The median 1-sigma continuum sensitivity for the IRS low-resolution modules is about 0.06 mJy from 6 to 15 microns and 0.4 mJy from 14 to 38 microns in 512 seconds of integration with low background. The median 1-sigma line sensitivity estimates for the short- and long-wavelength high-resolution modules are about 2.7x10^{-19} W/m^{2} and 8.5x10^{-19} W/m^{2}, respectively, for 512 seconds of integration with low background (see Figure 2.14 through Figure 2.18).

2.10.1 Equations for Calculating Sensitivity and Signal-To-Noise for Point Sources

This section illustrates the procedure by which the expected instrumental sensitivity and signal-to-noise ratios were calculated.

The background photocurrent from sky and telescope is given by

Equation 2.1

where h is the Planck constant and

Equation 2.2

where 2hc^{2 }= 1.19x10^{-4 }W micron^{2}, all lengths are in microns, f_{stray} is the ratio of total sky throughput to that of an ideal f/12 telescope, and are sky emissivities at temperatures T_{i}. We also define the following:

= the telescope throughput in cm^{2} sr.

=the spectral bandpass falling on one pixel (=/R) in cm. Note: while there are 2 pixels per resolution element, the slit width is also two pixels. Hence, is proportional to the slit width if the spectral plate scale is unchanged.

T_{i} = effective blackbody temperature of sky + telescope.

= the cold optical throughput for extended sources (i.e., omitting slit losses).

= the detector responsivity (electrons/photon) [=].

=quantum efficiency of the detector at the desired wavelength.

βG = the dispersion-gain product.

In the faint source limit, the total noise in electrons/pixel is the sum of the photocurrent shot noise (i_{sky}), read noise, and dark current shot noise. It is given by:

Equation 2.3

where i_{d} is the dark current contribution, RN is the read noise in electrons, t_{int} is the integration time in seconds, and is the dispersion-gain product for the dark observations.

The relationship between photocurrent/pixel and the incident flux density of a source in mJy is given by:

[electrons/pixel/mJy]

Equation 2.4

where 1 mJy = 10^{-26 }erg s^{-1} cm^{-2} Hz^{-1}, D is the telescope aperture (in cm^{2}), is?the slit throughput of a point source with angular diameter (without considering losses), and are throughputs accounting for losses in the telescope, alignment effects, and slit, respectively. The second factor of 4 in Equation 2.4 comes from dividing the point source flux over 2 pixels per spectral resolution element and 2 pixels spatial extent. Again assuming the faint source limit, the 1 staring point source continuum sensitivity (PSSC) in mJy at the full resolution of the spectrograph is:

[mJy]

Equation 2.5

where the factor accounts for the variation in PSF with wavelength and details of the point source extraction. Its value lies between 0.8 and 1.2. The 1 staring point source continuum sensitivity (in mJy) smoothed at the required resolution of the spectrograph (R_{req} = 50) is referred to as the PSSCS, and is given by

Equation 2.6

The spectra are smoothed from R to R_{req} and the S/N is assumed to increase as the square root of the number of pixels averaged. Suitable sub-pixel smoothing is assumed.

The 1 staring point source sensitivity (in W m^{-2}) for unresolved lines is then

Equation 2.7

where the factor 3x10^{-15 }= c (micron/s) (10^{-29 }W m^{-2} Hz^{-1}/mJy), PSSC is in units of mJy, and is in microns.