Dipole model of the Earth's magnetic field#Invariant latitude

The following equations describe the dipole magnetic field.{{cite book |last= Walt |first= Martin |title= Introduction to Geomagnetically Trapped Radiation |pages = 29–33 |publisher= Cambridge University Press |location = New York, NY |year= 1994 |isbn= 0-521-61611-5}}

First, define $B\_0$ as the mean value of the magnetic field at the magnetic equator on the Earth's surface. Typically $B\_0=3.12\backslash times10^\{-5\}\backslash \; \backslash textrm\{T\}$.

Then, the radial and latitudinal fields can be described as

:$B\_r\; =\; -2B\_0\backslash left(\backslash frac\{R\_E\}\{r\}\backslash right)^3\backslash cos\backslash theta$

:$B\_\backslash theta\; =\; -B\_0\backslash left(\backslash frac\{R\_E\}\{r\}\backslash right)^3\backslash sin\backslash theta$

:$|B|\; =\; B\_0\backslash left(\backslash frac\{R\_E\}\{r\}\backslash right)^3\; \backslash sqrt\{1\; +\; 3\backslash cos^2\backslash theta\}$

where $R\_E$ is the mean radius of the Earth (approximately 6370 km), $r$ is the radial distance from the center of the Earth (using the same units as used for $R\_E$), and $\backslash theta$ is the colatitude measured from the north magnetic pole (or geomagnetic pole).

Image:Mplwp earth-magnetic-field.svg

It is sometimes more convenient to express the magnetic field in terms of magnetic latitude and distance in Earth radii. The magnetic latitude (MLAT), or geomagnetic latitude, $\backslash lambda$ is measured northwards from the equator (analogous to geographic latitude) and is related to the colatitude $\backslash theta$ by

:$\backslash lambda\; =\; \backslash pi/2\; -\; \backslash theta$.

In this case, the radial and latitudinal components of the magnetic field (the latter still in the $\backslash theta$ direction, measured from the axis of the north pole) are given by

:$B\_r\; =\; -\backslash frac\{2B\_0\}\{R^3\}\backslash sin\backslash lambda$

:$B\_\backslash theta\; =\; \backslash frac\{B\_0\}\{R^3\}\backslash cos\backslash lambda$

:$|B|\; =\; \backslash frac\{B\_0\}\{R^3\}\; \backslash sqrt\{1\; +\; 3\backslash sin^2\backslash lambda\}$

where $R$ in this case has units of Earth radii ($R\; =\; r/R\_E$).

Invariant latitude is a parameter that describes where a particular magnetic field line touches the surface of the Earth. It is given by{{cite book |last1= Kivelson |first1= Margaret |last2= Russell |first2= Christopher |title= Introduction to Space Physics |pages = 166–167 |publisher= Cambridge University Press |location = New York, NY |year= 1995 |isbn= 0-521-45714-9}}

:$\backslash Lambda\; =\; \backslash arccos\backslash left(\backslash sqrt\{1/L\}\backslash right)$

or

:$L\; =\; 1/\backslash cos^2\backslash left(\backslash Lambda\backslash right)$

where $\backslash Lambda$ is the invariant latitude and $L$ is the L-shell describing the magnetic field line in question.

On the surface of the earth, the invariant latitude ($\backslash Lambda$) is equal to the magnetic latitude ($\backslash lambda$).

• Geomagnetic pole
• Dipole
• International Geomagnetic Reference Field (IGRF)
• Magnetosphere
• World Magnetic Model (WMM)
• Dynamo theory

{{reflist}}

• [https://web.archive.org/web/20161203215110/http://ccmc.gsfc.nasa.gov/models/modelinfo.php?model=Tsyganenko%20Magnetic%20Field Instant run of Tsyganenko magnetic field model] from NASA CCMC
• [http://geo.phys.spbu.ru/~tsyganenko/modeling.html Nikolai Tsyganenko's website] including Tsyganenko model source code

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Category:Geomagnetism

Category:Magnetic field of the Earth

Category:Space physics