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We will use the following designations and relations:
- wavelength in Å,
-- instrumental function width of the spectrograph in Å,
-- 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,
-- number of counts per 1 Å at the continuous spectrum level,
-- pixel size in Å,
-- number of counts per pixel,
-- line half-width at half intensity,
-- line depth resolved by the spectrograph,
-- line equivalent width,
-- sum of read-out noise and dark count per 1 pixel,
()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: ( and
). Let the
width of the pixel
exceed the line width
and the
instrumental function width
. 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
, the relative measurement error of equivalent width
measurement in the case of unresolved line is proportional to
. If
the number of counts per 1 Å is such that
, then
the equivalent width measurement error is proportional to
. Thus if the
spectral resolution is managed to be increased (i.e. to decrease
)without loss in the number of counts falling at 1 Å, the equivalent width
measurement error is then proportional to
. This holds for case
``B'': (
and
), which is a
condition satisfied most frequently at large diameter telescopes.
Case ``C'' ( and
). The
pixel width
is smaller than the line width
and
less than or equal to the instrumental function width
. 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 (
, i.e. it
is now proportional to (
. For
low read-out levels, i.e. at
, the equivalent width
determination error does not already depend on
, while at a low
signal level,
, the error increases with decreasing
. 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 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
. We illustrate this statement in Fig.1.
Here on
a logarithmic scale the spectral resolution estimated for the condition is laid off as abscissa, the number of counts per 1 Å
is plotted as ordinate. The inclined lines show the conditions
satisfied for two types of the first home-made CCDs used in our
observations -- one of
pixels with a read-out noise of
and the other of
with a
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
is no
longer dependent on
, i.e. it makes sense to increase the
spectral resolution in this region. For the
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
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
(i.e.
increasing R) will be of paramount importance.