Astronomical Spectroscopy - Physics - University of Cincinnati

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ratio of the focal lengths of the camera and the collimator, i.e., L cam /L coll . This is a good
approximation if all of the angles in the spectrograph are small, but if the collimator-tocamera
angle is greater than about 15 degrees one should include a factor of r, the “grating
anamorphic demagnification”, where r = cos(t + φ/2)/cos(t − φ/2), where t is the grating
tilt and φ is collimator-camera angle (Schweizer 1979) 5 . Thus the projected size of the slit
on the detector will be WrL cam /L coll , where W is the slit width. This projected size should
be equal to at least 2 pixels, and preferably 3 pixels.
The spectral resolution is characterized as R = λ/∆λ, where ∆λ is the resolution element,
the difference in wavelength between two equally strong (intrinsically skinny) spectral
lines that can be resolved, corresponding to the projected slit width in wavelength units.
Values of a few thousand are considered “moderate resolution”, while values of several tens
of thousands are described as “high resolution”. For comparison, broad-band filter imaging
has a resolution in the single digits, while most interference-filter imaging has an R ∼ 100.
The free spectral range δλ is the difference between two wavelengths λ m and λ (m+1) in
successive orders for a given angle θ:
δλ = λ m − λ m+1 = λ m+1 /m. (7)
For conventional spectrographs that work in low order (m=1-3) the free spectral range is
large, and blocking filters are needed to restrict the observation to a particular order. For
echelle spectrographs, m is large (m ≥ 5) and the free spectral range is small, and the orders
must be cross-dispersed to prevent overlap.
Real spectrographs do differ in some regards from the simple heuristic description here.
For example, the collimator for a conventional long-slit spectrograph must have a diameter
that is larger than would be needed just for the on-axis beam for a point source, because it
has to efficiently accept the light from each end of the slit as well as the center. One would
like the exit pupil of the telescope imaged onto the grating, so that small inconsistencies in
guiding etc will minimize how much the beam “walks about” on the grating. An off-axis
paraboloid can do this rather well, but only if the geometry of the rest of the system matches
it rather well.
5 Note that some observing manuals give the reciprocal of r. As defined here, r ≤ 1.