def ion_sound_speed(*ignore,
T_e=0 * units.K,
T_i=0 * units.K,
gamma_e=1,
gamma_i=3,
ion='p',
z_mean=None):
r"""
Returns the ion sound speed for an electron-ion plasma.
Parameters
----------
T_e : ~astropy.units.Quantity, optional
Electron temperature in units of temperature or energy per
particle. If this is not given, then the electron temperature
is assumed to be zero. If only one temperature is entered, it
is assumed to be the electron temperature.
T_i : ~astropy.units.Quantity, optional
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 : string, 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 : ~astropy.units.Quantity
The ion sound speed in units of meters per second.
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.
plasmapy.utils.PhysicsError
If an adiabatic index is less than one.
units.UnitConversionError
If the temperature is in incorrect units.
UserWarning
If the ion sound speed exceeds 10% of the speed of light, or
if units are not provided and SI units are assumed.
Notes
-----
The ion sound speed :math:`V_S` is approximately given by
.. math::
V_S = \sqrt{\frac{\gamma_e Z k_B T_e + \gamma_i k_B T_i}{m_i}}
where :math:`\gamma_e` and :math:`\gamma_i` are the electron and
ion adiabatic indices, :math:`k_B` is the Boltzmann constant,
:math:`T_e` and :math:`T_i` are the electron and ion temperatures,
:math:`Z` is the charge state of the ion, and :math:`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 (e.g., frequency independent).
When the electron temperature is much greater than the ion
temperature, the ion sound velocity reduces to
:math:`\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)
<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>
"""
if ignore:
raise TypeError("All arguments are required to be keyword arguments "
"in ion_sound_speed to prevent mixing up the electron "
"and ion temperatures. An example call that uses the "
"units subpackage from astropy is: "
"ion_sound_speed(T_e=5*units.K, T_i=0*units.K, "
"ion='D+')")
try:
m_i = atomic.ion_mass(ion)
if z_mean is None:
# warnings.warn("No z_mean given, defaulting to atomic charge",
# PhysicsWarning)
try:
Z = atomic.integer_charge(ion)
except AtomicError:
Z = 1
else:
# using average ionization provided by user
Z = z_mean
except AtomicError:
raise ValueError("Invalid ion in ion_sound_speed.")
if not isinstance(gamma_e, (float, int)):
raise TypeError("The adiabatic index for electrons (gamma_e) must be "
"a float or int in ion_sound_speed")
if not isinstance(gamma_i, (float, int)):
raise TypeError("The adiabatic index for ions (gamma_i) must be "
"a float or int in ion_sound_speed")
if not 1 <= gamma_e <= np.inf:
raise utils.PhysicsError(
"The adiabatic index for electrons must be between "
"one and infinity")
if not 1 <= gamma_i <= np.inf:
raise utils.PhysicsError(
"The adiabatic index for ions must be between "
"one and infinity")
T_i = T_i.to(units.K, equivalencies=units.temperature_energy())
T_e = T_e.to(units.K, equivalencies=units.temperature_energy())
try:
V_S_squared = (gamma_e * Z * k_B * T_e + gamma_i * k_B * T_i) / m_i
V_S = np.sqrt(V_S_squared).to(units.m / units.s)
except Exception:
raise ValueError("Unable to find ion sound speed.")
return V_S
def ion_sound_speed(T_e, T_i, gamma_e=1, gamma_i=3, ion='p+', z_mean=None):
r"""
Return the ion sound speed for an electron-ion plasma.
Parameters
----------
T_e : ~astropy.units.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 : ~astropy.units.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 : ~astropy.units.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 : ~astropy.units.Quantity
The ion sound speed in units of meters per second.
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.
~plasmapy.utils.PhysicsError
If an adiabatic index is less than one.
~astropy.units.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 :math:`V_S` is approximately given by
.. math::
V_S = \sqrt{\frac{\gamma_e Z k_B T_e + \gamma_i k_B T_i}{m_i}}
where :math:`\gamma_e` and :math:`\gamma_i` are the electron and
ion adiabatic indices, :math:`k_B` is the Boltzmann constant,
:math:`T_e` and :math:`T_i` are the electron and ion temperatures,
:math:`Z` is the charge state of the ion, and :math:`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
:math:`\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>
"""
m_i = atomic.ion_mass(ion)
Z = grab_charge(ion, z_mean)
for gamma, particles in zip([gamma_e, gamma_i], ["electrons", "ions"]):
if not isinstance(gamma, (float, int)):
raise TypeError(
f"The adiabatic index gamma for {particles} must be "
"a float or int")
if gamma < 1:
raise utils.PhysicsError(
f"The adiabatic index for {particles} must be between "
"one and infinity")
T_i = T_i.to(u.K, equivalencies=u.temperature_energy())
T_e = T_e.to(u.K, equivalencies=u.temperature_energy())
try:
V_S_squared = (gamma_e * Z * k_B * T_e + gamma_i * k_B * T_i) / m_i
V_S = np.sqrt(V_S_squared).to(u.m / u.s)
except Exception:
raise ValueError("Unable to find ion sound speed.")
return V_S
def ion_sound_speed(T_e,
T_i,
n_e=None,
k=None,
gamma_e=1,
gamma_i=3,
ion='p+',
z_mean=None):
r"""
Return the ion sound speed for an electron-ion plasma.
Parameters
----------
T_e : ~astropy.units.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 : ~astropy.units.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.
n_e : ~astropy.units.Quantity
Electron number density. If this is not given, then ion_sound_speed
will be approximated in the non-dispersive limit
(:math:`k^2 \lambda_{D}^2` will be assumed zero). If n_e is given,
a value for k must also be given.
k : ~astropy.units.Quantity
Wavenumber (in units of inverse length, e.g. per meter). If this
is not given, then ion_sound_speed will be approximated in the
non-dispersive limit (:math:`k^2 \lambda_{D}^2` will be assumed zero).
If k is given, a value for n_e must also be given.
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 : ~astropy.units.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 : ~astropy.units.Quantity
The ion sound speed in units of meters per second.
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.
~plasmapy.utils.PhysicsError
If an adiabatic index is less than one.
~astropy.units.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.
~astropy.units.UnitsWarning
If units are not provided, SI units are assumed.
PhysicsWarning
If only one of (k, n_e) is given, the non-dispersive limit
is assumed.
Notes
-----
The ion sound speed :math:`V_S` is given by
.. math::
V_S = \sqrt{\frac{\gamma_e Z k_B T_e + \gamma_i k_B T_i}{m_i (1 + k^2 \lambda_{D}^2)}}
where :math:`\gamma_e` and :math:`\gamma_i` are the electron and
ion adiabatic indices, :math:`k_B` is the Boltzmann constant,
:math:`T_e` and :math:`T_i` are the electron and ion temperatures,
:math:`Z` is the charge state of the ion, :math:`m_i` is the
ion mass, :math:`\lambda_{D}` is the Debye length, and :math:`k` is the
wavenumber.
In the non-dispersive limit (:math:`k^2 \lambda_{D}^2` is small) the
equation for :math:`V_S` is approximated (the denominator reduces
to :math:`m_i`).
When the electron temperature is much greater than the ion
temperature, the ion sound velocity reduces to
:math:`\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
>>> 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 203155.0764042 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 203155.03286794 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 310.31329069 m / s>
>>> ion_sound_speed(T_e=5e6*u.K, T_i=0*u.K, n_e=n, k=k_1)
<Quantity 203155.03286794 m / s>
>>> ion_sound_speed(T_e=500*u.eV, T_i=200*u.eV, n_e=n, k=k_1, ion='D+')
<Quantity 229585.96150738 m / s>
"""
m_i = atomic.particle_mass(ion)
Z = _grab_charge(ion, z_mean)
for gamma, particles in zip([gamma_e, gamma_i], ["electrons", "ions"]):
if not isinstance(gamma, (numbers.Real, numbers.Integral)):
raise TypeError(
f"The adiabatic index gamma for {particles} must be "
"a float or int")
if gamma < 1:
raise utils.PhysicsError(
f"The adiabatic index for {particles} must be between "
"one and infinity")
T_i = T_i.to(u.K, equivalencies=u.temperature_energy())
T_e = T_e.to(u.K, equivalencies=u.temperature_energy())
# Assume non-dispersive limit if values for n_e (or k) are not specified
klD2 = 0.0
if (n_e is None) ^ (k is None):
warnings.warn(
"The non-dispersive limit has been assumed for "
"this calculation. To prevent this, values must "
"be specified for both n_e and k.", PhysicsWarning)
elif n_e is not None and k is not None:
lambda_D = Debye_length(T_e, n_e)
klD2 = (k * lambda_D)**2
try:
V_S_squared = (gamma_e * Z * k_B * T_e +
gamma_i * k_B * T_i) / (m_i * (1 + klD2))
V_S = np.sqrt(V_S_squared).to(u.m / u.s)
except Exception:
raise ValueError("Unable to find ion sound speed.")
return V_S