The Formulae

This program will be used by observers for scheduling their observations with LUCI. The following fixed parameters and user-defined parameters are used to calculate two auxiliary values:
E = Tatm . Ttel . Tinst . Tfilt . QE (1)
F = F0 . 10-$\scriptstyle {\frac{{\mathrm{mag}}}{{2.5}}}$ (2)

F0 : flux density for Vega at $ \lambda$ = 550 nm
    F0 = 3.56 . 10-11$ {\frac{{\mathrm{W}}}{{\mathrm{m^2\cdot nm}}}}$ [FLUX95]
mag : magnitude of the object
QE : quantum efficiency of the detector
T atm : transmission of the atmosphere
T tel : transmission of the telescope
T inst : transmission of the instrument (without any filter and grating)
T filt : transmission of the filter used

The number of photons that are detected per second can be calculated with (1), (2) and the following formula [ESO-HP]:

Table 1: Formulae for calculating the number of photons from the source
Observing mode point source extended source
Imaging $ {\frac{{N}}{{\tau}}}$ = $ {\frac{{{\mathrm F} \cdot \Delta_{\mathrm i} \cdot {\mathrm E} \cdot {\mathrm S} }}{{\mathrm P}}}$ $ {\frac{{N}}{{\tau}}}$ = $ {\frac{{{\mathrm F} \cdot \Delta_{\mathrm {i}} \cdot {\mathrm E} \cdot {\mathrm S} \cdot \Omega_{\mathrm i}}}{{\mathrm P}}}$
Spectroscopy $ {\frac{{N}}{{\tau}}}$ = $ {\frac{{{\mathrm F} \cdot \Delta_{\mathrm s} \cdot {\mathrm E} \cdot {\mathrm S} }}{{\mathrm P}}}$ $ {\frac{{N}}{{\tau}}}$ = $ {\frac{{{\mathrm F} \cdot \Delta_{\mathrm {s}} \cdot {\mathrm E} \cdot {\mathrm S} \cdot \Omega_{\mathrm s}}}{{\mathrm P}}}$

N : number of photons   $ \Delta_{{\mathrm i}}^{}$ : filter band width
P : energy of one photon   $ \Delta_{{\mathrm s}}^{}$ : spectral resolution
S : light-collecting area   $ \Omega_{{\mathrm i}}^{}$ : scale in imaging mode
$ \tau$ : exposure time   $ \Omega_{{\mathrm s}}^{}$ : scale in spectroscopy mode

In the near-infrared regime the SNR for the exposure time DIT is given by the formula:

SNRDIT = $\displaystyle {\frac{{N_{\mathrm {DIT}}}}{{\sqrt{N_{\mathrm {DIT}} + {\mathrm n...
...\mathrm{sky}} + {\mathrm{DARK}_{\mathrm {DIT}}} + {\mathrm{RON}}^2 \right )}}}}$ (3)

DARK DIT : dark current for 1 DIT
n pix : number of integration pixels1
Nsky : sky signal for 1 DIT
RON : readout noise
SNRDIT : signal-to-noise ratio for 1 DIT

In imaging mode the user will be asked for the signal-to-noise ratio for the exposure time $ \tau$ ( SNR$\scriptstyle \tau$). The ETC calculates the necessary exposure time to achieve this SNR$\scriptstyle \tau$ (see also formula 3):

$\displaystyle \tau$ = NDIT . DIT   and   SNR$\scriptstyle \tau$ = SNRDIT . $\displaystyle \sqrt{{N_{\mathrm{DIT}}}}$ (4)
$\displaystyle \rightarrow$ $\displaystyle \tau$ = $\displaystyle \left(\vphantom{ \frac{\mathrm {SNR_\tau}}{\mathrm {SNR_{DIT}}} }\right.$$\displaystyle {\frac{{\mathrm {SNR_\tau}}}{{\mathrm {SNR_{DIT}}}}}$$\displaystyle \left.\vphantom{ \frac{\mathrm {SNR_\tau}}{\mathrm {SNR_{DIT}}} }\right)^{2}_{}$ . DIT (5)

$ \tau$ : total exposure time
DIT : detector integration time
NDIT : number of detector integrations
SNR$\scriptstyle \tau$ : signal-to-noise ratio for exposure time $ \uptau$
SNRDIT : signal-to-noise ratio for one DIT
SNR : signal-to-noise ratio for an exposure time of 1sec

Andre Germeroth 2016-11-04