Maxwellian_speed_3D

plasmapy.formulary.distribution.Maxwellian_speed_3D(

v,

T,

particle: str | int | integer | Particle | CustomParticle | Quantity = 'e-',

v_drift: float | Quantity = 0,

vTh=nan,

units: str = 'units',

*,

mass_numb=None,

Z=None,

)

Probability distribution function of speed for a Maxwellian distribution in 3D.

Return the probability density function for finding a particle with speed components vx, vy, and vz in m/s in an equilibrium plasma of temperature T which follows the 3D Maxwellian distribution function. This function assumes Cartesian coordinates.

Parameters:

• v (Quantity) – The speed in 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.

• v_drift (Quantity) – The drift speed 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 speed^-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_speed_1D

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:

\[f = 4 π v^{2} (π v_{Th}^2)^{-3/2} \exp(-v^{2} / v_{Th}^2)\]

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_speed_3D(v=v, T=30000 * u.K, particle="e-", v_drift=0 * u.m / u.s)
<Quantity 2.60235...e-18 s / m>