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Summation over all atoms (that is, over all Doppler velocities) is a summation of many such profiles centered on the various frequencies v' of the thermal distribution. This smears the two distributions together.

Formally, the damping function k2'(v) is weighted with the amplitude k1(v') of the thermal distribution at frequency v' and then integrated over all v'. The net profile f (v) is thus m f (v) a

(Combined profile) (11.32)

x exp

2a 2

which is called the convolution of the two functions. When the central part of the damping profile is narrow, the convolution simply maps out the central portion of the thermal curve, while the damping curve is mapped at off-center positions large compared to the thermal width. If the damping width is down by a factor of 100 or 1000, as it sometimes is, the wings will be at such a low level they will not affect the line shape.

Turbulent motions and collisional broadening Lines can also be Doppler broadened by rapid motions of clouds of emitting or absorbing atoms; this is known as bulk turbulent motion. The velocities of the several clouds are not necessarily thermally distributed so the shape of the line may differ from that quoted above. Bulk motion will significantly affect the line shapes when the bulk velocities approach or exceed the thermal velocities.

Broadened lines can also arise from collisions between the gas atoms which will de-excite the atoms before they would decay naturally. The atom is in its initial state for even a shorter time than in its unperturbed state, and this further broadens the (quantum-mechanical) damping profile according to (29). The damping profile is thus a measure of the number of collisions and hence of the density of the gas. (The number of collisions varies as particle density squared n2.)

In turn, the density n is related to the pressure in the region where the line is being formed, as P = nkT where k is the Boltzmann constant. This relation follows directly from the ideal gas law PV = ¡lRT where ¡l is the number of moles in a sample of volume V and where R (= N0k) is the universal gas constant (N0 is Avogrado's number). Thus, for gases of about the same temperature, measurement of collisional broadening yields the relative pressures. This turns out to be useful for stellar classification when comparing different stars of the same photospheric temperatures. Thus

Collisional broadening ^ Pressure in photosphere (T ~ constant) (11.33) Collisional broadening is sometimes called pressure broadening.

Saturation and the curve of growth

The curve of growth (Fig. 16) of a spectral line describes the measured strength (equivalent width, EW) as a function of the number N of absorbing (or emitting) atoms along the line of sight (atoms/m2). Consider the absorption case (Figs. 14a,c,e). When the strength of the line is weak (Fig. 14a), the atoms along the line of sight are so few they block only a small portion of the beam. An increase in the number of atoms removes a proportional amount of radiation so that the curve grows linearly with N. As more atoms are added, eventually there are sufficient numbers of low velocity atoms to completely absorb the photons near the central frequency. At this point, the specified line is saturated, and the addition of more atoms has only a small effect; the EW is increases only very slowly with N. Figure 11.16. The curve of growth or the variation of the equivalent width (EW) with the column density of absorbing atoms N(atoms/m2), for the absorption case. The flattening at intermediate N is due to the saturation of the central part of the absorption line (Fig. 14e). On this log-log plot, the linear dependence for weak lines and the (approximate) square root dependence for intense lines appear as slopes of 1 and 1/2 respectively.

Figure 11.16. The curve of growth or the variation of the equivalent width (EW) with the column density of absorbing atoms N(atoms/m2), for the absorption case. The flattening at intermediate N is due to the saturation of the central part of the absorption line (Fig. 14e). On this log-log plot, the linear dependence for weak lines and the (approximate) square root dependence for intense lines appear as slopes of 1 and 1/2 respectively.

As more and more atoms are added, the weak wings due to collisional broadening finally become important (Fig. 14e). At this stage the EW begins to increase again, approximately as N 1/2. This is a slower rate of increase in EW than for the unsat-urated state. The N1/2 dependence is not derived here. The growth of an emission line (Fig. 14 b,d,f) is largely similar to the absorption case.

The shape of the curve of growth will vary depending on the relative widths of the thermal and damping terms. Knowledge of the curve of growth enables one to determine column densities of different elements in stellar atmospheres and hence the chemical compositions. Telescopes Mastery

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