The Formulae

This program will be used by observers for scheduling their observations with LUCI. The following fixed parameters

• telescope (transmission, light-collecting area, reflectivity, ...)
• transmission of the instruments (entrance window, lenses, ...)
• quantum efficiency (QE) of the detector

and user-defined parameters

• object (geometry, spectrum, magnitude)
• camera (scale)
• filter
• grating, slit width (spectroscopic mode)
• atmospheric conditions (airmass, water vapor, sky background, seeing)
• adaptive optics (is loop closed, Strehl ratio)
• exposure parameters (detector integration time, total exposure time)
• signal-to-noise ratio

are used to calculate two auxiliary values:

E = T_atm ^. T_tel ^. T_inst ^. T_filt ^. QE (1)

$${\frac{{\mathrm{mag}}}{{2.5}}}$$ F = (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:

$${\frac{{N_{\mathrm {DIT}}}}{{\sqrt{N_{\mathrm {DIT}} + {\mathrm n... ...\mathrm{sky}} + {\mathrm{DARK}_{\mathrm {DIT}}} + {\mathrm{RON}}^2 \right )}}}}$$ (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_$\tau$). The ETC calculates the necessary exposure time to achieve this SNR_$\tau$ (see also formula 3):

$$\tau \tau \sqrt{{N_{\mathrm{DIT}}}}$$ = (4)

$$\rightarrow \tau \left(\vphantom{ \frac{\mathrm {SNR_\tau}}{\mathrm {SNR_{DIT}}} }\right. {\frac{{\mathrm {SNR_\tau}}}{{\mathrm {SNR_{DIT}}}}} \left.\vphantom{ \frac{\mathrm {SNR_\tau}}{\mathrm {SNR_{DIT}}} }\right)^{2}_{}$$ = (5)

$\tau$	:	total exposure time
DIT	:	detector integration time
NDIT	:	number of detector integrations
SNR$\tau$	:	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