Curve of growth

The focus on productivity in growing systems requires a time component in the study of ecosystem responses. The response of productivity to stress must therefore be considered in three dimensions (Fig. 6). This figure illustrates the effects of a stress at any particular time on the classic sigmoid curve of growth (productivity). Positive production will occur only if the stress is less than the ultimate stress and the residual strain (permanent productivity reduction) will be seen as a lowering of the growth curve below the upper boundary (the z dimension in Fig. 6). 

The radial velocities have been computed with the low resolution set-up (more spectral lines, no telluric line), using a cross-correlation technique. When excluding the seven outliers, the peak in centered at 83.0 0.4kms 1 with a dispersion of 1.9 0.2kms 1. Lithium abundance is being determined using Li i 6707.8 A. We used the B — V index to determined the ([3]), and the curve of growth from [7] to derive AT(Li). 

We now consider in somewhat more detail a simplified approach based on the curve of growth . For this, we ignore fine details of the observed line profile and use the equivalent width (EW) defined in Fig. 3.4, WA = f RdX or Wv = f Rdv, where R(AX) or R(Av) is the relative depression below the continuum at some part of the line. The curve of growth is a relationship between the equivalent width of a line and some measure of the effective number of absorbing atoms. Equivalent... 

To develop the curve of growth, we shall roughly approximate the convolution integral in Eq. (3.33) by treating the two broadenings separately, i.e. we take... 

Cosmic abundances of elements and isotopes Table 3.3. Exponential curves of growth... 

Fig. 3.10. Generic exponential curves of growth, normalized so that abscissa and ordinate are the same in the linear regime.

Fig. 3.11. Curves of growth for interstellar lines towards f Oph, observed with the Copernicus satellite. This method of plotting shows the dependence on b along the flat portion and its disappearance as the square-root portion is approached (provided all lines have the same value of yX2). After Spitzer and Jenkins (1975). Copyright by Annual Reviews, Inc.

A form of the curve of growth more relevant to stellar (as opposed to interstellar) absorption lines is derived from work by E. A. Milne, A. S. Eddington, M. Min-naert, D. H. Menzel and A. Unsold. In the Milne-Eddington model of a stellar photosphere, the continuum source function (equated to the Planck function in the LTE approximation) increases linearly with continuum optical depth rA and there is a selective absorption i]K, in the line, where rj(Av), the ratio of selective to continuous absorption, is a constant independent of depth given by... 

This is the Menzel-Minnaert-Unsold interpolation formula (often used assuming Roo = 1). It gives a better approximation to stellar absorption-line profiles (which are definitely not flat-bottomed) than does the exponential formula the shape of the corresponding curve of growth is much the same, but its use leads to a b-parameter that is about 25 per cent higher for the same observational data. Denoting the central value of p by po, the Doppler part of the curve is given by... 

Fig. 3.12. Simple (exponential) curve of growth for low-excitation Fe I lines with wavelengths between 4000 and 8700 A at the centre of the solar disk, with 6>ex = 1.00, b = lkms-1 (assuming Roo = 1), a = 0.02. Equivalent widths are from Moore, Minnaert and Houtgast (1966). gf -values are from furnace measurements by the Oxford group (Blackwell et al. 1986 and references therein).

Correcting gj Mi/H to the total abundance (mostly the first ion) using Saha s equation (Eq. 3.7), one can obtain an estimate of a metallic abundance M/H good to about 0.2 dex, provided that the linear part of the curve of growth is well defined and the value of j0n judiciously chosen. 

Ax = / — x is the ionization potential from the lower state of the line and 0.75 eV is the electron detachment potential of H. [M+/H] = [M/H] + [v], where x is the degree of ionization which changes negligibly while it is close to one, and the electron pressure cancels out. A9 can be identified with A9f obtained by optimally fitting neutral lines with different excitation potentials to one curve of growth (see Fig. 3.13), or deduced from red-infrared colours. As a refinement, a small term [0] should be added to the rhs of Eq. (3.59) to allow for an increase of the weighting function integral towards lower effective temperatures. 

Fig. 3.13. Differential curve of growth for Fe i in p, Cas relative to the Sun, after Catchpole, Pagel and Powell (1967).

The curve-of-growth shift for singly ionized lines (or for neutral lines of elements with large ionization potentials like C, N, O) is given by... 

Fig. 3.14. Curve-of-growth shifts for individual lines of individual neutral elements in /u. Cas, plotted against Ax. The solid line shows the corresponding relation for Fe i, so that the offsets give directly the differential abundance changes [M/Fe]. Adapted from Catchpole, Pagel and Powell (1967).

Fig. 3.15. Curve-of-growth shifts (relative to the Sun) for forbidden [O i], permitted O i and C i and singly ionized metals in the spectrum of Arcturus, adapted from Gasson and Pagel (1966).

Fig. 3.16. Curve-of-growth shifts (corrected by a factor (1 + ax) for Rayleigh scattering opacity) for singly ionized metals in the giant HD 122563 ([Fe/H] = -2.6), against excitation potential. The line joining plotted points for Ti n, Fe n and Cr n defines a locus from which one can judge the presence or absence of additional deficiencies in the heavier elements. Adapted from Pagel (1965).

Fig. 3.42. Depletion below solar abundances of elements in the H I gas towards f Ophiuchi plotted against atomic mass number in (a) and condensation temperature in (b), based in part on the curve of growth shown in Fig. 3.11. Vertical boxes indicate error bars. The dotted curve in the left panel represents an A-1/2 dependence expected for non-equilibrium accretion of gas on to grains in the ISM. The condensation temperature gives a somewhat better, though not perfect, fit, suggesting condensation under near-equilibrium conditions at a variety of temperatures either in stellar ejecta or in some nebular environment. Note the extreme depletion of Ca ( Calcium in the plane stays mainly in the grain ). After Spitzer and Jenkins (1975). Copyright by Annual Reviews, Inc.

The Na I D-lines have wavelengths and oscillator strengths A,i = 5896 A, /i = 1 /3, and X2 = 5889 A, f2 — 2/3. In a certain interstellar cloud, their equivalent widths are measured to be 230 mA and 370 mA respectively, with a maximum error of 30 mA in each case. Assuming a single cloud with a Gaussian velocity dispersion, use the exponential curve of growth to find preferred values of Na I column density and b, and approximate error limits for each of these two parameters. (Doublet ratio method.)... 

Use the solar Fe I curve of growth in Fig. 3.12 to deduce the solar iron abundance, using 9i0n = 0.9 and given that Fe I has an ionization potential of 7.87 eV and that the partition function of Fe II is 42. Compare the result with the one in Table 3.4. 

H. N. Russell analyzes solar spectrum with theoretical transition probabilities and eye estimates of line intensities. Notes predominance of hydrogen (also deduced independently by Bengt Stromgren from stellar structure considerations) and otherwise similarity to meteorites rather than Earth s crust. M. Minnaert et al. introduce quantitative measurements of equivalent width, interpreted by the curve of growth developed by M. Minnaert, D. H. Menzel and A. Unsold. 

This is best done graphically, using Table 3.3 (a = 0.001). The data are just barely compatible with the lines being on the linear part of the curve of growth, in which case the EW of Di is 200 mA and the column density is 2.0 x 1012cm-2 from Eq. (3.38). In this case, the equivalent width of D2 could be at most about 1 Doppler width,... 

Figure 4. Curves of growth rate for Si as function of the Cl2lH2 ratio predicted from equilibrium calculations (Reproduced with permission from reference 7. Copyright 1978 Elsevier.)...

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