Introduction

As with airspeed, an aircraft’s altitude is a fundamental parameter in both piloting and engineering because it directly influences air density and, in turn, aerodynamic performance and propulsion characteristics. Pilots are primarily concerned with altitude relative to mean sea level or above the ground, particularly during takeoff, landing, and terrain avoidance. The choice of altitude reference (datum) is essential, as all aircraft must operate using a common standard to ensure safe air traffic control separation and collision avoidance. Pilots must also anticipate aircraft performance, which depends strongly on altitude and ambient air temperature, both of which determine the local air density.

From an engineering perspective, however, the primary quantity of interest is the air density at the flight condition, with altitude serving mainly as an indirect parameter for its determination. Measurements of static pressure and temperature define the thermodynamic state of the atmosphere at a given altitude. An altimeter, being fundamentally a pressure-measuring instrument, provides a measure of the local static pressure, while the outside air temperature may be measured directly in flight. Therefore, a clear understanding of the basis of altitude measurement and how it is used, in conjunction with pressure and temperature, to determine air density, is essential in engineering practice, especially as it relates to aircraft performance.

What is an Altimeter?

Like the airspeed indicator (ASI), an altimeter is a pressure-measuring (pneumatic) instrument. An altimeter responds to ambient static pressure and must be connected to the aircraft’s static pressure source, i.e., the static port. However, unlike an airspeed indicator, which measures the difference between total (ram) pressure and static pressure, an altimeter has only a single pressure connection and measures static pressure alone. The mechanical design of an altimeter is shown schematically in Figure 1. Although it appears relatively simple, it is a delicate, precise instrument that requires careful, repeated calibration for use in aviation practice.

Inside the altimeter, the expansion and contraction of an evacuated aneroid wafer stack are transmitted through a system of levers and gears and displayed by the rotational movement of pointers on a dial, much like that of a clock. The altimeter typically has three pointers: one indicating hundreds of feet, another indicating thousands of feet, and a third indicating tens of thousands of feet. Notice from Figure 1 that the altimeter also has a setting window called the Kollsman window. On the lower side of the instrument is a knurled adjustment knob that allows the pilot to set the reference pressure displayed in the Kollsman window. Therefore, the indicated altitude will depend on the reference pressure set in this window.

Reading an Altimeter

Reading an altimeter is straightforward, but for most people it requires some practice. The longest, thinnest pointer indicates tens of thousands of feet and completes one full revolution for every 100,000 feet of altitude change. The shortest and widest pointer indicates thousands of feet and completes one full revolution for every 10,000 feet. The intermediate-length pointer indicates hundreds of feet and completes one full revolution for every 1,000 feet. Accordingly, the numbered markings on the dial represent 100-foot increments, with smaller tick marks typically corresponding to 20-foot increments. An altimeter can be read to within about ±10 feet.

In the example shown in Figure 2, the 100-foot pointer is just past the number 4 (indicating approximately 400 feet), the 1,000-foot pointer lies between 1 and 2, and the 10,000-foot pointer is near zero. Therefore, the indicated altitude is approximately 1,410 feet relative to the reference pressure set in the Kollsman window, which, in this case, is set to standard sea-level pressure of 29.92 inches of mercury (in Hg). The reference reading may also be in milibars (hectopascals).

All aviation altimeters are calibrated to conform to the pressure variation defined by the International Standard Atmosphere (ISA). The mechanical design of a typical general aviation altimeter allows an accuracy on the order of ±20 feet, which is adequate for piloting purposes. However, for engineering applications, more precise calibration is required to account for mechanical and installation errors, as is the case with the ASI. Such calibration is performed in a controlled pressure environment, such as a vacuum tank, and is referenced against known pressure standards or a calibrated instrument. The resulting corrections are usually provided in tabular or chart form.

Review of ISA Pressure & Density Variations

Before proceeding with the formal definitions of pressure altitude and density altitude, it is helpful to summarize the previously derived relations for the International Standard Atmosphere (ISA).

ISA Temperature Variations

The temperature in the ISA is given by

(1)  

or

(2)  

Here, is expressed in feet (ft), and is the standard atmospheric lapse rate. This relation is valid only within the troposphere, i.e., up to approximately 36,000 ft. The lapse rate is 3.57F per 1,000 ft, or equivalently 0.00357R per ft.

In SI units, the temperature variation is

(3)  

or

(4)  

where is in meters (m), and the lapse rate is 6.5C per 1,000 m. This value is equivalent to 1.981C per 1,000 ft, giving

(5)  

with in feet (ft). For aeronautical applications, altitude is usually expressed in feet. It is also convenient to define the nondimensional temperature ratio

(6)  

ISA Pressure Variations

The pressure variation in the ISA is

(7)  

where is in feet (ft). In USC units,

(8)  

The gas constant for air is ft·lb·slug·R in USC units and J kg K in SI units. The exponent is

(9)  

which is nondimensional and independent of the unit system.

If is expressed in meters (m), then

(10)  

where

(11)  

Under standard MSL conditions, lb/ft or 101.325 kPa.

ISA Density Variations

From the equation of state,

(12)  

If the local temperature follows the ISA lapse relation, i.e., , then

(13)  

and so

(14)  

where is in feet (ft). If is in meters (m), then

(15)  

Under standard MSL conditions, slug/ft or 1.225 kg/m.

Pressure Altitude

As previously described, an altimeter is a calibrated pressure gauge, and its calibration is based on the International Standard Atmosphere (ISA). Therefore, it is appropriate to use the altimeter to determine the static air pressure at which the aircraft is flying, expressed as pressure altitude. By definition, the pressure altitude, , is the altitude in the ISA corresponding to the prevailing ambient static pressure. The advantage of using pressure altitude as a reference is that it depends only on the local ambient pressure and is independent of temperature.

Using the relationship for pressure in the ISA given by Eq. 10, the pressure altitude, , expressed in feet (ft), is

(16)  

where in USC units, so that

(17)  

The measurement of pressure altitude from an altimeter is subject to static position error (SPE) and mechanical (instrument) error (IEC), both of which must be determined through calibration. The SPE, which has been discussed at length in the previous chapter, arises because the static pressure measured at the aircraft’s static ports may differ from the true freestream static pressure owing to local flow disturbances caused by the airframe. Like the airspeed indicator, the altimeter senses static pressure slightly higher or lower than ambient, resulting in a corresponding error in the indicated altitude.

To account for this error, the SPE is determined in flight by measuring the true static pressure using a trailing-cone apparatus positioned sufficiently far behind the aircraft to be outside the influence of the aircraft’s flow field. The difference between this reference pressure and that measured at the static ports is then converted into an altitude correction, . Because the static position error varies continuously with flight conditions (e.g., airspeed, angle of attack, and configuration), this correction must be mapped across the operational flight envelope and is usually represented as a curve fit to provide a smooth, continuous relationship. A typical calibration at a single altitude is shown in Figure 3.

Similarly, the IEC is obtained by calibrating the altimeter against a reference instrument in an evacuated chamber to determine , as illustrated in Figure 4. The results are usually presented in tabular form, although curve fits may also be used. These errors are generally small but are not negligible for engineering purposes.

The corrected pressure altitude is then

(18)  

where is the indicated pressure altitude. These corrections are usually small enough to be neglected for piloting purposes but must be included in engineering analyses, particularly those involving aircraft and engine performance.

Density Altitude

While pressure altitude can be determined directly from a suitably calibrated altimeter, it is ultimately the air density at the aircraft’s altitude that governs its aerodynamic and engine performance. Therefore, determining air density is essential for engineering analysis and performance evaluation.

By definition, the density altitude, , is the altitude in the ISA corresponding to the prevailing ambient density. The value of the density altitude (again expressed in feet) may be obtained by rearranging Eq. 14 to give

(19)  

Therefore,

(20)  

where is in feet.

Relating Density Altitude & Pressure Altitude

As previously explained, pressure altitude, ,  can be read directly from an altimeter calibrated according to the ISA using Eq. 17, provided that the reference pressure is set to the standard MSL value. This is accomplished by setting the Kollsman window to 29.92 in Hg (or 1013.2 mbar), corresponding to the ISA MSL pressure of 2,116.4 lb/ft or 101.325 kPa.

The value of can then be read directly from the altimeter, assuming that the instrument is properly calibrated and that mechanical and static position errors have been corrected. Unlike pressure altitude, density altitude must be determined from pressure altitude and the local ambient temperature. In this context, “nonstandard” refers to a temperature that deviates from the ISA value at the same pressure altitude. Because density varies inversely with temperature at a given pressure, an increase in temperature decreases density. Therefore, at higher-than-standard temperatures, the density altitude exceeds the pressure altitude, whereas at lower-than-standard temperatures, it is lower.

Density Altitude Calculation

Recall that pressure and density are related through the equation of state, so the density ratio in the ISA can be written as

(21)  

where is the local standard temperature according to the ISA, given by

(22)  

where is the pressure altitude.

If the pressure altitude and the outside air temperature, , are both available, then by using Eqs. 17 and 20 and some algebra, the corresponding density altitude can be determined from

(23)  

Because , then

(24)  

This result shows that Eq. 24 can be used to determine the density altitude for any given pressure altitude and outside air temperature. It can also be seen that if the local temperature equals the standard ISA temperature at the same pressure altitude, i.e., , then . All temperatures in these equations must be expressed as absolute values.

Armed with these equations, it is possible to examine variations in density altitude with temperature for different pressure altitudes, as shown in Figure 5 (SI units). The diagonal lines represent constant-pressure altitudes; however, unlike some simplified versions of this chart, they are not straight because density depends nonlinearly on temperature and pressure. The blue line corresponds to ISA conditions, for which . For example, if and 6,300 m, then following the chart gives an estimated value of of approximately 8,200 m.

The same type of presentation is shown in Figure 6 (USC units). In this case, F per 1,000 ft (or 0.00357R per ft). For example, if F and 17,500 ft, then the chart gives an estimated value of of approximately 23,000 ft.

Interpolation is generally required when using these charts. While they provide a useful visual representation of the effects of nonstandard temperature, they should not be used in place of the ISA equations for engineering calculations.

Density Ratio Calculation

Another useful metric, obtained from pressure altitude and outside air temperature, is the density ratio, , which is often required in aircraft and engine performance analysis. At a given pressure altitude, the pressure ratio is fixed by the ISA pressure-altitude relation, i.e.,

(25)  

The density ratio then follows from the equation of state as

(26)  

where must be expressed as an absolute temperature.

Therefore, if is in feet and is in , then

(27)  

Alternatively, if is in feet and is in , then

(28)  

It is also useful to recall that the equation of state may be written in nondimensional form as

(29)  

which provides a convenient relationship between pressure, density, and temperature.

Measuring Outside Air Temperature

In aviation terminology, the outside air temperature (OAT) refers to the temperature of the air surrounding an aircraft in flight. This value is assumed to be unaffected by the aircraft’s motion through the air and is often referred to as the static air temperature. For engineering purposes, the OAT must be measured to determine air density.

A simple OAT gauge may be used at low airspeeds, typically below a Mach number of 0.3 (i.e., within the incompressible flow regime), as shown in Figure 7, to provide a reasonably accurate measurement of the static air temperature. In this application, the probe is shielded from direct airflow to minimize temperature errors caused by ram heating and solar radiation. The probe typically protrudes through the aircraft’s windshield or fuselage near the cockpit, exposing it to the ambient air, while the instrument head is mounted inside for convenient reading.

At higher airspeeds, compressibility effects make accurate temperature measurement more challenging. In this case, the total air temperature (TAT) is defined as the static air temperature plus the temperature rise associated with the deceleration of the airflow. TAT probes are designed to measure this total temperature and provide signals for cockpit indications and aircraft systems, as illustrated in Figure 8.

The possibility of ice formation complicates the design of a TAT probe, as it does for Pitot probes. Therefore, TAT probes are equipped with heaters. The probe geometry and airflow path must be carefully designed so that the measured temperature accurately reflects the true total temperature, free from bias from the heating system. Several manufacturers specialize in TAT probes for aircraft applications.

Practical Use of Density Altitude

The ISA relations are widely used to standardize aircraft performance and to evaluate flight-test data. In many situations, however, a rapid estimate of density altitude is useful for piloting. For example, aircraft operating manuals commonly present performance data as a function of density altitude.
As previously discussed, pressure altitude can be obtained directly from the altimeter (with the appropriate reference pressure setting). In contrast, density altitude must be determined from the pressure altitude together with the local outside air temperature at that altitude.

Linearization of the ISA Equations

From the variation in density altitude with temperature at a given pressure altitude, it is evident that the relationship is nearly linear. This observation suggests that an approximation to Eq. 24 may be useful for rapid estimates of density altitude, particularly for piloting purposes. Such linearizations, however, are not appropriate for flight-test or engineering analysis.

To linearize Eq. 24, it may be expanded as a Taylor series about the ISA standard temperature at a given pressure altitude, retaining only the first-order term to give

(30)  

where is the ISA standard temperature at the given , i.e.,

(31)  

Evaluating the coefficient gives

(32)  

so that

(33)  

when and are in meters and temperatures are in C.
Alternatively, in mixed units,

(34)  

where and are in feet and temperatures are in C.
Finally, in USC units,

(35)  

so that

(36)  

where and are in feet and temperatures are in F.

Therefore, the linearized ISA relations show that density altitude exceeds pressure altitude by approximately 118.6 ft per C or 65.9 ft per F when the temperature exceeds the standard ISA value at a given pressure altitude. These results are approximate but sufficiently accurate for moderate deviations from ISA conditions. However, for engineering purposes, these approximations should not be used in place of the full ISA relations, i.e., Eqs. 17, 20, and 24.

Effects of Altitude on Propulsion Performance

Altitude significantly influences the performance of aircraft propulsion systems because it determines ambient air density. As altitude increases, air density decreases, reducing the mass flow rate of air through the propulsion system. Because thrust and power generation depend fundamentally on accelerating a mass of air, this reduction in density leads to a corresponding degradation in propulsion performance. For air-breathing engines, including piston engines, turboprops, turbofans, and turbojets, the thrust or power output depends on the mass flow rate of air entering the engine. This mass flow rate may be expressed approximately as

(37)  

where is the local ambient air density, is a characteristic inlet or disk area, and is the flow velocity, i.e., true airspeed. At higher altitudes, the reduced value of leads directly to a reduction in and so to reduced thrust or shaft power.

For propeller-driven aircraft, which are fundamentally power-limited, available engine power decreases with altitude as a result of reduced air density, leading to lower engine performance. Consequently, the rate of climb decreases and the takeoff distance increases. For normally aspirated piston engines, the available power decreases approximately in proportion to the air density. Turbocharged engines mitigate this effect by maintaining higher intake pressure, thereby preserving power up to a critical altitude. For jet-powered aircraft, which are fundamentally thrust-limited, thrust decreases with altitude primarily because of the reduced mass flow rate through the engine. However, this reduction is partly offset by the decrease in aerodynamic drag at higher altitudes, thereby enabling efficient cruise at both high subsonic and supersonic speeds.

In addition to density effects, altitude also influences propulsion performance through temperature. Higher-than-standard temperatures further reduce air density, increasing the density altitude and degrading both engine performance and aerodynamic efficiency. This effect is particularly significant during takeoff and climb, where high density altitude can substantially reduce the excess thrust or power available, compromising aircraft performance. Therefore, the concept of density altitude provides a direct and practical means of assessing propulsion performance. Regardless of the geometric altitude, the aircraft behaves as if operating at the corresponding density altitude, which governs both the aerodynamic forces and the propulsion system’s performance.

Reference Pressures for Piloting

To compensate for variations in atmospheric pressure, the pilot must continuously adjust the Kollsman window setting. In the U.S., the altimeter is typically set to the local mean sea level (MSL) pressure, ensuring that all aircraft in the vicinity operate relative to a common MSL reference. In this case, when the aircraft is on the ground, the altimeter should read approximately the field elevation, which also serves as a check of proper instrument operation. Air traffic control, weather stations, and other sources provide the local MSL pressure.

In other countries, altimeter settings known as QFE and QNH are used. The QFE is the reference pressure set in the Kollsman window, such that the altimeter indicates height above a specific reference elevation, usually the airfield elevation. In this case, the altimeter reads zero when the aircraft is on the ground. The QNH is the reference pressure that causes the altimeter to indicate altitude above mean sea level and is the standard setting used in the U.S.

Above 18,000 ft in the U.S., pilots set the altimeter to the standard MSL pressure of 29.92 in Hg. The corresponding altitude is then referred to as a flight level (FL). For example, flight level 230 (FL230) corresponds to a pressure altitude of 23,000 ft. At these higher altitudes, all aircraft operate relative to the same pressure datum to ensure consistent vertical separation.

Over oceanic and remote regions, where horizontal separation distances may be relatively small, vertical separation may be as little as 1,000 ft. This requirement further reinforces the need for all aircraft to reference a common pressure datum.

Altimetry in the Stratosphere

As will now be apparent, a pressure altimeter is a fundamental instrument in aviation for determining an aircraft’s altitude by measuring ambient static pressure. The relationship between pressure and altitude is defined by the International Standard Atmosphere (ISA), which specifies how pressure, temperature, and density vary with altitude under standard conditions. This model includes a linear temperature lapse rate in the troposphere and an isothermal layer in the lower stratosphere.

Mechanical altimeters designed for use at high altitudes, including the stratosphere, contain aneroid capsules that expand and contract in response to ambient pressure. This motion is transmitted through a system of levers, gears, and cams to rotate a needle on the altimeter dial. Because the pressure-altitude relationship in the ISA is nonlinear and piecewise defined, the internal gearing must be designed to reproduce the full ISA calibration, including:

1. The troposphere, where temperature decreases linearly with height, i.e.,

   (41)  

   where K/m is the standard lapse rate and K is the sea-level temperature. The corresponding pressure-altitude relationship is

   (42)  

   where J kg K is the specific gas constant for air.

2. The lower stratosphere, which is modeled as isothermal, i.e.,

   (43)  

   The corresponding pressure decays exponentially with altitude, i.e.,

   (44)  

   where Pa is the pressure at 11,000 m.

These ISA pressure-altitude relationships are used directly to calibrate the altimeter mechanism. The gearing and dial spacing are designed to reproduce the ISA profile, including the transition at the tropopause (approximately 11 km). Although the instrument is mechanical, its response is tailored to match the piecewise ISA model across both atmospheric layers.

If the altimeter mechanism were instead based on a single lapse-rate model extrapolated above 11 km, rather than transitioning to the isothermal formulation, it would underestimate altitude in the stratosphere. For example, at an altitude of 20,000 m, the ISA pressure is approximately Pa. If the tropospheric lapse-rate relation were incorrectly extended to this altitude, this pressure would correspond to an indicated altitude of about 18,900 m, an error of about 1,100 m.

In practice, such errors do not occur in properly calibrated instruments. High-altitude aircraft, including supersonic vehicles, rely on accurate altimetry to maintain vertical separation. Their altimeters, whether mechanical or digital, are designed to follow the complete ISA model. Therefore, when operating at FL600 (60,000 ft), the indicated altitude corresponds to the correct pressure altitude, consistent with air traffic control requirements and the aircraft’s certified flight envelope.

Electronic Flight Displays (EFD)

Advances in electronics and software engineering have significantly transformed the instrumentation used for modern aircraft with the introduction of Electronic Flight Displays (EFDs) and Electronic Flight Information Systems (EFIS), which are often referred to as “glass cockpits.” Glass cockpits replace traditional analog gauges and instruments with digital displays, providing pilots with a more intuitive and information-rich interface, an example being shown in Figure 9.

EFDs use the same types of instrument inputs as traditional analog gauges; however, their processing systems differ fundamentally. An Air Data Computer (ADC) first receives the Pitot and static pressure inputs. The ADC computes the difference between total and static pressures and uses equations to generate the information required to display airspeed on the PFD. Outside air temperatures are also monitored, entered into various components within the software, and displayed on the PFD screen. These are the essential elements of an EFIS:

1. Pitot-Static Inputs: EFDs still receive pitot-static inputs, in which the pitot tube measures total pressure, and the static ports measure static pressure. As with a traditional mechanical instrument, static pressure enables altitude calculation, whereas dynamic pressure provides the information needed to calculate airspeed.
2. Analog-to-Digital Converter (ADC): The pitot-static inputs are converted into digital signals by an Analog-to-Digital Converter (ADC). The ADC converts the analog signals from the pitot-static system into a digital format for processing by the EFD’s software and electronic system.
3. Air Data Computer (ADC): The ADC, a component of the EFD system, receives digitized pitot-static inputs. It computes the difference between total and static pressure to determine the airspeed. All necessary corrections for SPE can be included in the software. The ADC then generates the information required to display airspeed on the Primary Flight Display (PFD).
4. Outside Air Temperature (OAT) Monitoring: EFDs also measure OAT. This temperature data is essential for various components within the aircraft systems, such as engine performance calculations and fuel management. The OAT and TAT are displayed on the PFD screen, providing pilots with real-time temperature information.

Summary & Closure

The measurement of an aircraft’s altitude during flight is fundamental to both piloting and engineering. However, as with airspeed, the specific type of altitude must be clearly defined. A pressure altimeter provides the pressure altitude, , which depends only on ambient static pressure according to the International Standard Atmosphere (ISA). In contrast, the density altitude, , is a measure of the air density at the flight condition and directly affects both aerodynamic forces and powerplant performance.

Because density altitude cannot be measured directly, it must be determined from pressure altitude and outside air temperature. When the temperature exceeds the standard ISA value at a given altitude, the density altitude increases, corresponding to a reduction in air density and a degradation in aircraft performance. For this reason, density altitude, not pressure altitude, is the primary parameter governing aircraft performance.