Plasma Density Models — WPIT Environment_mod API

The plasma density functions in Environment_mod implement four peer-reviewed empirical models for electron number density in Earth’s inner magnetosphere. Two models supply equatorial densities as a function of L-shell, while two others extend a given equatorial density along the full magnetic field line as a function of magnetic latitude. Together these models allow construction of complete three-dimensional density profiles needed for wave dispersion and wave-particle interaction calculations.

import WPIT.Environment_mod as env

# Quick equatorial density at L = 4 using the Sheeley model
ne_mean, ne_min, ne_max = env.density_equ_sheeley(4.0)
print(f"Ne ≈ {ne_mean:.1f} cm⁻³  [{ne_min:.1f} – {ne_max:.1f}]")

`density_equ_sheeley`

Returns the mean, minimum, and maximum equatorial electron number density at a given L-shell using the empirical Sheeley et al. (2001) model. This model is based on CRRES satellite observations and provides a simple power-law description valid across the plasmasphere and trough. The mean density is:

\bar{N}_e = 1390 \left(\frac{3}{L}\right)^{4.8}\ \text{cm}^{-3}

The ±1σ bounds are:

N_e^{\min/\max} = \bar{N}_e \mp 440 \left(\frac{3}{L}\right)^{3.6}\ \text{cm}^{-3}

Reference: Sheeley, B. W., Moldwin, M. B., Rassoul, H. K., and Anderson, R. R. (2001). “An empirical plasmasphere and trough density model: CRRES observations.” J. Geophys. Res., 106(A11), 25631–25641. doi:10.1029/2000JA000286.

Parameters

L_arg

float

required

L-shell value (dimensionless). The model is empirically constrained to the CRRES observation range; typical application domain is

L \approx 2

L \approx 7

Returns

ne_mean

float

Mean equatorial electron number density in cm⁻³.

ne_min

float

Lower statistical bound of equatorial electron number density in cm⁻³.

ne_max

float

Upper statistical bound of equatorial electron number density in cm⁻³.

Example

import numpy as np
import WPIT.Environment_mod as env

# Single L-shell evaluation
ne_mean, ne_min, ne_max = env.density_equ_sheeley(4.0)
print(f"L=4 → Ne = {ne_mean:.1f} cm⁻³  (range: {ne_min:.1f} – {ne_max:.1f})")

# Density profile from L=2 to L=7
L_vals = np.linspace(2, 7, 100)
means, mins, maxs = env.density_equ_sheeley(L_vals)

import matplotlib.pyplot as plt
plt.fill_between(L_vals, mins, maxs, alpha=0.3, label="±1σ range")
plt.plot(L_vals, means, label="Mean")
plt.xlabel("L-shell")
plt.ylabel("Ne (cm⁻³)")
plt.yscale("log")
plt.legend()
plt.show()

Output densities are in cm⁻³. Multiply by

10^6

to convert to SI units (m⁻³) for use with omega_plasma and other SI-based frequency functions.

`density_equ_carpenter_anderson`

Calculates the equatorial electron number density according to the Carpenter & Anderson (1992) ISEE/whistler model. This is a more detailed empirical model that accounts for the plasmasphere, plasmapause transition region, and outer plasma trough, with dependencies on geomagnetic activity (Kp index), season, magnetic local time (MLT), and solar activity. The model divides the magnetosphere into three zones:

Saturated plasmasphere ( $2.25 \leq L \leq L_{ppi}$ ): density depends on day of year and sunspot number.
Plasmapause transition ( $L_{ppi} \leq L \leq L_{ppo}$ ): sharp density gradient whose width depends on MLT.
Plasma trough ( $L_{ppo} \leq L \leq 8$ ): low-density outer region with MLT asymmetry.

The inner plasmapause boundary is placed at

L_{ppi} = 5.6 - 0.46 K_{p,\max}

. Reference: Carpenter, D. L., and R. R. Anderson (1992). “An ISEE/whistler model of equatorial electron density in the magnetosphere.” J. Geophys. Res.: Space Physics, 97(A2), 1097–1108.

Parameters

Lsh

float

required

L-shell value (dimensionless). Valid domain:

2.25 \leq L \leq 8

Kpmax

float

required

Maximum Kp geomagnetic activity index in the preceding 24 hours (dimensionless, range 0–9). Higher Kp contracts the plasmapause to lower L-shells.

day

int

required

Day of year (integer, 1–365). Used to apply the seasonal modulation of plasmaspheric density.

mlt

float

required

Magnetic local time in decimal hours (0–24). Controls the width of the plasmapause transition and the trough density.

float

required

13-month smoothed average sunspot number (dimensionless). Represents solar cycle phase; typical range 0–200.

Returns

ne_eq

float

Equatorial electron number density in cm⁻³.

Example

import WPIT.Environment_mod as env

# Quiet-time conditions, noon MLT, mid-year, moderate solar activity
ne = env.density_equ_carpenter_anderson(
    Lsh=4.0,
    Kpmax=2.0,    # quiet geomagnetic conditions
    day=180,      # mid-year (~June)
    mlt=12.0,     # local noon
    Rb=80.0       # moderate solar activity
)
print(f"Ne = {ne:.1f} cm⁻³")

# Active conditions: plasmapause moves inward
ne_active = env.density_equ_carpenter_anderson(
    Lsh=4.0,
    Kpmax=6.0,    # active conditions
    day=180,
    mlt=12.0,
    Rb=80.0
)
print(f"Ne (active) = {ne_active:.2f} cm⁻³")

The model is defined for

2.25 \leq L \leq 8

. Evaluating outside this range may produce unphysical results. The plasmapause outer boundary

L_{ppo}

is determined numerically as the intersection of the plasmapause and plasma trough density profiles.

`density_FL_denton`

Calculates the electron number density at a point along a magnetic field line using the Denton et al. (2002) power-law model. Given an equatorial reference density, the model scales density with magnetic latitude using a

\cos^{-4}(\lambda)

relationship derived from field-aligned magnetospheric observations. The field-line density profile is:

N_e(\lambda) = N_{e,\text{eq}} \cdot \cos^{-4}(\lambda)

Reference: Denton, R. E., Goldstein, J., and Menietti, J. D. (2002). “Field line dependence of magnetospheric electron density.” Geophys. Res. Lett., 29(24), 2205. doi:10.1029/2002GL015963.

Parameters

ne0_arg

float

required

Equatorial electron number density in the same units as the desired output (typically cm⁻³ when using outputs from density_equ_sheeley or density_equ_carpenter_anderson). This is the reference density at

\lambda = 0

lamda_arg

float

required

Magnetic latitude in radians. The density diverges as

\lambda \to \pm 90°

; limit usage to latitudes well below the loss cone mirror latitude.

Returns

n_e_tmp

float

Electron number density at the specified latitude. Units match those of ne0_arg.

Example

import numpy as np
import WPIT.Environment_mod as env

L = 4.0

# Get equatorial density from Sheeley model (cm⁻³)
ne_eq, _, _ = env.density_equ_sheeley(L)

# Density at 30° latitude using Denton model
lamda = np.radians(30.0)
ne_30 = env.density_FL_denton(ne_eq, lamda)
print(f"Ne at λ=30°: {ne_30:.2f} cm⁻³")

# Full field-line profile (0° to 50°)
latitudes = np.linspace(0, np.radians(50), 200)
ne_profile = env.density_FL_denton(ne_eq, latitudes)

import matplotlib.pyplot as plt
plt.plot(np.degrees(latitudes), ne_profile)
plt.xlabel("Magnetic latitude (°)")
plt.ylabel("Ne (cm⁻³)")
plt.title(f"Denton density profile, L = {L}")
plt.show()

The Denton model assumes a

\cos^{-4}\lambda

latitude dependence, consistent with flux tube volume conservation in a dipole field under the assumption of 94% H⁺, 5.4% He⁺, and 0.6% O⁺ composition. The output units are the same as the input ne0_arg — no internal unit conversion is performed.

`density_ozhogin`

Calculates the equatorial electron density and the field-line electron density at a given magnetic latitude using the Ozhogin et al. (2012) IMAGE RPI empirical model. Unlike the Denton model, Ozhogin provides its own empirical equatorial density formula as a function of L-shell, as well as a field-line profile based on a cosine power law. The equatorial density is:

\log_{10} N_{e,\text{eq}} = 4.4693 - 0.4903\, L

The field-line density at latitude

\lambda

is:

N_e(\lambda) = N_{e,\text{eq}} \cdot \left[\cos\!\left(\frac{\pi}{2} \cdot \frac{\lambda}{\lambda_{\text{inv}}}\right)\right]^{-0.75}

where

\lambda_{\text{inv}} = \arccos(1/L)

is the field-line invariant latitude computed in the source. Reference: Ozhogin, P., et al. (2012). “Field-aligned distribution of the plasmaspheric electron density: An empirical model derived from the IMAGE RPI measurements.” J. Geophys. Res.: Space Physics, 117(A6). doi:10.1029/2011JA017330.

Parameters

L_arg

float

required

L-shell value (dimensionless). The model’s empirical log-linear equatorial density formula is derived from IMAGE RPI observations, typically valid for

L \approx 1.5

L \approx 5

lambda_arg

float

required

Magnetic latitude in radians at which to evaluate the field-line density.

Returns

neqtmp

float

Equatorial electron number density in cm⁻³, calculated from the Ozhogin log-linear L-shell formula.

n_lamtmp

float

Electron number density at the specified magnetic latitude in cm⁻³.

Example

import numpy as np
import WPIT.Environment_mod as env

L = 3.5

# Equatorial and off-equatorial density at λ = 25°
ne_eq, ne_lat = env.density_ozhogin(L, np.radians(25.0))
print(f"Equatorial Ne: {ne_eq:.2f} cm⁻³")
print(f"Ne at λ=25°:   {ne_lat:.2f} cm⁻³")

# Compare Ozhogin profile with Denton profile
latitudes = np.linspace(0, np.radians(45), 200)
ne_eq_oz, _ = env.density_ozhogin(L, 0.0)

oz_profile = [env.density_ozhogin(L, lam)[1] for lam in latitudes]
denton_profile = env.density_FL_denton(ne_eq_oz, latitudes)

density_ozhogin returns two values: the equatorial density and the density at the requested latitude. Both are in cm⁻³. The first return value is independent of lambda_arg — it is purely a function of L-shell. The cosine argument reaches zero at lambda_arg = arccos(1/L); avoid evaluating at or beyond that latitude.

Environment Module

Wave Properties Module

WPI Module

Landau Damping Module

Plasma Density Models — WPIT Environment_mod API

`density_equ_sheeley`

Parameters

Returns

Example

`density_equ_carpenter_anderson`

Parameters

Returns

Example

`density_FL_denton`

Parameters

Returns

Example

`density_ozhogin`

Parameters

Returns

Example

Build docs developers (and LLMs) love

Environment Module

Wave Properties Module

WPI Module

Landau Damping Module

Documentation Index

​density_equ_sheeley

​Parameters

​Returns

​Example

​density_equ_carpenter_anderson

​Parameters

​Returns

​Example

​density_FL_denton

​Parameters

​Returns

​Example

​density_ozhogin

​Parameters

​Returns

​Example

Build docs developers (and LLMs) love

`density_equ_sheeley`

Parameters

Returns

Example

`density_equ_carpenter_anderson`

Parameters

Returns

Example

`density_FL_denton`

Parameters

Returns

Example

`density_ozhogin`

Parameters

Returns

Example