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.
Calculate the signal-to-noise ratio, S/N (SNR) for a given exposure time. It is necessary to specify the exposure time in seconds.

2. Select the source type:

Point source

In this case, it is necessary to select the size of the detector area in which the signal from the astronomical object will be calculated. By default, a detector area of 5 × 5 pixels is used, which contains 80% of the signal from the source according to the point spread function (PSF).

The user is offered a choice to set the area size of 1 × 1, 2 × 2, 3 × 3, 5 × 5, ..., 101 × 101 pixels. Using the PSF, the amount of energy from the astronomical object, E_f, contained in the specified pixel area is calculated.

Extended source

It is necessary to specify the characteristic diameter of the source in arcseconds and the size of the detector area in which the signal from the astronomical object will be calculated. By default, a detector area of 5 × 5 pixels is used, which contains 80% of the signal from the source according to the point spread function (PSF).

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 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_pix = n_x × n_y – number of pixels along the x, y axes on which a certain level of incoming signal is achieved. By default, this level corresponds to 80% of the signal for a 5×5 pixel area. However, the user is given the opportunity to change the size of this area and, accordingly, the level of incoming signal;

N_bin - total number of binned pixels obtained during the readout from the CCD (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 problems is calculated as follows:

For photometry of a point source:

$$C = E_f \frac{1}{hc} \int_{\lambda} F_{\lambda} A_{eff}(\lambda) QE(\lambda) d\lambda$$

Where:

A_eff(λ) = A × Q_tel(λ) × Q_cam(λ) × T(λ) – effective telescope area, cm²;

A = (SH/100) × (πD²/4) – working telescope area, cm²;

SH – obscuration coefficient, %;

D – diameter of the telescope primary mirror, cm;

Q_tel(λ) – telescope throughput;

Q_cam(λ) – camera throughput;

T(λ) – filter throughput;

QE(λ) – quantum efficiency of the detector;

E_f – fraction of energy contained in N_pix. When the user selects the pixel area from which the signal will be taken (1×1, 2×2, 3×3, 5×5 pixels), using the point spread function, PSF, E_f contained in this area is calculated;

F_λ - flux from the astronomical object, [erg/(cm² s Å)], specified and normalized;

h - Planck's constant, erg s;

c – speed of light, Å/s.

For photometry of an extended source:

$$C = (p_{sx} \cdot scale)(p_{sy} \cdot scale) N_{pix} \frac{1}{hc} \int_{\lambda} I_{\lambda} A_{eff}(\lambda) QE(\lambda) d\lambda$$

Where:

p_sx, p_sy – pixel size in the x and y direction, respectively;

scale – image scale, arcsec/micron;

I_λ – spectral energy distribution of the extended astronomical object from one square arcsecond (surface brightness), erg/(cm² s Å arcsec²).

Calculation of the sky background is performed similarly as for 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:

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