Dieter Egger @ FESG TUM
Transformations between reference frames
Reference frames supply the base to define positions and velocities by means of coordinates. In 3-dimensional space, for example, you need 3 coordinates to determine a position. The collection of coordinates is called a vector. Only vectors referring to the same reference frame may be combined in a meaningful manner. You may characterize reference frames by three quantities:
  • reference point (a center body or a virtual center point, the origin)
  • reference plane (ecliptic, equator, horizon, ..., the x-y-plane)
  • reference direction (vernal equinox, Greenwich, South, ..., x-axis)

We mostly use (right-handed) rectangular coordinate systems (euclidean systems),

The following reference frames are common in celestial mechanics and astronomy:

Name Origin X-Y-Plane X-Axis
Heliocentric ecliptical system Sun Ecliptic Vernal Equinox
Geocentric ecliptical system Earth Ecliptic Vernal Equinox
Space-fixed equatorial system Earth Equator Vernal Equinox
Earth-fixed equatorial system Earth Equator Greenwich/Eq.
Topocentric equatorial system Observer || Equator || Greenwich/Eq.
Topocentric horizon system Observer Horizon South

Further distinctions concerning equatorial systems are required for high precision transformations. Due to the gravitational pull of mainly the Sun and the Moon, the Earth's rotation axis is tumbling in space. Therefore, the Earth-fixed reference frames are moving disorderly in space as well. Well known are the main effects like nutation (period of approx. 18 years) and precession (period of approx. 26000 years). Less known are the short term variations which displace the rotation axis with respect to the Earth's surface and increase or decrease the speed of rotation.
If the reference frames are in mutual motion, small "corrections" have to be added to the transformed vectors due to the finite speed of light (aberration and light time correction, in the solar system these may amount to some tens of arcseconds).

To further increase the precision of transformations, even more tiny "corrections" have to be added, which take into account the presence of gravitational fields (some milliarcseconds to less than two arcseconds in the solar system).

Concerning accuracy, remember: One arcsecond is the angle enclosing a dollar coin in a distance of about 4 kilometers (!).

The Transform applet uses today's noon as the actual epoch and assumes an observer at the north pole for the sake of simplicity. The aforementioned "corrections" are being neglected.

When choosing polar coordinates instead of rectangular ones, be sure of entering decimal degrees (dd.dddd) for the angles. In the case of polar coordinates, r represents the distance according to the chosen unit, ph shows the angle between the x-axis and the projection of the vector onto the x-y-plane and th is the angle enclosed by the z-axis and the vector.

Transformations are performed by leaving the cursor in one of the textfields which are going to represent the input and hitting RETURN, or, by selecting a reference frame.

Relationships between reference frames:

Frame1 Frame2 relationship
Heliocentric ecliptical system Geocentric ecliptical system Heliocentric position of the Earth
Geocentric ecliptical system Space-fixed equatorial system Mean obliquity of ecliptic
Space-fixed equatorial system Earth-fixed equatorial system Sidereal time
Earth-fixed equatorial system Topocentric equatorial system Geocentric position of observer
Topocentric equatorial system Topocentric horizon system Longitude and latitude of observer

Mutual motions are taking place between

Frame1 Frame2 Motion
Heliocentric ecliptical system Geocentric ecliptical system Earth orbiting the Sun
Earth-fixed equatorial system Topocentric equatorial system Observer rotating about the Earth's rotation axis


Dieter Egger,  last update 2002-02-26