ion_sound_speed

plasmapy.physics.parameters.ion_sound_speed(T_e, T_i, gamma_e=1, gamma_i=3, ion='p+', z_mean=None)

Return the ion sound speed for an electron-ion plasma.

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.

  • gamma_e (float or int) – 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 or int) – 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.

  • ion (str, optional) – Representation of the ion species (e.g., 'p' for protons, 'D+' for deuterium, or ‘He-4 +1’ for singly ionized helium-4), which defaults to protons. If no charge state information is provided, then the ions are assumed to be singly charged.

  • 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

Quantity

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 is in incorrect units.

Warns
  • RelativityWarning – If the ion sound speed exceeds 5% of the speed of light.

  • ~astropy.units.UnitsWarning – If units are not provided, SI units are assumed.

Notes

The ion sound speed \(V_S\) is approximately given by

\[V_S = \sqrt{\frac{\gamma_e Z k_B T_e + \gamma_i k_B T_i}{m_i}}\]

where \(\gamma_e\) and \(\gamma_i\) 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, and \(m_i\) is the ion mass.

This function assumes that the product of the wavenumber and the Debye length is small. In this limit, the ion sound speed is not dispersive. In other words, it is frequency independent.

When the electron temperature is much greater than the ion temperature, the ion sound velocity reduces to \(\sqrt{\gamma_e k_B T_e / m_i}\). Ion acoustic waves can therefore occur even when the ion temperature is zero.

Example

>>> from astropy import units as u
>>> ion_sound_speed(T_e=5e6*u.K, T_i=0*u.K, ion='p', gamma_e=1, gamma_i=3)
<Quantity 203155.0764042 m / s>
>>> ion_sound_speed(T_e=5e6*u.K, T_i=0*u.K)
<Quantity 203155.0764042 m / s>
>>> ion_sound_speed(T_e=500*u.eV, T_i=200*u.eV, ion='D+')
<Quantity 229586.01860212 m / s>