Week 2 (Oct 10, 12) In-Class Problems

Problem 1: Maxwellians

In thermodynamic equilibrium, the three-dimensional velocity distribution of free electrons in a gas is given by the Maxwellian distribution, which is written as: 

$f\left(v\right)\:dv=\:4\pi\left(\frac{m}{2\pi kT}\right)\:^{\frac{3}{2}}\:e^{\frac{-mv^2}{2kT}}\:v^2dv$

which gives the probability, per unit velocity, of finding the electron with a speed near . 

Let's imagine an HII region with an electron temperature,  $T_e\:=\:10^4K\:$. 

First, find the minimum velocity an electron needs to be moving to ionize a Hydrogen atom (e.g. assuming the gas is in kinetic equilibrium). 

Next, integrate the Maxwellian to compute the fraction of electrons that have sufficient energy (or, are moving at high enough velocities) to keep hydrogen atoms ionized. You will probably want to write a program or use an integral solver - but this IS solvable analytically too with use of integral tables (old school). 

What can you conclude about how the HII region is ionized based on this calculation? 

Now, imagine the hot halo gas surrounding a galaxy with $T_e\:=\:10^6K\:$.

Repeat your calculation, above, to determine whether electron collisions are able to keep this "coronal gas" ionized. 

Problem 2: The Dust Collector

The ISM in the immediate vicinity of the Sun has $n_H\:=\:0.22\:cm^{-3}$. Relative to this local ISM, the sun is moving with $v_W\:=\:26\:\pm1\:km\:s^{-1}$. 

Assume the gas has He/H (by number) = 0.1, and contains dust particles with total mass equal to 0.5% of the mass of the gas. Suppose the dust particles have radius $a\:=\:0.1\mu m\:$and density $\rho\:=\:2\:g\:cm^{-3}\:$, and we wish to design a spacecraft to collect them for further study.