Optimization of spectroscopic observations

Let us first find out which way -- rise in spectral resolution with some loss in the S/N ratio or increase in the S/N ratio with a fixed spectral resolution -- is better for the purpose of reduction of equivalent width determination errors.

We will use the following designations and relations:
$\lambda$ - wavelength in Å,
$\rm \Delta S$ -- instrumental function width of the spectrograph in Å,
$\rm R=\lambda/\Delta S$ -- spectral resolution of the device,
N -- number of counts for 1 Å per 1 s at the continuous spectrum level,
t -- exposure time in seconds,
$\rm Nt$ -- number of counts per 1 Å at the continuous spectrum level,
$\rm \Delta A$ -- pixel size in Å,
$\rm Nt\Delta A$ -- number of counts per pixel,
$\Delta \lambda$ -- line half-width at half intensity,
$\rm r$ -- line depth resolved by the spectrograph,
$\rm Ntr\Delta\lambda$ -- line equivalent width,
$\rm e$ -- sum of read-out noise and dark count per 1 pixel,
($\rm Nt\Delta A+e^2$)^1/2 -- statistical error of measurement of the number of counts per each pixel.

Consider three versions of the relation between the characteristics of the device of the light detector and the spectral line parameters.

Case A: ($\rm \Delta A\gt 2\Delta\lambda$ and $\rm \Delta A\gt\Delta S$). Let the width of the pixel $\rm \Delta A$ exceed the line width $2\Delta\lambda$ and the instrumental function width $\rm \Delta S$. The relative line intensity is then measured as the ratio of the number of counts in the pixel falling at the given line to the number of counts in the adjacent pixel falling at the continuous spectrum. Since the equivalent width equals $\rm Ntr\Delta\lambda$, the relative measurement error of equivalent width measurement in the case of unresolved line is proportional to $\rm (r\Delta\lambda)^{-1}\left[\Delta A(Nt)^{-1}+e^2(Nt)^{-2}\right]^{1/2}$. If the number of counts per 1 Å is such that $\rm \Delta A\gg e^2/Nt$, then the equivalent width measurement error is proportional to $\rm (r\Delta\lambda)^{-1}\left[\Delta A(Nt)^{-1}\right]^{1/2}$. Thus if the spectral resolution is managed to be increased (i.e. to decrease $\rm \Delta A$)without loss in the number of counts falling at 1 Å, the equivalent width measurement error is then proportional to $\rm R^{-1/2}$. This holds for case ``B'': ($\rm \Delta A\gt 2\Delta\lambda$ and $\Delta S\gt\Delta A$), which is a condition satisfied most frequently at large diameter telescopes.

Case ``C'' ($\rm \Delta A<2 \Delta \lambda$ and $\rm \Delta S\gt\Delta A$). The pixel width $\rm \Delta A$ is smaller than the line width $\rm 2\Delta \lambda$ and less than or equal to the instrumental function width $\rm \Delta S$. For simplicity, take the line profile to be of triangular shape. Then in comparison with case ``A'' the relative equivalent width measurement error will increase by the multiplicand ($\rm 2\Delta \lambda/\Delta A)^{1/2}$, i.e. it is now proportional to ($\rm \Delta \lambda/2)^{-1/2}r^{-1}\left[(Nt)^{-1}+e^2/\Delta A(Nt)^2\right]^{1/2}$. For low read-out levels, i.e. at $\rm e^2<Nt\Delta A$, the equivalent width determination error does not already depend on $\rm \Delta A$, while at a low signal level, $\rm e^2\gt Nt\Delta A$, the error increases with decreasing $\rm \Delta A$. Thus for the low S/N ratio spectra it makes no special sense to realize case ``C''.

So, in order to improve the equivalent width measurement accuracy it is more advantageous to increase the spectral resolution until the signal level is equal to read-out noise instead of increasing the exposure time at a specified spectral resolution. It goes without saying that the limit at which further increase in spectral resolution becomes unprofitable -- for different groups of astrophysical objects with comparable $\rm \Delta \lambda$and for different characteristics of the detector (read-out noise and linear pixel-size) -- is at different R values. Hence it follows that a large telescope needs to be equipped with several spectral devices differing in $\rm R=\lambda/2\Delta A$. We illustrate this statement in Fig.1.

 

Figure: To the choice of optimum spectral resolution (for designations see the text)

Here on a logarithmic scale the spectral resolution estimated for the condition $\rm \Delta S=2\Delta A$ is laid off as abscissa, the number of counts per 1 Å is plotted as ordinate. The inclined lines show the conditions $\rm \Delta A=e^2/Nt$ satisfied for two types of the first home-made CCDs used in our observations -- one of $580\times 530$ pixels with a read-out noise of $\rm 26e^-$ and the other of $1160\times 1040$ with a $\rm 6.5e^-$ read-out noise. The vertical arrows indicate the R values corresponding to different devices placed at the Nasmyth2 focus (Z - the moderate resolution echelle spectrograph (Klochkova and Panchuk, 1991), M - the Main Stellar Spectrograph, the Schmidt camera of 1:2.5 with a CCD (Panchuk, 1995), L -- high resolution echelle spectrograph (two modifications -- Panchuk et al., 1993; Klochkova, 1995). For convenience of comparing different systems we assumed everywhere a two-pixel resolution at the wavelength of 5500 Å, this is actually the upper estimate of R. The values of R do not pretend to be accurate here, since Fig. 1 is intended to illustrate the idea but not accurate estimates. The parameters of an echelle spectrograph with a large collimated beam diameter (E) manufactured in 1997 and the parameters of the prime focus echelle spectrograph (P) are also noted. The numbers in these notations indicate the year of introduction of this type of observations. The upper left corner of Fig. 1 corresponds to the region where the equivalent width determination error for the CCD of $530\times 580$ is no longer dependent on $\rm \Delta A$, i.e. it makes sense to increase the spectral resolution in this region. For the $1040\times 1160$ CCD the similar region takes up a greater part of the figure, and only in the lower right corner at given levels of the signal $\rm Nt$ accumulated in the band 1Å wide it makes no sense to increase spectral resolution. So, in the optimization of spectrum line measurement the parameters of the CCDs and those of the spectrographs possess equal rights, and as the read-out noise of the CCDs approaches a physical limit, the problem of reasonable (remaining in the frames of case ``B'') decreasing of $\rm \Delta A$ (i.e. increasing R) will be of paramount importance.