The following reference information was taken from: http://mtp.jpl.nasa.gov/notes/sat/sat.html

The relationship between static air temperature (SAT or T_s), and total air temperature (TAT or T_t), was given by:

(1)

where is the ratio of the specific heats at constant pressure and volume, and is the Mach Number, which is the ratio of the aircraft's true air speed (TAS or va) , and the local velocity of sound v_s .
Now the velocity of sound (to first order) depends only on the static air temperature ( ), where all the quantities have been previously defined except the Gas Constant, R , which for dry air is 287.053 J kg^-1 K ^-1 .

Measuring Temperature

We need to be clear about what is meant by the various temperatures:

• Static Air Temperature (SAT or T_s): This is the physical temperature of the air which the aircraft is flying through. It is also know as the outside air temperature (OAT); it is the temperature that we need to determine.
• Total Air Temperature (TAT or T_t): This is the temperature that would be measured by a probe if all of the kinetic energy of the air resulting from the aircraft's motion was absorbed. Because this is impossible, it can never be measured!
• Recovery Temperature (T_r): This is the temperature that the total air temperature is approximated by because of the incomplete recovery of the kinetic energy of the air by the temperature probe.
• Measured Temperature (T_m): As the name implies, this is the temperature that is actually measured by the aircraft's temperature probe. It differs from the recovery temperature, T_r, because of parasitic heating or cooling of the temperature sensor.

The figure to the right shows the relationship between these temperatures. As might be expected, T_r is always less than T_t, and both these approach Ts as the Mach Number approaches zero. All of these temperatures can be be related to the flight speed, but T_m is more involved. As shown by the cyan and white vertical box, T_m may be higher or lower than T_t and T_r due to sensor design, location, and parasitics for any particular flight condition(that is, speed and attitude). In extreme case such as for vortex-producing probe designs or for severe weather conditions, T_m may actually be less than T_s. For a well-designed temperature probe, however, it is possible to achieve T_m > 0.995 T_t.

The parameter that relates the T_r to T_t and T_s is called the recovery factor ( r ), which is defined by:

(19)

Using this definition in Equation (1) we obtain:

(20)

For most total temperature sensors, the recovery factor varies with Mach Number. It is now more common to use the recovery correction defined by:

(21)

Typically, the recovery correction varies with Mach Number below M = 1, but is constant for higher Mach Numbers. The recovery factor is related to the recovery correction by:

(22)

Based on wind tunnel measurements for a specific total air temperature probe, a chart for the dependence of the recovery correction on Mach Number can be created. This in turn can be used in Equation (22) to obtain the recovery factor (r), which in turn, by assuming that T_m very closely equals T_r , allows Equation (20) to be solved for T_s. How well all this can be done depends on how well the total temperature probe is characterized for a number of other important effects:

Time Constant: How fast the probe responds depends on the heat capacity of the sensor parts

• Thermal Conduction: If the aircraft body is much warmer than the probe (for example, in the sunlight on the ground), heat may be transferred to the probe.
• Thermal Radiation: If the probe is very warm as it might be on the ground, it may radiate some of it's energy away, thus appearing cooler than it actually is.
• Airflow Direction: Although TAT probes are designed to be insensitive to the direction of the air stream entering the probe, there nevertheless remains some dependence, especially for large attitude excursions.
• Self-Heating: Some probes require a small constant current to flow through them in order to generate a voltage proportional to resistance (which depends on temperature). This current can increase the temperature of the sensing element.
• De-icing: Some probes have de-icing elements which when activated, can cause heat to flow toward the sensing element.

Generally the probe manufacturer will provide tables or equations which can be used in the Air Data Computer (ADC) to correct for these effects.

This document provides additional information on the subject of air temperature measurement on aircraft: http://www.grc.nasa.gov/WWW/BGH/stagtmp.html