Spectral Line with Continuous Absorption

Abstract

Spectral lines are in general embedded in a continuous background created by photoionizations and recombinations, and free-free emission. In cool stars such as the Sun the dominant source of continuous absorption is the negative hydrogen ion H⁻, an ion with a single bound state of low binding energy. In this Chapter we show how a continuous absorption and a continuous emission modify the results presented in Chap. 7 for spectral lines formed with complete frequency redistribution.

Appendix E: Kernels for Spectral Lines with a Continuous Absorption

In this Appendix we calculate the inverse Laplace transforms k _β(ν, β), k ₁(ν, β) and k ₂(ν, β) of the kernel ${\bar K}(\tau ,\beta )$ and of the kernels L ₁(τ, β) and L ₂(τ, β). The kernels are defined by

$$\displaystyle \begin{aligned} {\bar K}(\tau,\beta)\equiv\frac{1}{n(\beta)}\frac{1}{2} \int_{-\infty}^{+\infty} \varphi^2(x) E_1[\tau(\varphi(x)+ \beta)]\,dx, {} \end{aligned} $$

(E.1)

$$\displaystyle \begin{aligned} L_1(\tau,\beta)\equiv\frac{1}{ F(\beta)}\int_0^{\infty}\varphi(x) E_1[-\tau(\varphi(x)+ \beta)]\,dx, {} \end{aligned} $$

(E.2)

$$\displaystyle \begin{aligned} L_2(\tau,\beta)\equiv\frac{1}{ F(\beta)}\int_0^{\infty}\frac{\varphi(x)}{\varphi(x) +\beta} E_2[-\tau(\varphi(x)+ \beta)]\,dx, {} \end{aligned} $$

(E.3)

and their inverse Laplace transforms by

$$\displaystyle \begin{aligned} {\bar K}(\tau,\beta)=\int_0^\infty k_\beta(\nu,\beta)\,\mathrm{e} ^{-\nu|\tau|}\,d\nu = \int_0^\infty g_\beta(\xi,\beta)\,\mathrm{e} ^{-|\tau|/\xi}\,\frac{d\xi}{\xi}, {} \end{aligned} $$

(E.4)

$$\displaystyle \begin{aligned} F(\beta)L_n(\tau,\beta)=\int_0^\infty k_n(\nu,\beta)\,\mathrm{e} ^{-\nu|\tau|}\,d\nu=\int_0^\infty g_n(\xi,\beta)\,\mathrm{e} ^{-|\tau|/\xi}\,\frac{d\xi}{\xi}, {} \end{aligned} $$

(E.5)

with n = 1, 2. We recall that E ₁ and E ₂ are the first and second exponential integrals. The functions n(β) and F(β) are defined in Eqs. (8.8) and (8.11). For simplicity we assume that τ is positive. For β = 0, $\bar K(\tau ,\beta )$ reduces to

$$\displaystyle \begin{aligned} {K}(\tau)=\frac{1}{2} \int_{-\infty}^{+\infty} \varphi^2(x) E_1[\tau\varphi(x)]\,dx. {} \end{aligned} $$

(E.6)

As shown in Sect. 2.2.3, it can be written as

$$\displaystyle \begin{aligned} K(\tau)=\int_0^\infty g(\xi)\exp(-\frac{|\tau|}{\xi})\,\frac{d\xi}{\xi}=\int_0^\infty k(\nu)\mathrm{e} ^{-\tau\nu}\,d\nu, {} \end{aligned} $$

(E.7)

where

$$\displaystyle \begin{aligned} g(\xi)= \int_{y(\xi)}^\infty \varphi^2(u)\,du, {} \end{aligned} $$

(E.8)

and

(E.9)

Here, the notation denotes the inverse function of φ. The functions k(ν) and g(ξ) are plotted in Figs. 5.1 and 5.2 and are related by

$$\displaystyle \begin{aligned} k(\nu)=\frac{1}{\nu}g(\frac{1}{\nu}), \quad \nu\in[0,\infty[. {} \end{aligned} $$

(E.10)

We now follow the method described in (Ivanov 1973, Chapter VII) to obtain an expression of k _β(ν, β) and of the other Laplace transforms, in terms of k(ν) or g(ξ). We start with k _β(ν, β). The idea is to write the derivative of $\bar K(\tau ,\beta )$ with respect to τ in terms of the derivative of K(τ). Using the definition

$$\displaystyle \begin{aligned} E_1(t)\equiv\int_t^\infty\frac{\mathrm{e} ^{-t^{\prime}}}{t^{\prime}}\,dt^{\prime}, {} \end{aligned} $$

(E.11)

and

$$\displaystyle \begin{aligned} \frac{d E_1(at)}{dt}=-\frac{\mathrm{e} ^{-at}}{t},\quad a\ \mbox{constant}, {} \end{aligned} $$

(E.12)

we can write

$$\displaystyle \begin{aligned} \frac{d {\bar K}(\tau,\beta)}{d\tau}=\frac{\mathrm{e} ^{-\beta\tau}}{n(\beta)}\frac{dK(\tau)}{d\tau}=-\frac{1}{n(\beta)} \int_0^\infty \nu\,k(\nu)\,\mathrm{e} ^{-(\nu+\beta)\tau}\,d\nu. {} \end{aligned} $$

(E.13)

An integration over τ leads to

$$\displaystyle \begin{aligned} {\bar K}(\tau,\beta)=\frac{1}{n(\beta)}\int_0^\infty \frac{\nu}{\nu +\beta}k(\nu)\,\mathrm{e} ^{-(\nu+\beta)\tau}\,d\nu. {}\end{aligned} $$

(E.14)

Introducing the variable ν ^′ = ν + β, we obtain

$$\displaystyle \begin{aligned} k_\beta(\nu,\beta)=\frac{1}{n(\beta)}(1-\frac{\beta}{\nu})k(\nu-\beta),\,\, \nu\in[\beta,\infty[, {}\end{aligned} $$

(E.15)

$$\displaystyle \begin{aligned} g_\beta(\xi,\beta)=\frac{1}{n(\beta)}g(\frac{\xi}{1-\beta\xi}), \quad \xi\in [0,1/\beta]. {} \end{aligned} $$

(E.16)

The function k _β(ν, β) is shifted by a quantity β. It is zero for ν < β and for ν > β, it is smaller than k(ν) by a factor (1 − β∕ν)∕n(β). The function g _β(ξ, β) is zero for ξ > 1∕β and has a constant value for ξ ∈ [0, 1∕(ϕ(0) + β)]. The function k _β(ν, β) is drawn in Fig. E.1 for the Doppler profile and different values of β.

Fig. E.1

The function k _β(ν, β) for the Doppler profile and different values of the parameter β, ratio of the continuous absorption coefficient to the line absorption coefficient. The function k _β(ν, β) is zero up to ν = β. It is defined in Eq. (8.17)

The same method can be used to calculate k ₁(ν, β), the inverse Laplace transform of L ₁(τ, β). We introduce

$$\displaystyle \begin{aligned} L_0(\tau)\equiv \frac{1}{2}\int_{-\infty}^{+\infty}\int_0^1\varphi(x)\mathrm{e} ^{-\tau\varphi(x)/\mu}\,\frac{d\mu}{\mu}\,dx= \int_0^{\infty}\varphi(x)E_1[\tau\varphi(x)]\,dx. {} \end{aligned} $$

(E.17)

This leads to the following exponential representation,

$$\displaystyle \begin{aligned} L_0(\tau)=\int_0^\infty k_0(\nu)\mathrm{e}^{-\nu\tau}\,{d\nu} =\int_0^\infty g_0(\xi)\mathrm{e} ^{-\tau/\xi}\frac{d\xi}{\xi}, {} \end{aligned} $$

(E.18)

where g ₀(ξ) is defined almost as in Eq. (E.8), except that φ ²(x) is replaced by φ(x). The derivatives with respect to τ of L ₁(τ, β) and L ₀(τ, β) are related by

$$\displaystyle \begin{aligned} \frac{d L_1(\tau,\beta)}{d\tau}=\frac{\mathrm{e} ^{-\beta\tau}}{F(\beta)} \frac{d L_0(\tau)}{d\tau}. {} \end{aligned} $$

(E.19)

The integration of Eq. (E.19) over τ leads to

$$\displaystyle \begin{aligned} k_1(\nu,\beta)=(1-\frac{\beta}{\nu})k_0(\nu-\beta),\quad \nu\ge\beta, {} \end{aligned} $$

(E.20)

$$\displaystyle \begin{aligned} g_1(\xi,\beta)=g_0(\frac{\xi}{1-\beta\xi}),\quad \nu\le 1/\beta. {} \end{aligned} $$

(E.21)

Equations (E.20) and (E.21) are identical to Eqs. (8.17) and (E.16), except for the factor 1∕n(β). The functions k ₁(ν, β) and g ₁(ξ, β) are zero for ν < β and ξ > 1∕β, respectively, and are related to k ₀(ν) and g ₀(ξ) exactly as k _β(ν, β) and g _β(ξ, β) are related to k(ν) and g(ξ).

The function k ₂(ν, β) is easily derived from

$$\displaystyle \begin{aligned} \frac{L_2(\tau,\beta)}{d\tau}= - L_1(\tau,\beta), \quad \tau\ge 0. {} \end{aligned} $$

(E.22)

One obtains

$$\displaystyle \begin{aligned} k_2(\nu,\beta)=\frac{1}{\nu}k_1(\nu,\beta)=\frac{1}{\nu}(1-\frac{\beta}{\nu}) k_0(\nu-\beta),\quad \nu\ge \beta, {} \end{aligned} $$

(E.23)

$$\displaystyle \begin{aligned} g_2(\xi,\beta)=\xi g_1(\xi,\beta)=\xi g_0(\frac{\xi}{1-\beta\xi}),\quad \xi\in[0,1/\beta]. {} \end{aligned} $$

(E.24)

These functions are zero for ν < β and ξ > 1∕β, respectively.