As an object moves through a gas, the gas molecules near the object
are disturbed and move around the object.
Aerodynamic
forces are generated between the gas and the object and the
magnitude of these forces depend on many factors associated
with the object and the gas. The
speed of the object relative to the gas
introduces many significant effects.
We characterize the speed of the object by
a non-dimensional number called the
Mach number;
the Mach number is the ratio of the speed of the
object to the
speed of sound
in the gas. The speed of "sound" is actually the
speed of transmission for small,
isentropic
disturbances in the gas.
The physical state of the gas
depends
on the Mach number of the object.
In our discussions, we will use the Mach number of the object and
the Mach number of the flow interchangeably.
If we travel with the object as it moves through the air, the
air moves past the object at the speed of the object. So, the
Mach number of the object and the Mach number of the flow are
the same number.

For a moving flow of gas,
there are several different values for the
temperature of the gas.
The static temperature
is the temperature of the gas if it had no ordered motion
and was not flowing.
From
kinetic theory,
static temperature is related to the average kinetic energy
of the random motion of the molecules of the gas. The value
of the static temperature of air depends on the
altitude.
For a moving flow, there is a dynamic temperature
associated with the kinetic energy of ordered motion of the flow
in the same way that static temperature is related to the average
kinetic energy of the random motion of the molecules.
The total temperature is
the sum of the static temperature and the dynamic temperature.
and the value of total temperature depends on the
Mach number of the flow.
If the moving flow is isentropically brought to
a halt on the body, we measure the stagnation temperature.
The stagnation temperature is important because it is the temperature
that occurs at a stagnation point on the object.
Because the total temperature does not change through a
shock wave, the stagnation temperature and
and the total temperature have the same value at a stagnation point.

In the process of slowing the flow, the gas is heated due to
the kinetic energy of flow. The amount of the heating depends
on the
specific heat capacity
of the gas.
If the specific heat capacity is a constant value, the gas is said
to be calorically perfect and if the specific heat capacity
changes, the gas is said to be calorically imperfect.
At subsonic and low supersonic
Mach numbers, air is calorically perfect.
But under low hypersonic conditions, air is
calorically imperfect.
Derived flow variables, like the
speed of sound and the
isentropic flow relations
are slightly different for a calorically imperfect gas
than the conditions predicted for a calorically perfect gas
because some of the energy of the flow excites the vibrational
modes of the diatomic molecules of nitrogen and oxygen in the air.

On the figure, we have plotted the value of stagnation temperature
for a standard day atmosphere as a function of altitude
and Mach number. There are two sets of lines on the figure
because of the inclusion of
real gas effects.
The solid line is the computed stagnation temperature for
a calorically perfect gas and the dashed line is
the computed stagnation temperature for a calorically
imperfect gas.
At the lower Mach numbers, below Mach 3, the
values of stagnation temperature are the same, because the
temperature is not high enough to excite the vibrational modes.
But beginning around Mach 3, real gas effects become increasingly
important with increasing Mach number.

For the perfect gas, the stagnation temperature is derived from the
isentropic total temperature
equation:

Tt = T * [1 + M^2 * (gamma-1)/2]

where Tt is the total temperature, T is the static
temperature at a given altitude, M is the Mach number,
and gamma is the ratio of specific heats for a calorically
perfect gas and has a constant value of 1.4.

For the calorically imperfect gas, the ratio of specific heats is not a constant but a
function of the static temperature.
Mathematical models
based on a simple harmonic vibrator have been developed.
The details of the analysis were given by Eggars in
NACA Report 959.
A synopsis of the report is included in
NACA Report 1135.
Solving the integrated
energy equation
with the assumption of variable specific heat, one obtains an equation
for the Mach number in terms of the ratio of the static and total (stagnation)
temperature:

where gamma is the ratio of specific heats for a perfect gas, theta is
a thermal constant equal to 5500 degrees Rankine, and gam is the ratio of
specific heats including a correction for the vibrational
modes:

This equation must be solved iteratively
to obtain a value for the total temperature.
Here is a Java calculator that solves the calorically imperfect gas equation:

To change input values, click on the input box (black on white),
backspace over the input value, type in your new value. Then hit
the red COMPUTE button to
send your new value to the program.
You will see the output boxes (yellow on black)
change value. You can use either Imperial or Metric units and you can input either the Mach number
or the speed by using the menu buttons. Just click on the menu button and click
on your
selection.
If you are an experienced user of this calculator, you can use a
sleek version
of the program which loads faster on your computer and does not include these instructions.
You can also download your own copy of the program to run off-line by clicking on this button:

As a test computation, set the Mach number to 5 and the altitude to 50,000 feet.
Notice that the calculated total temperature for the calorically imperfect
gas is less than the value for the perfect gas. The difference in temperature
is caused by the excitation of the vibrational modes; some of the energy of
the flow has been used to alter the state of the molecules.