# The resolution R

The relative resolution (hereinafter referred to simply as resolution R) is a key figure that characterizes the still resolved smallest wavelength interval (we call it „delta lambda“) . The resolution R is then

R = lambda / delta lambda

where lambda is the actual wavelength. Because delta lambda for gratings is about constant R itself depends on lambda. For example, if lambda = 6563 Angstrom and delta lambda = 0.45 Angstrom, R = 6563 / 0.45 = 14584. For lambda = 4861 is then provided R = 10802 (with the same delta lambda).

Delta lambda, the smallest resolved interval in the spectrum, can be calculated even with simspec. Even better, of course, is the experimental control of the spectrograph. But how?

In the case of a slit spectrograph the solution is simple. Here we find the slit shown in the light of neon calibration lines on the CCD chip. And the slit width is well known (by measurement with the laser method, see here). Is our slit spektrograph a Littrow type spectrograph (such as the Lhires III), then the slit mapped 1:1 to the CCD chip. With 9 µm x 9 µm pixels a 40 µm slit should be displayed over 4 pixels (good focus of the slit on the CCD). Multiplication of the 4 pixels with the known dispersion of the spectrograph (from neon calibration spectra or from stellar spectra identified by measuring known lines) results the smallest wavelength interval resolved in the spectrum.

Example: 9um pixels, 0.11 Angstrom / Pix dispersion , 4 Pix imaged slit width —> delta lambda = 0.44 Angstrom.

We have now solved by calculation. But we should check the result experimentally: We take a neon calibration frame and measure the FWHM of the slit images (FWHM = Full Width at Half Height). This can be done in VSpec, of course, in MIDAS, it automatically gets delivered in SMS. If the focus of the slit on the CCD is good, you will find agreement between theory and measurement.

In the case of a slitless spectrograph the absolute resolution is defined by the seeing disk of the observed star in the focus of the collimator lense of the spectrograph. Is that star picture 40 um in diameter, the result is the same resolution as in the example above with the slit spectrograph with 40 µm slit. However, now is the achievable resolution limited by the seeing, and thus by the weather. Different seeing conditions lead to different resolutions R with the same equipment.

For slitless spectrographs one can use terrestrial lines to measure the absolute resolution. Well suited are the „water lines“, they are very sharp, only a few hundredths of angstroms wide. They will reflect the lower resolution of our spectrograph, and therefore they will be broadened and smeared. Thus we see in reality, not the water lines in the original line profile, but the original line profile is folded with the profile of our optical apparatus and the size of seeing. By measuring the FWHM of the water lines, the real R is determined. The real R depends not only on the instrumental profile but also of the variable size of the seeing disk of the star and the actual goodness of the focus of the star in the focus planes of telescope and spectrographs collimator.
Examples:

A non-calibrated Neon raw spectrum is plotted, measured with a Sigma 1603ME on Lhires III, 2400 g / mm grating, 38 µm slit. No corrections done. Wavelength around 5800 Å. The measured FWHM are registered with Midas (command ‚center / gaus‘, these models the lines as a Gaussian profile, and computes the FWHM). The FWHM of the lines amount to about 5.3 pixels. With the known dispersion of 0.113 A / pixel is delta lambda = 0.63 Angstrom, and R = 5800/0.63 = 9274.

The graph below shows two spectra of the spectroscopic binary Mizar A. It’s the broad and strong H alpha line.

The upper spectrum was obtained with the LHIRES III , 2400 g / mm grating, 40 µm slit. R ca 13,000, dispersion = 0.11 A / pixel. The water lines are 0.5 angstrom wide (= spectral resolution).

The bottom line represents a recording with a 1200 g / mm grating (0.47 A / Pix, 1.5 A resolution, R = 4375) at a different time.

The influence of the resolution R is nice to see. The resolution difference is mainly evident in the water lines. Both R suffice to resolve the Doppler splitting of the line core of the binaries H alpha lines.

A calculation of the resolution R for different slit widths for LHIRES III with 2400 lines / mm grating shows the diagram.

As expected R increases exponentially with nascent narrow slit.