14.6: Curve of Growth of the Equivalent Width

While we have discussed the most important aspects of the formation of spectral lines, we have said little about the most important contributor to the appearance of the line in the spectrum - the abundance of the atomic species giving rise to the line. Obviously the more absorbers present in the atmosphere, the stronger the associated spectral line will appear. However, the quantitative relationship between the abundance and the equivalent width is not simple and is worthy of some discussion. Although most contemporary determinations of elemental abundances rely on detailed atmospheric modeling with the abundance as a parameter to be determined from comparison with observation, the classical picture of the relation between the equivalent width and the abundance is quite revealing about what to expect from such models. That classical quantitative relationship is known as the curve of growth. Some students have wondered what is growing in the curve of growth. The answer is that the equivalent width increases or "grows" with increasing abundance.

a. Schuster-Schwarzschild Curve of Growth

To create a curve of growth, we must relate the equivalent width to the atomic abundance. This requires some model of the atmosphere in which the atoms reside. For purposes of illustration, we take the simplest model possible. In Chapter 13 we set up the equation of radiative transfer for line radiation [equation \ref{13.1.6}], and we solved it for some special cases. For the Schuster-Schwarzschild atmosphere, this led to a line profile given by equation \ref{13.2.8}: \[r_\nu=\frac{F_\nu}{F_c}=\left(1+\frac{\sqrt{3} \tau_0}{2}\right)^{-1}\label{14.5.1}\]

The definition of \(\tau_0\) allows us to write \[\tau_0=\int_0^{\tau_0} d t_\nu=\int_0^{x_0} \kappa_\nu \rho d x=\int_0^{x_0} n_i S_\nu d x=\left\langle S_\nu\right\rangle \int_0^{x_0} n_i d x=N_i\left\langle S_\nu\right\rangle\label{14.5.2}\]

where \(\mathrm{N_i}\) is the column density of the atom giving rise to the line and \(<\boldsymbol{S}_{\nu}>\) is the line absorption coefficient averaged over depth. Since for this simple model the atmospheric conditions are considered constant throughout the cool gas, we drop the average-value symbols for the remainder of this section. We have already seen [equation \ref{14.3.27}] that for many atomic lines the atomic line absorption coefficient, including the effects of radiation damping, collisional damping, and Doppler broadening, can be written as \[S_\nu=S_0 H(a, u)\label{14.5.3}\]

where \(H(a,u)\) can be either the Voigt or normalized Voigt function depending on what constants have been absorbed into \(\boldsymbol{S}_0\). Thus the line profile for the Schuster-Schwarzschild atmosphere is \[r_\nu=\left(1+\frac{\sqrt{3} \tau_0}{2}\right)^{-1}=\left[1+\frac{\sqrt{3} S_0 H(a, u) N_i}{2}\right]^{-1}\label{14.5.4}\]

To relate this to the equivalent width, equation \ref{14.5.4} must be integrated over the frequencies contained the line so that \[W_\lambda=2 \int_0^{\infty} \frac{\sqrt{3} \tau_0 / 2}{1+\sqrt{3} \tau_0 / 2} d \lambda=2 \int_0^{\infty} \frac{\left[\sqrt{3} S_0 H(a, u) N_i / 2\right] d \lambda}{1+\sqrt{3} S_0 H(a, u) N_i / 2}\label{14.5.5}\]

It is convenient to express the frequency-dependent optical depth in the line in terms of the optical depth at the line center \(\chi_0\) so that \[\tau_0(\nu)=\frac{\chi_0 H(a, u)}{H(a, 0)}\label{14.5.6}\]

From equations \ref{14.5.2}, and \ref{14.5.3} \[\chi_0=S_0 H(a, 0) N_i \approx \frac{\sqrt{\pi} e^2 f_{j k} N_i \lambda_0}{m_e v_0 c}\label{14.5.7}\]

Consider the case where the damping constant is small compared to Doppler broadening so that \(a < 0.2\). Then the Doppler core will dominate the line profile, and we can write the optical depth in the line as \[\tau_0(\Delta \lambda)=\chi_0 e^{-\left(\Delta \lambda / \Delta \lambda_d\right)^2}=\chi_0 e^{-\xi^2}\label{14.5.8}\]

where \(\xi \equiv \Delta \lambda / \Delta \lambda_{\mathrm{d}}\). Substitution into equation \ref{14.5.5} yields \[W_\lambda=\sqrt{3} \Delta \lambda_d \chi_0 \int_0^{\infty}\left(e^{\xi^2}+\frac{\sqrt{3} \chi_0}{2}\right)^{-1} d \xi\label{14.5.9}\]

The integral can be expanded in a series so that \[\begin{aligned}
\int_0^{\infty}\left(e^{\xi^2}+\frac{\sqrt{3} \chi_0}{2}\right)^{-1} d \xi \approx & \int_0^{\infty}\left[e^{-\xi^2}-\frac{\sqrt{3} \chi_0}{2} e^{-2 \xi^2}+\frac{3 \chi_0^2}{4} e^{-3 \xi^2}\right. \\
& \left.+\cdots+(-1)^k\left(\frac{\sqrt{3}}{2}\right)^k \chi_0^k e^{-(k+1) \xi^2}\right] d \xi
\end{aligned}\label{14.5.10}\]

But \[\int_0^{\infty} e^{-k \xi^2} d \xi=\frac{1}{2} \sqrt{\frac{\pi}{k}}\label{14.5.11}\]

Thus, we can write the equivalent width in the line as \[W_\lambda=\frac{\sqrt{3 \pi}}{2} \chi_0 \Delta \lambda_d\left[1-\chi_0 \sqrt{\frac{3}{8}}+\frac{\chi_0^2 \sqrt{3}}{4}-\frac{9 \chi_0^3}{8}+\cdots\right]\label{14.5.12}\]

This, then, represents the first part of the curve of growth, and the equivalent width is indeed directly proportional to \(\chi_0\) and hence the abundance \(\mathrm{N_i}\). This is a commonsense result that simply says that the number of photons removed from the beam is proportional to the number of atoms doing the absorbing, so that section of the curve of growth is known as the linear section.

However, the seeds of difficulties are apparent in the higher-order terms in equation \ref{14.5.12}. As the number of absorbers increases, we would expect that some atoms high in the atmosphere to be "shadowed" by atoms lower in the atmosphere. When all the photons at a given frequency have been absorbed, then the further addition of atoms that can absorb at those frequencies will make no change in the equivalent width. When this happens, the line is said to be saturated. As the optical depth in the line center \(\chi_0\) increases, the term in brackets will fall below unity and the curve of growth will increase more slowly than the linear growth. For \(0 \leq \chi_0 \leq 0.5\), the series may be terminated after the first term. However, for larger values of (\chi_0\), a somewhat different expression of the integral on the left hand side of equation \ref{14.5.10} is in order. If we make the transformation \[\frac{\sqrt{3} \chi_0}{2}=e^b \quad\quad \xi=\zeta^{1 / 2}\label{14.5.13}\]

the equivalent width becomes \[W_\lambda=\Delta \lambda_d \int_0^{\infty} \frac{\zeta^{-1 / 2} d \zeta}{1+e^{\zeta-b}}=2 \Delta \lambda_d \sqrt{b}\left(1-\frac{\pi^2}{24 b^2}-\frac{\pi^4}{384 b^4}-\cdots\right)\label{14.5.14}\]

If \(\chi_0>55\), all but the lead term of the approximation may be ignored. However, in the region where \(0.5<\chi_0<55\), the series given by either equation \ref{14.5.12} or equation \ref{14.5.14} must be used. From the lead term of equation \ref{14.5.14} it is clear that as the Doppler core saturates, the equivalent width grows very slowly as \[\frac{W_\lambda}{\lambda} \propto \sqrt{\ln N_i}\label{14.5.15}\]

This is known as the "flat" part of the curve of growth.

As the abundance increases still further, a significant number of atoms will exist that can absorb in the damping wings of the line and the equivalent width will again begin to increase, but at a rate that will depend on the damping constant appropriate for the line (see Figure 14.7).

Once more we will need a different representation of the optical depth that is appropriate for the damping wings of the line. From the definition of the dimensionless variables of the Voigt function [see equation \ref{14.3.23}] \[d \nu=\Delta \nu_d d u\label{14.5.16}\]

so that we can rewrite equation \ref{14.5.5} with the aid of equation \ref{14.5.6} to obtain \[W_\nu=2 \Delta \nu_d \int_0^{\infty} \frac{d u}{1+2 H(a, 0) /\left[\sqrt{3} \chi_0 H(a, u)\right]}\label{14.5.17}\]

The Voigt function as given in equation \ref{14.3.26} can be approximated for large u as \[H(a, u) \approx \frac{a}{\pi} \int_{-\infty}^{+\infty} \frac{e^{-y^2}}{u^2} d y=\frac{a}{\sqrt{\pi} u^2}\label{14.5.18}\]

For modest values of the damping parameter \(a\), \(H(a,0)\) is near unity so that \[W_\nu \approx\left(\frac{\sqrt{2} \Delta \nu_d}{c}\right)\left(3^{1 / 4} \pi^{3 / 4} \sqrt{\chi_0 a}\right)\label{14.5.19}\]

So for large abundances the curve of growth will again increase in a manner that depends on the square root of the damping constant as well as the square root of the abundance. Except for the separation brought about by the growth of the damping wings of the line, the curve of growth is a single-valued function of \(\mathrm{W}_\lambda / \Delta \lambda_{\mathrm{d}}\) versus the optical depth at the line center \(\chi_0\). Both these parameters are dimensionless, so for this model a single curve satisfies all problems. However, it is worth remembering that the Schuster-Schwarzschild model is correct for scattering lines only, and very few spectral lines that go into abundance calculations are scattering lines. Thus, the classical curve of growth can give only very approximate results even if it is calculated exactly.

Figure 14.7 shows the curve of growth for the classical Schuster-Schwarzschild model atmosphere.

b. More Advanced Models for the Curve of Growth

There are several ways to improve the accuracy of the curve of growth. First, we could use a more accurate solution to the equation of radiative transfer such as the Chandrasekhar discrete ordinate method. The use of the equations of condition on the boundary values [equation \ref{10.2.31}] enables us to obtain a profile of the form \[r_\nu=\frac{\sum_{\alpha=1}^{n-1} L_{c,+\alpha}+Q_c}{\sum_{\alpha=1}^{n-1} L_{+\alpha} e^{-k_\alpha \tau_0(\nu)}+L_{-\alpha} e^{+k_\alpha \tau_0(\nu)}+\tau_0(\nu)+Q}\label{14.5.20}\]

The behavior of the optical depth could then be substituted into equation \ref{14.5.20} and from there into equation \ref{14.5.5}, thereby relating the equivalent width to the optical depth at the line center. However, this would only improve the details of the radiative transfer without improving the model itself. Since we know that the errors of the two-stream (Eddington) approximation are of the order of 12 percent, this is a small improvement indeed for the additional work involved.

A significant improvement could be made by using the Milne-Eddington model atmosphere. Here the line profile is given by equation \ref{13.2.29}, where the frequency dependence is entirely contained in the behavior of \(\mathscr{L}_\nu\), \(\varepsilon_\nu\), and \(\eta_\nu\) with frequency. In addition, the parameters \(a\) and \(b\) which describe the surface temperature and temperature gradient need to be specified. Laborious as the task of constructing these more sophisticated curves of growth is, it was done by Marshall Wrubel^25-27 in a series of papers. Although the additional parameters required by the model are annoying, the improvement in the representation of the star by these models is usually worth the effort. It is probably not worth the trouble to generate more sophisticated classical models than these. Direct modeling by a model atmosphere code is the appropriate approach, for one can remove virtually all the assumptions required for the classical models so that the accuracy is largely determined by the accuracy of the atomic constants characterizing the line.

c. Uses of the Curve of Growth

Determination of Doppler Velocity and Abundance We already indicated that the curve of growth can be used to estimate stellar abundances. However, it is possible (in principle) to learn a great deal more about the conditions in the atmosphere of the star from the curve of growth. Imagine that we have measured equivalent widths for a collection of lines that all arise from the same lower level for which the atomic parameters and damping constants are accurately known. Further suppose that the values for the lines span a reasonable range of the curve of growth. Thus we have empirical values for \(W\left(\lambda_{\mathrm{i}}\right) / \lambda_{\mathrm{i}}\) and \(\sqrt{\pi} e^2 f_i \lambda_i /\left(m_e c\right)\). The second of these two quantities, which we call \(\mathrm{x_i}\) is given by \[\log \chi_i=\log \chi_0+\log \frac{\nu_0}{N}\label{14.5.21}\]

Thus, a plot of \(\log \mathrm{X_i}\) versus. \(\log \left[\mathrm{W}\left(\lambda_{\mathrm{i}}\right) / \lambda_{\mathrm{i}}\right]\) will yield an empirical curve of growth that differs from the theoretical curve given in Figure 14.7 by a shift in both the ordinate and the abscissa. Since the lines all arise from the same lower level, N is the same for all points. The horizontal shift then specifies \(\log(\mathrm{v_0/N})\), while the vertical shift specifies \(\log(\mathrm{c/v_0})\). Thus both the abundance and the Doppler velocity are determined independently. To the extent that the kinetic temperature is known, we know the microturbulent velocity. If the span of the curve of growth is large enough to determine a, an average value of \(\Gamma_\mathrm{c}\) may also be found.

Determination of the Excitation Temperature Consider the situation where, in addition to the information given above, we know the equivalent widths for a number of lines arising from different states of excitation. Further assume that LTE holds so that the populations of those excited states are given by the Boltzmann formula. Then \[\log \chi_i=\log \chi_0+\log \frac{\mathrm{v}_0}{N}-\log \frac{g_i e^{-\epsilon_i /(k T)}}{U(T)}\label{14.5.22}\]

We have already determined \(\mathrm{v_0}\), so we may correct the observed equivalent widths so that the observed values are brought into correspondence with the theoretical ordinate of the curve of growth \(\mathrm{W}_\lambda / \Delta \lambda_{\mathrm{d}}\). The horizontal points will now miss the theoretical curve of growth by an amount \[\aleph_i=\log \chi_i-\left(\log \chi_0+\log \frac{\mathrm{v}_0}{N}\right)+\log \frac{g_i e^{-\epsilon_i /(k T)}}{U(T)}\label{14.5.23}\]

or \[\log \frac{g_i e^{-\epsilon_i /(k T)}}{U(T)}+\text { const }=\aleph_i\label{14.5.24}\]

Since everything about the lines in equation \ref{14.5.24} is known, only the constant and the temperature are unknowns, and they can be determined by least squares.

Important parameters concerning the structure of a stellar atmosphere can be estimated from the classical curve of growth. Not only can the abundance of the elements that make up the atmosphere be measured, but also the turbulent velocity and excitation temperature can be roughly determined. However, to use the classical curve of growth is to make some very restrictive assumptions. The assumption that the parameters determining the lines are independent of optical depth is a poor assumption and is usually the reason that the excitation temperature does not agree with the effective temperature. In addition, the thickness of the atmosphere is probably not the same for all the lines used. Finally, the lines are usually not scattering lines. Nevertheless, the method should be used prior to undertaking any detailed analysis in order to set the ranges for the expected solution. Any sophisticated analysis that produces answers wildly different from those of the curve of growth should be regarded with suspicion.

Finally, sooner or later, we must be wary of the assumption of LTE. In the upper layers of the atmosphere, the density will become low enough that collisions will no longer occur frequently enough to overcome the nonequilibrium effects of the radiation field, and the level populations of the various atomic states will depart from that given by the Saha-Boltzmann ionization-excitation formula. This will particularly affect the strong spectral lines that are formed very high up in the atmosphere. In the next chapter, we survey what is to be done when LTE fails.