Maxwellian_velocity_2D

Probability distribution function of velocity for a Maxwellian distribution in 2D.

Return the probability density function for finding a particle with velocity components vx and vy in m/s in an equilibrium plasma of temperature T which follows the 2D Maxwellian 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.
T (Quantity) – The temperature, preferably in kelvin.
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 in units convertible to m/s.
vy_drift (Quantity, optional) – The drift velocity in y-direction 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.

Returns:

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

Return type:

Quantity

Raises:

TypeError – A parameter argument is not a Quantity and cannot be converted into a Quantity.
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.

See also

Maxwellian_1D

Notes

In 2D, the Maxwellian velocity distribution function describing the distribution of particles with speed \(v\) in a plasma with temperature \(T\) is given by:

\[f = (π v_{Th}^2)^{-1} \exp \left [-(\vec{v} - \vec{V}_{drift})^2 / v_{Th}^2 \right ]\]

where \(v_{Th} = \sqrt{2 k_B T / m}\) is the thermal speed.

Examples

>>> import astropy.units as u
>>> v = 1 * u.m / u.s
>>> Maxwellian_velocity_2D(
...     vx=v,
...     vy=v,
...     T=30000 * u.K,
...     particle="e-",
...     vx_drift=0 * u.m / u.s,
...     vy_drift=0 * u.m / u.s,
... )
<Quantity 3.5002...e-13 s2 / m2>