# Radiation Lecture 24

<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\hf{\frac12} \def\^{\hat } \def\.{\dot } \usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \subsection*{ Cosmic Masers }

$OH$ and $H_20$ masers can occur in dusty, star-forming regions which are cold enough for these molecules to form. The the dust's black-body radiation in the infrared band creates is absorbed by these molecules and a population inversion is established. When maser emission is caused by via stimulated emission, these clouds can get very bright (brightness temperatures $\sim 10^{14}K$). Temperatures this high cannot be thermal, so we know a maser when we see one. \par Generally, masers are useful for tracing the galaxy's magnetic field (emission lines are Zeeman split), and for following disks of gas and dust around stars in star-forming regions. In order to detect them, we need clouds which are moving uniformly together, and have velocity coherence both $\perp, \|$ to our line-of-sight through a disk of rotation gas. In general, the intensity we observe depends on the path length through the masing cloud, so we like long path lengths with the same velocity.

\subsection*{ How masers Work }

Consider a molecule with two rotational energy levels. Observing a homogeneous slab of this molecule, the intensity we receive is given by the familiar: $$I_\nu=S_\nu(1-e^{-\tau_\nu})$$ where $S_\nu={j_\nu\over\alpha_\nu}$. $j_\nu$ and $\alpha_\nu$ are given by: $$\begin{aligned}j_\nu&=\overbrace{n_2\ato}^{per time}\overbrace{h\nu}^{per E} \overbrace{\phi(\nu)}^{per\ Hz}\overbrace{\inv{4\pi}}^{per steradian}\\ \alpha_\nu&={h\nu\over4\pi}\phi(\nu)[\overbrace{n_1\bot}^{abs}-\overbrace{ n_2\bto}^{stim\atop emis}]\\ \end{aligned}$$ Thus, our source function looks like: $$S_\nu={n_2\ato\phi(\nu)h\nu\inv{4\pi}\over{h\nu\over4\pi}\phi(\nu)[n_1\bot- n2\bto]}$$ Then since $g_1\bot=g_2\bto$ and ${\ato\over\bto}={2h\nu^3\over c^2}$, we have: $$S_\nu={2h\nu^3\over c^2}\inv{{n_1g_2\over g_1n_2}-1}$$ A population is said to be inverted when $n_1g_2<g_1n_2$ (not $n_1<n_2$). $n_1\over g_1$ is an expression for the population per degenerate sub-level in energy level 1. If ${n_1\over g_1}<{n_2\over g_2}$, then $S_\nu<0$, and we have a maser. Expressed in terms of the excitation temperature (${n_2\over n_1} ={g_2\over g_1}e^{-{h\nu\over kT_{ex}}}$), we have: $$S_\nu={2h\nu^3\over c^2}\inv{e^{h\nu\over kT_{ex}}-1}$$ which is less than 0 when $T_{ex}<0$. We can express the optical depth of this slab to maser radiation as: $$\tau_\nu=\alpha_\mu L ={h\nu\over4\pi}\phi(\nu)\bto\left[{n_1g_2\over g_1}-n_2\right] ={h\nu\over4\pi}\phi(\nu)\bto n_2\left[{n_1g_2\over g_1n_2}-1\right]$$ If ${n_2\over g_2}>{n_1\over g_1}$, then $\tau_\nu<0$. Now you might think we're talking nonsense with a negative source function and a negative optical depth, but we're not. Recall that the intensity is: $$I_\nu=S_\nu(1-e^{-\tau})$$ If $\tau\to-\infty$, then $I_\nu\to-S_\nu e^{\tau_\nu}$, so $I_\nu\gg1$. On the other hand, if $\tau<0$ and $|\tau|\ll1$, then $I_\nu\to S_\nu\tau_\nu$, which is the product of two negatives = positive. This should be convincing you that $I_\nu$ is always positive, and therefore, actually manifested.

\subsection*{ Maser Species }

$OH$ mases at around 18 cm, and is found around AGB (asymptotic-giant-branch) stars in star-forming regions and around the galactic nucleus. AGB's are important because they have lots of dust. The two masing transitions are from $1667\to1612MHz$ and $1720\to1665MHz$.\par $H_2O$ mases at $1.35 cm$ in a transition to its ground rotational state. Note that an order-of-magnitude calculation of $\Delta E={\hbar\over 2I}$ for $H_2O$ gives us an estimate of $\sim 1 mm$, which is incorrect. The correct transitional energy is caused by a slight degeneracy in water molecules.\par $SiO$ mases at $3.4 mm$ in its ground vibrational state, and is typically found in star-forming regions and around AGB stars.\par Other molecules mase, and there is even potential for detecting atomic lasers around massive stars ($L\sim10^5 L_\odot$). Hydrogen transitions from $10\to9$ have been observed (at $55\mu m$), and Stielnitski 1996 claims to have observed a population inversion in atomic H.

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