16.2: Illuminated Stellar Atmospheres

Although the majority of stars exist in binary systems, and many of them are close binary systems, little attention has been paid to the effects on a stellar atmosphere, by the illuminating radiation of the companion. The general impact of this illumination on the light curve of an eclipsing binary system has usually been lumped under the generic term "reflection effect". This is a complete misnomer, for most of any incident energy is absorbed by the atmosphere and then reradiated. Only that fraction which is scattered could, in any sense, be considered to be reflected and then only that fraction that is scattered in the direction of the observer.

a. Effects of Incident Radiation on the Atmospheric Structure

The presence of incident radiation introduces an entirely new set of parameters into the problem of modeling a stellar atmosphere. Basically the incident radiation will be absorbed in the atmosphere, causing local heating. The amount of this heating will depend on the intensity, direction, and frequency of the incident radiation as well as the fraction of the visible sky covered by the source. This local heating can totally alter the atmospheric structure causing the appearance of the illuminated star to change greatly. However, if radiative equilibrium is to apply, all the radiation that falls on the star must eventually emerge. The interaction of the incident radiation with the atmosphere will cause its spectral energy distribution to be significantly altered. In any event, the total emergent flux must simply be the sum of the incident flux and that which would be present in the unilluminated star. In addition to this being the common sense result, it is a consequence of the linearity of the equation of radiative transfer.

Before we can even formulate an equation of radiative transfer appropriate for an illuminated atmosphere, we must know the angular distribution of the incident energy. For simplicity, let us assume that the incident radiation comes from some known direction (\(\theta_0\), \(\phi_0\)) in the form of a plane wave (see Figure 16.1).

Figure 16.1 shows radiation incident on a plane-parallel atmosphere. The radiation can be specified as coming from a specific direction indicated by the coordinates \(\theta_0\), \(\phi_0\).

The specific intensity incident on the atmosphere can then be characterized by \[I_{i, 0}(\mu, \phi)=4 \pi F_i \delta\left(\phi-\phi_0\right) \delta\left(\mu-\mu_0\right)\label{16.1.1}\]

where \(F_\mathrm{i}\) is the incident flux and the \(\delta(\mathrm{x_i})\) is the Dirac delta function indicating the direction of the beam. Along an optical path \(\tau_\nu/\mu_0\), this radiation will be attenuated by \(e^{-\tau_\nu / \mu_0}\). Chandrasekhar¹ makes a distinction between the direct transmitted intensity and that part of the radiation field (the diffuse field) that has been scattered at least once. The contribution to the diffuse radiation field from the attenuated incident radiation field is \[S_i=\left(1-\epsilon_\nu\right)\left(\frac{F_i}{4}\right) e^{-\tau_\nu / \mu_0}\label{16.1.2}\]

where \(\varepsilon_\nu\) is the fraction of the intensity absorbed in a differential length of the optical path and has the same meaning as in equation \ref{10.1.8}. This can be regarded as the scattering contribution to the diffuse source function from the attenuated incident radiation field. Thus the equation of radiative transfer for the illuminated atmosphere is \[\mu \frac{d I_\nu}{d \tau_\nu}=I_\nu-\epsilon_\nu B_\nu-\left(1-\epsilon_\nu\right) J_\nu-\left(1-\epsilon_\nu\right)\left(\frac{F_i}{4}\right) e^{-\tau_\nu / \mu_0}\label{16.1.3}\]

By applying the classical solution to the equation of transfer, we can generate an integral equation for the diffuse field source function, as we did in (Section 10.1) so that \[S_\nu\left(\tau_\nu\right)=\epsilon_\nu B_\nu\left[T\left(\tau_\nu\right)\right]+\frac{1}{2}\left(1-\epsilon_\nu\right) \int_0^{\infty} S_\nu(t) E_1\left|t-t_\nu\right| d t-\left(1-\epsilon_\nu\right)\left(\frac{F_i}{4}\right) e^{-\tau_\nu / \mu_0}\label{16.1.4}\]

This is a Schwarzschild-Milne integral equation of the same type as we considered in Chapter 10 it and can be solved in the same manner. The only difference is that the term that makes the equation inhomogeneous has been augmented by the last term on the right in equation \ref{16.1.4}. Thus the presence of incident radiation will even make a pure scattering atmosphere inhomogeneous and subject to a unique solution.

Modification of the Avrett-Krook Iteration Scheme for Incident Radiation. Buerger² has

solved this equation, using the ATLAS

atmosphere program for a variety of cases. He found that the standard Avrett-Krook iteration scheme described in Section 11.4 would no longer lead to a converged atmosphere. This is not surprising since the Eddington approximation was used to obtain the specific perturbation formulas (see equations \ref{12.4.17}, \ref{12.4.25}, and \ref{12.4.29}, and the approximation J(0) = ½F(0) will no longer be correct. Buerger therefore adopted a modification of the Avrett-Krook scheme due to Karp³ which replaces the ad hoc assumption of equations \ref{12.4.14} and \ref{12.4.20} with \[J_\nu^{(1) \prime}(t)=a F_\nu^{(1)^{\prime}}(t)\label{16.1.5}\]

which implies that \[J_\nu^{(1)}(t)=a F_\nu^{(1)}(t)\label{16.1.6}\]

He continues by making the plausible assumption that perturbed flux variations have the same frequency dependence as the zeroth-order flux variations so that \[F_\nu^{(1)}=\frac{F^{(1)} F_\nu^{(0)}}{F^{(0)}}\label{16.1.7}\]

Following the same procedure as Karp, Buerger² finds the perturbation formulas for \(\tau^{(1)}\) and \(\mathrm{T}^{(1)}\) analogous to equations \ref{12.4.21} and \ref{12.4.29}, respectively, to be \[\begin{aligned}
& \tau^{(1)^{\prime}}+\tau^{(1)} \int_0^{\infty} \frac{k_\nu^{(0)^{\prime}}}{k_\nu^{(0)}} \frac{F_\nu^{(0)}}{F^{(0)}} d \nu=1-\frac{\mathbf{F}}{F^{(0)}}-\frac{a F^{(0)}}{\int_0^{\infty} k_\nu^{(0)} F_\nu^{(0)} d \nu} \\
& T^{(1)}(t) \int_0^{\infty} k_\nu^{(0)}(\tau) \epsilon_\nu^{(0)}(t) \dot{B}_\nu^{(0)} d \nu=-a\left[1-\frac{\mathbf{F}}{F^{(0)}}\right] \int_0^{\infty} k_\nu^{(0)}(t) \epsilon_\nu^{(0)}(t) F_\nu^{(0)} d \nu \\
& +\int_0^{\infty} k_\nu^{(0)}(t) \epsilon_\nu^{(0)}(t)\left[J_\nu^{(0)}(t)-B_\nu^{(0)}(t)+\frac{1}{4} F_\nu^{(i)} e^{-\tau_\nu / \mu_0}\right] d \nu \\
& +\tau^{(1)} \int_0^{\infty}\left[k_\nu^{(0)}(t) \epsilon_\nu^{(0)}(t)\right]^{\prime}\left[J_\nu^{(0)}(t)-B_\nu^{(0)}(t)\right]+\left\{k_\nu^{(0)}(t)\left[1-\epsilon_\nu^{(0)}(t)\right]\right\}^{\prime} \\
& \times \frac{1}{4} F_\nu^{(i)} e^{-\tau_\nu / \mu_0} d \nu+\tau^{(1)^{\prime}} \int_0^{\infty} k_\nu^{(0)}(t) \epsilon_\nu^{(0)}(t)\left[J_\nu^{(0)}(t)-B_\nu^{(0)}(t)+\frac{1}{4} F_\nu^{(i)} e^{-\tau_\nu / \mu_0}\right] d \nu
\end{aligned}\label{16.1.8}\]

where \[\mathbf{F}=\frac{\sigma T_e^4}{\pi}+\mu_0 \int_0^{\infty} F_\nu^{(i)} e^{-\tau_\nu / \mu_0} d \nu\label{16.1.9}\]

While these equations do provide a convergent iteration algorithm, the rate of convergence can be quite slow if the amount of incident radiation is large and differs greatly from that which the illuminated star would have in the absence of incident radiation.

If event that the incident flux has an energy distribution significantly different from that of the illuminated star and large compared to the stellar flux, the Eddington approximation will fail rather badly at some frequencies and in that part of the atmosphere where the majority of the incident flux is absorbed. Under these conditions, the Avrett-Krook iteration scheme will fail badly. Steps must be taken to directly estimate those parameters obtained from the Eddington approximation. In some instances it is possible to use results from a previous iteration.

Initial Temperature Distribution As with any iteration process, the closer the initial guess, the faster a correct answer will be found. In Chapter 12, we suggested that the model-making process could be initiated with the unilluminated gray atmosphere temperature distribution in the absence of anything better. In the case of incident radiation, such a choice will yield an initial temperature distribution that departs rather badly from the correct one. It is tempting to suggest that a suitable first guess could be obtained by merely scaling up the gray atmosphere temperature distribution so as to include the incident flux in the definition of the effective temperature. Thus, \[\frac{\sigma T_e^4}{\pi}=\frac{\sigma T_e^4(0)}{\pi}+F_i\label{16.1.10}\]

where \(T_e(0)\) is the effective temperature and the temperature distribution is \[T^4(\tau)=\frac{3}{4} T_e^4[\tau+q(\tau)]\label{16.1.11}\]

Here \(q(\tau)\) is the Hopf function defined in Chapter 10 [equation \ref{10.2.33}]. For the Eddington approximation, \[q(\tau)=Q_0=\frac{2}{3}\label{16.1.12}\]

However, there are two effects that make this a poor choice. First, if the sky of the illuminated star is partially filled with the illuminating star, then much of the incident radiation will enter at a grazing angle and preferentially heat the upper atmosphere. This will raise the surface temperature over what would normally be expected from a gray atmosphere and will flatten the temperature gradient. The more radiation that enters from grazing angles, the greater this effect will be. This effect may also be understood by considering how radiation can ultimately escape from the atmosphere. If the source of radiation is a close companion filling much of the sky, then a smaller and smaller fraction of the sky is "black" representing directions of possible escape. As the possible escape routes for photons diminish, the atmosphere will heat up since the photons have nowhere to go. Anderson and Shu⁴ suggest that this may be compensated for by \[Q_0 \approx \frac{2}{3}\left[\frac{1+\Omega_0 /(2 \pi)}{1-\Omega_0 /(2 \pi)}\right]\label{16.1.13}\]

where Ω₀ is the solid angle subtended by the source of radiation in the sky. As Ω₀ → 0, we recover the Eddington approximation's value for Q₀. As Ω0 increases, so does the value of Q₀ implying that the surface temperature will rise relative to the effective temperature. In the limit, the atmosphere will approach being isothermal.

The second aspect of incident radiation that can markedly affect the temperature distribution is the frequency distribution. If the source of the incident radiation contains a large quantity of ionizing radiation and is incident on a relatively cool star, then almost all the radiation will be absorbed by the neutral hydrogen of the upper reaches of the atmosphere, raising the local temperature significantly. Similarly, if the incident radiation is highly penetrating radiation, such as x-rays, the heating may occur much lower in the atmosphere. Since this energy must be reradiated, this will result in a general elevation of the entire atmospheric temperature distribution. This dependence of the atmosphere's temperature gradient on the frequency distribution makes predicting the outcome extremely difficult since the process is highly nonlinear. That is, the local heating may change the opacity, which will in turn change the heating.

In general, incident radiation will preferentially heat the surface layers and a modification like that suggested by Anderson and Shu⁴ should be used for the initial guess. An additional approach that some investigators have found useful is to slowly increase the amount of incident radiation during the iteration processes from an initially low value until the desired level is reached. The rate at which this can be done depends on the spectral energy distribution of the incident radiation and always tends to lengthen the convergence process.

b. Effects of Incident Radiation on the Stellar Spectra

If the amount of this illuminating radiation is large, then the heating of the upper atmosphere will cause a decrease in the temperature gradient. As we have seen in Chapter 13, the strength of a pure absorption line will depend strongly on the temperature gradient in the atmosphere. Thus, aside from changes in the environment of the radiation field caused by the external heating, we should expect spectral lines to appear weaker simply because of the change in the source function gradient. The change in line strength may cause an attendant change in the spectral type. Although such changes should be phase dependent in a binary system, as different aspects of the illuminated surface are presented to the observer, such changes can be confused with the contribution to the light by the companion.

Although Milne⁵ first laid the foundations for estimating these effects in 1930, very little has been done in the intervening half-century to quantify them. The atmospheres investigated by Buerger² did indeed show the general decrease in the equivalent widths of the lines investigated due to heating of the higher levels of the atmosphere. However, his investigation was restricted to point-source illumination which tends to minimize the upper-level heating of the atmosphere. A more realistic modeling of the atmosphere of an early-type binary system by Kuzma⁶ showed that the effects can be quite pronounced. Local changes of the atmospheric structure can produce spectral changes of several spectral subtypes with the integrated effect being as large as several tenths of a subtype with phase.

The effects of the incident radiation on the spectra become deeply intertwined with the structure. The inclusion of metallic line blanketing brought about a reversal of the effects of non-LTE on the hydrogen lines in that, in the upper levels of the atmosphere, the departure coefficients for the first two levels of hydrogen (\(b_1\) and \(b_2\) went from about 0.6 to 1.5 when the effects of line blanketing were included. Similar anomalies occurred for the departure coefficients for the higher levels as well. This fact largely serves to show the interconnectedness of the phenomena which earlier we were able to consider separately. This results primarily from the dominance of the interactions of the gas with the radiation field over interactions with itself. That is, the radiation field and its departures from statistical equilibrium begin to determine completely the statistical equilibrium of the gas. The illuminating radiation has a spectral energy distribution significantly different from that of the illuminating star, the departures from LTE may be very large indeed.