In some situations it may be desirable to go through each step of the calculation. One example would be the case of a source with strong emission lines, where one wants to estimate the contribution of the line(s) to the signal. This could include the case of a strong emission line which happens to fall in the wing of a desired filter's bandpass. To facilitate such calculations, in this section we provide recipes for determining the signal to noise ratio or exposure time by hand.
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 Fj in Jansky/pixel. Please refer to Appendix A to calculate the fraction of flux that will lie in the central pixel for any camera/filter combination, and Appendix B for unit conversions
Then the electron count in that pixel due to the continuum source
is the transmittance of the entire optical train up to the detector,
excluding the filters;
is the detector quantum efficiency;
is the unobscured area of the primary;
where h is Planck's constant and the wavelength. The quantities Fj,
opt,
det and
filt 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 by:
where E is defined as before. In this case, the detector quantum efficiency and filter transmission are determined for the wavelength 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 > 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. It is:
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 Table 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. Appendix A 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 c and
l for each filter are denoted as
and
and are listed in Table 9.1, 9.2, and 9.3 for a detector temperature of 77.1 K. 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 band-pass another correction factor may be needed which can be estimated from filter transmission curves in Appendix A. Or one can input a user supplied spectrum into the web-based ETC to estimate the S/N for an emission line in the wings (see Section 9.3).
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:
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