Calculator
Formula used
The classical Mach number is the ratio of flow speed to the local sound speed: M = v / cs.
If you compute sound speed from temperature using an ideal-gas model, the calculator uses: cs = √( γ kB T / ( μ mH ) ), where kB is Boltzmann’s constant and mH is the hydrogen mass.
In relativistic mode, it uses M = (Γ v) / (Γs cs), with Lorentz factors Γ = 1/√(1−β²) and β = v/c.
How to use this calculator
- Choose a calculation mode to solve for Mach, velocity, or sound speed.
- Enter the known values with units. Use km/s for typical space plasmas.
- Pick a sound speed source: direct input or compute from temperature.
- Set γ and μ for your gas composition and thermodynamics.
- Optionally enable relativistic mode for very fast flows.
- Press Calculate. Use the export buttons to save your results.
Example data
| Scenario | Velocity (km/s) | Sound speed (km/s) | Mach (classic) | Interpretation |
|---|---|---|---|---|
| Solar wind near 1 AU | 400 | 50 | 8 | Strongly supersonic flow |
| Molecular cloud turbulence | 2 | 0.2 | 10 | Highly compressible turbulence |
| Supernova remnant shock | 10,000 | 10 | 1000 | Extreme shock strength |
| Warm ionized medium | 20 | 12 | 1.67 | Moderately supersonic flow |
Values are illustrative. Real sound speeds depend on temperature, μ, and γ.
Mach number in astrophysical flows
1) Why Mach number matters
Mach number compares a flow speed to the local sound speed, highlighting where compressibility, shocks, and heating become important. In space plasmas and interstellar gas, the same velocity can be subsonic or highly supersonic depending on temperature and composition.
2) Sound speed from temperature
For an ideal gas, the adiabatic sound speed is cs = √(γkBT/μmp). At T = 10,000 K with γ = 5/3 and μ ≈ 0.61 (fully ionized), cs is about 15 km/s. At T = 100 K with μ ≈ 2.33 (molecular), cs is about 0.6 km/s.
3) Choosing γ for common gases
γ depends on degrees of freedom. Monatomic ideal gas uses γ = 5/3, while a diatomic gas often uses γ ≈ 7/5 at moderate temperatures. If cooling or microphysics is complex, treat γ as an effective model parameter.
4) Mean molecular weight μ in practice
μ converts temperature into sound speed. Typical values: μ ≈ 0.61 for fully ionized solar‑abundance gas, μ ≈ 1.27 for neutral atomic gas, and μ ≈ 2.33 for molecular clouds. Smaller μ increases sound speed and lowers Mach for the same velocity.
5) Solar wind and heliosphere
Near 1 AU, solar wind speeds of about 300–800 km/s often exceed ion sound speeds of tens of km/s. Mach numbers of several to >10 help explain strong compressions at planetary bow shocks and larger heliospheric structures.
6) Molecular clouds and star formation
Cold molecular gas (T ≈ 10–30 K) has sound speed around 0.2–0.4 km/s. Observed linewidths of 1–5 km/s imply Mach numbers of roughly 3–25. Supersonic turbulence can build dense filaments, then dissipate through shocks.
7) Supernova remnants and strong shocks
Young remnants can drive shocks of thousands to ~10,000 km/s. Against pre‑shock sound speeds near 10 km/s in warm gas, Mach numbers can reach hundreds to thousands, producing extreme compression and heating to millions of kelvin. Such shocks can also amplify magnetic fields and accelerate cosmic rays.
8) When to consider relativistic effects
If v is a significant fraction of c (for example ≥0.1c), a classical Mach estimate can be misleading. The optional relativistic mode uses β = v/c and a sound‑speed fraction to form a fast‑flow Mach‑like ratio for jets.
FAQs
1) What is a typical Mach number for the solar wind?
Near Earth, Mach numbers are often several to above 10, because flow speeds of hundreds of km/s exceed ion sound speeds of a few tens of km/s.
2) Which γ should I use?
Use γ = 5/3 for monatomic gas or many hot plasmas. Use γ ≈ 7/5 for diatomic gas at moderate temperatures. If cooling or physics is complex, γ is an effective parameter.
3) How do I pick μ?
Common choices are μ ≈ 0.61 (fully ionized), μ ≈ 1.27 (neutral atomic), and μ ≈ 2.33 (molecular). μ affects sound speed directly, so pick the best match for your gas phase.
4) Is this the magnetosonic Mach number?
No. This is the hydrodynamic Mach number based on the gas sound speed. In magnetized plasmas, shocks are often described using the fast magnetosonic speed instead.
5) Can I enter velocity in km/s?
Yes. Choose the unit selector next to the velocity field. The calculator converts to m/s internally and reports results in both m/s and km/s.
6) What if my velocity is negative?
Mach number is conventionally a magnitude. The calculator uses the absolute value for Mach while your sign still indicates direction for interpretation.
7) What does M < 1 imply?
Subsonic flow (M < 1) is weakly compressible and typically does not form shocks from small disturbances. Supersonic flow (M > 1) readily forms shocks and strong compressions.