ion_sound_speed¶

plasmapy.formulary.
ion_sound_speed
(T_e: Unit(‘K’), T_i: Unit(‘K’), ion: plasmapy.particles.particle_class.Particle, n_e: Unit(‘1 / m3’) = None, k: Unit(‘1 / m’) = None, gamma_e=1, gamma_i=3, z_mean=None)¶ Return the ion sound speed for an electronion plasma.
Aliases:
cs_
 Parameters
T_e (
Quantity
) – Electron temperature in units of temperature or energy per particle. If this is not given, then the electron temperature is assumed to be zero.T_i (
Quantity
) – Ion temperature in units of temperature or energy per particle. If this is not given, then the ion temperature is assumed to be zero.ion (
Particle
) – Representation of the ion species (e.g.,'p'
for protons,'D+'
for deuterium, or ‘He4 +1’ for singly ionized helium4). If no charge state information is provided, then the ions are assumed to be singly charged.n_e (
Quantity
) – Electron number density. If this is not given, then ion_sound_speed will be approximated in the nondispersive limit (\(k^2 λ_{D}^2\) will be assumed zero). Ifn_e
is given, a value fork
must also be given.k (
Quantity
) – Wavenumber (in units of inverse length, e.g. m^{1}). If this is not given, then ion_sound_speed will be approximated in the nondispersive limit (\(k^2 λ_{D}^2\) will be assumed zero). Ifk
is given, a value forn_e
must also be given.gamma_e (
float
orint
) – The adiabatic index for electrons, which defaults to 1. This value assumes that the electrons are able to equalize their temperature rapidly enough that the electrons are effectively isothermal.gamma_i (
float
orint
) – The adiabatic index for ions, which defaults to 3. This value assumes that ion motion has only one degree of freedom, namely along magnetic field lines.z_mean (
Quantity
, optional) – The average ionization (arithmetic mean) for a plasma where the a macroscopic description is valid. If this quantity is not given then the atomic charge state (integer) of the ion is used. This is effectively an average ion sound speed for the plasma where multiple charge states are present.
 Returns
V_S – The ion sound speed in units of meters per second.
 Return type
 Raises
TypeError – If any of the arguments are not entered as keyword arguments or are of an incorrect type.
ValueError – If the ion mass, adiabatic index, or temperature are invalid.
PhysicsError – If an adiabatic index is less than one.
UnitConversionError – If the temperature, electron number density, or wavenumber is in incorrect units.
 Warns
RelativityWarning
– If the ion sound speed exceeds 5% of the speed of light.UnitsWarning
– If units are not provided, SI units are assumed.PhysicsWarning
– If only one ofk
orn_e
is given, the nondispersive limit is assumed.
Notes
The ion sound speed \(V_S\) is given by
\[V_S = \sqrt{\frac{γ_e Z k_B T_e + γ_i k_B T_i}{m_i (1 + k^2 λ_{D}^2)}}\]where \(γ_e\) and \(γ_i\) are the electron and ion adiabatic indices, \(k_B\) is the Boltzmann constant, \(T_e\) and \(T_i\) are the electron and ion temperatures, \(Z\) is the charge state of the ion, \(m_i\) is the ion mass, \(λ_D\) is the Debye length, and \(k\) is the wavenumber.
In the nondispersive limit (\(k^2 λ_D^2\) is small) the equation for \(V_S\) is approximated (the denominator reduces to \(m_i\)).
When the electron temperature is much greater than the ion temperature, the ion sound velocity reduces to \(\sqrt{γ_e k_B T_e / m_i}\). Ion acoustic waves can therefore occur even when the ion temperature is zero.
Example
>>> from astropy import units as u >>> n = 5e19*u.m**3 >>> k_1 = 3e1*u.m**1 >>> k_2 = 3e7*u.m**1 >>> ion_sound_speed(T_e=5e6*u.K, T_i=0*u.K, ion='p', gamma_e=1, gamma_i=3) <Quantity 203155... m / s> >>> ion_sound_speed(T_e=5e6*u.K, T_i=0*u.K, n_e=n, k=k_1, ion='p', gamma_e=1, gamma_i=3) <Quantity 203155... m / s> >>> ion_sound_speed(T_e=5e6*u.K, T_i=0*u.K, n_e=n, k=k_2, ion='p', gamma_e=1, gamma_i=3) <Quantity 310.31... m / s> >>> ion_sound_speed(T_e=5e6*u.K, T_i=0*u.K, n_e=n, k=k_1, ion='p') <Quantity 203155... m / s> >>> ion_sound_speed(T_e=500*u.eV, T_i=200*u.eV, n_e=n, k=k_1, ion='D+') <Quantity 229585... m / s>