See also a short writeup on extended source calibration issues with the IRS.
Note: the correction curves presented here have been superseded by new calibrations produced by the SSC.
Flux calibrating extended sources with the IRS is problematic for several reasons.
This leads to two correction factors: an aperture loss correction function (ALCF) and a slit loss correction function (SLCF), which correct orthogonal losses that occur along both directions along the slit: spatial (ALCF) and spectral (SLCF). Both forms of flux loss are peculiar to point sources. Since point sources form the main calibration standard, the exact form and magnitude of these two loss terms are implicit in the pipeline. Only point sources which suffer these exact losses will produce unbiased fluxed spectra.
The ALCF is necessary to account for the narrowing, point-source-optimized extraction which the SSC uses, the form of which feeds back into the flat field and FLUXCON tables via the extracted standard star spectra. These standard stars, together with their flux models, form the primary calibration basis. Essentially, the ALCF is a secondary loss imposed by the exact form of the narrowing extraction aperture used by the SSC: all point-sources (and only point-sources) extracted with exactly this aperture (and only this aperture) will result in unbiased flux estimates, having been stripped of precisely the "correct" fraction of their flux as a function of wavelength to match the pipeline's assumptions. Any other aperture will result in biased flux estimates, typically biased high for larger extraction widths. The ALCF correction could be obviated if the SSC performed a parallel "extended-source-optimized" extraction of the calibration standards, and propagated that extraction through to the flat-field and FLUXCON tables. In fact, such an extraction could be used for point sources too, and would be far less sensitive to errors in object centroiding. Its only drawback is the admission of more noise from adjacent sky pixels, which, for bright calibrator stars, is non-issue.
The ALCF can be measured, as described below, using standard stars. Ideally two independent measurements of the ALCF would agree:
Thus far I've only used method #2.
The SLCF is a more fundamental, and less easily approached correction. It results from the unavoidable loss of flux through the slit, which is related to the changing size of the telescope+instrument foreoptics-delivered PSF as a function of wavelength. For a point source, assumed to be accurately centered on the slit, the obtained slit-loss is implicitly accounted for in the pipeline, and removed, to obtain the true stellar flux of calibration standards forced to match their model spectra. All well-centered point sources will thus have the same wavelength dependence of slit-loss, and the standard pipeline will recover an unbiased flux estimate for them.
Extended sources, by definition, cannot be well-centered. Arbitrary extended sources suffer both diffraction losses from within the beam, and diffraction gains from emission which, in the absence of diffraction, would have fallen outside the beam. For a spatially uniform extended source, the losses and gains exactly cancel each other out, such that, to recover an unbiased flux, you must multiply by an SLCF which is simply the fraction of the PSF which the slit admits, as a function of wavelength. In general, spectra of extended sources with structure will suffer a net gain or loss, depending on the exact distribution of that structure. What's more, due to the effects of the expanding PSF, the physical region probed changes as a function of wavelength. With spectral cubes, this problem can be mitigated by convolving the image formed at each wavelength to a common reference, thus using knowledge of what is outside the extraction aperture. A similar correction can be made using imaging in spanning wavelengths, and an assumed form of the spectrum.
This figure shows calibrator star hr6348, with the pipeline extractions for all 3 orders in red, green & blue, and a simple constant aperture extractions in black. The model spectrum, in purple, is well matched by the pipeline, as expected. The FLUXCON and tuning coefficients were divided into the extracted spectrum to arrive at this flux. Since the entirety of the flux is contained in this straight-sided aperture, shown below, the spectrum is insensitive to changes (shifts, size change) of the aperture:
This overlap mismatch is a direct result of neglecting the ALCF. Taking box statistics on matching portions of SL1 and SL2 and dividing by the relevant FLUXCON values easily demonstrates the mismatch without any special extraction assumptions. But the discrepancy between this straight extraction and the pipeline is not a constant ratio. What we need is some factor which excludes flux at short wavelength but not at longer wavelengths -- the expanding aperture is a good candidate. If I choke the SL1 aperture down at short wavelengths to look like:
I get a spectrum like:
That certainly does it! If this is the true cause of the disparity, it represents a real problem in the pipeline. In the attempt to cut down on unnecessary noise from summing too much sky, the narrowing aperture has introduced a wavelength-dependent aperture-loss function: 40% of the light is being dropped at short wavelengths! The problem, aside from missing flux, is that such a spectrum is highly sensitive to slight changes in aperture centering. Ideally, the aperture would be large enough that the resulting spectrum is insensitive to minor changes in position. Presumably, the aperture loss function introduced by this style of aperture, which ranges from 0% to 40%, has been propagated through and taken out in the flat field to force pipeline extractions of calibrator stars to match their models. Is so, only spectra which are extracted in this lossy way can be accurately calibrated: extended sources must be corrected, or, better, use calibration products which do not impose a point-source extraction aperture.
The best was to assess and correct this order mismatch (aside from removing the dependency on the exact details of the extraction from the flat and fluxcon calibrations) is to measure the ALCF, which depends on wavelength. The best way to do this is using the exact extraction parameters and calibration stars which, in the pipeline, went into the creation of the flat and FLUXCON tables. The star would be extracted with the standard, expanding aperture, and then with a much larger, flux-unbiased aperture (after suitable background subtraction, of course). Not having access to the exact pipeline extraction setup, I approximate the ALCF as follows: extract background-subtracted calibrator stars using a very large aperture (as large as possible without running into negative flux for in-slit pairs, using slit-to-slit pairs for background-subtraction when possible), apply FLUXCON, and then divide the result into the model spectrum. If ALCF=1 everywhere, I should recover the correct, unbiased spectrum. Otherwise I get some function necessary to bring spectra into proper flux units, which varies by order: the ALCF.
Here's what I get for SL:
Each curve is a quadratic fit to an individual star, and the solid curve is the aggregate fit. The two stars are actually systematically offset from one another, which may point to an error in the flux model. The factor of 1.4 is reproduced again, but it's hard to understand why SL1 remains so high throughout the order, since the aperture is expanding. The fact that SL1 has stitching problems with LL may be a sign that something is wrong.
LL looks similar, and the 15% mismatch is detected. A proper measurement would include many more stars, would attempt to smooth out (or explicitly remove) residual fringing, and could probably determine the ALCF within 5% at all wavelengths.
Verifying these results using the pipeline extractor, with nominal expanding aperture, and a larger, open, straight-sided aperture, would be highly advised.
Applying these results to NGC 7331 results in a spectrum which looks like:
Which shows ISOCAM photometry (black), compared to snythetic aperture photometry through the known filter transmission function (red), extracted over the same regions. So we are still about 30\% high. There are two potential sources of this discrepancy: the unimplemented SLCF, or an incorrect beam profile (currently, quite incorrectly, taken to be the half-power width).
The slit loss correction cannot be measured, except with an extended source of known flux intensity. Since calibration standards are almost always point sources (stars, planets, etc.), this correction is necessarily model-dependent. For spatially uniform extended sources, or extended sources with spatial variation for which external information in the form of imaging or spectral mapping can be used to match the PSF at every wavelength, the "slit loss correction function" (SLCF) is simply the fraction of the PSF delivered by the telescope + instrument foreoptics which is accepted by the slit.
An attempt to estimate the SLCF for well-centered point sources, using a model PSF which accounts only for the primary and central obscuration, and the fixed geometric dimensions of the slit was made by Sloan et al. This does not include diffraction terms at the slit, and utilizes a crude estimate of the Spitzer+IRS PSF delivered at the slit, but nonetheless illustrates the principle issues, and can be used as a first-order correction.
At the overlap wavelengths between SL and LL, the slit width goes from 3.7" (SL) to 10" (LL), with a concomitant jump in the fraction of the PSF passed by the slit. Applying this correction, in addition to the ALCF, yields: