Maxwellian_velocity_3D¶

plasmapy.formulary.
Maxwellian_velocity_3D
(vx, vy, vz, T, particle='e', vx_drift=0, vy_drift=0, vz_drift=0, vTh=nan, units='units')¶ Probability distribution function of velocity for a Maxwellian distribution in 3D.
Return the probability density function for finding a particle with velocity components
vx
,vy
, andvz
in m/s in an equilibrium plasma of temperatureT
which follows the 3D Maxwellian distribution function. This function assumes Cartesian coordinates. Parameters
vx (Quantity) – The velocity in xdirection units convertible to m/s.
vy (Quantity) – The velocity in ydirection units convertible to m/s.
vz (Quantity) – The velocity in zdirection units convertible to m/s.
T (Quantity) – The temperature, preferably in Kelvin.
particle (str, optional) – Representation of the particle species (e.g.,
'p'
for protons,'D+'
for deuterium, or'He4 +1'
for \(He_4^{+1}\) (singly ionized helium4)), which defaults to electrons.vx_drift (Quantity, optional) – The drift velocity in xdirection units convertible to m/s.
vy_drift (Quantity, optional) – The drift velocity in ydirection units convertible to m/s.
vz_drift (Quantity, optional) – The drift velocity in zdirection units convertible to m/s.
vTh (Quantity, optional) – Thermal velocity (most probable) in m/s. This is used for optimization purposes to avoid recalculating
vTh
, for example when integrating over velocityspace.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
 Raises
TypeError – A parameter argument is not a
Quantity
and cannot be converted into aQuantity
.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 3D, the Maxwellian speed distribution function describing the distribution of particles with speed \(v\) in a plasma with temperature \(T\) is given by:
\[f = (\pi v_{Th}^2)^{3/2} \exp \left [(\vec{v}  \vec{V}_{drift})^2 / v_{Th}^2 \right ]\]where \(v_{Th} = \sqrt{2 k_B T / m}\) is the thermal speed.
See also
Example
>>> from astropy import units as u >>> v=1 * u.m / u.s >>> Maxwellian_velocity_3D(vx=v, ... vy=v, ... vz=v, ... T=30000 * u.K, ... particle='e', ... vx_drift=0 * u.m / u.s, ... vy_drift=0 * u.m / u.s, ... vz_drift=0 * u.m / u.s) <Quantity 2.0708...e19 s3 / m3>