Mach Number Calculator (Speed & Temperature)

Introduction: Mach number is a ratio, and the denominator moves

Mach number is speed divided by the local speed of sound. That sounds like a trivial division, and the trap is entirely in the denominator: the speed of sound is not a constant, it depends on the temperature of the air the object is actually moving through, and in the atmosphere that varies by more than 15% between the ground on a warm day and the tropopause. The same true airspeed of 250 m/s is Mach 0.73 at sea level on a mild day and Mach 0.85 at 11 km. Nothing about the aircraft changed.

That is why an aviation Mach calculator that only accepts a temperature in Celsius is asking for the number people are least likely to have. What a pilot or a student usually has is an altitude. This page therefore takes either: type the outside air temperature if you know it, or give an altitude and let the International Standard Atmosphere supply the temperature.

It also offers both of the speed-of-sound models in common use, and shows what the choice costs you. The linear approximation is tuned around 0 °C and drifts as you move away from it; the ideal-gas form is exact for a perfect diatomic gas at any temperature. At the tropopause the two disagree by about 2 m/s, which is 0.7% — small, but the wrong side of small if you are near a critical Mach number.

How to use the Mach number calculator

1. Enter the speed relative to the air, in whichever unit you have it. Metres per second, kilometres per hour, miles per hour and knots are all accepted and converted internally. Use true airspeed, not ground speed: wind changes the ground speed and not the Mach number.
2. Give a temperature or an altitude. Temperature mode takes Celsius, Fahrenheit or Kelvin. Altitude mode applies the International Standard Atmosphere lapse rate of 6.5 K per kilometre up to 11 km and the constant −56.5 °C stratospheric temperature above it, and tells you the temperature it derived.
3. Choose the speed-of-sound model. The ideal-gas form is the default and is the one to use at altitude. The linear approximation is offered because it is what most textbooks and many other calculators use, and because seeing both makes the size of the difference obvious.
4. Read the regime, not just the number. The result names the flow regime and, more usefully, tells you how far you are from the next boundary in both Mach and in your own speed units.

The formulas, and where they differ

Mach number is defined as the ratio of speed to the local speed of sound:

For an ideal gas, the speed of sound follows from the adiabatic bulk modulus and depends only on absolute temperature, not on pressure or density independently:

with in kelvin. The familiar near-surface shortcut is the first-order Taylor expansion of that square root about 0 °C:

Because it is a tangent line to a square root, it is exact at 0 °C and increasingly optimistic on both sides — it over-predicts the speed of sound, and therefore under-predicts Mach, at every temperature except the one it was fitted at.

In altitude mode the temperature comes from the International Standard Atmosphere, which is a piecewise linear profile:

Plain-text formula: speedOfSound = sqrt(1.4 * 287.05 * tempKelvin) for the ideal-gas model, or 331.3 + 0.606 * tempCelsius for the linear one; mach = speedMetresPerSecond / speedOfSound; and in altitude mode tempKelvin = altitude <= 11000 ? 288.15 - 0.0065 * altitude : 216.65.

Worked example: 250 m/s at 15 °C, and the same speed at altitude

An aircraft moving at 250 m/s through air at 15 °C:

1. Ideal gas: a = √(1.4 × 287.05 × 288.15) = 340.29 m/s.
2. Mach: 250 ÷ 340.29 = M 0.735, comfortably subsonic.
3. The linear approximation gives a = 331.3 + 0.606 × 15 = 340.39 m/s and M 0.7345 — a difference of 0.0002 in Mach, which is negligible this close to 0 °C.

Now fly the same 250 m/s at 11,000 m, a typical airliner cruise altitude. The standard atmosphere puts the temperature at 288.15 − 0.0065 × 11,000 = 216.65 K, or −56.5 °C. The speed of sound falls to 295.07 m/s and the same true airspeed becomes M 0.847 — transonic, in the regime where wave drag rises sharply. The aircraft did not speed up; the air got colder.

Here the two models genuinely part company. The linear approximation gives 297.06 m/s at −56.5 °C, which is 1.99 m/s or 0.68% too high, and therefore a Mach number of 0.841 instead of 0.847. Six thousandths of a Mach is irrelevant in a classroom and is not irrelevant when the critical Mach number of the wing is 0.85.

The error is smallest in the range the approximation was designed for and grows in both directions, which is the signature of a tangent-line fit. For weather-range temperatures it does not matter. For anything above about 8 km it does.

How to read the regime

• Subsonic (M < 0.8): compressibility effects are usually small.
• Transonic (0.8 ≤ M < 1.2): mixed subsonic and supersonic flow over the same body; local shocks form and drag rises rapidly.
• Supersonic (1.2 ≤ M < 5): sustained supersonic flow, dominated by shock waves and expansion fans.
• Hypersonic (M ≥ 5): strong shocks, aerodynamic heating, and real-gas chemistry such as dissociation.

These boundaries are conventions, not physics. Nothing changes discontinuously at exactly M 0.8; the transonic label marks the region where a body moving below the speed of sound can still have supersonic flow somewhere over it, which for a typical wing begins somewhere in the 0.7 to 0.8 range and depends entirely on the shape.

Assumptions and limitations

Dry air only. Humidity raises the speed of sound slightly, because water vapour is lighter than the nitrogen and oxygen it displaces. The effect is a few tenths of a percent at most in ordinary conditions, which is smaller than the difference between the two models offered here.

Temperature is the only atmospheric input. For an ideal gas the speed of sound genuinely does depend only on temperature, so this is not an approximation for the gas model — but it does mean altitude enters only through temperature, and the standard atmosphere is a model, not a forecast. A real day can be 15 K away from ISA at cruise altitude.

Air-relative speed, not ground speed. Mach number is defined against the air the object is moving through. A 100 knot tailwind changes ground speed by 100 knots and Mach number not at all.

Not valid for real-gas hypersonics. Above roughly M 5, temperatures behind shocks become high enough for vibrational excitation, dissociation and ionisation, at which point γ is no longer 1.4 and the ideal-gas speed of sound stops being the right quantity.

Local, not free-stream. The temperature entered is treated as the ambient temperature of the undisturbed air. Air near a surface, behind a shock, or in an exhaust plume is at a very different temperature and a very different local Mach number.

The regime boundaries are labels. They are conventional ranges for describing flow behaviour, not design limits, and the critical Mach number of any particular airframe is a property of that airframe.

Common questions about Mach number

Sources and constants. The ideal-gas speed of sound uses a ratio of specific heats of 1.4 and a specific gas constant for dry air of 287.05 J/(kg·K), giving a = √(γRT) with temperature in kelvin; these are the standard values for dry air treated as a perfect diatomic gas. The linear approximation a = 331.3 + 0.606T with T in degrees Celsius is the conventional near-surface shortcut and is the first-order expansion of the exact expression about 0 °C. The International Standard Atmosphere temperature profile is 288.15 K at sea level with a lapse rate of 6.5 K per kilometre to the tropopause at 11,000 m, then constant at 216.65 K to 20,000 m. Unit conversions are exact: 1 knot = 1,852 m/h, 1 mile = 1,609.344 m. The flow-regime boundaries at Mach 0.8, 1.2 and 5 are descriptive conventions rather than physical thresholds, and the critical Mach number of any particular airframe is a property of that airframe. Humidity, real-gas effects above about Mach 5, and any departure of the actual atmosphere from the standard profile are outside this model.