$\star$ The signal on noise

The number of photons $T_p$ is written:

$$N_{Sn}^\gamma = \phi_{Sn}^\gamma \frac{\pi}{4} D_m^2 T_p$$ (7.1)

where $\phi_{Sn}^\gamma$ is the flux of the supernova.

The number of blows of ADC actually recorded by the camera depends on a certain number of parameters like the gain of electronics of acquisition, the quantum effectiveness of camera CCD, the transmission of the filter and the transmission of optics. These various contributions can have a strong dependence in wavelength (in particular filters and to a lesser extent quantum effectiveness). For the moment we will neglect these effects of color, we will return there abundantly thereafter. We thus consider here whom the transmission of the instrument can be described by only one number $\epsilon_{filtre}$ that we will take equal to his average value in the filter considered . Lastly, one includes in this term the attenuation by the atmosphere.

One can thus write:

$$\overline{ N_{Sn}^{e^-}} = \epsilon_{filtre} ~{ N^\gamma_{Sn}} = \epsilon_{filtre} \phi_{Sn}^\gamma \frac{\pi}{4} D_m^2 T_p$$ (7.2)

where $\overline{ N_{Sn}^{e^-}}$ is the average number of recorded electrons. The noise associated with measurement is mainly Poissonnian. The nonPoissonnian noise of reading or shot effect is generally negligible compared to the noise of photon induced by the bottom of sky in the case of the filters à.large.bande busy for the images which we consider ( i.e. with long durations). The various contributions in our case are the flux of the supernova, the flux of the subjacent galaxy and the flux of the night bottom of sky .

These various components not being correlated, one can write the variance of our flux like:

$$Var_{flux_{Sn}} = (\sigma_{tot}^{e^{-}})^2=\overline{N_{tot}^{e^-}} = \overline{N_{Sn}} + \overline{N_{Gal}} + \overline{N_{ciel}}$$ (7.3)

For major research, fluxes of the studied objects (supernova and galaxy host) are weak compared to fluxes of the bottom of sky, their contribution to the noise is thus negligible.

Finally, one can write the fluctuations of the flux of the supernova like:

$${\sigma_{tot}^{e^-}}^2 = \epsilon_\lambda ~~\frac{\pi}{4} ~D^2_m ~T_p \pi r^2 \beta^\gamma_{ciel}$$ (7.4)

where $r$ the ray (angular) in which is integrated the flux of the supernova. We will see in the continuation, that for the aperture photometry (integration of the flux of an object in a ray centered on its position), there is an optimal aperture. This ray is proportional to the spreading out of star over the CCD, its width with middle height referred by the term `` seeing '' which we will precisely define in this chapter, it characterizes the quality of image delivered by the combination telescopes, site, instrument and detector.

One can thus express $\sigma_{seeing}$) like:

$$r = n_{opt} \sigma_{seeing}$$ (7.5)

One can finally write the signal report/ratio on noise like:

$$\left(\frac{S}{N}\right) = \frac{\sqrt{\epsilon_{filtre}}}{2n_{opt}\sigma_{seeing}} ~~\frac{\phi_{Sn}^\gamma}{\beta_{ciel}^{1/2}}~~\sqrt{Tp}~D_m$$ (7.6)

One thus has:

$$\left( \frac{S}{N}\right) \propto T_p^{1/2}~\sigma_{seeing}^{-1} ~D_m$$ (7.7)

The signal on noise is thus directly proportional to the size of the telescope. The seeing him also will play a significant role. Variations of seeing from one site to another sometimes which can be of a factor 4 (of $2' '$).