5 The change in intensity of optically thick UV FeII lines in the presence of weak nonresonant absorption (a Spherical Blob)

Let us consider a gas cloud (or blob) model in the form of a spherical region of radius R, which contains FeII ions, which, all over the volume of the region, are subject to homogeneous excitation within the limits of the Doppler profile and, hence, emit spontaneously within the limits of the Doppler line width. The optical thickness of the region, with reference to the line center of the $1\rightarrow2$ transition, $\tau_{12}^{0}=\kappa_{\rm {sc}}R$ (notation in Johansson & Letokhov 2001b), will vary over the range $10^{-1}\ldots10^{2}$, where $\kappa_{\rm {sc}}=\sigma_{12}N_{1}$ is the coefficient of resonance scattering per unit length, $\tau_{12}^{0}$ is the radiative transition cross section with reference to the line center, and N₁ is the concentration of the FeII ions in the state 1. The coefficient of nonresonant (continuous) absorption per unit length, $\kappa_{\rm {abs}}$, will vary relative to  $\kappa_{\rm {sc}}$ within the limits of $\beta=\kappa_{\rm {abs}}/\kappa_{\rm {sc}}=0\ldots0.1$. Thus, at high $\beta$ and $\kappa_{\rm {sc}}R$ values ( $\kappa_{\rm {sc}}R\beta>1$) nonresonant absorption per unit physical size is high, but where $\kappa_{\rm {sc}}R\beta\ll1$nonresonant absorption can be substantial only for a repeated scattering of radiation, which increases the optical path length.

5.1 Basic radiation transport equations

The equation for steady-state radiation intensity $J_{\omega}$ inside a blob at the frequency $\omega$ with a spherical scattering diagram has the form (Sobolev 1949a)

$$\left ( \mathbf{\Omega}\cdot\nabla+\kappa_{\rm {sc}}\left( \mathb... ...}\right) J_{\omega}\left( \vec{r},\mathbf{\Omega}\right) =~~~~~~~~~~~~~~~~~~~~~$$

$$\frac{1}{4\pi}\int {\rm d}\omega^{\prime}p\left( \omega,\omega^{\... ...\vec{r},\mathbf{\Omega}\right) {\rm d}\mathbf{\Omega}+I_{0}\left( \omega\right)$$ (26)

where the spontaneous emission intensity of FeII at level 2 is described by the term

$$I_{0}\left( \omega\right) =\frac{A}{4\pi c}\kappa_{\rm {sc}}\... ...ac{1}{\int\kappa_{\rm {sc}}\left( \omega\right) {\rm d}\omega}$$ (27)

and the normalization corresponds to the radiation density (in photons/cm³). In this equation, $J_{\omega}\left( \vec{r}% ,\mathbf{\Omega}\right) {\rm d}\omega {\rm d}\vec{r}{\rm d}\mathbf{\Omega}$ is the number of photons contained in the volume d$\vec{r}$, having their frequencies in the interval d$\omega$ and moving within the limits of the solid angle d $\mathbf{\Omega}$. The factor $p\left( \omega,\omega^{\prime}\right)$ is the probability that a photon of frequency $\omega$ will be emitted after the atom has interacted with a photon of frequency $\omega^{\prime}$, $\mathbf{\Omega}$ is the unit vector of the photon motion direction, and $\kappa_{\rm {sc}}\left( \omega\right)$ and $\kappa_{\rm {abs}}$ are the coefficients of resonance scattering and nonresonant absorption per unit length at the frequency $\omega$. In the subsequent discussion we assume that the frequency dependence of the scattering coefficient is due to the Maxwellian atomic velocity distribution and has the form

$$\kappa_{\rm {sc}}\left( \omega\right) =\kappa_{\rm {sc}}\exp\... ...ft( \omega-\omega_{0}\right) }{\Delta \omega_{\rm D}}\sqrt{2}.$$ (28)

Here we use the Doppler width definition, which is adopted in the astrophysical literature (Mihalas 1978). This definition differs only by a factor of $\left( 4\ln2\right) ^{-1/2}$ from the definition used in optical spectroscopy (Svelto 1989).

The transport Eq. (26) differs from the commonly used one in two aspects. Firstly, intending to describe radiation in spherical blobs, we consider a three-dimensional, spherically symmetric problem, whereas it is usually a two-dimensional, plane problem that is subject to consideration. Secondly, we assume that the frequency dependence of the radiation source, $I_{0}\left( \omega\right)$, is defined by the Doppler profile.

Since the optical thickness we consider for the $1\rightarrow2$ UV transition in FeII is not very large ( $\tau_{12}^{0}a<1$), scattering takes place inside the Doppler core (Mihalas 1978). In that case, photon scattering is completely noncoherent, i.e., the scattering probability $p(\omega, \omega^{\prime} )$has the following form:

$$p\left( \omega,\omega^{\prime} \right) =A\kappa_{\rm {sc}}\left( \omega\right).$$ (29)

In this approximation, Eq. (7) adopts the form

$$\left( \mathbf{\Omega}\cdot\nabla+\kappa_{\rm {sc}}\left( \omega\... ...hbf{\Omega}\right) =\frac{A}{4\pi}\kappa_{\rm {sc}}\left( \omega\right)~~~~~~~~$$

$$\times \left( \int {\rm d}\omega^{\prime }\kappa_{\rm {sc}}\left(... ... \vec{r},\mathbf{\Omega}\right) {\rm d}\mathbf{\Omega}+\frac{1}{c}W_{0}\right).$$ (30)

Equation (30) describes the transfer of UV radiation inside the blob. To fully formulate the problem, it is necessary to specify the boundary conditions at the edge of the blob, which assumes no incoming flux into the blob. However, it seems to be more convenient to formulate the problem in another way, where we assume that radiation propagates in an infinite space, but scattering occurs only at r<R. One can easily see that also in this case there is no incoming flux on the surface of the blob, and by virtue of that fact, the solutions of the different approaches to the same problem coincide inside the blob. Thus, we have to solve the equation

$$\left( \mathbf{\Omega}\cdot\nabla+\kappa_{\rm {sc}}\left( \omega\... ...right) J_{\omega}\left( \vec{r},\mathbf{\Omega}\right) =~~~~~~~~~~~~~~~~~~~~~~~$$

$$\left\{ \begin{array}[c]{c}% \frac{A}{4\pi}\kappa_{\rm {sc}}\left... ...mega}+\frac{1}{c}W_{0}\right) ,r<R\\ 0,\quad\quad\quad r>R. \end{array}\right.$$ (31)

If we do not consider the distribution of photons over the propagation directions, we then can exclude from Eq. (31) the dependence on the variable $\mathbf{\Omega}$. This can be achieved by taking the Fourier transform in terms of spatial coordinates and then averaging over propagation directions. As a result, we obtain the following integral equation for the spectral density of photons in space, $\bar{J}(\vec{r},\omega)=\int {\rm d}\mathbf{\Omega}J_{\omega}\left( \vec{r},\mathbf{\Omega}\right)$:

$$\bar{J}(\vec{r},\omega)=\frac{A}{4\pi}\kappa_{\rm {sc}}\left( \om... ...) +\kappa_{\rm {abs}}\right) \left\vert \vec{r}% -\vec{r}^{\prime}\right\vert }$$

$$\times\left\{ \int {\rm d}\omega^{\prime}\kappa_{\rm {sc}}\left( ... ...) \bar{J} (\vec{r}^{\prime},\omega^{\prime})+\frac {1}{c}W_{0}\right\}\cdot~~~~$$ (32)

As already noted, this equation correctly describes the true photon density inside the blob only. In the case of spherical symmetry of the problem, Eq. (32) may be reduced to the form (Sobolev 1949b):

$$r\bar{J}(r,\omega)=\frac{A\kappa_{\rm {sc}}^{{}}\left( \omega\rig... ...t \left( \kappa_{\rm sc}\left( \omega\right) +\kappa_{\rm {abs}}\right) \right)$$

$$\times\left( \int {\rm d}\omega^{\prime} \kappa_{\rm {sc}}\left( ... ...ar{J}(r^{\prime} ,\omega^{\prime} )+\frac{1}{c}W_{0}\right) {\rm d}r^{\prime}~~$$ (33)

where E₁ is the integral exponential function (Abramovitz 1964).

For the total photon density in space, $J^{*} (r)=\int {\rm d}\omega^{\prime} \kappa_{\rm {sc}} \left( \omega^{\prime} \right) \bar{J} (r,\omega^{\prime} )$ , we have from Eq. (33)

$$rJ^{\ast}(r)=\frac{A}{2}\int\limits_{-R}^{R}r^{\prime} \left( J^{\ast }(r^{\prime} )+\frac{1}{c}W_{0}\right) {\rm d}r^{\prime} ~~~~~~~~~~~~~~~$$

$$\times\int\limits_{{}} {\rm d}\omega\kappa_{\rm {sc}}^{2}\left( \... ...left( \kappa_{\rm {sc}}\left( \omega\right) +\kappa_{\rm {abs}}\right) \right).$$ (34)

Introducing the effective photon density (resonance-line source function)

$$Q\left( r\right) =r\left( J^{\ast}(r)+\frac{1}{c}W_{0}\right)$$ (35)

we may represent the final integral equation in the form

$$Q(r)=\int\limits_{-R}^{R}Q\left( r^{\prime} \right) {\rm d}r^... ...ime} K(\left\vert r-r^{\prime} \right\vert )+r\frac{W_{0}}{c}%$$ (36)

where

$$K(\left\vert r-r^{\prime} \right\vert )=\frac{A}{2}\int {\rm d}\o... ...\left( \kappa_{\rm {sc}}\left( \omega\right) +\kappa_{\rm {abs}}\right) \right)$$ (37)

and $-R<r,r^{\prime} <R$.

Given the solution of Eq. (37), we can find the spectral photon density $\bar{J} (r,\omega)$ and the total photon density P(r) at any point inside the blob:

$$r\bar{J}(r,\omega) \frac{A\kappa_{\rm {sc}}^{{}}\left( \omega\right) }{2}$$

$$\times\int\limits_{-R}^{R}E_{1}\left( \left\vert r-r^{\prime} \ri... ...+\kappa_{\rm {abs}}\right) \right) Q\left( r^{\prime} \right) {\rm d}r^{\prime}$$ (38)

$$P(r)=\int {\rm d}\omega\bar{J}(r,\omega)=\frac{A}{2r}\int\limits_{-R}^{R}Q\left( r^{\prime}\right) {\rm d}r^{\prime} ~~~~~~ ~~~~~~~~~~~$$

$$\times \int {\rm d}\omega^{\prime}\kappa_{\rm {sc}}^{{}}\left( \o... ...ppa_{\rm {sc}}\left( \omega^{\prime}\right) +\kappa_{\rm {abs}}\right) \right).$$ (39)

At the edge of the blob $\left( r=R\right)$ we have

$$\bar{J}(R,\omega)=\frac{A\kappa_{\rm {sc}}^{{}}\left( \omega\right) }{2R} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$

$$\times\int\limits_{-R}^{R}E_{1}\left( \left\vert R-r^{\prime} \ri... ...+\kappa_{\rm {abs}}\right) \right) Q\left( r^{\prime} \right) {\rm d}r^{\prime}$$ (42)

$$P(R)=\int {\rm d}\omega\bar{J}(R,\omega)=\frac{A}{2R}\int\limits_{-R}^{R}Q\left( r^{\prime}\right) {\rm d}r^{\prime}~~~~~~~~~~$$

$$\times \int {\rm d}\omega^{\prime}\kappa_{\rm {sc}}^{{}}\left( \o... ...ppa_{\rm {sc}}\left( \omega^{\prime}\right) +\kappa_{\rm {abs}}\right) \right).$$ (41)

5.2 Numerical solution of the integral equation

The integral equation (Eq. (26)) can be solved either numerically or by the approximate analytical methods (Sobolev 1949a; Ivanov 1973). In this work, we solve this integral equation numerically, as we are interested in the radiation characteristics at a wide variety of values of the scattering, absorption, etc. parameters, and analytical solutions are usually applicable only within limited parameter regions. The dimensionless form of Eq. (36) is as follows:

$$\tilde{Q}(x)=\frac{\Lambda}{2\sqrt{2\pi}}\int\limits_{-1}^{1}\tilde{Q}\left( x^{\prime}\right) {\rm d}x^{\prime}~~~~~~~~~~~~~~~~~~~~~~~~~$$

$$\times\int\limits_{-\infty}^{\infty}{\rm d}\xi\alpha ^{2}\left( \... ...\left\vert x-x^{\prime}\right\vert \tilde{\Lambda}\left( \xi\right) \right) +x,$$ (42)

where $-1<x,x^{\prime}<1$. When obtaining Eqs. (36) to (42), we have introduced the following notations:

$$\tilde{Q}(x)=\frac{cQ(r)}{RW_{0}};x=\frac{r}{R};\Lambda=\kappa_{\rm {sc}}^{{}}R; ~~~~~~~~~~~$$

$$\tilde{\Lambda}\left( \xi\right) =\Lambda\left[ \alpha\left( \xi\... ...eta\right] ,\alpha\left( \xi\right) =\exp\left( -\frac {\xi^{2}}{2}\right)\cdot$$ (43)

The integral transport Eq. (42) is often solved by approximating the kernel with a set of exponents (Avrett & Hummer 1965). This approach, however, imposes restrictions upon the form of the right-hand side of Eq. (26) and fails to describe the singular character of the kernel adequately enough.

In the present work, we use the Galerkin method (Kalitkin 1978) and expand the solution Q(x) of Eq. (42) into a trigonometric series:

$$Q(x)=\alpha x+\sum\limits_{m=1}^{\infty}y_{m}\sin\pi {m}x.$$ (44)

The choice of expansion (28) is due to the odd character of Q(x) and the fact that $Q(\pm1)\neq0$ at the boundaries of the region. Substituting expression (28) into (21), we get an infinite set of linear equations for $y_{\rm n}$:

$$y_{m}-\sum\limits_{n}A_{{mn}}y_{n}=\alpha b_{m}-\frac{2\left( -1\right) ^{n}% }{\pi n}\left( 1-\alpha\right)$$ (45)

where

$$A_{{mn}}=\frac{\Lambda}{\sqrt{2\pi}}\int\limits_{-\infty}^{\i... ...rm d}\xi\alpha ^{2}\left( \xi\right) G_{{mn}}\left( \xi\right)$$ (46)

$$b_{m}=\frac{\Lambda}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}{\rm d}\xi\alpha ^{2}\left( \xi\right) H_{m}\left( \xi\right)$$ (47)

and

$$G_{{mn}}\left( \xi\right) =\frac{1}{2}\int\limits_{-1}^{1}{\rm d}... ...t( \left\vert x-x^{\prime}\right\vert \tilde{\Lambda }\left( \xi\right) \right)$$ (48)

$$H_{m}\left( \xi\right) =\frac{1}{2}\int\limits_{-1}^{1}{\rm d... ...^{\prime}\right\vert \tilde{\Lambda}\left( \xi\right) \right).$$ (49)

The integrals G_mn and H_m can be expressed in terms of integral exponential functions of complex variable:

$$\frac{\left( -1\right) ^{m+n}m}{\pi^{2}n\left( m^{2}-n^{2}\right)... ...\right) }\right) -E_{1}\left( 2\tilde{\Lambda}\left( \xi\right) \right) \right.$$

$$+\Re\left( E_{1}\left( 2\tilde{\Lambda}\left( \xi\right) +2\pi ni\right) \right) \bigg\} +(m\Leftrightarrow n)$$ (50)

$$\frac{1}{2\pi^{2}n^{2}}\left\{ \ln\left( 1+\frac{\pi^{2}n^{2}}{\t... ... }\right) -2\pi n\arctan\frac {\tilde{\Lambda}\left( \xi\right) }{\pi n}\right.$$ G_nn =

$$+\pi^{2}n-\frac{\pi^{2}n^{2} }{\tilde{\Lambda}^{2}\left( \xi\righ... ...}^{-2\tilde {\Lambda}\left( \xi\right) }\right) \Bigg\} -\frac{1}{\pi^{2}n^{2}}$$

$$\times\left\{ E_{1}\left( 2\tilde{\Lambda}\left( \xi\right) \righ... ... E_{1}\left( 2\tilde{\Lambda}\left( \xi\right) +2\pi ni\right) \right) \right\}$$ (51)

$$\frac{\left( -1\right) ^{n}}{\left( \pi n\right) ^{3}}\left\{ -\f... ...t) }\right) +\pi n\arctan\frac{\tilde{\Lambda}\left( \xi\right) }{\pi n}\right.$$ H_m =

$$-\left.\frac{\pi^{2}n}{2}+\frac{\pi^{2}n^{2}}{2\tilde{\Lambda}^{2... ...t( \xi\right) }\right\} +\frac{\left( -1\right) ^{n}}{\left( \pi n\right) ^{3}}$$

$$\times\left\{ E_{1}\left( 2\tilde{\Lambda}\left( \xi\right) \righ... ...}\left( 2\tilde{\Lambda}\left( \xi\right) +2\pi ni\right) \right) \right\}\cdot$$ (52)

The analytical calculation of integrals (50)-(52) in terms of coordinates allows one to avoid the difficulties associated with the singular character of the kernel. The integrals in terms of the dimensionless frequency $\xi$ are computed numerically. The infinite system (45) can be reduced to a finite-dimensional one by simply eliminating the higher harmonics. In our numerical computations, we included up to 700 harmonics. In doing so, we determined the parameter $\alpha$ in this system so as to minimize oscillations at the point x=0. The absence of oscillations corresponded to the optimum choice of $\alpha=\tilde{Q}\left( 1\right)$. With the parameter $\alpha$ determined in this way, the Fourier series was summed by the Fejer method (Courant & Hilbert 1962). As a result, we obtained a smooth solution of Eq. (42). Figure 6 shows the solution of Eq. (42) at typical parameter values ( $\Lambda=100,\beta=0.001$).

Figure 6: Source function Q(x) at $\tau _{12} =\kappa _{\rm {sc}} R=100,$ and $\beta =\kappa _{\rm {abs}} /\kappa _{\rm {sc}}=0.001$.

With integral Eq. (42) solved, the spectral radiation density and photon density were found by direct numerical integration in accordance with expressions (38)-(41).

To answer the question about the effect of continuous absorption on the intensity of the spectral line observed, it is sufficient to calculate the dependence of the spectrum-integrated radiation intensity at the boundary of the spherical region on the parameters $\kappa_{\rm {sc}}R$ and $\beta =\kappa _{\rm {abs}} /\kappa _{\rm {sc}}$. We will, however, present a much greater mass of numerical data obtained, for they may also prove to be of interest in understanding the origin and evolution of spectral lines inside the blob.

Figures 7 and 8 present the dependences of the spectral radiation density at the center of the spherical blob on the transition frequency and optical density $\tau_{12} =\kappa_{\rm {sc}} R$ at various values of the nonresonance absorption $\beta =\kappa _{\rm {abs}} /\kappa _{\rm {sc}}$.

Figure 7: Spectral radiation density at the center of the blob, $\bar{J} (r=0,\omega)$, as a function of the optical density in the absence of nonresonance absorption $(\tau_{12}=\kappa_{\rm {sc}} R=0.5\ldots200,$ $\beta =\kappa _{\rm {abs}} /\kappa _{\rm {sc}} =0.000$), normalized to the spectral density value at the center of the blob in the absence of scattering.

Figure 8: Spectral radiation density at the center of nebulae, $\bar{J} (r=0,\omega)$, as a function of the optical density $\tau _{12}$ $(\tau_{12} =\kappa_{\rm {sc}} R=0.5\ldots200)$ at various nonresonance absorption coefficients ( $\beta =\kappa _{\rm {abs}} /\kappa _{\rm {sc}}$), normalized to the spectral density value at the center of the nebula in the absence of scattering: a) $\beta =0.001;$ b) $\beta =0.005;$ c) $\beta =0.01;$ d) $\beta =0.1$.

It can be seen from these figures that when absorption is absent ($\beta =0$) (Fig. 7), the line both broadens monotonically and increases in intensity as the optical density grows higher. When absorption is nonzero (Fig. 8), the line broadens with increasing optical density, but its intensity reaches a maximum and then decreases. As absorption grows higher (Fig. 8), the line intensity already starts diminishing at low optical density values. Such a behavior of both the broadening of the line and the reduction of its intensity with increasing absorption is quite understandable: the line broadening is associated with non-coherent scattering in each scattering event, whereas the reduction of the line intensity at the center of the blob is due to the fact that the photons, which formerly came to the center of the blob from its intermediate layers, can no longer reach the center because of absorption.

Figures 9 and 10 present the dependences of the spectral radiation density at the edge of the blob on the frequency $\omega$ and optical density $\tau _{12}$ at various nonresonance absorption cross section values. It can be seen from these figures that at low absorption values and sufficiently high $\tau _{12}$values there is the formation of the well-known doublet line structure (Figs. 9, 10a), which vanishes as absorption grows higher (Figs. 10b, c, and d). This is explained quite naturally: only those photons can reach the boundary of the blob which come from inside the thin layer next to its surface, wherein the number of scattering events is small and so is, accordingly, the spectral diffusion effect.

Figure 11 illustrates the dependence of the total photon density (normalized to the photon density in the absence of scattering) at the center of the blob on the scattering cross section and blob radius at various absorption coefficients. It can be seen that even here, as in the case of spectral radiation density, the photon density in the absence of absorption increases in a monotonic fashion as a result of the diffusive confinement effect. At nonzero absorption values the photon density first grows slightly higher and then decreases substantially.

A similar situation is also obtained for the photon density on the surface of the blob, which governs the line intensity observed (Fig. 12). In this case, however, even in the absence of absorption, the photon density slowly tends to 1, i.e., the photon density on the surface of the major blob in the absence of scattering, following a slight increase. The most important feature of Fig. 12 is the fact that the photon density rapidly drops with increasing absorption coefficient as a result of the optical path length growing longer in the absorbing medium.