N_ron = sqrt( npix . RON^2)
= sqrt( n_pix) . RON
(all the individual read-out-noises add quadratically). If we read the chip NDIT times,
N_ron = sqrt ( n_pix . NDIT ) . RON
4.3. Adding noises and total signal to noise ratio
Random, uncorrelated noises (such those described) add quadratically, so, the generic formula for the noise is
|N_total = sqrt( N^2 + N_sky^2 + N_dark^2 + N_ron^2 )
We can play with the formula, introducing the various definitions, to get the generic formula for the SNR:
S/N = S/N_tot = S / sqrt ( S + Sky + Dark + N_ron^2 )
|S/N = s.t / sqrt ( s.t + n_pix . sky . t + n_pix . dark . t + n_pix . NDIT . RON^2 )
5. Special cases -- in practice
The generic S/N equation is used in details in the Exposure Time
Calculators. However, in practice, it is useful to consider some
special cases, to understand the behavior of the instrument, and why
one can/should make many short or one long exposure.
5.1.- Bright source
Let's first consider the case of a bright star. In that case, the
signal of the star is so bright that all the other sources of noises
we can remove them from the equation:
S/N = s.t / sqrt ( s.t ) = sqrt( s . t)
S/N === sqrt ( t )
( === means "varies proprortionally to").
standard stars in imaging and in spectro. Note that in this case, the
problem is usually not to reach a good S/N, but not to saturate the
5.2- Sky Noise Dominated case
After this warm up, let's consider a more useful case: a faint
star with a bright sky background: sky >> s . The S/N equation becomes:
S/N = s.t / sqrt ( n_pix . sky . t)
S/N === sqrt ( t ) / sqrt (n_pix)
First aspect: if n_pix is smaller, S/N is larger. As n_pix === seeing^2, one sees that good seeing is critical
Second aspect: the signal-to-noise increases as the square root of the TOTAL time t = NDIT.DIT.
This means that the S/N will not be affected if we take 1
exposure of 3000s or 10 exposures of 300s (as long as each of the
individual images is sky-noise dominated). In many cases it IS
advantageous to split the exposure time in many exposures.
- broad band images in the visible (typical
value: DIT = 300-600s, Sky = 5000adu = 10000 e, RON = 3 e).
Taking many images improves the flat field of the final image
- low resolution spectro in the visible (EFOSC, FORS,
EMMI-RILD...). As soon as the sky reaches a few hundred e, one can
split. Advantages: get rid of the cosmic rays.
- Infra red imaging: the sky is so high that it would saturate the
detector very quickly (few s to few 10s in JHK, few ms in thermal IR).
There is no other choice than to split the total exp. with many short
DIT. This is not a problem for the final S/N, since we are sky noise
Counter applications: This is not
case in high resolutions spectroscopy, or in narrow band imaging, since
the sky is then very low. Do not split in short DITs, cf next section.
5.3- Read-out-noise dominated case
In case the sky --and the dark-- are very small, they do not dominate
in the general S/N equation. There are even cases where they are very
small; in that case, the RON dominates:
S/N = s.t / sqrt ( n_pix . NDIT . RON^2 )
S/N === 1/sqrt(n_pix) . 1/sqrt(NDIT)
First aspect: 1/sqrt(n_pix): there
is here a way to cheat. n_pix is the number of pixel read. We can
decrease this number by binning the detector (i.e. reading only once
for 2x2=4 pixels). There is a gain of sqrt(2x2) = 2
in S/N! Of course, there is a price: one loses some of the resolution:
spacial resolution in case of imaging, beware of keeping at least 2
(binned) pix across the seeing) or spectral resolution in case of
spectro, beware of keeping at least 2 (binned) pix across one spectral
Second: 1/sqrt(NDIT): one should keep the NDIT as small as
possible, meaning keeping DIT as long as possible. In case of CCDs, the
real limit becomes the number of cosmic rays. In practice, DIT_max is
about 45min, but in some cases, it is worth making it even longer (e.g.
if the observer does not care about cosmic rays).
- narrow band imaging (sky = a couple of electrons), typical exposure times can be as long as 1h.
- Echelle spectro: the sky light is so much dispersed that it does not count anymore. Keep the exp. time as long as possible.