13.2: Terms and Definitions Relating to Spectral Lines

a. Residual Intensity, Residual Flux, and Equivalent Width

Now that the notion of a continuum has been established, we can use it to provide a normalization of the spectrum so that the resulting line strength is measured in units of the continuum (see Figure 13.1). Some authors call this normalized flux as the residual intensity; however, all that can be observed from stars (other than the sun) is the flux of radiation emitted from all points on the stellar surface in the direction of the observer. Only for the sun can the specific intensity of a particular part of the stellar disk be directly observed. For that reason, we reserve the term residual intensity for the normalized intensity spectra obtainable from the sun, and we use the term residual flux to describe the normalized spectra from stars. Thus, in terms of the emergent intensity and flux of the line and continuum, we have \[\begin{aligned}
f_\nu(\mu) & \equiv \frac{I_\nu(\mu, 0)}{I_c(\mu, 0)} \equiv \text { residual intensity } \\
r_\nu & \equiv \frac{F_\nu(0)}{F_c(0)} \equiv \text { residual flux }
\end{aligned}\label{13.1.1}\]

After the wavelength of the center of the line, probably the most common quantity used to describe a spectral line is the equivalent width. For absorption lines, this is the width of a rectangular shaped "line", completely black at the center, that absorbs the same number of photons as the spectral line of interest (see Figure 13.2).

Figure 13.1 shows the shape of a spectral line as it might be observed in units of the absolute flux in the spectrum (panel a). Panel b depicts the same line after normalization by the continuum flux.

We may formally express this definition by \[W_\lambda \equiv \int_0^{\infty}\left(1-r_\lambda\right) d \lambda\label{13.1.2}\]

It is customary to write integrals of this type as ranging from 0 to 4 largely for convenience. What is meant in reality is that the integral should cover those wavelengths for which \((1-r_\lambda)\) is significantly different from zero. As long as the line is relatively narrow (that is, \(\delta \lambda \ll \lambda_0\)), \[\frac{W_\lambda}{\lambda_0} \approx \frac{W_\nu}{\nu_0}\label{13.1.3}\]

Figure 13.2 shows the equivalent 'black' line profile (double cross-hatched area) appropriate for a specific spectral line (single cross-hatched area). The areas of the two profiles are equal and so the rectangular one can be characterized by its width alone. Such equivalent widths are usually measured in Ångstroms.

b. Selective (True) Absorption and Resonance Scattering

It is useful to divide the processes by which a photon interacts with an atom into two idealized cases. The first is called true, or pure absorption. In this instance, the emission of photons is completely uncorrelated to the previous absorption of photons. In a sense, any photon that is emitted has "lost all memory of what it was". Such a process is called a true absorption process, and it can occur in stellar atmospheres when an atom suffers numerous collisions between the time that the photon is absorbed and reemitted. These collisions can de-excite or further excite the atom, and with the assumption of LTE, these collisions will be randomly distributed so that any subsequently emitted photon is, indeed, totally uncorrelated with the one that was absorbed. Spectral lines formed in this manner are known as pure absorption lines. They exist in stellar spectra since the enhanced opacity provided by the line implies that the line will become optically thick higher in the atmosphere where it is generally cooler. Since it is cooler, then the source function is smaller and the emergent intensity is less than that of the neighboring continuum.

The second type of process is called resonance scattering and it results in the loss of photons in a specialized and indirect manner. Here the photon has a "perfect memory" of its origin. The emitted photon is completely correlated in frequency with the absorbed photon. In Chapter 9, we described such a process as a coherent scattering and lines for which this is true are known as scattering lines. In contrast to the case of pure absorption, a scattering line is formed when the emitted photon is created so soon after the prior absorption that there is no time for the atom to be perturbed by collisions, and the probability of a transition to the prior state is very great. Such cases occur from those states that have very short lifetimes and only one lower level to which the electron can jump. The resonance line transitions meet all these conditions, and hence any resonance line is likely to be a strong scattering line. However, it is possible for any strong line to behave as a scattering line if the probability of returning to the previous state is very large. The same photon that was absorbed is then reemitted, with no net energy exchange with the atom. This is essentially the condition for an interaction to be termed a scattering. The scattering process does not directly result in any loss of energy from the radiation field, but by changing the direction of the photon the process lengthens the stay of the photon in the atmosphere making it subject to destruction by continuum absorption processes.d

Thus, we can divide spectral lines into two types; the pure absorption lines, where the absorbed energy of the photon is fully shared with the gas, and the resonance scattering lines where it is not. In Chapter 9 we showed that nature is really more complicated than this and in reality most lines can be viewed as a mixture of these two extreme states. However, the radiative transfer of these two kinds of lines is quite different, and understanding the behavior of these two limiting cases will provide a comprehensive basis for understanding the behavior of spectral lines in general. The different behavior of these two processes is clearly seen by noting that the energy of a pure absorption process is shared immediately with the gas while that of a resonance scattering processes is not. Scattering is a fully conservative process and therefore cannot, by itself, result in the destruction of photons. However, scattering does change the direction of a photon, thereby increasing the distance that the photon must travel through the atmosphere before escaping into interstellar space. Any process that lengthens the path of a photon through the atmosphere also enhances the probability that the photon will be absorbed by some other process such as continuum absorption. Thus, the redirection of line photons that results from resonant scattering also produces a net loss of these photons relative to those in the neighboring continuum. This, then, is the origin of the resonance scattering lines in stellar spectra.

c. Equation of Radiative Transfer for Spectral Line Radiation

It is customary to denote the part of the mass extinction coefficient that results from pure absorption processes by the letter \(\kappa\), while the part that results from scattering is represented by the Greek letter σ. Those photon interactions that occur as a result of absorptions within the line are subscripted by the letter \(\nu\). Since the continuum processes generally vary quite slowly across a spectral line, we omit the subscript \(ν\) entirely. Thus, \[\begin{aligned}
\kappa &\equiv \text{the pure absorption coefficient for the continuum}\\
\sigma &\equiv \text{the scattering coefficient for the continuum}\\
\sigma_\nu &\equiv \text{the resonance scattering coefficient for the line}\\
\kappa_\nu &\equiv \text{the pure absorption coefficient for the line}\\
\ell_\nu &\equiv \kappa_\nu+\sigma_\nu=\text{the line extinction coefficient}
\end{aligned}\label{13.1.4}\]

Since the origin of the equation of radiative transfer was dealt with extensively in Section 9.2, we provide only a brief derivation here. Basically we balance the energy passing in and out of a differential volume along a specific path through the atmosphere. If we do this for a plane-parallel atmosphere where we keep the line and continuum processes separate, we get \[\begin{aligned}
\frac{\mu}{\rho} \frac{d I(x, \mu)}{d x}= & -\left[\left(\kappa_\nu+\kappa\right)+\left(\sigma+\sigma_\nu\right)\right] I_\nu+j_\nu \\
& +\frac{1}{4 \pi} \int_{4 \pi}\left(\sigma+\sigma_\nu\right) I_\nu\left(\mu^{\prime}, \phi^{\prime}, x\right) d \Omega
\end{aligned}\label{13.1.5}\]

where the first term on the right-hand side represents the energy lost from the beam. The second and third terms on the right-hand side denote the contributions to the beam within the differential volume. The first of these is just due to thermal emission, while the second results from scattering by both line and continuum processes. By making the usual identification between the Planck function and the processes of thermal emission and absorption, the equation of radiative transfer for line radiation becomes \[\mu \frac{d I_\nu\left(\mu, \tau_\nu\right)}{d \tau_\nu}=+I_\nu\left(\mu, \tau_\nu\right)-\frac{\left(\kappa+\kappa_\nu\right) B_\nu}{\kappa+\kappa_\nu+\sigma+\sigma_\nu}-\frac{\left(\sigma+\sigma_\nu\right) J_\nu}{\kappa+\kappa_\nu+\sigma+\sigma_\nu}\label{13.1.6}\]

where the optical depth in the line \(\tau_\nu\) is given by \[d \tau_\nu=-\left(\kappa+\kappa_\nu+\sigma+\sigma_\nu\right) \rho d x\label{13.1.7}\]