kappa_velocity_3D

plasmapy.physics.distribution.kappa_velocity_3D(vx, vy, vz, T, kappa, particle='e', vx_drift=0, vy_drift=0, vz_drift=0, vTh=nan, units='units')

Return the probability density function for finding a particle with velocity components v_x, v_y, and v_z`in m/s in a suprathermal plasma of temperature `T and parameter ‘kappa’ which follows the 3D Kappa distribution function. This function assumes Cartesian coordinates.

Parameters
  • vx (Quantity) – The velocity in x-direction units convertible to m/s.

  • vy (Quantity) – The velocity in y-direction units convertible to m/s.

  • vz (Quantity) – The velocity in z-direction units convertible to m/s.

  • T (Quantity) – The temperature, preferably 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.

  • vx_drift (Quantity, optional) – The drift velocity in x-direction units convertible to m/s.

  • vy_drift (Quantity, optional) – The drift velocity in y-direction units convertible to m/s.

  • vz_drift (Quantity, optional) – The drift velocity in z-direction 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.

Returns

f – Probability density in Velocity^-1, normalized so that: \(\iiint_{0}^{\infty} f(\vec{v}) d\vec{v} = 1\)

Return type

Quantity

Raises
  • TypeError – The parameter arguments are not Quantities and cannot be converted into Quantities.

  • UnitConversionError – If the parameters is not in appropriate units.

  • ValueError – If the temperature is negative, or the particle mass or charge state cannot be found.

Notes

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

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

where \(v_{Th},\kappa\) is the kappa thermal speed and \(A_\kappa = \frac{1}{2 \pi (\kappa v_{Th},\kappa^2)^{3/2}} \frac{\Gamma(\kappa + 1)}{\Gamma(\kappa - 1/2) \Gamma(3/2)}\) is the normalization constant.

As \(\kappa\) approaches infinity, the kappa distribution function converges to the Maxwellian distribution function.

See also

kappa_velocity_1D(), kappa_thermal_speed()

Example

>>> from plasmapy.physics import kappa_velocity_3D
>>> from astropy import units as u
>>> v=1 * u.m / u.s
>>> kappa_velocity_3D(vx=v,
... vy=v,
... vz=v,
... T=30000 * u.K,
... kappa=4,
... particle='e',
... vx_drift=0 * u.m / u.s,
... vy_drift=0 * u.m / u.s,
... vz_drift=0 * u.m / u.s)
<Quantity 3.7833988e-19 s3 / m3>