System Throughput and SNR / Exposure Time Estimation

We now consider estimation of count rates for objects with stellar, power law, and emission line spectra.

where *t* is the exposure time in seconds, the *QT* integral is given in Table 6.1, and *AB _{nu}* is given in Table 6.2 as a function of spectral type and wavelength for some example spectral energy distributions. The quantity

where Fn is the flux in erg cm^{-2} s^{-1} Hz^{-1}.

Equation 6.1 may be trivially rewritten to give the count rate Robject in units of e^{-} s^{-1} pixel^{-1} for a target with a stellar spectrum as:

where S* _{nu}* is the flux in ergs cm

if the integrands are weighted by a source with spectral index a in the definition of lambda. See also Koornneef, J., et al. "Synthetic Photometry and the Calibration of the Hubble Space Telescope" in Highlights of Astronomy (7, 833, J.-P. Swings Ed (1983). Combining the above equations gives

where F is the emission line flux in units of ergs cm^{-2} s^{-1}, and lambda is the wavelength of the line in Angstroms. The quantity QT is the (system + filter) quantum efficiency at the wavelength of the line, which can be determined from inspection of the figures in section A.2, "Passbands including the System Response", on page 211. For lines near the maxima of the filter transmission curves, it should be sufficient to use QTmax from Table 6.1. Note that the integrated filter efficiency is not relevant for the signal calculation.

In cases where the width of the line approaches that of the filter, it will be necessary to convolve the line shape and filter bandpass using either the SYNPHOT or XCAL programs.

For example, *H*_{a} emission at 6563Å, with total source flux *F*=10^{-16} erg s^{-1} cm^{-2}, observed through the F656N filter (total system throughput *T*=0.104 from "F656N, F658N, F673N, F675W, F702W, F785LP" on page 217), will produce a target count rate R_{object}=0.155 e^{-} s^{-1} integrated over the entire source.

**6.3.1**- Count Rates for Stellar Sources**6.3.2**- Count Rates for Power Law Sources**6.3.3**- Count Rates for Emission Line Sources