Dipole field model functions
The dip contains functions corresponding to the dipole field model.
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The dipole model of magnetic field. |
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Magnetic field value at the magnetic equator. |
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Approximation of the integral function T related to the bounce period, derived in the dipole approximation. |
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Approximation of the integral function Y related to the second adiabatic invariant, derived in the dipole approximation. |
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Equatorial loss-cone pitch angle, \(\alpha_{lc}\), in a dipole field at the surface of the planet, as a function of L-shell. |
Functions
- rbamlib.models.dip.B(r, mlat=0, planet='Earth')
The dipole model of magnetic field.
- Parameters:
r (float or ndarray) – The distance from the center of the planet, in planet’s radii.
mlat (float or ndarray, default = 0) – The magnetic latitude (MLAT), or geomagnetic latitude, is measured northwards from the equator in radians
planet (str, default = 'Earth') – The planet.
- Returns:
The magnetic field strength, in Gauss.
- Return type:
float or ndarray
- rbamlib.models.dip.B0(r, planet='Earth')
Magnetic field value at the magnetic equator.
Simplified calculation of the magnetic field at the magnetic equator.
- Parameters:
r (float or ndarray) – The distance from the center of the planet, in planet’s radii.
planet (str, default = 'Earth') – The planet.
- Returns:
The magnetic field strength, in Gauss.
- Return type:
float or ndarray
- rbamlib.models.dip.T(al)
Approximation of the integral function T related to the bounce period, derived in the dipole approximation.
- Parameters:
al (float or ndarray) – Equatorial pitch angle, in radians.
- Returns:
Value of T
- Return type:
float or ndarray
Notes
See Schulz and Lanzerotti [1974].
\[T( \alpha ) \approx T_0 - \frac{1}{2}(T_0 - T_1) \cdot \left( \sin( \alpha ) + \sin( \alpha)^1/2 \right)\]
- rbamlib.models.dip.Y(al)
Approximation of the integral function Y related to the second adiabatic invariant, derived in the dipole approximation.
- Parameters:
al (float or ndarray) – Equatorial pitch angle, in radians.
- Returns:
Value of Y
- Return type:
float or ndarray
Notes
See Schulz and Lanzerotti [1974].
\[Y( \alpha ) \approx 2(1 - \sin( \alpha ))T_0 + (T_0 - T_1) \cdot \left( \sin( \alpha ) \cdot \ln( \sin( \alpha ) ) + 2 \sin( \alpha ) - 2 \sqrt{ \sin( \alpha ) } \right) Y(0) = 2 \cdot T(0)\]
- rbamlib.models.dip.al_lc(L)
Equatorial loss-cone pitch angle, \(\alpha_{lc}\), in a dipole field at the surface of the planet, as a function of L-shell.
- Parameters:
L (float or ndarray) – McIlwain L-shell
- Returns:
Equatorial loss-cone pitch angle \(\alpha_{lc}\) in radians.
- Return type:
float or ndarray
Notes
Deriviation (based on chapter 3.4 of Roederer and Zhang [2016])
Magnetic moment \(\mu\) conservation:
\[\mu = \frac{m v_\perp^2}{2B} = \text{const} \;\Rightarrow\; \sin^2\alpha_{lc} = \frac{B_{eq}}{B_{lc}}.\]\(m\) - mass, \(v_\perp\) - velocity, \(B\) - magnetic field, \(\alpha\) - pitch angle
Dipole magnitude:
\[\begin{split}B(r,\lambda) = B_0\left(\frac{R}{r}\right)^3 \sqrt{1+3\sin^2\lambda}, \\ B_{eq} = B_0 / L^3, (\lambda=0,\; r=L R)\end{split}\]\(R\) - planet radius, \(\lambda\) - magnetic latitude
Field line & mirror field:
\[\begin{split}\begin{aligned} r &= L R \cos^2\lambda, \\ r &= R \Rightarrow \cos^2\lambda_{lc} = \frac{1}{L_{lc}}, \quad \sin^2\lambda_{lc} = 1-\frac{1}{L_{lc}}, \\ B_{lc} &= B_0\sqrt{4-\frac{3}{L}}. \end{aligned}\end{split}\]Loss-cone:
\[\sin^2\alpha_{lc} = \frac{B_{eq}}{B_{lc}} = \frac{1}{L^3\sqrt{4-\frac{3}{L}}}, \qquad \alpha_{lc} = \arcsin\left[ \frac{1}{L^{3/2}(4-3/L)^{1/4}} \right].\]Examples
>>> np.rad2deg(al_lc(4.0)) 5.34184...