11 Atmospheric Properties

Introduction

The performance of any atmospheric flight vehicle is affected by the density of the air in which it is flying. The pressure, temperature, and density of the air in the atmosphere are all functions of altitude. In particular, air density affects the aerodynamic performance of an aircraft in terms of its lift and drag but also the thrust and/or power output from its engines. The aerodynamic forces on a launch vehicle are also crucial during its flight within the lower parts of the atmosphere, as well as spacecraft in the high atmosphere during reentry. Finally, the readings on aircraft instruments, such as the altimeter and the airspeed indicator, which are pneumatic, also depend on the local properties found in the atmosphere.

It should be appreciated that the properties of the atmosphere vary from place to place over the Earth, from day to day, and even from hour to hour. Therefore, one problem that arises immediately is developing a means of standardizing the properties of the atmosphere for use in various types of measurements and for both engineering and aviation purposes. This issue is why defining a standard atmospheric model, called the International Standard Atmosphere (ISA), is necessary.

Objectives of this Lesson

  • Know more about the composition and properties of the Earth’s atmosphere as a function of altitude or “height” from the surface.
  • Understand why a “standard” atmospheric model needs to be defined for use in engineering analysis and other areas of practical aviation.
  • Be able to calculate air properties in the International Standard Atmosphere (ISA) using the ISA equations, and also read the relevant properties from a table.

Composition of the Atmosphere

Aeronautical engineers are concerned mainly about the properties of the air in the lower atmosphere where aircraft fly, which is called the troposphere, as shown in the figure below. In this regard, the relevant properties are pressure, temperature, density, viscosity, and the speed of sound. Air represents a mixture of several gases by volume, namely nitrogen (78%), oxygen (21%), noble gases (0.9%), carbon dioxide (less than 0.1%), and other trace gases. The natural winds and turbulence in the lower atmosphere keep these gases well mixed. Other materials naturally present in the atmosphere include water vapor, dust particles, viruses, bacteria, etc. Most dust particles are of volcanic origin, and because of their tiny sub-micron size, they remain suspended for a very long time.

 

The composition of the Earth’s atmosphere. Notice that the troposphere is flattened somewhat towards the poles of the Earth. The atmosphere naturally forms into a series of strata or “shells” where the air has different characteristics and properties.

Astronautical engineers are concerned more about the atmospheric properties above the tropopause, and into the stratosphere and mesosphere. Above the tropopause, the temperature and other properties of the atmosphere change differently compared to those in the troposphere. At higher altitudes above 300,000 ft (90 km), the different gases also begin to settle or separate into shells according to their respective densities, namely oxygen (heaviest), helium, and hydrogen (lightest). The edge of space is usually defined as 100 km (about 62 miles or 328,000 ft), which is called the Kármán Line. Although the Kármán line is an arbitrary boundary, it is widely recognized and used as a standard in the aerospace industry. It is also the reference altitude used by the Fédération Aéronautique Internationale (FAI) to define the edge of space.

General Properties of the Atmosphere

The figure below shows the variations in temperature, pressure, and density in the atmosphere. Both pressure and density decrease quickly and asymptotically with height above mean sea level (MSL), which is perhaps not so surprising because the pressure is related to the weight of air directly above; there is less weight of air above when going higher into the atmosphere.

 

Variations of pressure, temperature and density in the Earth’s atmosphere. Note: Not to be used for quantitative evaluations.

The temperature in the totality of the atmosphere is not quite so easy to describe. There are regions where the temperature increases with height and regions where the temperature decreases with height. The temperature decreases approximately linearly with height in the troposphere (below 36,000 ft or 12,000 m), although this rate can vary from day to day and place to place. The region above the troposphere is called the stratosphere, and in this region, temperature increases with increasing height, which is because of the manner in which solar radiation affects this part of the atmosphere containing the ozone layer.

Development of the ISA model

The International Standard Atmosphere (ISA) model is an average representation of an ideal atmosphere. It is based on thermodynamic relationships (equation of state) and assumes that the atmosphere is absent of water vapor, wind, and turbulence. The ISA has been established to provide a common reference standard of the lower atmosphere (i.e., in the troposphere) in terms of pressure, temperature, density, and other properties such as viscosity and speed of sound.

The ISA is important for the aerospace industry because it provides a standardized reference for calculating and testing aircraft and engine performance. With the ISA, engineers can predict and compare the performance of different aircraft designs under the same set of standard atmospheric conditions, making it easier to compare the performance of different aircraft. In addition to its use in performance calculations, the ISA also serves as a reference for instrument calibration. By calibrating instruments such as altimeters and airspeed indicators relative to the ISA, all such instruments will show identical readings under the same atmospheric conditions, regardless of the actual conditions on any given day or time.

It is essential to recognize that the ISA model is not a meteorological model of actual atmospheric conditions, e.g., from barometric pressure changes or wind conditions. Neither does it account for humidity effects; the air in the ISA model is assumed to be dry, clean, and of constant composition. However, humidity effects are typically accounted for in the flight vehicle performance or engine analysis by applying a correction factor after obtaining the pressure and density from the ISA model, as discussed later.

The International Organization for Standardization (ISO) publishes the ISA model as International Standard ISO 2533:1975. The International Civil Aviation Organization (ICAO) has also adopted the ISA model, which is used universally throughout all aspects of civil aviation. A document on the ISA standard can also be obtained from NOAA.

Tables of ISA properties are published for discrete altitude values incremented in a few hundred feet or meters in many sources, including almost every textbook on aerodynamics and aircraft performance. These ISA tables can be used to find appropriate values at intermediate altitudes by using linear interpolation, recognizing that interpolation is a straightforward process but is also approximate. A much better approach, in general, is to calculate the properties for any specified altitude using the equations of the ISA, which will now be determined.

Equations of the ISA in the Troposphere

The equations of the ISA can be established starting from the hydrostatic equation, which has already been introduced, i.e., using

(1)   \begin{equation*} \frac{dp}{dh} = -\rho g \end{equation*}

Remember that the hydrostatic equation is an ordinary differential equation that relates the change in pressure dp in a fluid with respect to a change in vertical height dh. Normally, when dealing with the atmosphere, the symbol h is used instead of z to represent “height” or altitude. In this case, the height is measured relative to mean sea level or MSL.

The temperature T in the standard atmosphere is assumed to be linearly decreasing function of altitude, which is a good statistical approximation to the actual average temperature variation in the atmosphere, and can be expressed by

(2)   \begin{equation*} T = T_0 - B h = 59 - 3.57(h/1000)\quad \hbox{in units of $^{\circ}$F} \end{equation*}

where h in this case is expressed in feet (ft) and B is a constant and is known as the standard atmospheric lapse rate. This temperature lapse equation has validity only to the limits of the troposphere or about 36,000 ft, which is called the tropopause. Thereafter, the temperature stays constant up to about 65,000 ft.

The value of B is 3.57^{\circ}F per 1,000 ft of altitude or 0.00357^{\circ}F per ft = 0.00357^{\circ}R per ft. In practice, lapse rates will change with the level of humidity. The dry adiabatic lapse rate is about 5.5 ^{\circ}F per 1,000 ft of altitude, and the moist lapse rate ranges varies between 2–3 ^{\circ}F per 1,000 ft of altitude; the ISA model uses a lapse rate that is between these two other lapse rates.

In SI units, the atmospheric temperature lapse equation can be written as

(3)   \begin{equation*} T = T_0 - B h = 15 - 6.5(h/1000) \quad \hbox{in units of $^{\circ}$C} \end{equation*}

where h is expressed in meters (m) and the lapse rate in this case is -6.5^{\circ}C per 1,000 m or 1.983^{\circ}C per 1,000 ft giving

(4)   \begin{equation*} T = 15 - 0.001981h = 15 - B h \quad \hbox{in units of $^{\circ}$C} \end{equation*}

where B = 0.00198 is the standard atmospheric lapse rate in ^{\circ}C per foot and the altitude h is in feet; it is standard practice to measure altitude in feet for aeronautical and aviation purposes although, if needed, the conversion is 1 meter = 3.28084 feet.

Often the ratio of the ISA temperature to the MSL temperature is used, which is given the symbol \theta, i.e.,

(5)   \begin{equation*} \theta = \frac{T}{T_0} \end{equation*}

The other equation needed to relate temperature, pressure and density is the thermodynamic equation of state, which has also been previously introduced, i.e.,

(6)   \begin{equation*} p = \rho R T \end{equation*}

In summary, three equations are now available:

1. The hydrostatic equation, which is an ordinary differential equation, i.e.,

(7)   \begin{equation*} \frac{dp}{dh} = -\rho g \end{equation*}

2. An equation for the linear temperature variation or lapse with height, i.e.,

(8)   \begin{equation*} T = T_0 - B h \end{equation*}

3. The thermodynamic equation of state, i.e.,

(9)   \begin{equation*} p  = \rho R T \end{equation*}

Substituting the latter two equations (Eqs. 8 and 9) into the hydrostatic equation (Eq. 7) gives

(10)   \begin{equation*} \frac{dp}{dh} = -\rho g = - \frac{p g}{R T} = - \frac{p g}{R (T_0 - B h)} \end{equation*}

which is the governing equation that now needs to be solved. By means of separation of variables and integrating between MSL (h = 0, p = p_0) and the height of interest, h where the pressure is p gives

(11)   \begin{equation*} \int_{p_0}^{p} \frac{dp}{p} = - \int_{0}^{h} \frac{g}{R (T_0 - B h)} dh = - \frac{g}{R} \int_{0}^{h} \frac{dh}{T_0 - B h} \end{equation*}

Under standard MSL conditions the value of p_0 is 2116.4 lb/ft^2 or 101,325 N/m^{^2}.

Therefore, the resulting pressure in the ISA, p, relative to the value at MSL, p_0, becomes

(12)   \begin{equation*} \frac{p}{p_0} = \delta & = & \left( 1 - 6.876\times 10^{-6} h \right)^{g/RB} = \left( 1 - 6.876\times 10^{-6} h \right)^{5.252} \end{equation*}

where h is measured in feet. Remember that R is the gas constant for air, which in USC units is 1716.49 lb ft slug^{-1} ^{\circ}R and in SI units it is 287.057 J kg^{-1} K^{-1}. Notice that the value of g/RB is evaluated using

(13)   \begin{equation*} \frac{g}{R B} = \frac{32.174}{(1716.49) \, (0.00357)} = 5.252 \end{equation*}

which is a non-dimensional parameter.

The corresponding density \rho in the ISA can be calculated by invoking the thermodynamic equation of state. The pressure and density are related using

(14)   \begin{equation*} \frac{\rho}{\rho_0} = \frac{p}{p_0} \left( \frac{T_0}{T} \right) \end{equation*}

Therefore, if the local temperature corresponds to the standard local temperature in the ISA, i.e., T_0 - B h then

(15)   \begin{equation*} \frac{\rho}{\rho_0} = \frac{p}{p_0} \left( \frac{T_0}{T_0 - B h} \right) = \frac{p}{p_0} \left( \frac{1}{1 - \displaystyle{\frac{B h}{T_0}} } \right) \end{equation*}

so then the local density in the ISA relative to MSL is

(16)   \begin{equation*} \frac{\rho}{\rho_0} = \sigma = \left( 1 - 6.876\times 10^{-6} h \right)^{(g/RB) - 1} = \left( 1 - 6.876\times 10^{-6} h \right)^{4.252} \end{equation*}

Values of the ISA at standard MSL conditions are useful to have on hand, and these are given in the table below.

Table of ISA Properties at MSL.
Property Symbol SI units USC units
Pressure p_0 1.01325\times10^{5}Nm^{-2} 2116.4 lb/ft^2
Density \rho_0 1.225 kg m^{-3} 0.002378 slugs ft^{-3}
Temperature T_0 288.15 K 518.67 R
Dynamic viscosity \mu_0  1.789\times10^{-5} kg m^{-1} s^{-1} 3.737\times10^{-7}slug ft^{-1} s^{-1}
Speed of sound a_0 340.3 m s^{-1} 1116.47 ft s^{-1}
Gas constant R 287.057 J kg^{-1} K^{-1} 1716.59 ft lb slug^{-1}R^{-1}

 

Example # 1 – Calculating the Properties of the ISA Using MatLab

Calculate the properties of the ISA using the equations developed in this lessons. You may do this in SI or USC units, but it is best to try to calculate the properties in both sets of units.

The equations for the ISA are relatively easy to program in Matlab; below is a short piece of code used to calculate such properties in the troposphere.

g=9.81; % acceleration under gravity in m/s^2
B=-0.0065; % temp lapse rate in ISA in Kelvin/meter
R=287.05; % gas constant for air
T_sl=288.15; % ISA MSL standard temperature in Kelvin
p_sl=1.01325e+5; % standard pressure at sea level in Pascals
rho_sl=p_sl/(R*T_sl) % use ideal gas law to get density
a_sl=sqrt(1.40*p_sl/rho_sl) % speed of sound at sea level
z=linspace(0,10000,1000); % calculate ISA properties up to the tropopause
T_z=T_sl+B.*z;
p_z=p_sl.*((1+B.*z./T_sl).^(-g/(R*B)));
rho_z=p_z./(R.*T_z);
mu_z=1.458e-6.*sqrt(T_z)./(1.0+0+110.40./T_z); % Sutherland’s law
a_z=sqrt(1.40.*p_z./rho_z); % speed of sound

The outcomes can be plotted as follows:

  • Variation of density in the ISA measured in SI units.
  • Variation of pressure in the ISA measured in SI units.
  • Variation of temperature in the ISA measured in SI units.
  • Variation of viscosity in the ISA measured in SI units.

Correction Factors for Humidity

The changes in air density from changes in humidity tend to be small but they can still have an impact in some applications. To account for this, a correction factor is applied to account for the change in air density with varying levels of humidity. This correction can range from 1–2% and is important to consider in applications where air density plays a more important role, such as aviation, aerospace engineering, and meteorology.

It should be appreciated that the addition of humidity to air will decrease its density. The reason is that air comprises mainly nitrogen molecules and oxygen molecules, i.e., N2 and O2, with air being composed of 78% nitrogen and 20% oxygen. Nitrogen has an atomic weight of 14, so an N2 molecule has an atomic weight of 28. Oxygen has an atomic weight of 16, so an O2 molecule has an atomic weight of 32. Therefore, the overall atomic weight of air is about 28. Water, which is an H2O molecule, has an atomic weight of 18. Therefore, the air’s mass per unit volume (i.e., the density) will decrease as water vapor is added, with the conclusion that moist air will be less dense than dry air.

There are different methods to determine the correction factor for the effects of humidity on air density, such as using wet and dry bulb temperature measurements or finding the dew point. These methods allow for accurate correction of the small but non-negligible impact that humidity can have on air density. In aviation and wind tunnel measurements, it is important to correct for the effects of humidity on density, especially in flight testing and performance measurements, because small errors in air density can accumulate and result in inaccuracies.

Other Atmospheric Models

Various other extended atmospheric models are used to represent the characteristics of the atmosphere above the tropopause and into space. One purpose of using these models is to help predict the orbital decay of satellites from atmospheric drag.

Approximate Model for the Troposphere

In aerodynamics, air density is frequently calculated to determine aircraft and engine performance. In the lower atmosphere where many general aviation aircraft normally fly (say, below 20,000 ft or 6,000 m), the standard value of air density can be closely approximated by the equation

(17)   \begin{equation*} \frac{\rho}{\rho_0} = \exp \left( \frac{-0.0297 h}{1000} \right) \end{equation*}

where h is expressed in feet and \rho_0 = 0.002378 slugs ft^{-3}. In SI units the corresponding equation is

(18)   \begin{equation*} \frac{\rho}{\rho_0} = \exp \left( \frac{ -0.0296 h}{304.8} \right) \end{equation*}

where h is expressed in meters and \rho_0 = 1.225 kg/m^3.

ICAO extended ISA model

The ICAO has published an extended ISA model as “ICAO Standard Atmosphere” (Document 7488-CD). It has the same model as the ISA in the troposphere but extends the altitude coverage to 80 km (262,500 ft). The temperature and pressure values defined by ICAO for the extended model are given in the table below.

Values used for the ICAO extended ISA model.
Altitude (km) Altitude (ft) Temp. (^{\circ}C) Press. (hPa) Lapse Rate (^{\circ}C/1,000 ft)
0 MSL 15.0 1013.25 -1.98 (Troposphere)
11 36,000 -56.5 226.00 0.00 (Stratosphere)
20 65,000 -56.5 54.70 -0.3 (Stratosphere)
32 105,000 -44.5 8.68

NLR and Air Force Models

The U.S. Naval Research Laboratory (NLR) has developed a model of the Earth’s atmosphere from Earth’s surface to space. One use of this model is to help engineers predict the orbital decay of satellites from atmospheric drag. The U.S. Air Force Space Command and Space Environment Technologies have also developed a model for the Earth’s atmosphere from 120 km to 2,000 km.

Tables of ISA Properties

The equations of the International Standard Atmosphere (ISA) have now been introduced. All aerospace engineers must understand the ISA properties and how the ISA is used for both engineering and other aviation purposes. In addition, it is necessary to practice to be able to apply the equations of the ISA to calculate temperature, pressure, and density (for both standard and non-standard conditions) and in appropriate engineering units, i.e., USC and SI. Handy online ISA calculators are also available.

It is also necessary to be able to read values of pressure, temperature, and density from tables of ISA properties, which are listed for discrete values of altitude in almost every aerodynamics textbook; an example is shown below. These tables can be used with linear interpolation methods to find approximate values at intermediate altitudes. For example, the table below gives ISA properties from -1,000 feet to 65,000 feet in 1,000-foot intervals. Recall that \sigma is density divided by MSL density, \delta is the pressure divided by MSL pressure, and \theta is the temperature divided by MSL temperature. Notice that the temperature T in the table below is in units of degrees Rankine (^{\circ}R), pressure p is in units of pounds per square foot (lb ft^{-2}), density \rho is in units of slugs per cubic foot (slug ft^{-3}), a is the speed of sound in feet per second (ft s^{-1}), and the viscosity \mu is in units of slugs per foot-second (slug ft^{-1}s^{-1} \times 10^6).

Table of ISA values in USC units in steps of 1,000 feet.
Altitude h \sigma \delta \theta T p \rho a \mu
-1 1.0296 1.0367 1.0069 522.2 2193.8 0.0024472 1120.3 0.376
0 1 1 1 518.7 2116.2 0.0023769 1116.5 0.374
1 0.9711 0.9644 0.9931 515.1 2040.9 0.0023081 1112.6 0.372
2 0.9428 0.9298 0.9863 511.5 1967.7 0.0022409 1108.7 0.37
3 0.9151 0.8963 0.9794 508 1896.7 0.0021752 1104.9 0.368
4 0.8881 0.8637 0.9725 504.4 1827.7 0.0021109 1101 0.366
5 0.8617 0.8321 0.9656 500.8 1760.9 0.0020482 1097.1 0.364
6 0.8359 0.8014 0.9588 497.3 1696 0.0019869 1093.2 0.362
7 0.8107 0.7717 0.9519 493.7 1633.1 0.001927 1089.3 0.36
8 0.7861 0.7429 0.945 490.2 1572.1 0.0018685 1085.3 0.358
9 0.7621 0.7149 0.9381 486.6 1512.9 0.0018113 1081.4 0.355
10 0.7386 0.6878 0.9313 483 1455.6 0.0017555 1077.4 0.353
11 0.7157 0.6616 0.9244 479.5 1400.1 0.0017011 1073.4 0.351
12 0.6933 0.6362 0.9175 475.9 1346.2 0.001648 1069.4 0.349
13 0.6715 0.6115 0.9107 472.3 1294.1 0.0015961 1065.4 0.347
14 0.6502 0.5877 0.9038 468.8 1243.6 0.0015455 1061.4 0.345
15 0.6295 0.5646 0.8969 465.2 1194.8 0.0014962 1057.4 0.343
16 0.6092 0.5422 0.8901 461.7 1147.5 0.001448 1053.3 0.341
17 0.5895 0.5206 0.8832 458.1 1101.7 0.0014011 1049.2 0.339
18 0.5702 0.4997 0.8763 454.5 1057.5 0.0013553 1045.1 0.337
19 0.5514 0.4795 0.8695 451 1014.7 0.0013107 1041 0.335
20 0.5332 0.4599 0.8626 447.4 973.3 0.0012673 1036.9 0.332
21 0.5153 0.441 0.8558 443.9 933.3 0.0012249 1032.8 0.33
22 0.498 0.4227 0.8489 440.3 894.6 0.0011836 1028.6 0.328
23 0.4811 0.4051 0.842 436.7 857.2 0.0011435 1024.5 0.326
24 0.4646 0.388 0.8352 433.2 821.2 0.0011043 1020.3 0.324
25 0.4486 0.3716 0.8283 429.6 786.3 0.0010663 1016.1 0.322
26 0.433 0.3557 0.8215 426.1 752.7 0.0010292 1011.9 0.319
27 0.4178 0.3404 0.8146 422.5 720.3 0.0009931 1007.7 0.317
28 0.4031 0.3256 0.8077 419 689 0.000958 1003.4 0.315
29 0.3887 0.3113 0.8009 415.4 658.8 0.0009239 999.1 0.313
30 0.3747 0.2975 0.794 411.8 629.7 0.0008907 994.8 0.311
31 0.3611 0.2843 0.7872 408.3 601.6 0.0008584 990.5 0.308
32 0.348 0.2715 0.7803 404.7 574.6 0.000827 986.2 0.306
33 0.3351 0.2592 0.7735 401.2 548.5 0.0007966 981.9 0.304
34 0.3227 0.2474 0.7666 397.6 523.5 0.000767 977.5 0.302
35 0.3106 0.236 0.7598 394.1 499.3 0.0007382 973.1 0.3
36 0.2988 0.225 0.7529 390.5 476.1 0.0007103 968.7 0.297
37 0.2852 0.2145 0.7519 390 453.9 0.000678 968.1 0.297
38 0.2719 0.2044 0.7519 390 432.6 0.0006463 968.1 0.297
39 0.2592 0.1949 0.7519 390 412.4 0.0006161 968.1 0.297
40 0.2471 0.1858 0.7519 390 393.1 0.0005873 968.1 0.297
41 0.2355 0.1771 0.7519 390 374.7 0.0005598 968.1 0.297
42 0.2245 0.1688 0.7519 390 357.2 0.0005336 968.1 0.297
43 0.214 0.1609 0.7519 390 340.5 0.0005087 968.1 0.297
44 0.204 0.1534 0.7519 390 324.6 0.0004849 968.1 0.297
45 0.1945 0.1462 0.7519 390 309.4 0.0004623 968.1 0.297
46 0.1854 0.1394 0.7519 390 295 0.0004407 968.1 0.297
47 0.1767 0.1329 0.7519 390 281.2 0.0004201 968.1 0.297
48 0.1685 0.1267 0.7519 390 268.1 0.0004005 968.1 0.297
49 0.1606 0.1208 0.7519 390 255.5 0.0003817 968.1 0.297
50 0.1531 0.1151 0.7519 390 243.6 0.0003639 968.1 0.297
51 0.146 0.1097 0.7519 390 232.2 0.0003469 968.1 0.297
52 0.1391 0.1046 0.7519 390 221.4 0.0003307 968.1 0.297
53 0.1326 0.0997 0.7519 390 211 0.0003153 968.1 0.297
54 0.1264 0.0951 0.7519 390 201.2 0.0003006 968.1 0.297
55 0.1205 0.0906 0.7519 390 191.8 0.0002865 968.1 0.297
56 0.1149 0.0864 0.7519 390 182.8 0.0002731 968.1 0.297
57 0.1096 0.0824 0.7519 390 174.3 0.0002604 968.1 0.297
58 0.1044 0.0785 0.7519 390 166.2 0.0002482 968.1 0.297
59 0.0996 0.0749 0.7519 390 158.4 0.0002367 968.1 0.297
60 0.0949 0.0714 0.7519 390 151 0.0002256 968.1 0.297
61 0.0905 0.068 0.7519 390 144 0.0002151 968.1 0.297
62 0.0863 0.0649 0.7519 390 137.3 0.000205 968.1 0.297
63 0.0822 0.0618 0.7519 390 130.9 0.0001955 968.1 0.297
64 0.0784 0.059 0.7519 390 124.8 0.0001864 968.1 0.297
65 0.0747 0.0562 0.7519 390 118.9 0.0001777 968.1 0.297

 

Example #2 – Evaluating the Atmospheric Properties on Mars

The Viking landers have measured the atmospheric properties of Mars, and engineers have determined that the atmosphere has a linear thermal gradient layer from z = 0 to 40 km and an isothermal layer from z = 40 km to 80 km. The lapse rate B in the temperature gradient layer is 2^{\circ}K/km. If the pressure and temperature at the surface of Mars are 230^{\circ}K and 750 Pa, respectively, then determine:

  1. The density of the Martian atmosphere at its surface.
  2. The pressure and density of the Martian atmosphere at 20 km and 40 km.
  3. The temperature, pressure and density of the Martian atmosphere at 60 km.

Hint: The specific gas constant on Mars is 188.92 J kg^{-1} ^{\circ}K^{-1} and the gravitational acceleration is 3.8 ms-2

  1. Assume the validity of an ideal gas and so use the gas-specific equation of state, i.e.,

        \[ p = \rho R T \quad \mbox{or} \quad \rho = \frac{p}{R \, T} \]

    where R on Mars is 188.92 J kg^{-1}^{\circ}K^{-1}. At the surface of Mars with T = 230^{\circ}K and p =750 Pa then

        \[ \rho_0 = \frac{p_0}{R \, T_0} = \frac{750.0}{188.92 \times 230} = 0.01726 \mbox{ kg/m$^3$}\]

  2. In the non-isothermal layer with the constant temperature lapse rate (and so the linearly decreasing temperature gradient) then at 20 km from the surface of Mars the temperature is

        \[ T_{\rm 20} = T_0 + B z = 230 - (2 \times 20 ) = 190 \mbox{ $^{\circ}$K} \]

    The pressure change with a linear thermal gradient with temperature lapse B between two heights z_1 and z_2 is

        \[ \frac{p_2}{p_1} = \bigg( \frac{T_2}{T_1} \bigg)^{-g/RB} = \bigg( \frac{T_1 - B (z_2 - z_1)}{T_1} \bigg)^{-g/RB} \]

    The exponent g/RB for Mars is

        \[ \frac{g}{R \, B} = \frac{ 3.8}{188.92 \times 2/1000} = 10.0572 \]

    Therefore, the pressure at 20 km is

        \[ p_{20} = p_0 \bigg( \frac{T_1 - B (z_2 - z_1)}{T_1} \bigg)^{-10.0572} = 750.0 \bigg( \frac{230 - 2 \times 20}{230} \bigg)^{-10.571} = 109.79 \mbox{ Pa} \]

    Using the equation of state the for the density at 20 km then

        \[ \rho_{20} = \frac{p_{20}}{R T_{20}} = \frac{109.79}{188.92 \times 190} = 0.00305 \mbox{ kg/m$^3$} \]

    At 40 km then

        \[ T_{40}= 230 - 2 \times 40 = 150 \mbox{ $^{\circ}$K} \]

    so the pressure at 40 km on Mars is

        \[ p_{40} = 750.0 \bigg( \frac{150}{230} \bigg)^{-10.0572} = 10.188 \mbox{ Pa} \]

    Using the equation of state for the density at 40 km then

        \[ \rho_{40} = \frac{p_{40}}{R T_{40}} = \frac{10.188}{188.92 \times 150} = 0.000359 \mbox{ kg/m$^3$} \]

  3.  For the isothermal layer then

        \[ \frac{p_2}{p_1} = \exp\bigg( \frac{ -g (z_2 - z_1)}{R T_0}\bigg) \]

    So for an altitude of 60 km then

        \[ T_{60} = T_{40} = 150 \mbox{ $^{\circ}$K} \]

    The pressure at 60 km will be

        \[ p_{60} = p_{40} \, \exp\bigg( \frac{ -g (h_{60} - h_{40})}{R \, T_{40}} \bigg) \]

    and inserting the values gives

        \[ p_{60} = 10.188 \, \exp\bigg( \frac{-3.8 \times 1000 (60-40)}{188.92 \times 150}\bigg) = 0.697 \mbox{ Pa} \]

    Finally, using the equation of state then the density at 60 km will be

        \[ \rho_{60} = \frac{p_{60}}{R T_{60}} = \frac{0.697}{188.92 \times 150} = 0.000024 \mbox{ kg/m$^3$} \]

Satellite Drag

The outer fringes of the Earth’s atmosphere extend from about  200 miles (320 km) to about 1,200 miles (2,000 km). Although the density here is much lower than in the troposphere and stratosphere, the residual atmosphere is still sufficient to create a drag force on anything in Low Earth Orbit (LEO), which is really close to the Earth. Indeed, ff the Earth were the size of an apple, then then LEO zone would be about the thickness of the skin on the apple! Satellites in LEO experience some aerodynamic drag causing their orbits to decay gradually, as illustrated in the figure below. Both the International Space Station (ISS) and the Hubble Space Telescope (HST) operate in LEO.

Satellites in low Earth orbit will encounter drag from the residual atmosphere causing their orbits to decay.

Estimating the orbital decay of objects in LEO is essential but challenging because of the constant spatial and temporal variations in what is left of the atmosphere at these altitudes. Atmospheric density models used to predict orbital decay are semi-empirical and usually consider variations in orbital latitude, season, tides, solar radiation and flares, geomagnetic activity, etc. When the orbital decay is too large and reaches an orbital maintenance threshold, satellites can be boosted back into their initial orbits by using onboard thrusters. To this end, sufficient fuel must be estimated accurately and then carried onboard for the projected life of the satellite.

Summary & Closure

When dealing with the performance of flight vehicles in the Earth’s atmosphere, it is essential to know (or estimate) the air density in which the vehicle is flying. The power and thrust from air-breathing powerplants are also affected by the air density in which they operate, their performance generally diminishing with increasing altitude and/or increasing temperature. To address these challenges, aeronautical engineers must have a deep understanding of the properties of the air in the troposphere and how these properties may impact the performance of their aircraft. They use this knowledge to make informed decisions about the design of the aircraft and to conduct tests to ensure that the aircraft meets its performance and safety requirements.

The International Standard Atmosphere (ISA) model forms a basis for standardization. The ISA is used universally in engineering and practical aviation, including as a basis for instrumentation calibration. The International Organization for Standardization (ISO) publishes the ISA model as ISO 2533:1975. The International Civil Aviation Organization (ICAO) has also adopted the ISA model, which is used universally throughout aviation. There are also various other extended atmospheric models to represent the characteristics of the atmosphere above the tropopause and into space, which may be needed for a variety of reasons including estimates of the drag on satellites in low Earth orbit. It may also be necessary to know the properties of the atmospheres on other planets, such as for probes or landers.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • Program the ISA equations and plot the temperature, pressure, and density as a function of altitude in U.S. customary (USC) units.
  • Use the ISA tables to estimate air density at a pressure altitude of 5,500 ft where the temperature is 10^{\circ}F above standard. Confirm the result using the ISA equations.
  • On a given day, the air temperature at 2,000 feet above mean sea level is 5^{\circ}C. What is this temperature value compared to the ISA temperature?
  • In which atmospheric layer is most of the water vapor contained?
  • Why does the temperature of the atmosphere increase with increasing altitude within the stratosphere?

Additional Online Resources

To improve your understanding of the ISA, navigate to some of these online resources:

  • High-definition video about the motion of the atmosphere.
  • Take a ride on a U-2 spy-plane through the atmosphere to the edge of space.
  • Try out this handy online ISA calculator.
  • A lecture video on the ISA from Delft University.
  • A YouTube video on the structure of the atmosphere for pilots.