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Discussion on Strange bump in solar spectrum taken with home-made spectrograph made me wonder: How do astronomers calibrate the intensity scale of their spectrometers? I mean, how to take in consideration the characteristics of the particular apparatus (diffraction grating, etc.) employed in obtaining a stellar spectrum, in order to get the correct intensities of the various wavelengths recorded with it?

I think you get me right, but just to be more clear: maybe a certain diffraction grating lets "pass" more yellow that green or red light, so the raw spectrum shows a "bump" in the yellow region in relation with their neighbors colors. So, to get the correct spectrum, one should "attenuate" somehow that yellow part.

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    $\begingroup$ Generally I assume it would be with the emission of a well known substance, like hydrogen. That’s only a guess though $\endgroup$
    – Topcode
    Commented Oct 30, 2022 at 3:43
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    $\begingroup$ @Topcode what's the emission "intensity" of hydrogen? $\endgroup$
    – uhoh
    Commented Oct 30, 2022 at 13:25
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    $\begingroup$ @uhoh don’t know, but I assume someone has figured out what a given amount of hydrogen with a given energy will produce. $\endgroup$
    – Topcode
    Commented Oct 30, 2022 at 16:23
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    $\begingroup$ @Topcode what about all the pixels (i.e. most of them!) between the hydrogen lines where there's no emission? How will their intensity response be calibrated? A broadband and spectrally smooth thermal source like a blackbody radiator is much better for this than an emission spectrum from atomic transitions. The line source is more applicable to spectral calibration but there are still nonlinearities that cause trouble between the sparsely distributed peaks of a single electron atom's transitions. $\endgroup$
    – uhoh
    Commented Oct 31, 2022 at 2:50

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This can be done in a number of ways. There is a "theoretical" approach, where the transmission and reflection characteristics of all the components are measured in the lab and put together to get an overall, wavelength-dependent efficiency. This isn't used much apart from in system design.

An on-site method is to measure the spectra of spectrophotometric standard stars, using a very wide slit, so that all the light is captured. These standard stars have measured fluxes as a function of wavelength that can then be used to calibrate the response of the spectrometer. A correction must also be made for atmospheric extinction, that might involve observing standards at the same elevation as the target, or at a range of elevations to actually assess the extinction. The history of how spectrophotometric standards were established is long and complicated.

Standard software packages take the spectrum of the standard star to obtain a response function that can be applied to other spectra. Algorithmically, the observed spectrum is rebinned into intervals that match a table of calibrated standard star fluxes in the same intervals. The standard star calibration data is divided by the rebinned spectrum, taking account of the ratio of exposure times and any estimate of relative extinction, to get the conversion in flux per binned count. Any observed spectrum can be multiplied by (an interpolated and possibly smoothed version) of this to put a flux scale on an observed spectrum. A good description is included in the help file for the old IRAF package sensfunc.

It is rarely the case that spectra have an accurate absolute flux calibration. Usually, stars are observed using slits or fibers to improve spectral resolution or improve wavelength calibration. In such circumstances, a variable amount of light actually enters the spectrograph, associated with turbulence in the atmosphere (not a problem for HST/JWST) and prevents accurate absolute flux calibration.

Sometimes all you want to do is remove the shape of the wavelength response. This can also be done by observing the spectrophotometric standard stars, even if the absolute flux level is uncertain. The absolute flux level could then be established if you have broad-band photometry of the source at a similar wavelength range (providing the source isn't variable).

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    $\begingroup$ I would argue that it's fairly common (not "rarely the case") to have "on-site" spectrophotometric calibration, at least if observing conditions (e.g. "clear" or ideally "photometric") permit. The point is to observe the spectrophotometric standard star with the same instrument and observing setup (e.g., slit width, air mass, etc.) as used for the target observation. This is routinely done as part of the standard calibrations for modern spectrographs on large telescopes. $\endgroup$
    – Peter Erwin
    Commented Oct 31, 2022 at 12:39
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    $\begingroup$ @PeterErwin Absolute flux calibration is rarely attempted because seeing is variable and if it is attempted then nobody would believe it even at the 10% level unless observing with a slit much larger than the seeing. It is also impossibe to do properly for fiber spectroscopy, which comprises of at least half the observing time on spectrographs and the vast majority of spectra taken. $\endgroup$
    – ProfRob
    Commented Oct 31, 2022 at 12:43
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    $\begingroup$ @PeterErwin a more common procedure if you need fluxed spectra but also need to observe through a slit or fiber is to correct for the wavelength response - as I described - and then use the known photometry of the star to put that onto an absolute flux scale (providing the object is not photometrically variable). $\endgroup$
    – ProfRob
    Commented Oct 31, 2022 at 12:45
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    $\begingroup$ @PeterErwin but that still doesn;t work very well for fibers which have differing charcteristics ldepending on how they are bent and are badly affected by atmospheric dispersion. $\endgroup$
    – ProfRob
    Commented Oct 31, 2022 at 12:48
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    $\begingroup$ If you want really accurate absolute calibration, sure. But that doesn't stop, e.g., SDSS from producing spectrophotometrically calibrated spectra from their fiber-based observations, even if they have to warn you about the uncertainties. $\endgroup$
    – Peter Erwin
    Commented Oct 31, 2022 at 12:52

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