests, wind tunnel results, and aircraft design to a common reference. It provides mean values of pressure, temperature, density, etc., as functions of altitude, based on experimental data plus a mathematical model.
    • Main purpose: to give reference tables (Apps. A and B) and to show how those tables are constructed and used for aerospace analyses.

  • Preview and motivation (Preview Box)

    • Before studying flight through air, one should know atmospheric properties: temperature, density, pressure.
    • Properties change with altitude; questions arise: how do they change with altitude? how to determine? These are addressed in this chapter.
    • The standard atmosphere provides a reasonable average reference for design, testing, and analysis, not exact real-time conditions.
    • The standard tables come from empirical measurements (balloons, sounding rockets) combined with a mathematical model; regions of the atmosphere are modeled with isothermal (constant T) and gradient (linearly varying T) layers.
    • Appendix A (SI units) and Appendix B (English engineering units) tabulate temperature, pressure, and density at various altitudes.
    • Road map of Chapter 3: derive definitions and the hydrostatic equation, then explain how the standard atmosphere is constructed, and finally define altitude types derived from the tables.

  • Important definitions and scope

    • Six altitudes used in the text: absolute altitude, geometric altitude, geopotential altitude, pressure altitude, temperature altitude, density altitude.
    • The standard atmosphere in this chapter is for Earth; methods also apply to other planets but Earth is the main focus.
    • Several standard atmospheres exist (e.g., ARDC 1959). For practical purposes differences below 30 km are insignificant; the book uses the 1959 ARDC model.

3.1 Definition of Altitude

  • Geometric altitude, h_G
    • Definition: distance above the Earth's surface.
  • Absolute altitude, h_a
    • Definition: distance above the center of the Earth (distance to Earth's center along the radius). If r is Earth’s radius, ha = hc + r.
  • Relationship to gravity variation
    • Local gravitational acceleration g varies with altitude: g = g0 (1 + hG/r)^{-2} ≈ g0 when hG is small.
    • This gravity variation is important for accurate atmospheric modeling but is often simplified in standard atmosphere calculations by using a constant g0.
  • Geopotential altitude, h
    • A fictitious altitude introduced to simplify hydrostatic integration by assuming constant gravitational acceleration g0.
    • It differs from h_G by a small amount, especially at higher altitudes; the geopotential altitude is the variable used in the standard-atmosphere equations (Eqs. 3.7–3.14).
  • Practical note on altitude usage
    • In this book, equations are derived with geopotential altitude h, while the tabulated standard-atmosphere values in Apps. A and B are given in terms of geometric altitude h_G. The two are related by a specific transformation (Eq. 3.6).

3.2 Hydrostatic Equation

  • Foundational model: force balance on a thin fluid element at rest.
  • Consider a small air element with bottom pressure p and top pressure p + dp, height dh_G, and density ρ.
  • For the element to be in equilibrium (no acceleration), the vertical forces balance:
    • Upward: pressure on bottom face, p × 1 × 1
    • Downward: pressure on top face, (p + dp) × 1 × 1, plus weight ρ g dh_G
  • Hydrostatic equation (exact form):
    • dpdhG=−ρg\frac{dp}{d h_G} = - \rho g
  • When using the equations of state and assuming g constant (g ≈ g0), this becomes:
    • dpdh=−ρg0\frac{dp}{d h} = - \rho g_0
  • Key point about g and h
    • The variable h in the hydrostatic equation is geopotential altitude, while h_G is geometric altitude. The two are connected via the relation that accounts for the variation of g with altitude:
    • The geopotential-altitude formulation simplifies the mathematics, but one must relate back to geometric altitude when comparing with tabulated Earth values.
  • Practical assumption
    • For many standard-atmosphere calculations, g is taken as constant equal to g0 to obtain closed-form expressions.

3.3 Relation Between Geopotential and Geometric Altitudes

  • Goal: relate p(h) to p(h_G).
  • Start from the hydrostatic equation in geopotential form and the definition of gravity variation, then divide Eq. (3.3) by Eq. (3.2) to obtain a link between h and h_G.
  • Core relationship (exact form for Earth’s gravity variation):
    • h=r h<em>Gr+h</em>Gh = \frac{r\, h<em>G}{r + h</em>G}
    • Equivalently, solving for h_G:
    • hG=r hr−hh_G = \frac{r\, h}{r - h}
  • Interpretation
    • h is geopotential altitude (a convenient fictitious altitude for calculations with g ≈ g0).
    • h_G is geometric altitude (the actual height above sea level).
    • For low altitudes (hG << r), h ≈ hG (differences begin to appear at higher altitudes; above ~65 km the difference exceeds 1%).
  • Practical note
    • The standard-atmosphere tables use hG as the primary altitude in Apps. A and B; whenever p(h) is computed, one can convert to p(hG) using Eq. (3.6).
  • Quick numerical example (Earth): r ≈ 6.3568 × 10^6 m; h_G = 7 km leads to h ≈ 6.9923 km (≈ 0.1% difference).

3.4 Definition of the Standard Atmosphere

  • Keystone idea: a defined variation of temperature with altitude, T(h), based on experimental evidence.

  • Temperature distribution T(h) is piecewise linear (Fig. 3.4): a series of straight lines, some isothermal (vertical segments) and some with a slope (gradient layers).

  • Once T(h) is defined, pressure p(h) and density ρ(h) follow from hydrostatics and the equation of state.

  • Key constants and base values (sea level, geopotential altitude base h = 0):

    • Sea-level standard values:
    • Ps=1.01325×105 N/m2P_s = 1.01325 \times 10^{5}\ \text{N/m}^2
    • ρs=1.2250 kg/m3\rho_s = 1.2250\ \text{kg/m}^3
    • Ts=288.16 KT_s = 288.16\ \text{K}

  • Structure of the standard atmosphere (Earth, ARDC 1959 model used in this book)

    • The temperature profile is drawn as Fig. 3.4 with several layers:
    • Layer 1 (gradient): 0 to 11 km with lapse rate a1 = -6.5 × 10^{-3} K/m; T decreases from 288.16 K at sea level to 216.66 K at 11 km.
    • Layer 2 (isothermal): 11 to 25 km with T = 216.66 K (constant with height).
    • Layer 3 and beyond (gradient layers): additional gradient layers with lapse rates a2 = +3.0 × 10^{-3} K/m, a3 = -4.5 × 10^{-3} K/m, a4 = +4.0 × 10^{-3} K/m, etc., up to the highest altitude tabulated.
    • The base of the modules and the exact base temperatures/pressures at each boundary are determined from the isothermal/gradient relations below.

  • How the numbers are obtained

    • The numbers in Tables A (SI) and B (English engineering) come from integrating the hydrostatic equation with the defined T(h) and using the equation of state, with g approximated as g0.
    • This results in a complete table of p(h), ρ(h), and T(h) versus altitude h (geopotential altitude is used in the derivation; geometric altitude is used in the tables).

  • Construction procedure (summary)

    • Step 1: Define T(h) as piecewise linear across altitude bands with specified base values and lapse rates.
    • Step 2: Use hydrostatics and the equation of state to obtain p(h) and ρ(h) in each layer via the appropriate integrals.
    • Step 3: Relate geopotential altitude h to geometric altitude h_G using Eq. (3.6) to connect to the tabulated heights.
    • Step 4: Build the table across all altitudes; identify base temperatures, pressures, and densities for each layer (as in App. A and App. B).

  • Practical interpretation and cautions

    • The standard atmosphere is a reference, not a precise predictor of actual conditions at a given time/location.
    • It is used to reduce data to a common reference and to perform preliminary design and back-of-the-envelope calculations.
    • The concept of “standard altitude” (geometric) is the altitude corresponding to standard pressure/temperature in the tables.

  • Isothermal and gradient layers: mathematical forms

    • Isothermal layers (temperature constant):
    • Given base values at altitude h1: P1, ρ1, T1, with h1 as the base geopotential altitude, and T = T1 constant, the variations are:
    • P=P<em>1exp⁡(−g</em>0RT<em>1(h−h</em>1))P = P<em>1 \exp\left(-\frac{g</em>0}{R T<em>1}(h - h</em>1)\right)
    • ρ=ρ<em>1exp⁡(−g</em>0RT<em>1(h−h</em>1))\rho = \rho<em>1 \exp\left(-\frac{g</em>0}{R T<em>1}(h - h</em>1)\right)
    • (g0, R are constants; h is geopotential altitude.)
    • Gradient layers (temperature varies linearly with height):
    • Define T(h) = T1 + a (h - h1), where a is the lapse rate for the layer.
    • The pressure-density relation in a gradient layer (integrated with g0 and R constants) yields:
    • Pressure:
      • P(h)=P<em>1(T</em>1T(h))g0aRP(h) = P<em>1 \left( \frac{T</em>1}{T(h)} \right)^{\frac{g_0}{a R}}
    • Density:
      • ρ(h)=ρ<em>1(T</em>1T(h))g0aR+1\rho(h) = \rho<em>1 \left( \frac{T</em>1}{T(h)} \right)^{\frac{g_0}{a R} + 1}
    • where T(h) = T1 + a (h - h1).
    • The lapse rate a and base values (T1, P1, ρ1, h1) are layer-specific and come from the defined Fig. 3.4 and the base conditions at layer boundaries.

3.5 Pressure, Temperature, and Density Altitudes

  • Concept
    • Pressure altitude: the altitude in the standard atmosphere corresponding to the actual ambient pressure.
    • Temperature altitude: the altitude in the standard atmosphere corresponding to the actual ambient temperature.
    • Density altitude: the altitude in the standard atmosphere corresponding to the actual density (ρ).
  • How to determine from data
    • Use Apps. A or B to locate the standard altitude that matches the observed P, T (and then derive density via the equation of state).
  • Example 3.4 (illustrative procedure)
    • Given ambient pressure P and ambient temperature T, determine:
    • Pressure altitude: find standard altitude hG such that P(hG) = P, i.e., P equals the standard pressure at that altitude.
    • Temperature altitude: find standard altitude hT such that T(hT) = T.
    • Density altitude: compute ρ from P = ρ R T, then interpolate to find the standard altitude giving that density.
  • Example 3.5–3.7 (additional insights)
    • Example 3.5 (test data): If pressure and density altitudes are given, derive the temperature at flight altitude using the equation of state and the standard tables.
    • Example 3.6–3.7 illustrate interpolation and the ambiguity of temperature altitude (three different altitudes can yield T = 240 K from the same T(h) curve), and the issues in selecting a unique temperature altitude.

3.6 Historical Note: The Standard Atmosphere

  • Toussaint’s formula (1920) for the temperature decrease with height (in Celsius, geopotential altitude in meters):
    • T=15−0.0065 h(C)T = 15 - 0.0065\, h \quad (\text{C})
  • Adoption and validation
    • Adopted by several countries for airplane performance testing up to about 10 km (33,000 ft).
    • Gregg (1922) provided a mean annual atmospheric table for the United States based on extensive flight, balloon, and artillery data.
    • Diehl (1925) produced practical standard-atmosphere tables in metric and English units, with means for temperature, pressure, and density at various altitudes.
  • Evolution of the model
    • By the 1940s and 1950s, with rockets and spaceflight, standard-atmosphere tables were extended to higher altitudes and refined (notably the ARDC 1959 Standard Atmosphere).
    • The ARDC 1959 model is the one used in this textbook (Apps. A and B).
  • Historical context and practical relevance
    • The creation of standardized atmospheric data was essential for reliable aircraft design and performance prediction across the world.
    • The same physics and methods apply to planetary atmospheres (Venus, Mars, Jupiter) in later chapters.

3.7 Summary and Review

  • Core takeaways
    • The standard atmosphere provides a common reference for testing, wind-tunnel results, and design, enabling rational comparison and aggregation of data from diverse sources.
    • The construction uses hydrostatic equilibrium and the equation of state, with a defined T(h) profile consisting of isothermal and gradient layers.
    • The hydrostatic equation: dpdh<em>G=−ρg\frac{dp}{d h<em>G} = - \rho g; with the approximation g ≈ g0, this becomes dpdh=−ρg</em>0\frac{dp}{d h} = - \rho g</em>0.
    • In isothermal layers: P(h)=P<em>1exp⁡(−g</em>0RT<em>1(h−h</em>1))P(h) = P<em>1 \exp\left(-\frac{g</em>0}{R T<em>1} (h - h</em>1)\right) and ρ(h)=ρ<em>1exp⁡(−g</em>0RT<em>1(h−h</em>1))  .\rho(h) = \rho<em>1 \exp\left(-\frac{g</em>0}{R T<em>1} (h - h</em>1)\right) \;.
    • In gradient layers: T(h) = T1 + a (h - h1) with lapse rate a; then
    • P(h)=P<em>1(T</em>1T(h))g0aRP(h) = P<em>1 \left( \frac{T</em>1}{T(h)} \right)^{\frac{g_0}{a R}}
    • ρ(h)=ρ<em>1(T</em>1T(h))g0aR+1  .\rho(h) = \rho<em>1 \left( \frac{T</em>1}{T(h)} \right)^{\frac{g_0}{a R} + 1} \;.
    • The geopotential altitude h provides a convenient framework; the relationship between geopotential and geometric altitudes is
    • h=rh<em>Gr+h</em>GandhG=rhr−hh = \frac{r h<em>G}{r + h</em>G} \quad \text{and} \quad h_G = \frac{r h}{r - h}
    • The standard atmospheric tables (App. A and App. B) give values of P, ρ, T versus altitude, and are intended as reference for engineering work.
    • The standard atmosphere is a reference, not a precise real-time model; actual conditions vary with time and location.
  • Practical applications and connections
    • The standard atmosphere is crucial for predicting airplane performance (Ch. 6) and for comparing test data across facilities.
    • It can be programmed into calculators or software to enable quick engineering estimates (back-of-the-envelope calculations).
    • The material lays the groundwork for constructing model atmospheres for other planets (Ch. 8).
  • Closing note
    • The tables in Apps. A and B were derived from the hydrostatic equation and the defined temperature profile (Fig. 3.4) using a constant gravity approximation; the geopotential-altitude framework makes the math tractable while remaining physically meaningful for practical altitudes used in aviation.