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<latex> \documentclass[11pt]{article} \def\inv#1Template:1 \over \def\ddtTemplate:D \over dt \def\mean#1{\left\langle #1\right\rangle} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\wz{\omega_0} \def\ef{\vec E} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\bfieldTemplate:\vec B \def\apTemplate:A^\prime \def\xp{{x^{\prime}}} \def\yp{{y^{\prime}}} \def\zp{{z^{\prime}}} \def\tp{{t^{\prime}}} \def\upxTemplate:U x^\prime \def\upyTemplate:U y^\prime \def\e#1{\cdot10^{#1}} \def\sigfb{\sigma_{fb}} \def\sigbf{\sigma_{bf}} \def\hf{\frac12} \def\npo{n_{+,1}} \def\noo{n_{0,1}} \def\gpo{g_{+,1}} \def\goo{g_{0,1}} \usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document}

\subsection*{ Applications of Saha}

Recall that we had calculated the time for a proton to radiatively recombine with an $e^-$ as: $$t_{recomb}\sim\inv{n_e\mean{\sigfb v}}$$ Saha tells us that: $${\npo\over\noo}=\underbrace{\left({2\pi m_ekT\over h^2n_e^{2\over3}}\right)^{3\over2}}_{{translational\atop phase\ space}\atop factor} {2\gpo\over\goo}e^{-\chi_1\over kT}$$ where T is the temperature of matter.

\begin{itemize} \item T and z of recombination:\par COBE has measured the temperature of the CMB to be: $$T=3K(1+z)$$ Let's assume that just prior to recombination, photons and matter were in thermal equilibrium, so: $$T_\gamma\eval{recomb}=T_m\eval{recomb}$$ and also, the rate of photoionization of H by the photon gas should equal the rate of radiative recombination. These assumptions allow us to use the Saha Equation. Since ${\rho_0\over m_H}\sim{H^2\Omega_b\over Gm_H}$, $$n=n_0(1+z)^3$$ Therefore, if we set ${n_+\over n_0}=\hf$ (we define this at recombination time), then we can use Saha to solve for T, and then z. After recombination, there is no more Compton scattering to exchange energy between photons and matter, so $T_\gamma\ne T_m$. Since fractional ionization is determined by the ratio of the rates of ionization and recombination (of which radiative recombination is largest): $$\inv{n_e\mean{\sigfb v}}\eval{recomb}\sim{a\over\dot a}\eval{recomb}$$ That is, the timescale of radiative recombination is of order $2\e5yrs$.

\item Kramer's Opacity (used for stellar interiors and accretion disk interiors.\par This is the opacity due to the photoionization of metals. It is useful in a gas at thermal equilibrium which is as sufficiently high temperatures that H,He are nearly ionized, but metals retain their last few $e^-$. This formula is an approximate fitting formula (it doesn't account for the ionization edges of metals). It assumes the cosmic abundance of materials, and it uses Saha. It also uses $v\sim{kT\over h}$: $$\boxed{K_{Kramer}={\rho_{cgs}\over T^{3.5}\cdot3\e{23}{cm^2\over g}}}$$ where $\rho_{cgs}$ is the density of all gas (including metals). Thus: $$\rho K_{bf}=n_{metal}\sigbf$$ where $\rho,K_{bf}$ are for everything, $n_{metal}$ is the number of extinguishers, and $\sigbf$ is the cross-section for extinguishers. \def\nmet{n_{metals}} \begin{aligned}R_{photoion}&\propto\nmet n_{photons}\sigbf\\ &\propto\nmet\underbrace{\left({aT^4\over h\nu}\right)}_{whole\ field\over energy}\sigbf\\ &\propto\nmet{T^4\over T}\sigbf\\ \end{aligned} Note that we used $h\nu=kT$. Similarly, if $\alpha\sim T^{-\hf}$, then: \begin{aligned}R_{rad-recomb}&\propto n_e\nmet+\alpha\\ &\propto n_e\nmet+T^{-\hf}\\ \end{aligned} Setting these two results equal we get: $$\nmet\sigbf\propto n_e\nmet+T^{-3.5}$$ $$K_{bf}\propto{n_e\nmet\over\rho} T^{-3.5}$$ And since $n_e\sim\rho$ and $\nmet\propto\rho Z$: $$K_{bf}\propto\rho T^{-3.5}$$ \end{itemize}

\subsection*{ Instability Strip}

There is a strip on an HR diagram where stars will pulsate. This is the result of the exponential sensitivity of the ionization temperature ($e^{-\chi\over kT}$). The following describes the K mechanism for star pulsation.\par In a normal star, if the outer shell loses pressure, the work done by the collapse of the shell heats the star, increasing density and pressure. $K_{bf}$ affects how fast radiation escapes, and when energy finally does escape, equilibrium is restored and the star maintains its new size. However, in zones of partial ionization, the adiabatic increase in E goes into ionizing material. This liberates $e^-$ without raising the temperature. Since temperature is constant and $\rho$ increases, so $K_{bf}$ increases, and photons can't get out. This continues until He is fully ionized, and then the outer shell puffs up, and the cycle continues.

\subsection*{ Collisional Ionization Cross-Section}

For H: $$\sigma_{coll,bf}\sim\pi a_0^2\left({\chi\over E}\right)$$ for an $e^-$ hitting an H atom. Note that $\chi$ is the ionization energy, and E is the electron kinetic energy. The cross-sections for many atoms are uncalculated and unmeasured. For more information, take a look at (Bely and van Regemorter 1970 ARAA), (Bates 1962 Atomic and Molecular Processes), and (Jefferies 1968 Spectral Line Formation).

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