reference point (a central body or a virtual central point, the origin)
reference plane (ecliptic, equator, horizon, ..., the x-y-plane)
reference direction (vernal equinox, Greenwich, South, ..., x-axis)
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 referred to the same reference frame may be combined in a meaningful manner. You may characterize reference frames by three quantities:
reference point (a central body or a virtual central 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), where the y- and z-axis can easily be derived.
The following reference frames are often met 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 higher 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 lengthen or shorten
the duration of the daily revolution period.
If reference frames are in mutual motion, small "corrections" have to be added to the transformed vectors due to the limited velocity of light (aberration and light time correction, in the solar system some tens of arcseconds). |
These "corrections" are not necessary if the transformation process is subjected to
the rules of special relativity (Lorentz transformations). In this case, they
arise as a natural consequence.
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). You do not need the term "correction" if you follow the transformation rules of general relativity. In this case they are a natural ingredient of the theory. Concerning accuracy, remember: One arcsecond is the angle enclosing a dollar coin in a distance of about 4 kilometers (!). |
The Transform applet uses the actual epoch of today's date at noon 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 within the reference plane counted counter-clockwise from the x-axis to the projection of the vector onto the x-y-plane and th is the angle enclosed by the z-axis and the vector, counted from the z-axis.
Transformations are performed by leaving the cursor in one of the textfields and hitting RETURN, or, by choosing a reference frame. The left vector is transformed to the right one, when you change the left vector or when you change the selection of the reference frame at the right hand side and vice versa.
Enjoy the applet. (Remember: No Java, no applet)
The connections between the reference frames are
| Frame1 | Frame2 | Connection |
| 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 occur 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 |
If you transform the vector of a moving body, you still have to apply the light time correction, which expresses the fact, that the observed body is no longer at the observed position.
Dieter Egger,
1997-02-28