is called atmospheric refraction R.
Because the refractive index of air is somewhat larger than one, the light coming from outside the earths atmosphere is refracted towards the zenit (see figure 1).
Figure 1: Refraction and dispersion within earths
atmosphere
The difference between the true (outside earths atmosphere) zenith
distance z and the apperant zenith distance at the telescope
is called atmospheric refraction R.
Neglecting earths curvature and assuming small atmospheric refraction, one can derive from Snell's law, that the atmospheric refraction depends only on the refractive index of air n at the telescope and the zenith distance:
Because the refractive index of air depends on the wavelength
, the atmospheric refraction is wavelength dependent,
too. The difference between the atmospheric refraction for a given
wavelength and the atmospheric refraction for a reference
wavelenghth 
is called atmospheric dispersion 
.
The difference between the two refractive indices 
is called the disperion of air. This
dispersion of air depends not only on the wavelength, but also on the
conditions of the atmosphere. Beside the atmospheric pressure and the
temperature, also the relative humidity do have significant
contribution to the dispersion of air. All the results presented in
this article are based on the formula for the refractive index of air from
the paper ``Optical Refractive Index of Air: Dependence on Pressure,
Temperature and Composition'', Applied Optics Vol. 6, No. 1, 1967,
p.51 - 59 by J.C. Owens. Allthough the measurements considered for
the derivation of this formula are not covering the whole wavelength
range between 
and 
, the formula predicts the
dispersion of air with an estimated accuraccy of 
within this wavelength range (see also ``Auslegung und Bau einer
benutzerorientierten Nahinfrarot-Kamera für astronomische
Beobachtungen mit dem adaptiven Optik System ADONIS am 3.6 m Teleskop
der ESO'', Diplomarbeit at the Technical University of Munich, 1995 by
the author of this article). For the average meteorological
conditions in La Silla and a maximum zenith distance of 60 deg the
accuracy of this formula is sufficient to predict the atmospheric
dispersion within 5 marcsec. The average air pressure in La Silla
during the nights is 772.7 mbar, the average temperature is 11.5
deg C and the average humidity is 44.2 % (data from M. Sarazin,
ESO). For these average conditions and the maximum zenith distance of
60 deg the atmospheric dispersion is plotted in figure 2.
Figure 2: Atmospheric dispersion at a zenith distance
of 60 deg for average meteorological conditions in La Silla
The overall atmospheric dispersion in the wavelength range from to is about 280 marcsec.
The instrument to correct for the atmospheric dispersion (ADC) consists of two pair of prisms mountable in front of the fokal plane of ADONIS. Figure 3 illustrates the working principle of this ADC.
Figure 3: Working principle of an atmospheric
dispersion corrector
The combination of prisms made of ZnS (Multispectral) and ZnSe corrects the atmospheric dispersion down to an residual dispersion of less than 28 marcsec for the whole wavelength range and a zenith distance of 60 deg under average atmospheric conditions in La Silla (see figure 4).
Figure 4: Residual atmospheric dispersion after
correction by the ADC
The prism angles of the ADC were calculated such, that each of the pairs correct half of the atmospheric dispersion at 60 deg zenith distance and average atmospheric conditions without moving the center of light. Therefore turning the two pairs of prisms around the optical axis allows to correct for every atmospheric dispersion down to a zenith distance of 60 deg (see figure 5).
Figure 5: Correction of different atmospheric
dispersion by an appropiate rotation of the pairs of prisms
Allthough a combination of prisms made of KRS-5 and ZnSe would produce
a better correction, the first combination was selected, because no
manufacturer was found to do the antireflex coating on KRS-5. All the
calculations are based on an image scale of 
. The
geometrical properties (see figure 6) of the ADC are
summerized in table 1.
Figure 6: Geometrical properties of the SHARP II+ ADC
Table 1: Geometrical properties of the SHARP II+ ADC
The refractive indices of the ADC materials were taken from the product information ``Infrared Materials: ZnSe, ZnS Regular, ZnS Multispectral'' by II-VI Incorporated, 1991.
Figure 7 shows the direction of dispersion produced by the two pairs of prisms, as mounted in the ADONIS / SHARP II+ system, when seen along the optical axis towards the camera. In this figure the rotation angle of the rotary units are set to zero. A positive rotation angle will rotate the dispersion counter clockwise for the first pair of prisms and clockwise for the second one.
Figure 7: Direction of the dispersion produced by the
two pairs of prisms, as mounted in the ADONIS / SHARP II+ system,
when seen along the optical axis towards the camera
To calculate the rotation angles of the rotary units of the ADC for a given position of the object on the sky and for given meteorological conditions, one has to calculate the zenith distance and the direction and amplitude of the dispersion:
of the telescope, hour
angle h and declination 
of the observed object, the zenith
distance z is given by:
to the temperature in units of Kelvin T:
:
is then given by the
relative humidity rh (in %):
can be calculated from the atmospheric pressure p:
:
:
(in 
) the refraction of air 
:
(in rad) for the wavelength intervall from
to is then given by:
, the relative atmospheric dispersion
is then:
This relative dispersion defines the rotation angle 
between the
pairs of prisms:
Given the geometrical setup as shown in figure 7, the rotation angles of the two rotary units need to be x (the first rotary unit) and y (the second one) for the best correction of the atmospheric dispersion:
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