plasmapy.formulary.speeds.kappa_thermal_speed(T: Quantity, kappa, particle: str | Integral | Particle | CustomParticle | Quantity, method='most_probable', *, mass_numb: Real | None = None, Z: Real | None = None) Quantity[source]

Return the most probable speed for a particle within a kappa distribution.

Aliases: vth_kappa_

  • T (Quantity) – The particle temperature in either kelvin or energy per particle

  • kappa (float) – The kappa parameter is a dimensionless number which sets the slope of the energy spectrum of suprathermal particles forming the tail of the kappa velocity distribution function. kappa must be greater than 3/2.

  • particle (particle-like) – Representation of the particle species (e.g., 'p+' for protons, 'D+' for deuterium, or ‘He-4 +1’ for singly ionized helium-4).

  • method (str, optional) – Method to be used for calculating the thermal speed. Options are 'most_probable' (default), 'rms', and 'mean_magnitude'.

  • mass_numb (integer, optional) – The mass number corresponding to particle.

  • Z (real number, optional) – The charge number corresponding to particle.


V – Particle thermal speed.

Return type:


  • TypeError – The particle temperature is not a Quantity.

  • UnitConversionError – If the particle temperature is not in units of temperature or energy per particle.

  • ValueError – The particle temperature is invalid or particle cannot be used to identify an isotope or particle.

  • RelativityWarning – If the particle thermal speed exceeds 5% of the speed of light.

  • UnitsWarning – If units are not provided, SI units are assumed.


The particle thermal speed is given by:

\[V_{th,i} = \sqrt{(2 κ - 3)\frac{2 k_B T_i}{κ m_i}}\]

For more discussion on the 'mean_magnitude' calculation method, see PlasmaPy issue #186.


>>> import astropy.units as u
>>> kappa_thermal_speed(5 * u.eV, 4, "p")  # defaults to most probable
<Quantity 24467.87... m / s>
>>> kappa_thermal_speed(5 * u.eV, 4, "p", "rms")
<Quantity 37905.47... m / s>
>>> kappa_thermal_speed(5 * u.eV, 4, "p", "mean_magnitude")
<Quantity 34922.98... m / s>