Particle motion

The motion provides functions for representation of charged particle motion in electric and magnetic field.

Main Features:

  • Oscilation frequencies (gyro, bounce, and drift)

  • Angular Frequencies (gyro, bounce, and drift)

  • Motion period (gyro, bounce and drift)

  • Drift velocities calculation [TBD]

T_bounce(L, al, en[, planet, m])

Calculate the bounce period of a trapped charged particle in a dipolar magnetic field.

T_drift(L, en[, al, planet, m, q])

Calculate the drift period.

T_gyro([B, L, en, planet, q, m])

Calculate the gyro period of a charged particle in a magnetic field.

f_bounce(L, al, en[, planet, m])

Calculate the bounce frequency of a charged particle in a dipolar magnetic field.

f_drift(L, en[, al, planet, m, q])

Calculate the drift frequency.

f_gyro([B, L, en, planet, q, m])

Calculate the oscillation gyro (cyclotron) frequency for a charged particle.

omega_bounce(L, al, en[, planet, m])

Calculate the angular bounce frequency for a trapped charged particle in a dipolar magnetic field.

omega_drift(L, en[, al, planet, m, q])

Calculate the angular drift frequency.

omega_gyro([B, L, en, planet, q, m])

Calculate the angular gyro (cyclotron) frequency for a charged particle in a magnetic field.

Functions

rbamlib.motion.f_gyro(B=None, L=None, en=None, planet='Earth', q=4.803e-10, m=9.1e-28)

Calculate the oscillation gyro (cyclotron) frequency for a charged particle.

It accounts for the relativistic correction when the particle’s kinetic energy is provided.

Parameters:

  • B (float or ndarray, optional) – Magnetic field strength, in Gauss. Either B or L must be provided.

  • L (float or ndarray, optional) – McIlwain L-shell parameter (dimensionless). Used to compute B if it is not provided.

  • en (float or ndarray, optional) – Kinetic energy of the particle, in MeV. When provided, a relativistic correction via the Lorentz factor is applied.

  • planet (str, optional, default = 'Earth') – Name of the planet for which the magnetic field is calculated.

  • q (float, optional) – Particle charge, in statcoulombs (SGS units). Default is electrons.

  • m (float, optional) – Particle mass, in grams. Default is electrons.

Returns:

Gyro frequency, in Hertz (Hz).

Return type:

float or ndarray

Notes

For a non-relativistic particle, the gyro frequency is given by

\[f = \frac{|q| B}{2 \pi m c}\]

For a relativistic particle with kinetic energy (en), the Lorentz factor is

\[\gamma = 1 + \frac{en}{m \cdot c^2}\]

and the gyro frequency becomes [Roederer and Zhang, 2016]

\[f = \frac{|q| B}{2 \pi \gamma m c}\]
rbamlib.motion.omega_gyro(B=None, L=None, en=None, planet='Earth', q=4.803e-10, m=9.1e-28)

Calculate the angular gyro (cyclotron) frequency for a charged particle in a magnetic field.

Parameters:

  • B (float or ndarray, optional) – Magnetic field strength, in Gauss. Either B or L must be provided.

  • L (float or ndarray, optional) – McIlwain L-shell parameter (dimensionless). Used to compute B if it is not provided.

  • en (float or ndarray, optional) – Kinetic energy of the particle, in MeV. When provided, a relativistic correction via the Lorentz factor is applied.

  • planet (str, optional, default = 'Earth') – Name of the planet for which the magnetic field is calculated.

  • q (float, optional) – Particle charge, in statcoulombs (SGS units). Defaults is for electrons.

  • m (float, optional) – Particle mass, in grams. Defaults is for electrons.

Returns:

Angular gyro frequency, in radians per second (rad/s).

Return type:

float or ndarray

Notes

The angular gyro frequency is calculated from the standard gyro frequency f [Koskinen and Kilpua, 2022]:

\[\omega = 2 \pi f\]
rbamlib.motion.T_gyro(B=None, L=None, en=None, planet='Earth', q=4.803e-10, m=9.1e-28)

Calculate the gyro period of a charged particle in a magnetic field.

Parameters:

  • B (float or ndarray, optional) – Magnetic field strength, in Gauss. Either B or L must be provided.

  • L (float or ndarray, optional) – McIlwain L-shell parameter (dimensionless). Used to compute B if it is not provided.

  • en (float or ndarray, optional) – Kinetic energy of the particle, in MeV. When provided, a relativistic correction via the Lorentz factor is applied.

  • planet (str, optional, default = 'Earth') – Name of the planet for which the magnetic field is calculated.

  • q (float, optional) – Particle charge, in statcoulombs (SGS units). Defaults is for electrons.

  • m (float, optional) – Particle mass, in grams. Defaults is for electrons.

Returns:

Gyro period, in seconds (s).

Return type:

float or ndarray

Notes

The gyro period is defined as the inverse of the angular gyro frequency [Koskinen and Kilpua, 2022]:

\[T = \frac{2\pi}{\omega}\]
rbamlib.motion.f_bounce(L, al, en, planet='Earth', m=9.1e-28)

Calculate the bounce frequency of a charged particle in a dipolar magnetic field.

Parameters:

  • L (float or ndarray) – L-shell parameter (dimensionless).

  • al (float or ndarray) – Equatorial pitch angle of the particle, in radians.

  • en (float or ndarray) – Kinetic energy of the particle, in MeV.

  • planet (str, optional, Default = 'Earth'.) – Name of the planet.

  • m (float, optional) – Particle mass, in grams. Default is electrons.

Returns:

Bounce frequency, in Hertz (Hz).

Return type:

float or ndarray

Notes

The bounce frequency is defined as the inverse of the bounce period [Schulz and Lanzerotti, 1974].

\[f_{bounce} = \frac{1}{T_{bounce}}\]
rbamlib.motion.omega_bounce(L, al, en, planet='Earth', m=9.1e-28)

Calculate the angular bounce frequency for a trapped charged particle in a dipolar magnetic field.

Parameters:

  • L (float or ndarray) – L-shell parameter (dimensionless).

  • al (float or ndarray) – Equatorial pitch angle of the particle, in radians.

  • en (float or ndarray) – Kinetic energy of the particle, in MeV.

  • planet (str, optional, Default = 'Earth'.) – Name of the planet.

  • m (float, optional) – Particle mass, in grams. Default is electrons.

Returns:

Angular bounce frequency, in radians per second (rad/s).

Return type:

float or ndarray

Notes

The angular bounce frequency is obtained by Schulz and Lanzerotti [1974]:

\[\omega_{bounce} = \frac{2\pi}{T_{bounce}}\]

where \(T_{bounce}\) is the bounce period calculated by T_bounce function.

rbamlib.motion.T_bounce(L, al, en, planet='Earth', m=9.1e-28)

Calculate the bounce period of a trapped charged particle in a dipolar magnetic field.

Parameters:

  • L (float or ndarray) – L-shell parameter (dimensionless).

  • al (float or ndarray) – Equatorial pitch angle of the particle, in radians.

  • en (float or ndarray) – Kinetic energy of the particle, in MeV.

  • planet (str, optional, Default = 'Earth'.) – Name of the planet.

  • m (float, optional) – Particle mass, in grams. Default is electrons.

Returns:

Bounce period, in seconds (s).

Return type:

float or ndarray

Notes

The bounce period is calculated using the following Schulz and Lanzerotti [1974]:

\[\begin{split}T_{bounce} = \left( \frac{4r}{v} \right) T(\alpha) \\\end{split}\]

  • v is the effective particle velocity computed using the conversion of energy to momentum,

\[v = \frac{pc}{m\gamma}\]
\[r = L \cdot r_0\]

  • T(alpha) is a factor dependent on the equatorial pitch angle provided by rbamlib.models.dip.T.

rbamlib.motion.f_drift(L, en, al=1.5707963267948966, planet='Earth', m=9.1e-28, q=4.803e-10)

Calculate the drift frequency.

Parameters:

  • L (float or ndarray) – L-shell parameter.

  • en (float or ndarray) – Particle kinetic energy in MeV.

  • al (float or ndarray, optional, default=pi/2) – Pitch angle in radians.

  • planet (str, optional, default='Earth') – Name of the planet.

  • m (float, optional) – Particle mass in grams. Default is electrons.

  • q (float, optional) – Particle charge in statcoulombs (CGS units). Default is electrons.

Returns:

Drift frequency in Hertz (Hz).

Return type:

float or ndarray

Notes

The drift frequency is given by the combined equation [Schulz and Lanzerotti, 1974]:

\[f_{drift} = \left[ \frac{L}{2\pi} \cdot \frac{3 m c^3}{|q| B_0 r_0^2} \cdot \frac{\gamma^2 - 1}{\gamma} \right] \cdot \frac{6 - \frac{Y(\alpha)}{T(\alpha)}}{12}\]

  • \(\gamma\) is the relativistic Lorentz factor,

  • \(B_0\) is the magnetic field for the specified planet,

  • \(r_0\) is the planetary radius,

  • \(Y(\alpha)\) and \(T(\alpha)\) are functions from the dipole model.

rbamlib.motion.omega_drift(L, en, al=1.5707963267948966, planet='Earth', m=9.1e-28, q=4.803e-10)

Calculate the angular drift frequency.

Parameters:

  • L (float or ndarray) – L-shell parameter.

  • en (float or ndarray) – Particle kinetic energy in MeV.

  • al (float or ndarray, optional, default=pi/2) – Pitch angle in radians.

  • planet (str, optional, default='Earth') – Name of the planet.

  • m (float, optional) – Particle mass in grams. Default is electrons.

  • q (float, optional) – Particle charge in statcoulombs (CGS units). Default is electrons.

Returns:

Angular drift frequency in radians per second (rad/s).

Return type:

float or ndarray

Notes

The angular drift frequency is calculated using [Schulz and Lanzerotti, 1974]:

\[\omega_{drift} = 2\pi f_{drift}\]
rbamlib.motion.T_drift(L, en, al=1.5707963267948966, planet='Earth', m=9.1e-28, q=4.803e-10)

Calculate the drift period.

Parameters:

  • L (float or ndarray) – L-shell parameter.

  • en (float or ndarray) – Particle kinetic energy in MeV.

  • al (float or ndarray, optional, default=pi/2) – Pitch angle in radians.

  • planet (str, optional, default='Earth') – Name of the planet.

  • m (float, optional) – Particle mass in grams. Default is electrons.

  • q (float, optional) – Particle charge in statcoulombs (CGS units). Default is electrons.

Returns:

Drift period in seconds (s).

Return type:

float or ndarray

Notes

The drift period is the reciprocal of the drift frequency [Schulz and Lanzerotti, 1974]:

\[T_{drift} = \frac{1}{f_{drift}}\]