Week 3 (Oct 17,19) In-Class Problems

Due Oct 21, 2022 by 11:59pm
Points 10
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Problem 1: Hydrogen and Helium Strömgren Radii

Assume there is an O7V star inside a gas cloud of temperature $LaTeX: T\:=\:10^{4\:}K$ $T\:=\:10^{4\:}K$ with Hydrogen density of $LaTeX: n_H\:=\:5000\:cm^{-3}.\:$ $n_H\:=\:5000\:cm^{-3}.\:$

Assume the recombination case B coefficient for Hydrogen is $LaTeX: \alpha_{B,\:H_{_{ }}}=\:4\:\times10^{-13}\:cm^3s^{-1}\:$ $\alpha_{B,\:H_{_{ }}}=\:4\:\times10^{-13}\:cm^3s^{-1}\:$ at $LaTeX: T\:=\:10^{4\:}K$ $T\:=\:10^{4\:}K$ .

Assume the corresponding recombination coefficient for Helium is 2 times larger than that of Hydrogen, and that the Helium to Hydrogen ionizing ratio is 0.135.

Assume in this cloud that $LaTeX: \frac{n_{He}}{n_H}\:=\:0.1$ $\frac{n_{He}}{n_H}\:=\:0.1$ .

a) What relevant quantity do you need to know about the O7V star to understand the ionization equilibrium state of the HII region? (check out the lecture notes on Stromgren Spheres if you need to!).

b) What is the Hydrogen Strömgren Radius for this gas cloud in pc?

c) What is the Helium Strömgren Radius for this gas cloud in pc?

d) Estimate the amount of time in years it takes for these two spherical gas clouds to reach a steady-state solution.

e) Which one forms first? Why?

Problem 2: OII Spectroscopic Terms...

Use the following short PDF to remind yourself of electron orbitals and L-S Coupling:

SpectroscopicNotationCheatSheet.pdf Download SpectroscopicNotationCheatSheet.pdf

a) Determine the spectroscopic terms for OII (singly ionized oxygen) in the form: $LaTeX: ^{\left(2S+1\right)}L^p_J$ $^{\left(2S+1\right)}L^p_J$

Recall there are 20 unique combinations for the 3 electrons in the 2p shell.

A good first step would be to write down the combinations in the notation $LaTeX: \left(m_l,\:m_s\right)$ $\left(m_l,\:m_s\right)$ for each of the three electrons and compute their sums $LaTeX: \left(M_l,\:M_s\right)$ $\left(M_l,\:M_s\right)$ . The best way to organize this is in the form of a table.