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Figure 4.2. The line absorption profiles due to random thermal motions (Doppler) and finite energy width (Lorentz). Pressure broadening also has a Lorentz profile. The resulting Voigt profile is not a simple sum of the component profiles, although the properties of the Doppler core and damping wings are preserved.

Figure 4.2. The line absorption profiles due to random thermal motions (Doppler) and finite energy width (Lorentz). Pressure broadening also has a Lorentz profile. The resulting Voigt profile is not a simple sum of the component profiles, although the properties of the Doppler core and damping wings are preserved.

sharp; each has a finite energy width AE and lifetime At such that AE At = h. Transitions with shorter lifetimes, such as resonance transitions, have large transition probabilities and large energy widths, leading to broader profiles, which are characterised by the damping constant 7 (7 a transition probability). The width of the Lorentz profile is typically only 10~4A, but the broad damping wings of the Lorentz profile can make an important contribution to the opacity at wavelengths far from the line centre. For M dwarfs, we shall see that the Lorentz profile resulting from van der Waals interactions between neutral atoms and molecules will dominate the line profile except at the line centre.

A second important effect on the line profile, which usually dominates at wavelengths near the centre, or Doppler core, of the line, is a result of random motion of the atoms in the atmosphere leading to a Doppler effect on the wavelength at which the atom can absorb a photon. The wavelength is modified by an amount AAd, such that AAD/A = v/c, where the random velocity, v, depends on the mass of the atom, m, and the temperature, T; v = \J2kT/m. An M dwarf with a temperature of 3,000 K has a Doppler width (for hydrogen) AAD ~ 7kms_1, corresponding to 0.15 A at Ha.

The Lorentz and Doppler profiles are shown schematically in Figure 4.2, which plots the absorption coefficient, k (the probability for photon absorption), as a function of wavelength. The combined (Voigt) profile consists of a Doppler core and Lorentz damping wings, and may be calculated analytically for any transition with known atomic properties, at a given gas temperature (see Bohm-Vitense [B4]; Mihalas [M3]).

In the late 1920s, Minnaert introduced the concept of the line equivalent width to quantitatively measure the total absorption in the line. The equivalent width is the width in A of a perfectly dark line that subtends the same area under the stellar continuum as the actual line (see Figure 4.3). He then showed that a plot of equivalent width (EW) against the number (N) of absorbing atoms had a characteristic shape that was nearly universal. This famous diagram is known as the 'curve of growth', and is shown schematically in Figure 4.4. The curve of growth divides into three parts, which Schutz explained in terms of the line profile. When the line is 'optically thin' (r< 1; see Section 4.3), each atom added to the layer will absorb radiation, hence the EW rises linearly with the number of atoms for weak lines. In the flat part of the curve of growth, the core of the line, where the probability of absorption is high, has saturated; every available photon of that wavelength is already being absorbed, so the addition of more atoms does not change the amount of absorption. Thus the EW barely changes as N increases. Finally, when there are enough atoms present, the probability of being absorbed in the damping wings becomes significant and the EW begins to rise as the square root of N (the square root dependence comes from the explicit expression for the damping profile, see [B4], [M3]). Since the number of atoms in the absorbing layer of the star cannot be varied, the curve-of-growth method is applied by observing the equivalent widths of a distribution of weak and strong lines (corresponding to low and high transition probabilities, respectively), produced by a given atomic species such as Fe I. The curve of growth can then be calibrated in absolute units using solar measurements.

Figure 4.3. The area under the continuum of the absorption line (dotted) is the same as the area within the rectangle of width WA (the equivalent width) and height equal to the continuum flux.

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Figure 4.3. The area under the continuum of the absorption line (dotted) is the same as the area within the rectangle of width WA (the equivalent width) and height equal to the continuum flux.

Figure 4.4. A schematic curve of growth, showing the progression in equivalent width that an absorption line would experience as the number of atoms increases. In practice, one observes many lines of the same species with varying strength to map the curve of growth (it being impossible to vary the number of atoms on a star many parsecs away!).

Figure 4.4. A schematic curve of growth, showing the progression in equivalent width that an absorption line would experience as the number of atoms increases. In practice, one observes many lines of the same species with varying strength to map the curve of growth (it being impossible to vary the number of atoms on a star many parsecs away!).

The curve-of-growth method remained the principal tool for abundance analysis until well into the latter half of the twentieth century.

In the late 1930s a final important puzzle was solved by Wildt, who showed that the principal source of continuous opacity in the Sun and other cool stars (including the M dwarfs) is the ion. It had been known for some time that stars did not radiate as perfect black-bodies, since the Balmer jump was easily observable in hot stars. In particular, colour temperatures determined by matching observed flux gradients to a black-body spectrum were in conflict with the ionisation temperatures found from analysis of spectral lines, with the colour temperatures being generally hotter. A source of continuous opacity was required in the optical wavelength region to bring the colour temperatures into agreement. It was thought by many that this opacity was due to a large number of metal transitions providing a 'line blanketing' effect. The first crude model atmospheres were being developed in the 1930s; Biermann and Unsold had independently found that an atmosphere comprised of 2/3 metals and 1/3 hydrogen could provide the needed opacity. This result was sharply at odds with that of Payne, and later Russell, who had found hydrogen to be thousands of times more abundant than the metals.

The opacity due to the H~ ion, which consists of an electron loosely bound to a neutral hydrogen atom, resolved both the temperature and abundance discrepancies. The second electron can be ionised by photons with E > 0.7 eV, or A < 1.7 Thus any optical or near-infrared photon can ionise an H~ ion, providing a source of continuous opacity which depends on the hydrogen abundance (rather than requiring a large metal abundance).

The period from 1940 to the mid-1960s was marked by increasing sophistication of the atmospheric models for solar and hotter stars, including particularly the first approximate description of convection and its application to hot stars by Bohm and Bohm-Vitense. Little work was done on the M dwarfs, both for lack of observational material and because of the daunting task of describing the atomic and molecular opacity sources. We shall pick up the modelling discussion again in Section 4.5, after first reviewing the methods by which models are produced, and taking a closer look at the opacity sources. The complexity and sheer volume of the modern model results is rather daunting. They should be perused while keeping in mind the cautionary words of Armin Deutsch, who in 1966 wrote in a paper entitled Even Simpler Methods of Abundance Determination from Stellar Spectra [D1]: 'Of course, we must recognise that these methods produce results that are rough and not fully reliable. But this is preferable to the delusion that we can improve our results by adducing an inapplicable model, however sophisticated; or by processing irrelevant data, in quantities however vast.'

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