A good description of how to make transformations between the different coordinate systems can be found in a paper by M. A. Hapgood, "Space physics coordinate transformations: A user guide", in Planetary and Space Science, Vol. 40, No. 5, pp. 711-717, 1992.

See also a good description of space-based coordinate systems and transformatons at ESA's SPace ENVironment Information System (SPENVIS)

J2000 GCI

Geocentric inertial reference frame using the Julian Epoch of Jan. 1, 2000.

At the Julian Epoch,

X = First point of Aries (Vernal Equinox, i.e. from Earth to the Sun in the first day of Spring)

Z = Earth's mean spin axis

RTN - Radial Tangential Normal

Spacecraft centered coordinate system.

R = Sun to Spacecraft unit vector

T = (Omega x R) / | (Omega x R) |

where Omega is Sun's spin axis (in J2000 GCI)

N completes the right-handed triad

GSE - Geocentric Solar Ecliptic

X = Earth-Sun Line

Z = Ecliptic North Pole

GSM - Geocentric Solar Magnetospheric

X = Earth-Sun Line

Z = Projection of dipole axis on GSE YZ plane

HSEa - Heliocentric Solar Ecliptic (Inertial)

X = First poin tof Aries (Vernal Equinox, i.e. to the Sun from Earth in the first day of Spring)

Z = Ecliptic North Pole

HSEb - Heliocentric Solar Ecliptic (Earth Oriented)

X = Sun-Earth Line

Z = Ecliptic North Pole

HS - Heliocentric Solar

X = Intersection between solar equator and solar central meridian as seen from Earth

Z = North Pole of solar rotation axis