plasmapy.formulary.distribution.kappa_velocity_1D(v, T, kappa, particle: str | int | integer | Particle | CustomParticle | Quantity = 'e-', v_drift=0, vTh=nan, units='units', *, mass_numb=None, Z=None)[source]

Return the probability density at the velocity v in m/s to find a particle particle in a plasma of temperature T following the Kappa distribution function in 1D. The slope of the tail of the Kappa distribution function is set by ‘kappa’, which must be greater than \(1/2\).

  • v (Quantity) – The velocity in units convertible to m/s.

  • T (Quantity) – The temperature in kelvin.

  • kappa (Quantity) – 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 (str, optional) – Representation of the particle species(e.g., 'p for protons, 'D+' for deuterium, or 'He-4 +1' for \(He_4^{+1}\) (singly ionized helium-4)), which defaults to electrons.

  • v_drift (Quantity, optional) – The drift velocity in units convertible to m/s.

  • vTh (Quantity, optional) – Thermal velocity (most probable) in m/s. This is used for optimization purposes to avoid re-calculating vTh, for example when integrating over velocity-space.

  • units (str, optional) – Selects whether to run function with units and unit checks (when equal to "units") or to run as unitless (when equal to "unitless"). The unitless version is substantially faster for intensive computations.

  • mass_numb (integer, keyword-only, optional) – The mass number associated with particle.

  • Z (real number, keyword-only, optional) – The charge number associated with particle.


f – Probability density in velocity-1, normalized so that \(\int_{-∞}^{+∞} f(v) dv = 1\).

Return type:




In one dimension, the Kappa velocity distribution function describing the distribution of particles with speed \(v\) in a plasma with temperature \(T\) and suprathermal parameter \(κ\) is given by:

\[f = A_κ \left(1 + \frac{(\vec{v} - \vec{V_{drift}})^2}{κ v_{Th},κ^2}\right)^{-κ}\]

where \(v_{Th},κ\) is the kappa thermal speed and \(A_κ = \frac{1}{\sqrt{π} κ^{3/2} v_{Th},κ^2 \frac{Γ(κ + 1)}{Γ(κ - 1/2)}}\) is the normalization constant.

As \(κ → ∞\), the kappa distribution function converges to the Maxwellian distribution function.


>>> import astropy.units as u
>>> v = 1 * u.m / u.s
>>> kappa_velocity_1D(
...     v=v,
...     T=30000 * u.K,
...     kappa=4,
...     particle="e-",
...     v_drift=0 * u.m / u.s,
... )
<Quantity 6.75549...e-07 s / m>