In some situations it may be desirable to go through each step of the calculation. To facilitate calculations when the Web connection is not possible, in this section we provide recipes for determining the signal-to-noise ratio or exposure time by hand. These calculations refer to the brightest pixel (i.e., aperture of 1×1 pixel). These calculations do not take the count rate non-linearity into account described in
Section 4.2.4.
The first step in this process is to calculate the electrons/second/pixel generated by the source. For this we need to know the flux in the central pixel F
j in Jansky/pixel. Please refer to to calculate the fraction of flux that will lie in the central pixel for any camera/filter combination, and for unit conversions.
where h is Planck’s constant and
λ the wavelength. The quantities F
j,
γopt,
γdet and
γfilt are all frequency dependent. The expression for C
c has to be integrated over the bandpass of the filter, since some of the terms vary significantly with wavelength. It should be noted that to determine
Cc more accurately, the source flux
Fj should be included in the integral over the filter bandpass, since the source flux is bound to be a function of wavelength.
For an emission line with intensity Ilj (in W m
-2 pixel
-1) falling in the bandpass of the filter, the counts in e
-/s are given by
where E is defined as before. In this case, the detector quantum efficiency and filter transmission are determined for the wavelength
λ of the emission line. The total signal per pixel is the sum of the continuum and line signals calculated above, namely
Cs=Cc+Cl.
The source signal is superimposed on sky background and thermal background from warm optics. At
λ > 1.7
μm the background is often much brighter than the source. In such cases the observation is background limited, not read noise limited. There is little point in increasing the number of multiple initial and final reads when the observation is background-limited, though multiple exposures and dithering will help cosmic ray removal and correction of other effects such as persistence from previously-observed bright objects.
The other components of unwanted signal are read noise, Nr, and dark current,
Id (in e
-/s/pixel). By read noise, we mean the electronic noise in the pixel signal after subtraction of two reads (double correlated sampling).
where Cs, the count rate in e
-/sec/pixel, is the sum of
Cc plus
Cline,
B is the background in e
-/sec/pixel (also listed in Tables
9.1,
9.2, and
9.3),
Id is the dark current in e
-/sec/pixel and
Nr is the read-out noise, in e
-/pixel, for one initial and one final read. Although the effective
Nr can vary somewhat depending on the readout sequence, ETC considers a fixed value per camera of about 26 e
-.
It is important to note that in these equations, the flux to be entered (either F
j or I
lj or both) is
not the total source flux, but the flux falling on a pixel. In the case of an extended source this can easily be worked out from the surface brightness and the size of the pixel. For a point source, it will be necessary to determine the fraction of the total flux which is contained within the area of one pixel and scale the source flux by this fraction. For Camera 1 in particular, this fraction may be quite small, and so will make a substantial difference to the outcome of the calculation. gives the fraction of the PSF falling in the brightest pixel assuming a point source centered on the pixel, for each filter.
The signal-to-noise ratio evaluated by a fit over the full PSF for point sources would, of course, be larger than this central pixel SNR; this discrepancy will be largest for the higher resolution cameras and for the longest wavelengths.
The average values for ηc and
ελ for each filter are denoted as
and
and are listed in Tables
9.1,
9.2, and
9.3 for a detector temperature of 77.1 K. (Note that the above tables are not necessarily updated and the results in S/N could be off by ~20% (or more) in some cases. The Web based ETC, which is updated and provides more accurate results, should be used for any serious ETC calculation.) For estimating
we have assumed a source with an effective temperature of 5,000K, but the Web based ETC will take the spectral type chosen by the user to integrate over the bandpass. For emission lines in the wings of the filter bandpass, another correction factor may be needed which can be estimated from filter transmission curves in Appendix A.
Given a particular filter-detector combination and a requested target flux, there is an exposure time above which the detector starts to saturate. The WWW NICMOS ETC will produce this exposure time when it performs the requested estimation.
The other situation frequently encountered is when the required signal-to-noise is known, and it is necessary to calculate from this the exposure time needed. In this case one uses the same instrumental and telescope parameters as described above, and the required time is given by:
is the transmittance of the entire optical train up to the detector,
where h is Planck’s constant and λ the wavelength. The quantities Fj, γopt, γdet and γfilt are all frequency dependent. The expression for Cc has to be integrated over the bandpass of the filter, since some of the terms vary significantly with wavelength. It should be noted that to determine Cc more accurately, the source flux Fj should be included in the integral over the filter bandpass, since the source flux is bound to be a function of wavelength.For an emission line with intensity Ilj (in W m-2 pixel-1) falling in the bandpass of the filter, the counts in e-/s are given bywhere E is defined as before. In this case, the detector quantum efficiency and filter transmission are determined for the wavelength λ of the emission line. The total signal per pixel is the sum of the continuum and line signals calculated above, namely Cs=Cc+Cl.The source signal is superimposed on sky background and thermal background from warm optics. At λ > 1.7 μm the background is often much brighter than the source. In such cases the observation is background limited, not read noise limited. There is little point in increasing the number of multiple initial and final reads when the observation is background-limited, though multiple exposures and dithering will help cosmic ray removal and correction of other effects such as persistence from previously-observed bright objects.The other components of unwanted signal are read noise, Nr, and dark current, Id (in e-/s/pixel). By read noise, we mean the electronic noise in the pixel signal after subtraction of two reads (double correlated sampling).It is now possible to calculate the signal-to-noise ratio expected for an exposure of duration t seconds:where Cs, the count rate in e-/sec/pixel, is the sum of Cc plus Cline, B is the background in e-/sec/pixel (also listed in Tables 9.1, 9.2, and 9.3), Id is the dark current in e-/sec/pixel and Nr is the read-out noise, in e-/pixel, for one initial and one final read. Although the effective Nr can vary somewhat depending on the readout sequence, ETC considers a fixed value per camera of about 26 e-.It is important to note that in these equations, the flux to be entered (either Fj or Ilj or both) is not the total source flux, but the flux falling on a pixel. In the case of an extended source this can easily be worked out from the surface brightness and the size of the pixel. For a point source, it will be necessary to determine the fraction of the total flux which is contained within the area of one pixel and scale the source flux by this fraction. For Camera 1 in particular, this fraction may be quite small, and so will make a substantial difference to the outcome of the calculation. gives the fraction of the PSF falling in the brightest pixel assuming a point source centered on the pixel, for each filter.The signal-to-noise ratio evaluated by a fit over the full PSF for point sources would, of course, be larger than this central pixel SNR; this discrepancy will be largest for the higher resolution cameras and for the longest wavelengths.The average values for ηc and ελ for each filter are denoted as
is the detector quantum efficiency; 
is the filter transmittance;
is the unobscured area of the primary;