Connection star

Star connection , also Anhalt- or reference star called fixed stars with precisely known coordinates (RA / declination), the other in visual or photographic measurements celestial body to the "port" to an absolute coordinate system are used.

The method of connecting stars is a transformation of the measurement to the star coordinates. It has been used in astronomy and geodesy for a long time, but has undergone a change and generalization through EDP and astrometric satellites .

It is carried out either with high-precision fundamental stars (4100 stars in FK6 ) or - for example in the case of photographic star images - with data from a more comprehensive star catalog with up to a million stars.

Connection of visual and photographic measurements

Simplest case: star map

If a mobile star is observed - for example a (small) planet or a comet , it is necessary for documentation and for further calculations to determine its position in the starry sky.

This can be done freely with a star map , estimating the relative distances to neighboring stars and entering them into the map. This "connection" of a relative measurement in the system of celestial coordinates is limited in its accuracy to degrees or tenths of a degree; a few minutes of arc can be achieved with binoculars .

A frequent application is the entry of the traces of light from falling stars , fireballs or meteor streams in a star map in order to determine the so-called radians (vanishing point).

Visual astrometric measurements

A more precise connection method has long been in astronomy to determine Sternörtern used - probably since ancient times ( star catalog of Hipparchus ). For example , if a series of star coordinates is to be determined on a meridian circle , wall quadrant or passage instrument, the transit time of the stars in the meridian and the associated zenith distance are measured . However, measurements are always affected by (small) systematic effects. They can be taken into account more precisely if connection stars with known coordinates are also included in the measurement program.

If a connecting star shows that the calculated declinations δ of the stars are too large by dδ = 0.5 ", this value is attached to the newly determined stars. It can be composed of several causes, which can then be better modeled in the evaluation model: Telescope bending or temperature effects , small target errors by the observer, differential refraction or even “ hall refraction ” (small anomalies in the air stratification in the observatory dome ).

If two or more connecting stars show that dδ changes from 0.5 "to 0.9" in the course of two hours, you can (after checking for other sources of error) set the correction value to 0.7 "or even interpolate linearly.

When determining the right ascension , the accuracy of the transit time is crucial. Among other things, the reaction time of the observer plays a role here , which is also determined using connecting stars . It is amazingly constant - which is why it is also known as the “ personal equation ” - and is between 0.1 and 0.3 seconds, depending on the type of person.

When using a recording micrometer - a moving thread in the field of vision that closes electrical contacts when tracking the star - the personal equation sinks to less than 0.1 seconds.

Photographic and CCD measurements

The astrometry used in a large number of photographic images on film and photographic plates , and since about 1990 and CCD sensors and other semiconductor chips. Connection stars are also required here to determine the exact coordinates (at least three per photo plate). The same applies to satellite geodesy and to special recordings in space travel and photogrammetry .
(Note: In terrestrial measurement images or in remote sensing with satellites, the connection points are referred to as control points ).

The photographic connection process is also called plate reduction . It corresponds to a 2D coordinate transformation between the " image coordinates " ( x , y ) measured in the image and the star words (α, δ), the latter being first converted into "tangential coordinates " (ξ, η). These theoretical image coordinates are calculated as a "central perspective" ( gnomonic projection ) with focal length and spatial alignment of the camera .

If one compares the measured values x , y on the image with their theoretical values (ξ, η) for the photographed connecting stars, a transformation between the two coordinate systems is possible. The parameters of this transformation are then used the other way around, in order to convert all photographed objects from image to sky coordinates. Represented symbolically:

Inverse Transformation (x, y) ⇒ (ξ, η) ⇒ (α, δ)

Choice of transformation

The number of transformation parameters required is four to twelve and depends on several factors:

desired accuracy (or the planned effort),
Number of easily measurable connecting stars (mostly 10–50, at least four),
Distortion of optics, brightness of stars,
possible deformation of the photo carrier (not applicable with CCD; a glass plate is better than film).

The simplest transformation is linear - conformal :

ξ = A·x − B·y + C   und   η = B·x + A·y + D   (4 Parameter, mindestens 2 Sterne)

In the case of possible distortions due to angles deviating from 90 °, the affine transformation ("short turner") is better:

ξ = a·x + b·y + c   und   η = d·x + e·y + g   (6 Parameter, mindestens 3 Sterne)

In both cases, the focal length f of the camera is in the parameters A, B or a, b and the image center in C, D or c, g. If there are more stars available than absolutely necessary, an adjustment is made using the least squares method : the left-hand sides of the equations are expanded by an "improvement" ( residual ) v x , v y and their sum of squares is minimized (method of least squares invented by Carl Friedrich Gauss ).

In general - if more than eight stars are available for the connection - one selects the quadratic transformation ("long turner"):

ξ = A·x + B·y + C + D·x² + E·xy + F·y²   (η analog, 12 Parameter)

or a projective approach like that of photogrammetry . Once the 4–12 parameters have been determined, the reverse transformation is used to calculate the new coordinates (the unknown objects in the photo).

Notes on the coordinate system

As a rule, the connecting stars are in an absolute system (see fundamental star ). In the case of photographic measurements, it is sometimes also a relative one - for example, when individual objects in a star cluster are to be added to an existing data set .

The measurements of the astrometry satellite Hipparcos (based on a kind of scanner principle) were also relative , although they were able to significantly improve the fundamental system of the FK5 .

All places refer to the celestial equator (extension of the earth's equator) and the vernal equator . Due to a slow cone movement of the earth's axis (the 26,000-year precession ) and other effects, this coordinate system changes over time - that is, when connecting measurements using connecting stars, the point in time at which they were measured (the so-called epoch ) must be taken into account. The star coordinates of the respective epoch are extrapolated from the standard epoch J2000.0 (end of September 2004 would then be 2004.75). All projects and measurements before about 1990 still relate to the standard epoch 1950.0 and before 1930 to 1900.

literature

Albert Schödlbauer : Geodetic Astronomy . Verlag de Gruyter, Berlin and New York 2000, 640 pages, especially page 562 and the following.