Dipole model of Earth's magnetic field

Alternative formulation

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, ${\displaystyle \lambda }$ is measured northwards from the equator (analogous to geographic latitude) and is related to the colatitude ${\displaystyle \theta }$ by

${\displaystyle \lambda =\pi /2-\theta }$.

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

${\displaystyle B_r=-\frac{2B_0}{R^3}\sin \lambda }$

${\displaystyle B_{\theta }=\frac{B_0}{R^3}\cos \lambda }$

${\displaystyle |B|=\frac{B_0}{R^3}\sqrt{1+3{\sin }^2\lambda }}$

where ${\displaystyle R}$ in this case has units of Earth radii (${\displaystyle R=r/R_E}$).

Invariant latitude

Invariant latitude is a parameter that describes where a particular magnetic field line touches the surface of Earth. It is given by

${\displaystyle \mathrm{\Lambda }=\arccos \left(\sqrt{1/L}\right)}$

or

${\displaystyle L=1/{\cos }^2\left(\mathrm{\Lambda }\right)}$

where ${\displaystyle \mathrm{\Lambda }}$ is the invariant latitude and ${\displaystyle L}$ is the L-shell describing the magnetic field line in question.

On the surface of Earth, the invariant latitude (${\displaystyle \mathrm{\Lambda }}$) is equal to the magnetic latitude (${\displaystyle \lambda }$).