Electron density models

The ne provides electron density models.

Models:

Captenter and Anderson (1992)
Sheeley et al (2001)
Denton et al., (2002)
Denton et al., (2004)
Denton et al., (2006)

`CA1992`(L[, doy, R13, Lpp, MLT, Lppo])	Equatorial electron density \(n_e\) (cm⁻³) by Carpenter & Anderson [1992].
`D2002`(r, ne_eq[, L, Rmax, alpha])	Electron density along a field line by Denton et al. [2002] eq.
`D2004`(L[, Lpp, R13, a])	Denton et al. [2004] -style plasmaspheric electron density (equatorial or along field line).
`D2006`(L[, mlt, r])	Denton et al. [2006] plasmaspheric electron density model with MLT dependence.
`S2001`(L[, Lpp, kp, mask_invalid])	Equatorial electron density model of the plasmasphere and plasmatrough by Sheeley et al. [2001].

Functions

rbamlib.models.ne.CA1992(L, doy=None, R13=None, Lpp=None, MLT=None, Lppo=False)

Equatorial electron density \(n_e\) (cm⁻³) by Carpenter & Anderson [1992].

Includes plasmasphere (L < Lpp), plasmapause segment (Lpp < L ≤ Lppo), and plasma trough (L ≥ Lpp or Lppo) regimes, with optional seasonal (doy) and solar–cycle corrections (R13). If Lpp is not supplied the plasmasphere value is returned. If MLT is provided, the resulting array is shaped as (len(L), len(MLT)). Lppo is a flag which enables calculation of the plasmapause segment.

Lppo=False: \(L \le L_{pp}\) → plasmasphere; \(L > L_{pp}\) → extended trough. (No plasmapause segment; this is a fast mask by \(L_{pp}\).)
Lppo=True: solve an outer plasmapause \(L_{ppo}\) by intersection and add the plasmapause segment for \(L_{pp} < L \le L_{ppo}\).

Parameters:

L (float or 1D ndarray) – McIlwain L shell.
doy (int or ndarray, optional) – Day of year (1–366). If provided, seasonal terms are added in \(\log_{10} n_e\).
R13 (float or ndarray, optional) – 13-month smoothed sunspot number. If provided, solar–cycle term is added in \(\log_{10} n_e\).
Lpp (float or 1D ndarray, optional) – Plasmapause inner limit. If array, it is a function of MLT (one per column).
MLT (float or 1D ndarray, optional) – Magnetic local time in hours (0–24). If array, the output is 2-D of shape (len(L), len(MLT)). If omitted while Lpp is given, MLT=0 is assumed.
Lppo (bool, default=False) – If True, compute \(L_{ppo}\) by intersection and include the plasmapause segment. If False, skip the segment and use a simple plasmasphere/trough mask.

Returns:

ne – Equatorial electron density (cm⁻³). Shape:

(len(L),) if MLT is None or scalar,
(len(L), len(MLT)) if MLT is a 1-D array.

Return type:

ndarray

Notes

Plasmasphere (\(L \le L_{pp}\)):

\[\log_{10} n_e = -0.3145L + 3.9043.\]

Optional seasonal terms, using day of year \(\text{doy}=d\):

\[\Delta_{\text{season}}(d) = 0.15 \cos\left( \frac{2\pi(d+9)}{365} \right) - 0.075 \cos\left( \frac{4\pi(d+9)}{365} \right).\]

Optional solar–cycle term using 13-month smoothed sunspot number \(\text{R13}=\overline{R}\):

\[\Delta_{\text{solar}}(\overline{R}) = (0.00127 \overline{R} - 0.0635) \exp(-(L-2)/1.5)\]

The full expression with \(d\): and \(\overline{R}\): is

\[\log_{10} n_e = -0.3145 L + 3.9043 + \Delta_{\text{season}}(d) + \Delta_{\text{solar}}(\overline{R})\]

Plasma trough (\(L > L_{pp}\))

\[ \begin{align}\begin{aligned}n_e(L,MLT) = A(MLT) L^{-4.5} + (1 - \exp(-(L-2)/10)\\\begin{split}A(t) = \begin{cases} 5800 + 300 MLT, & 0 \le MLT < 6, \\ -800 + 1400 MLT, & 6 \le MLT \le 15 \end{cases}\end{split}\end{aligned}\end{align} \]

If \(MLT > 15\), it is kept at constant \(MLT=15\).

Plasmapause segment (\(L_{pp} < L \le L_{ppo}\), used only if Lppo=True)

Anchored to \(n_e(L_{pp})\) with MLT-dependent decade slope.

\[ \begin{align}\begin{aligned}n_e(L,MLT) = n_e(L_{pp}) \times 10^{ - (L - L_{pp}) / s(MLT)}\\\begin{split}s(MLT) = \begin{cases} 0.10, & 0 \le MLT < 6, \\ 0.10 + 0.01 (MLT-6), & 6 \le MLT \le 15 \end{cases}\end{split}\end{aligned}\end{align} \]

The outer plasmapause \(L_{ppo}\) is obtained by solving for the intersection between the plasmapause segment and the extended trough at the given MLT.

If \(MLT > 15\), it is kept at constant \(MLT=15\).

rbamlib.models.ne.S2001(L, Lpp=None, kp=None, mask_invalid=True)

Equatorial electron density model of the plasmasphere and plasmatrough by Sheeley et al. [2001].

Parameters:

L (float or ndarray) – McIlwain L-shell.
Lpp (float or ndarray, optional) – Plasmapause location. If supplied, \(L \le L_{\mathrm{pp}}\) uses the plasmasphere model and \(L > L_{\mathrm{pp}}\) uses the trough model. If None, all inputs default to the plasmasphere model.
kp (float or ndarray, optional) – Kp index. Uses for plasmatrough calculation. If None, the base model for plasmatrough density is returned
mask_invalid (bool, optional) – If True (default), values outside \(3 \le L \le 7\) are returned as NaN. If False, equations are evaluated without restriction.

Returns:

ne – Equatorial electron density, \(\mathrm{cm}^{-3}\).

Return type:

float or ndarray

Notes

Model’s valid range: \(3 \le L \le 7\).

Plasmasphere (\(L \le L_{pp}\), Eq. 6):

\[n_e(L) = 1390 (3/L)^{4.83}\]

Plasmatrough (\(L > L_{pp}\), Eq. 7):

Base model for plasmatrough density.

\[n_e(L,\mathrm{LT}) = 124 \left(\frac{3}{L}\right)^{4.0}\]

Plasmatrough density accounting for Gallagher et al. (1998) [Gallagher et al., 1998] local time of maximum density.

\[n_e(L,\mathrm{LT}) = 124 \left(\frac{3}{L}\right)^{4.0} + 36 \left(\frac{3}{L}\right)^{3.5} \cos\left( \frac{\pi}{12} \left[\mathrm{LT} - \left(7.7 \left(\tfrac{3}{L}\right)^{2.0} + 12\right)\right] \right)\]

\[\mathrm{LT} = 0.145 K_p^2 - 2.63 K_p + 21.86\]

rbamlib.models.ne.D2002(r, ne_eq, L=None, Rmax=None, alpha=None)

Electron density along a field line by Denton et al. [2002] eq. (1, 2, 3, 4).

Parameters:

r (array_like) – Geocentric distance along the field line in Earth radii.
ne_eq (array_like) – Equatorial electron density \(n_{e,eq}\) in cm⁻³.
L (array_like, optional) – McIlwain L-shell. Provide L if you do not pass Rmax.
Rmax (array_like, optional) – Maximum geocentric distance along the field line in Earth radii. Provide Rmax if you do not pass L.
alpha (array_like, optional) – Power-law exponent controlling the density variation \(\alpha\). This parameter can be used to drive other model of similar form.

Returns:

ne – Electron density \(n_e\) in cm⁻³.

Return type:

ndarray

Notes

The original data set used to derive these relations was limited to roughly \(2 \le n_{e,eq} \le 1500\ \text{cm}^{-3}\) and \(2 \lesssim L \lesssim 8.5\).

The density varies with geocentric distance \(r\) (along a field line) as a power law of the equatorial value \(n_{e,eq}\) (eq. 1):

\[n_e(r) = n_{e,eq} \left( \frac{R_{\max}}{r} \right)^{\alpha}\]

where \(R_{\max} \approx L R_{Earth}\) is the maximum geocentric distance on the field line (near the magnetic equator). Denton et al. parameterized the exponent \(\alpha\) as (eq. 2-4) as:

\[\alpha = 8 - 0.43\frac{R_{\max}}{R_{Earth}} - 3\log_{10}(n_{e,eq}) + 0.28[\log_{10}(n_{e,eq})]^2\]

which becomes (using \(L \approx R_{\max}/R_{Earth}\)) the commonly used form

\[\alpha = 8 - 0.43L - 3\log_{10}(n_{e,eq}) + 0.28[\log_{10}(n_{e,eq})]^2\]

Since \(r\) is expressed in Earth radii, we set \(R_{Earth} = 1\) in the calculations. Under this convention, the maximum geocentric distance along the field line is simply \(R_{\max} = L\). The use of \(R_{\max}\) reflects the original notation in Denton et al. (2002), but in practice it is equivalent to specifying \(L\).

rbamlib.models.ne.D2004(L, Lpp=None, R13=13.0, a=None)

Denton et al. [2004] -style plasmaspheric electron density (equatorial or along field line).

The plasmasphere (L <= Lpp or when no Lpp is provided) uses Denton et al. (2004) equation (5). The plasmatrough (L > Lpp) uses a power-law form (eq. 11).

Parameters:

L (float or ndarray) – McIlwain L-shell.
Lpp (float or ndarray, optional) – Plasmapause location. If supplied, \(L \le L_{\mathrm{pp}}\) uses the plasmasphere model and \(L > L_{\mathrm{pp}}\) uses the trough model. If None, all inputs default to the plasmasphere model.
R13 (float, optional) – 13-month smoothed sunspot number. If is not None, solar–cycle term is added in \(\log_{10} n_e\). The default value 13 corresponds to the Denton et al. (2004).
a (tuple of 4 floats or None, optional) –
Coefficients \((a_1,a_2,a_3,a_4)\):
- \(a_1, a_2\) for plasmasphere (eq. 5).
- \(a_3, a_4\) for plasmatrough (eq. 11).
If None, defaults are used: (3.78, -0.324, 3.77, -3.45).

Returns:

ne – Electron density in cm⁻³.

Return type:

float or ndarray

Notes

Plasmasphere (\(L \le L_{pp}\), Eq. 5)

The sunspot term uses 13-month smoothed sunspot number \(\text{R13}=\overline{R}\). If R13 is None the sunspot term is omitted.

\[\log_{10}(n_e) = a_1 + a_2 L + (0.00127 \overline{R} - 0.0635) \exp(-\tfrac{L-2}{1.5})\]

Plasmatrough (\(L > L_{pp}\), Eq. 11)

\[n_e = a_3 * L ^ {a_4}\]

rbamlib.models.ne.D2006(L, mlt=0.0, r=None)

Denton et al. [2006] plasmaspheric electron density model with MLT dependence.

Parameters:

L (float or ndarray) – McIlwain \(L\)-shell (dimensionless).
mlt (float or ndarray, optional) – Magnetic local time in hours (0–24). Default is 0.
r (float or ndarray, optional) – Geocentric distance along the field line in Earth radii. If provided, extend the equatorial value along the field line using D2002().

Returns:

ne – Electron density, cm⁻³. If r is provided, this is \(n_e(r)\); otherwise the equatorial value \(n_{e,eq}(L,MLT)\).

Return type:

float or ndarray

Notes

Denton et al. (2006) defines an empirical equatorial plasmasphere model with explicit MLT variation. Using \(\phi = 2\pi(\mathrm{MLT}-18)/24\), the two regimes are:

Eq. (2): inner plasmasphere (\(L \le 3.2\))

\[\log_{10} n_e = 5.25 - 0.82 L + 0.25 f(L) \cos\phi\]

with

\[\begin{split}f(L) = \begin{cases} \max(L-1.5, 0), & L \le 2.5, \\ 1, & L > 2.5 \end{cases}\end{split}\]

Eq. (3): outer plasmasphere (\(L > 3.2\))

\[\log_{10} n_e = 2.62 - 0.45 (L-3.2) + 0.12 (L-3.2)^2 + 0.25 \cos\phi\]

The full Denton 2006 distribution along a field line is obtained by combining the equatorial density above with the Denton power law (see D2002()):

\[n_e(r) = n_{e,eq}\left(\frac{R_{\max}}{r}\right)^{\alpha}, R_{\max} \approx L R_E\]

Densities are in cm⁻³, distances in Earth radii.