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# 11.6: Curve of Growth for Voigt Profiles

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

Our next task is to construct curves of growth for Voigt profiles for different values of the ratio of the lorentzian and gaussian HWHMs, $$l/g$$, which is

$\tag{11.6.1}\label{11.6.1}\frac{l}{g}=\frac{\Gamma \lambda_0}{4\pi V_\text{m}\sqrt{\ln 2}}=\frac{\Gamma \lambda_0}{V_\text{m}\pi \sqrt{\ln 65536}}=\frac{\Gamma \lambda_0}{34.841V_\text{m}},$

or, better, for different values of the gaussian ratio $$k_G =\frac{g}{l+g}$$. These should look intermediate in appearance between figures XI.3 and 5.

The expression for the equivalent width in wavelength units is given by Equation 11.3.4:

$\tag{11.3.4}\label{11.3.4}W=2\int_0^\infty \left [ 1-\text{exp}\left \{ -\tau(x)\right \}\right ]\,dx.$

combined with Equation 10.5.20

$\tag{10.5.20}\label{10.5.20}\tau (x) = Cl \tau (0)\int_{-\infty}^\infty \frac{\text{exp}\left [ -(ξ-x)^2\ln 2/g^2\right ]}{ξ^2+l^2}\,dξ.$

That is:

$\tag{11.6.2}\label{11.6.2}W=2\int_0^\infty \left ( 1-\text{exp} \left \{ -Cl\tau(0)\int_{-\infty}^\infty \frac{\text{exp}\left [ -(ξ-x)^2\ln 2/g^2 \right ]}{ξ^2+l^2}dξ\right \}\right )\,dx.$

Here

• $$x = \lambda − \lambda_0$$, $$l$$ is the Lorentzian $$\text{HWHM} = \lambda_0^2\Gamma/(4\pi c)$$ (where $$\Gamma$$ may include a pressure-broadening contribution),
• $$g$$ is the gaussian $$\text{HWHM} = V_\text{m}\lambda_0\sqrt{\ln 2} / c$$ (where $$V_\text{m}$$ may include a microturbulence contribution), and
• $$W$$ is the equivalent width, all of dimension $$L$$.

The symbol $$ξ$$, also of dimension L, is a dummy variable, which disappears after the definite integration. $$\tau(0)$$ is the optical thickness at the line centre. $$C$$ is a dimensionless number given by Equation 10.5.23 and tabulated as a function of gaussian fraction in Chapter 10. The reader is urged to check the dimensions of Equation \ref{11.6.2} carefully. The integration of Equation \ref{11.6.2} is discussed in Appendix A.

Our aim is to calculate the equivalent width as a function of $$\tau(0)$$ for different values of the gaussian fraction $$k_G = g/(l + g)$$. What we find is as follows. Let $$W^\prime = W \sqrt{\ln 2 }/g$$; that is, $$W^\prime$$ is the equivalent width expressed in units of $$g /\sqrt{ \ln 2}$$. For $$\tau(0)$$ less than about 5, where the wings contribute relatively little to the equivalent width, we find that $$W^\prime$$ is almost independent of the gaussian fraction. The difference in behaviour of the curve of growth for different profiles appears only for large values of $$\tau (0)$$, when the wings assume a larger role. However, for any profile which is less gaussian than about $$k_G$$ equal to about 0.9, the behaviour of the curve of growth (for $$\tau(0) > 5$$) mimics that for a lorentzian profile. For that reason I have drawn curves of growth in figure XI.6 only for $$k_G = 0.9, 0.99, 0.999\text{ and }1$$. This corresponds to $$l/g = 0.1111, 0.0101, 0.0010\text{ and }0$$, or to $$\Gamma \lambda_0 /V_m = 1.162, 0.1057, 0.0105\text{ and }0$$ respectively. $$\text{FIGURE XI.6}$$