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/m2 and 8.5x10-19 W/m2, 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 2hc2 = 1.19x10-4 W micron2, all lengths are in microns, fstray is the ratio of total sky throughput to that of an ideal f/12 telescope, and are sky emissivities at temperatures Ti. We also define the following:
= the telescope throughput in cm2 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.
Ti = 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 (isky), read noise, and dark current shot noise. It is given by:
Equation 2.3
where id is the dark current contribution, RN is the read noise in electrons, tint 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 cm2), 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 (Rreq = 50) is referred to as the PSSCS, and is given by
Equation 2.6
The spectra are smoothed from R to Rreq 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.
are sky emissivities at temperatures Ti. We also define the following:
= the telescope throughput in cm2 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, 
= 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.
is the dispersion-gain product for the dark observations.
[electrons/pixel/mJy]
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]
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 sensitivity (in W m-2) for unresolved lines is then 
is in microns.