7.2.3 Large-Aperture Tilt Correction

  The major axis of the large aperture in low-dispersion data is not precisely perpendicular to the direction of dispersion. The major axis is at an angle, called the $\omega$ angle, with the dispersion direction, which is slightly different for each of the three cameras. The geometry of the entrance apertures in relation to the image scan lines and dispersion directions as they appear in the low-dispersion SI is shown in Figure 7.6. The left side of each figure shows the intrinsic geometry with no corrections applied, the middle portion shows the resulting geometry after applying the aperture alignment correction only, and the right side shows the results after applying both alignment and tilt corrections. Note that the major axis of the large aperture in high-dispersion data is very nearly coincident with the direction of dispersion so no tilt correction is necessary.

Figure 7.6:  Geometry of the spectrograph apertures as they appear in the low-dispersion SI.

The error introduced by extracting extended spectral data (data that fills the large aperture) which has not been tilt corrected, by summing the flux perpendicular to the dispersion can be estimated as follows. The maximum tilt of the large aperture with respect to the dispersion direction occurs in the SWP camera, with an $\omega$ angle of 81$^\circ$. If the length of the large aperture is taken to be 14.2 pixels (based on a plate scale factor of 1.53 arcsec/pixel), then the displacement from the line perpendicular to the dispersion direction is 1.1 pixels at each end of the aperture, or approximately 1.9Å. Spectral features will show some broadening if extracted perpendicular to the dispersion ($\omega=90^{\circ}$), which will amount to a broadening of the base of the point spread function by 3.8Å. The overall effect is in reality not as severe as a simple convolution of the data with a 3.8Å wide slit, because the part of the signal extracted near the slit center is not really being degraded. A more precise approximation to the error is to think of the 10Å slit width as being convolved with a triangular function with FWHM of 2Å and a base width of 4Å, which will result in a degradation in resolution of 20-25%.

Since this is a significant effect, a correction is made to ``detilt'' the large-aperture data for observations that result in the detection of spatially extended spectra. This correction is obviously not necessary for point source spectra, nor is it appropriate for most trailed or multiple observations because in these types of observations the source is moved along the FES x-axis which is nearly perpendicular to the dispersion direction. Multiple spectra acquired along the major axis of the aperture, however, are tilt corrected. The tilt correction is not applied to any non-multiple spectrum having object class designation 10-57 (stellar sources), regardless of whether the spectrum is determined to be point or extended. Presumably a stellar spectrum measured to be extended is saturated.

The tilt correction is only applied to a region of the low-dispersion SI that extends for 18 image lines on either side of the predicted center of the large aperture, thus leaving the small-aperture data intact. The extent of the correction region was chosen so as to include the areas on either side of the aperture which are used for background subtraction in the subsequent extraction step of the image processing. The actual correction is accomplished by simply shifting each image line in the dispersion direction by an amount given by:

$$\Delta x = \Delta y \times \tan(\omega_t)$$

where $\Delta y$ is the number of spatial image lines between the center of the large aperture and the image line under consideration, and $\omega_t$ is the tilt angle of the aperture. Note that this correction is a projection of the aperture onto the spatial axis of the image, as opposed to a rotation about the center of the aperture, since the latter would result in a slight remapping of the spatial character of the data.