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4. Confronting the model with observational data

4.1. The CASPEC model

In the following the first principle description is tested against actual arc lamp exposures from an existing spectrograph, here CASPEC in service at ESO, La Silla since 1984. With the exception of the camera, this spectrograph employs only reflective surfaces, and the distortions of this camera are very small. As an initial guess for the parameters of the model one uses engineering parameters, as they were measured at the time of instrument commissioning in January 1984, namely:

4.1.1. Adjusting the CASPEC model

Arc lamp calibration exposures from 3 different epochs were compared with the model, each corresponding to different configurations (detectors, central wavelength and cross-dispersers). Because the limiting factor for the accuracy of our model will be the camera distortions, and since no Code V simulations are available to compare with, we assume a threshold accuracy of 1.5 pixel rms for the model.

For each well detected calibration line the x, y-positions are predicted by the model using its catalog wavelength and the order number. The rms residuals between the measured and predicted positions serve to estimate the goodness of the fit. One starts with the above initial values for the optical configuration. An initial guess for tex2html_wrap_inline1340 can be obtained by geometrical measurement on the frame. An initial guess for tex2html_wrap_inline1342 can be derived from the central wavelength by solving the dispersion equation of the cross-disperser. The following parameters are been kept at their nominal values: tex2html_wrap_inline1344 degrees; tex2html_wrap_inline1346 degrees; the grating constants; and the detector pixel size.

The model is iteratively refined by modifying the four parameters which are found to have the largest influence:

Near the optimal solution for each configuration, respectively epoch, the model matches the observational frames to within tex2html_wrap_inline1356 pixel rms (Fig. 3 (click here)). The neighborhood within which the condition of 1.5 pixel rms is fulfilled yields an uncertainty estimate for these four parameters. Table 1 presents the results. It is important to note, that the values estimated by this method on the exposures from different epochs and configurations are stable and consistent with the measured values. The value of tex2html_wrap_inline1352 is not a measurement of the focal length of the camera itself, but represents the distance between the camera lens and the detector plane instead. We note that the incidence angle on the echelle grating remained stable from 1984 to 1991. The nominal value is different for epoch 1994, and although this change is within the quoted error margin, it can actually be traced to a reassembly of the instrument in 1992. This is an indication, that the predictive power of our model might be better than the conservative error estimates based on an arbitrary value of 1.5 pixel.

Figure 3: Residuals between the measured positions of about 560 Th-Ar lines in the CASPEC observation C and the prediction from the model. The square indicates one pixel on the detector. Without a detailed analysis of the line centering methods it is not possible to relate the asymmetry of the distribution to systematics in either the model or the measurements. In any case the asymmetry is small in comparison to detector pixel size, i.e. 1/2 resolution element

Certainly, this technique for controlling an instrument configuration by comparison with a model needs further investigation. However, the example shows that an accurate determination of the instrument configuration can be obtained, provided that the model includes all the important optical effects. Thus the approach through analytical models is useful to monitor the instrument stability and to predict calibration solutions, if configuration control is imposed.


Observation A B C
Date Jul. 1984 Jan. 1991 Apr. 1994
Central Wav. 610 nm 490 nm 600 nm
Detector CCD#3 CCD#16 CCD#32
Cross Disperser 300. gr./mm 300. gr./mm 158. gr./mm
tex2html_wrap_inline1342 (deg.) tex2html_wrap_inline1362 tex2html_wrap_inline1364 tex2html_wrap_inline1366
tex2html_wrap_inline1368 (deg.) tex2html_wrap_inline1370 tex2html_wrap_inline1372 tex2html_wrap_inline1374
tex2html_wrap_inline1352 (m) tex2html_wrap_inline1378 tex2html_wrap_inline1380 tex2html_wrap_inline1382
tex2html_wrap_inline1340 (deg.) tex2html_wrap_inline1386 tex2html_wrap_inline1388 tex2html_wrap_inline1390
Table 1: Optical configuration of CASPEC as determined from a set of observations using the analytical model


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