Specify the Exposure Parameters

The user needs to make a choice on two points:

1. Specify the required mode of operation of the calculator and set the parameters of this mode:

Calculate the exposure time for a given S/N ratio. It is necessary to specify the S/N ratio, as well as the considered wavelength. The calculation of exposure time will be performed for the given wavelength. At the same time, calculation of S/N = f(λ) for all wavelengths from the working spectral range of the spectrograph with the calculated exposure time will also be performed.
Calculate the signal-to-noise ratio, S/N, for a given exposure time. It is necessary to specify the exposure time in seconds, as well as the considered wavelength. Calculation of S/N will be performed for the given wavelength. At the same time, calculation of S/N = f(λ) for all wavelengths from the working spectral range of the spectrograph with the specified exposure time will also be performed.

2. Select the source type:

Point source

When calculating the signal from the astronomical object, a detector area of size [2 pixels in the dispersion direction × 7 pixels in the direction perpendicular to the dispersion direction] is used. The fraction of energy from the source contained in 7 pixels (in the direction perpendicular to dispersion) is $E_{fs} = 80\%$.

Extended source

It is necessary to specify the characteristic diameter of the source in arcseconds. When calculating the signal from the astronomical object, a detector area of size [2 pixels in the dispersion direction × 7 pixels in the direction perpendicular to the dispersion direction] is used.

Calculation Methodology

Based on the specified initial parameters, as well as the known characteristics of the Spektr-UF, the signal-to-noise ratio from the astronomical object at a given exposure time $t$ is subsequently calculated:

$$S/N = \frac{Ct}{\sqrt{Ct + (B_{det} + B_{sky})N_{pix}t + \frac{N_{pix}}{N_{bin}}N_{read}R^2}}$$

or the exposure time required to achieve a given S/N ratio:

$$t = \frac{(S/N)^2 [C + N_{pix}(B_{det} + B_{sky})]}{2C^2} + \frac{\sqrt{(S/N)^4 [C + N_{pix}(B_{det} + B_{sky})]^2 + 4(S/N \cdot C)^2 \frac{N_{pix}}{N_{bin}} N_{read} R^2}}{2C^2}$$

Where:

$B_{sky}$ - sky background, $e^{-}/s/pixel$;

$B_{det}$ - detector dark current, $e^{-}/s/pixel$;

$N_{bin}$ - total number of binned pixels obtained during the readout of information from the CCD detector (by default $N_{bin} = 1$);

$N_{read}$ - number of detector readouts (by default $N_{read} = 1$);

$R$ – readout noise, $e^{-}$;

$C$ – useful signal from the astronomical object, $e^{-}/s$, which for different tasks is calculated as follows:

For spectroscopy of a point source:

$$C = E_{f}^{spec} \cdot F_{\lambda} \cdot \frac{A_{eff}^{spec} \cdot QE \cdot d \cdot \lambda}{hc}$$

Where:

$E_{f}^{spec}$ – fraction of energy contained in $N_{spix}$ in the direction perpendicular to dispersion;

$A_{eff}^{spec} = A \cdot Q_{tel}(\lambda) \cdot S_{spec}(\lambda)$, $cm^2$;

$A = \frac{SH}{100} \cdot \frac{\pi D^2}{4}$ – working telescope area, $cm^2$;

$SH$ – obscuration coefficient, %;

$D$ – diameter of the telescope primary mirror, $cm$;

$Q_{tel}(\lambda)$ – telescope throughput;

$S_{spec}(\lambda)$ – spectrograph efficiency;

$d$ – dispersion, $Å$/pixel;

$N_{\lambda pix}$ – number of pixels in the dispersion direction;

$F_{\lambda}$ - flux from the astronomical object, $[erg/(cm^2\ s\ Å)]$, specified and normalized;

$h$ - Planck's constant, $erg\ s$;

$c$ – speed of light, $Å/s$.

For spectroscopy of an extended source:

$$C = I_{\lambda} \cdot \frac{A_{eff}^{spec} \cdot QE \cdot d \cdot \lambda}{hc} \cdot scale_s \cdot W \cdot N_{spix} \cdot N_{\lambda pix}$$

Where:

$scale_s$ – image scale in the direction perpendicular to dispersion, $arcsec/pixel$;

$W$ - slit width in arcseconds.

$I_{\lambda}$ – spectral energy distribution of the extended astronomical object from one square arcsecond (surface brightness), $erg/(cm^2\ s\ Å\ arcsec^2)$.

Calculation of the sky background is performed similarly to expression for the extended source.

Note: It should also be noted that for an emission spectral line, when calculating the $S/N$ ratio, it is necessary to take into account both the flux in the spectral line, $C_{line}$, and the flux in the continuum, $C_{cont}$. The first expression in this case transforms to:

$$S/N = \frac{C_{line}t}{\sqrt{(C_{line} + C_{cont})t + (B_{det} + B_{sky})N_{pix}t + \frac{N_{pix}}{N_{bin}}N_{read}R^2}}$$