Systems

The following systems are referred to the true Earth equator or ecliptic of date, that is corrections for nutation and precession should be applied in transformations. We also give the bracket notation $<,>$ used by Hapgood (1992) (see Appendix).

• Geographic Coordinates $GEO$ (Hapgood, 1992)
  

  XY-plane:	True Earth equator of date.
  +X-axis:	Intersection of Greenwich meridian and Earth equator.
  Angles:	Geographic latitude and longitude (increasing westward) right handed,
  	in the sense of a planetographic system (see section 4.3).
  Transform:	$T(GEI_{T},GEO) = <\theta_{GMST},Z> = E(0^\circ,0^\circ,\theta_{GMST})$
  	where the $\theta_{GMST}$ is given by eqn.20.

• Geocentric Solar Ecliptic $GSE$ (Hapgood, 1992)
  

  XY-plane:	Earth mean ecliptic of date.
  +X-axis:	vector Earth-Sun of date.
  Transform:	$T(HAE_{D},GSE) = <\lambda_{geo}+180^\circ,Z> = E(0^\circ,0^\circ,\lambda_{geo}+180^\circ)$ with $\lambda_{geo}$ from eqn.36
  	and subtraction of solar position vector if necessary.
  	Also $T(GEI_{D},GSE_{D}) = T(GEI_{D},HAE_{D})^{-1} * T(HAE_{D},GSE)^{-1}.$

• Geocentric Solar Magnetospheric $GSM$ (Hapgood, 1992)
  

  +Z-axis:	projection of northern dipole axis on $GSE_D$ YZ plane.
  +X-axis:	vector Earth-Sun of date.
  Transform:	$T(GSE_D,GSM) = <-\psi,X> = E(0^\circ,-\psi,0^\circ)$ where $\psi = \arctan(y_e/z_e)$ and $\mathbf{Q}_e = (x_e,y_e,z_e)$
  	is the Earth dipole vector in GSE-coordinates.
  	This can be calculated from the geographic position $\mathbf{Q}_g$ given in eqn.22 by
  	$\mathbf{Q}_e= T(GEI_{D},GSE_{D}) * T(GEI_{D},GEO)^{-1} \mathbf{Q}_g.$

• Boundary Normal Coordinates $LMN$ (Russell and Elphic, 1978)¹⁴
  

  +Z-axis:	Normal vector to Earth Magnetopause.
  +Y-axis:	cross-product of +Z-axis and GSM-Z-axis.

  The normal vector may be determined by a model or by minimum-variance analysis of data.

• Solar Magnetic $SM$ (Chapman and Bartels, 1962)
  

  +Z-axis:	Northern Earth dipole axis of date.
  +Y-axis:	cross-product of +Z-axis and Earth-Sun vector of date.
  Transform:	$T(GSM,SM) = <-\mu,Y> = E(90^{\circ},-\mu,-90^{\circ})$
  	where $\mu = \arctan\frac{x_e}{\sqrt{y_e^2+z_e^2}}$ with $\mathbf{Q}_e$ given above.

  The longitude of this system is also called magnetic local time (MLT) increasing eastwards from the anti-solar ($0^h$) to the solar ($12^h$) direction.

• Geomagnetic $MAG$ (Chapman and Bartels, 1962)
  

  +Z-axis:	Northern Earth dipole axis of date.
  +Y-axis:	cross-product of Geographic North Pole of date and +Z-axis.
  Transform:	$T(GEO,MAG) = <\Phi_D-90^{\circ},Y>*<\lambda_D,Z> = E(\lambda_D+90^{\circ},90^{\circ}-\Phi_D,-90^\circ)$
  	where $\Phi_D$ and $\lambda_D$ are given in eqn.22.

  Geomagnetic latitude $\beta_m$ and longitude $\lambda_m$ (increasing eastward) refer to this system.

• Invariant magnetic shells $(B_d,L_d)$(McIlwain, 1966)
  These coordinates are used for functions of the magnetic field which are constant along the lines of force. For a position of radial distance $R$ from the dipole center and magnetic latitude $\beta_m$ in a dipolar field the magnetic field strength $B_d$ and equatorial distance $L_d$ of the line of forth are given by
  

  $$B_d = \frac{M}{R^3}\sqrt{1+3\sin^2\beta_m} \hspace{1cm} L_d = \frac{R}{\cos^2\beta_m},$$ (23)

  
  
  where $M$ is the magnetic moment of the dipole (see eqn.22 for Earth value). The offset between dipole center and gravity center ($\approx 500$ km for Earth) has been neglected (Kertz, 1969).
• Other magnetospheric coordinates ^{15 16 17}
  Many coordinate systems depend on a specific magnetic field model. For example Corrected Magnetic Coordinates ($CGM$)¹⁸are constructed by field line tracing. Magnetospheric Equatorial Coordinates ($GME$) use specific magnetotail models (Dunlop and Cargill, 1999). For a field model again $(B,L)$ coordinates may be derived for which particle drift shells can be defined (McIlwain, 1966). See the references for details.