Home » Spacecraft Environment » Debye length, plasma frequency and cyclotron frequency

# Debye length, plasma frequency and cyclotron frequency

The calculation of characteristic Debye length, plasma frequency and cyclotron frequency of electrons in the ionosphere, the plasmasphere, the outer magnetosphere and the solar wind.

```clc;clear all;
me   = 9.109E-31;            %[kg]           Electron rest mass
mp   = 1.673E-27;            %[kg]           Proton rest mass
eps0 = 8.8542E-12;           %[A*s/(V*m)]    Permittivity
e    = 1.602E-19;            %[C]            Elementary charge
Re   = 6.37E6;               %[m]            Earth’s radius
Md   = 8E15;                 %[T*m3]         Earth’s magnetic dipole moment```

## Ionosphere

```n   = 1E12;                     %[m-3]  Electron density
kTe = 0.1;                      %[eV]   Thermal energy
r   = Re + 3E5;                 %[km]   Radial distance from the Earth’s center
wp  = (e^2*n/(eps0*me))^0.5/(2*pi);  %[Hz]   Plasma frequency
lD  = (eps0*kTe/(n*e))^0.5;     %[m]    Debye length
B   = 2*Md/r^3;                 %[T]    Earth’s magnetic field
Fc  = e*B/(2*pi*me);            %[Hz]   Cyclotron frequency
fprintf('Debye length        %4.2d [m]\n',lD);
fprintf('Plasma frequency    %4.2d [Hz]\n',wp);
fprintf('Cyclotron frequency %4.2d [Hz]\n\n',Fc);```
```Debye length        2.35e-003 [m]
Plasma frequency    8.98e+006 [Hz]
Cyclotron frequency 1.51e+006 [Hz]```

## Plasmasphere

```n   = 1E7;                      %[m-3]  Electron density
kTe = 1;                        %[eV]   Thermal energy
r   = 5*Re;                     %[km]   Radial distance from the Earth’s center
wp  = (e^2*n/(eps0*me))^0.5/(2*pi);  %[Hz]   Plasma frequency
lD  = (eps0*kTe/(n*e))^0.5;     %[m]    Debye length
B   = 2*Md/r^3;                 %[T]    Earth’s magnetic field
Fc  = e*B/(2*pi*me);            %[Hz]   Cyclotron frequency
fprintf('Debye length        %4.2d [m]\n',lD);
fprintf('Plasma frequency    %4.2d [Hz]\n',wp);
fprintf('Cyclotron frequency %4.2d [Hz]\n\n',Fc);```
```Debye length        2.35e+000 [m]
Plasma frequency    2.84e+004 [Hz]
Cyclotron frequency 1.39e+004 [Hz]```

## Outer magnetosphere

```n   = 1E6;                      %[m-3]  Electron density
kTe = 50;                       %[eV]   Thermal energy
r   = 7*Re;                     %[km]   Radial distance from the Earth’s center
wp  = (e^2*n/(eps0*me))^0.5/(2*pi);  %[Hz]   Plasma frequency
lD  = (eps0*kTe/(n*e))^0.5;     %[m]    Debye length
B   = 2*Md/r^3;                 %[T]    Earth’s magnetic field
Fc  = e*B/(2*pi*me);            %[Hz]   Cyclotron frequency
fprintf('Debye length        %4.2d [m]\n',lD);
fprintf('Plasma frequency    %4.2d [Hz]\n',wp);
fprintf('Cyclotron frequency %4.2d [Hz]\n\n',Fc);```
```Debye length        5.26e+001 [m]
Plasma frequency    8.98e+003 [Hz]
Cyclotron frequency 5.05e+003 [Hz]```

## Solar wind

```n   = 1E6;                      %[m-3]  Electron density
kTe = 5;                        %[eV]   Thermal energy
wp  = (e^2*n/(eps0*me))^0.5/(2*pi);  %[Hz]   Plasma frequency
lD  = (eps0*kTe/(n*e))^0.5;     %[m]    Debye length
B   = 1E-8;                     %[T]    Earth’s magnetic field
Fc  = e*B/(2*pi*me);            %[Hz]   Cyclotron frequency
fprintf('Debye length        %4.2d [m]\n',lD);
fprintf('Plasma frequency    %4.2d [Hz]\n',wp);
fprintf('Cyclotron frequency %4.2d [Hz]\n\n',Fc);```
```Debye length        1.66e+001 [m]
Plasma frequency    8.98e+003 [Hz]
Cyclotron frequency 2.80e+002 [Hz]```

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## 2 Comments

1. David Fairbairn says:

Just a minor point: the ‘wp’ name for the plasma frequency implies slightly that this is an omega value, but given the divisor of 2 pi, it might actually be clearer to name it something like ‘fp’.

That is, if I’m understanding correctly!

• smallsat says:

I think you are right. It was a long ago I wrote this code and I forgot the reason I for using such a notion.
To be safe you always look to the units 🙂 Good luck!