Multum in parvo (Much in little)

An important difference between photometry and spectroscopy is spectral resolution, defined as

where AX is the smallest discernible unit of bandwidth or the smallest wavelength-resolution element of the instrument (photometer or spectrograph) which is being used. In broad-band photometry, R & 10, in intermediate-band, R & 20 - 50, and in narrow-band photometry, R < 500 typically. Spectroscopy implies that the details of the spectrum are important and R > 1000 usually, depending on the application and spectrum range. In highest resolution spectroscopy, R > 105. Such resolution is required for fine spectral analysis and for the highest precision radial velocity studies, e.g., extrasolar planet research.

There are differences in aims, instrumentation, techniques, and emphases of various kinds between photometry and spectroscopy, in addition to the resolution difference. We consider later the special case of spectrophotometry, where the purposes are closer to those of photometry. Spectral details are important for several very different kinds of studies, namely classification, line profile analysis, and radial velocity determination.

Spectral classification, a useful way to estimate temperature and luminosity, usually is done at relatively low spectral resolution. The criteria are the relative strengths of features which may be (possibly blended) absorption lines of various atomic and molecular species. Spectral lines vary in strength with temperature, which partly determines the relative populations of atoms in the lower states of the transition represented by the absorption line. Figure 2.4 shows the spectra of some binaries with high temperatures. Plots of relative flux versus wavelength are in Jacoby et al. (1984), who provide a useful "library of stellar spectra", covering a large range of spectral types. The analysis of spectralline profiles, on the other hand, requires

Fig. 2.4 Spectra of early spectral-type eclipsing binaries. (a) CX Canis Majoris, (b) TU Crucis, (c) AQ Monocerotis, and (d) DQ Velorum, from Fig. 2 in Milone (1986)

the highest resolution to determine abundances, observe Zeeman splitting (through magnetic fields), and model atmosphere parameters. Because this topic is beyond our present scope, we refer the reader to detailed accounts by Aller (1963), Mihalas (1965, 1978), and Gray (1992).

The measurement of a Doppler shift in a spectral line6 permits the determination of a radial velocity for that spectral line and may be carried out at intermediate resolutions. Higher resolutions are desirable for precise radial velocity work. The resolution is determined by the size of a resolution element of the detector, the scale of the spectrum at the detector, the projected slit width, and various optical characteristics of the spectrograph. The resolution defined by equation (2.2.1) depends on the wavelength and the size of the smallest useful spectral element. That size is determined by the dispersion properties of the grating. The grating resolution must at least be matched by the detector in order to be realized. The spectral scale is usually described in terms of the linear dispersion, or, more commonly, the reciprocal linear dispersion, dX w cos i'

ds mf where w is called the "width" of a rule or groove of the grating (more correctly, it is the separation of one groove from another), i' is the diffracted angle at the grating, m is the order of the spectrum, and f is the focal length of the spectro-graph camera which converges the (parallel) light emerging from the (plane) grating onto the detector at the output focal plane. The dispersion is also determined by the grating properties used to obtain the spectrum. The number of grooves, N, and the width of the grooved part of the grating, W, are related to w by the relation w = W/N. (2.2.3)

Equation (2.2.3) may be substituted into equation (2.2.2) in order to eliminate w. Finally, reflection gratings are usually blazed, so that the maximum reflected energy goes into the diffracted order of interest. For normal incidence at the grating, the blaze angle is achieved for wavelength w

m where ab is the blaze angle, the difference between the normals of the grating and the grooved surface.

6 Note that the radial velocities of spectral lines may be affected by blending with circumstellar matter.

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