Introduction to the simulation
The effects of the refraction are in principle easily seen in a telescope. In order that the different effects can described separately we developed a simulation program which take the atmosphere and the alignment of the hour axis into consideration. The extinction and the spectral absorption of the atmosphere are ignored as well as the telescope dependent effects like all kinds of abberations, tube deflection etc.
Figure 1.1:
Intensity
distribution of diffraction images at
= 575 nm for an opening D of 200 mm diameter. The smooth line is a
fit with the results D = 199.98 ± .02 mm and
= 575 ± 7 nm.
Figure 1.2:
Black body radiation at the temperature T = 5500 K, that corresponds to a star of spectral type G. The smooth line is a fit with the results T = 5498 ± 2K.
The
simulation is divided in three parts: The first generator produces
the black body wavelength distribution with an average temperature
corresponding to the spectral types O through N (fig. 1.2).
The second generator produces the spatial distribution which
corresponds to a Fraunhofer diffraction at circular opening with a
diameter D = 200 mm at the wavelength
= 575 nm (fig. 1.1,
1.5
and
1.7).
In
a second step the transformation from the true equatorial system,
that is the local hour angle
and the declination
of a star, to the azimuthal system takes place. In this system the
wavelength dependent refraction is added and then transformed back
to the telescope system. To these values the spatial distribution
is added, which are scaled with the wave length of the arriving
photon and the actual diameter of the opening. The settings of the
telescope determine the center of the gnomonic projection or in
other words the tangent plane simulates the focal plane.
In a third step the wave lengths are converted to RGB values and filled together with the spatial information into a three histograms. The efficiency is independent from the wave length and is set to 1. At the end the three color channels are stored as a picture.
In order to reduce the dispersion there are optionally a pair of Risley prism in the optical path as described in [2]. For the sake of simplicity the system has been implemented as a objective prism with the real glass data (Schott F4 and SK10) according to the set up described in [3].
The
air turbulences, which is a topic for itself, has been implemented
as a addition of a two dimensional Gaussian distributed value. The
standard deviation
is a measure for the seeing. The guiding system is not affected,
because the mean value of this spatial distribution is the
location of the star. Despite the fact that this treatment is not
really correct, it is only justified by the long exposure times,
it reproduces the seeing close to reality as far as this
investigation is concerned. The fig. 1.3
is
simulated after the description of J. Dragesco (table 1.1 in [1]).
The agreement between the results and the description cited in
tab. 1.1
is
good.
Perfect images, without visible distortion and little agitated
Complete rings, crossed by moving rings
Medium turbulence, diffraction rings broken, central spot
Strong turbulence, rings weak or absent
Image tending towards a planetary appearance
Table 1.1:
Table
1.1 cited from [1],
is the radius of the turbulences a is the radius of the
diffraction image.
Figure 1.3:
From
left to right: diffraction images of a star of spectral type G
with
= 0,1/4 a,1/2 a,a and 3/2 a. The stellar images are in good
agreement with the description in tab. 1.1.
In fig. 1.9 there are simulated stars of spectral type O - N displayed how they would look like seen through an ideal telescope and without the atmosphere. The absolute color values are only an approximation because the color is question how the wavelength is transformed to the RGB values and how they are finally displayed on the computer monitor. The movement of the intensity maxima from red to blue, that is the color of the star, is clearly visible and the diameter varies according the mean wavelength. This is only at first sight surprising because all photons with wavelength between 380 and 780 nm were accepted. The spectral distribution of the blackbody radiation at different temperatures shows, that a red giant (fig. 1.6) emits in red but the hotter stars (fig. 1.8) almost only in the blue. The sun belongs to the spectral type G (fig. 1.2) and emits in all areas of the spectrum about equally.
Figure 1.4:
Intensity
as a function of the radius and at the wavelengths
= 650 (a), 537 (b) and 410 nm (c).
Figure 1.5:
Radial intensity distribution of a diffraction image in the white light of the spectral type N for an opening of 200 mm.
Figure 1.6:
Black body radiation of the spectral type N (red).
Figure 1.7:
Radial intensity distribution of a diffraction image in the white light of the spectral type O for an opening of 200 mm.
Figure 1.8:
Black body radiation of the spectral type O (blue).
If
a star is observed without a filter then the photons are mixture
of all wavelengths. The diffraction image is normally described as
a intensity distribution which depends on the radius for fixed
wavelength. In fig. 1.4
these
function is plotted for three different wavelengths how it was
already presented in [5].
The diffraction image of a white source is a supposition of the
intensity functions at all wavelength and looks a in the beginning
a little bit strange (fig. 1.5,
1.7
and
1.10).
The colored edges of the central spot come into existence because
the resolution of a given aperture is better the shorter the
wavelength is. So the first minimum at
= 400 nm falls on the maximum at
= 537 nm (fig. 1.4),
that means that at is location the color impression us reddish
yellow etc. The observation trough a real telescope without a
narrow band filter shows no diffraction rings but a series of dark
and bright rings how it is seen in fig. 1.10
for
the spectral types O - N.
Figure 1.9:
Simulation of stars of spectral types N (upper left) through O (lower right) without atmosphere. The aperture is 200 mm. Length of the scale 2.5''.
Figure 1.10:
The colored rings are a result of the mixture of all wavelength. This figure is made brighter otherwise identical with fig. 1.9.
The results of the simulation can be displayed in two ways: the direct and the difference modus. The direct modus displays the results as an ordinary camera would record it. The sensitivity, the scale and the size of the detector could be adapted to a given problem.
In
fig. 1.11
there
are nine stars on a regular grid each separated by 0.5o.
The telescope rate is strictly sidereal and there is no movement
in the
direction. The resultant star trails are due to the refraction
alone and are clearly visible. This kind of photograph is normally
not desired because one like to see the stars as points and not as
trails. This mode allows to understand the effects of the
refraction easier and is therefore used.
At the beginning and at the end of this six hours long exposure the trails are most affected by the refraction and are the apparent position of the star is farther away from the true position. In the middle of the exposure, that is when the star field transits, the trails reach there lowest close to the true position. Even if the exposure of six hours might appear long the star trails remain longer than the critical length of 1'' even if the exposure time is significantly reduced.
The
difference mode is nothing else than a perfectly guided photograph
with a guide star in the center. That is the reason, why the star
trails relative to this guide star can be magnified. It is
possible to investigate the star trails e.g. as a function of the
spot in the focal plane or as a function of the spectral type. In
fig. 1.12
there
is the same grid displayed as in the previous figure, but the star
trails are now magnified by a factor 480. The scale bar at the
lower left is 2.5'' long in both directions. This photograph has
been guided with the star in the center at the wavelength
= 550 nm.
That
is the reason why only a single yellow dot is visible in the
center. The blue (
= 450nm) and the red trail (=650
nm) which belong to the star in the center make clear that even in
a restricted field around the center of about 0.5o the
star images will not appear as point like objects. The effects of
the refraction are easily visible in the case of the other eight
stars at the edge of the square despite the hour axis points
exactly to the celestial pole. The star trails do not describe a
circle around the center the shape. These irregular shapes depend
on the spot, the spectral type and the properties of the detector
among others.
This direct mode is suitable for quick quantitative check between a real and a simulated photograph. The direct mode needs a resolution which compares to a real photograph and needs therefore a high resolution of about 4000 pixels per 36 mm and that is about 300 MB computer memory.
It possible in both modes to display the diffraction image (fig. 1.9) or only individual colors (fig. 1.12 . The three color mode in the difference mode is in great number of cases an adequate way of displaying the results and has the advantage of low computer power consumption. The normalization of the brightness is chosen in a manner that the subtle details remain visible despite the stars are burnt on a real photograph.
Figure 1.11:
Direct
mode: Begin of the exposure at
= -3h, exposure time 6 hours,
= 0o. The hour axis points to the true celestial pole
and the telescope rate is strictly sidereal and the declination
axis is fixed, the diameter of the aperture is 200 mm and the
focal length 1260 mm. Length of the scale bar 0.5o in
both direction.
Figure 1.12:
Three colors in the difference mode. This diagram is nothing else the a perfectly guided photograph. Length of the scale bar is 2.5'', otherwise like fig. 1.11.
Of
course the virtual telescope must stand the test against the
virtual sky. In fig. 1.13
the
region around the double stars
Lyrae is displayed. The coordinates, spectral types and the
brightness of the stars is taken from Centre de Données
astronomiques de Strasbourg, Simbad [4].
The magnified results are displayed in fig. 1.14
for
an aperture D = 100 mm (top) and for an aperture D = 200 mm
(bottom). In the case of the smaller aperture the
1
Lyrae are not separated completely, but in with the larger one and
even the diffraction rings are visible.
Figure 1.13:
Lyrae
overview. The coordinates, spectral types and the brightness of
the stars are taken from Simbad [4].
Figure 1.14:
2
Lyrae (left),
1
Lyrae (right), resolution 0.2'' /pixel, diameter of the aperture D
= 100 mm (top), D = 200 mm (bottom). Both pairs are not completely
separated with an aperture of 100 mm.
Literature
[1] DRAGESCO, J.: High resolution astrophotography. Cambridge University Press, Cambridge, 1995.
[2] EISENHAUER, F.: Auslegung und Bau einer benutzerorientierten Nahinfrarot-Kamera für astronomische Beobachtungen mit dem adaptiven Optik System ADONIS am 3,6 m Teleskop der ESO. Diplomarbeit, Technische Universiät München, Physik-Departement, 1995.
[3] PRIEUR, J.-L.: Correction de la dispersion atmosphérique. http://webast.ast.obs-mip.fr/people/prieur/risley/index.html, Dez. 2001.
[4] STRASBOURG CENTRE DE DONNÉES ASTRONOMIQUES DE. simbad.u-strasbg.fr, 2002.
[5] WALLIS, B. D. und R. W. PROVIN: A manual of advanced celestial photography. Cambridge University Press, Cambridge, 1988.
Comments, questions, corrections: markus.wildi@one-arcsec.org