Sky

With the configuration previously specified, in the J band the sky magnitude is 16.2 so:

$$M_{J}=16.2\Longrightarrow Z_{J}=8.4989\Longrightarrow F=1.04... ...} ~J\cdot s^{-1}\cdot m^{-2}\cdot \mu m^{-1}\cdot arcsec^{-2}$$

$$\displaystyle\left(\frac{F}{P}\right)_{Obj}=6.60740258403\cdo... ...{3}~ph\cdot s^{-1}\cdot m^{-2}\cdot\mu m^{-1}\cdot arcsec^{-2}$$

Using the dispersion value $\Delta _{i}$, the telescope surface S, the exposure time T and the efficiency E as specified before, and taking in count theat the plate scale is 0.15 arcsec, we get

$$\Omega=0.0225~arcsec^{2}$$

So we get

$$N_{Sky}=\frac{F\cdot \Delta_{i}\cdot T\cdot E\cdot S\cdot\Omega}{P}= 2.71850132614\cdot~cnts$$ (2.16)

The ETC for ISAAC with the input parameter as specified above predicts $N_{ETC}=2.407837\cdot 10^{4}$ cnts.