# Galaxies Lecture 21

Today we’ll be talking about the Jean’s instability, ram-pressure stripping, and tidal forces (all topics pertaining to interstellar gas).

### Jean’s Instability

{\displaystyle {\begin{aligned}{d \over dt}\rho &=0\\\rho {d \over dt}{v}&=-\nabla p-\rho \nabla \phi \\\end{aligned}}\,\!}

We will then perturb ${\displaystyle \rho ,v,p}$, and ${\displaystyle \phi }$, keeping only first-order perturbations, and write down the differences between this perturbation and the steady-state solution:

{\displaystyle {\begin{aligned}{\frac {\partial }{\partial t}}{\rho _{1}}+\rho _{0}{\vec {\nabla }}\cdot {\vec {v}}_{1}+v_{0}{\vec {\nabla }}\rho _{1}&=0\\{\frac {\partial }{\partial t}}{v_{1}}=({\vec {v}}_{0}\cdot {\vec {\nabla }}){\vec {v}}_{1}+({\vec {v}}_{1}\cdot {\vec {\nabla }})v_{0}&={\rho _{1} \over \rho _{0}^{2}}\nabla p_{0}-{1 \over \rho _{0}}\nabla p_{1}-\nabla \phi _{1}\\\end{aligned}}\,\!}

We will then make the barotropic assumption that ${\displaystyle p=p(\rho )}$ only. Then we can say ${\displaystyle p_{1}=c^{2}\rho _{1}}$, where ${\displaystyle c}$ is the (isothermal) sound speed of the gas. Using this in our previous equations, we have:

${\displaystyle {\frac {\partial }{\partial t}}{v_{1}}=({\vec {v}}_{0}\cdot {\vec {\nabla }}){\vec {v}}_{1}+({\vec {v}}_{1}\cdot \nabla ){\vec {v}}_{0}=-c^{2}\nabla \left({\rho _{1} \over \rho _{0}}\right)-\nabla \phi _{1}\,\!}$

This equation, along with ${\displaystyle p_{1}=c^{2}\rho _{1}}$ and the two equations above, we have 4 equations. Assuming that ${\displaystyle {\vec {v}}_{0}=0}$, ${\displaystyle \rho _{0}=const}$, and ${\displaystyle \phi _{0}=0}$, our equations become:

{\displaystyle {\begin{aligned}{\frac {\partial }{\partial t}}{\rho _{1}}+\rho _{0}{\vec {\nabla }}\cdot {\vec {v}}_{1}&=0\\{\frac {\partial }{\partial t}}{v_{1}}&=-c^{2}\nabla {\rho _{1} \over \rho _{0}}-\nabla \phi _{1}\\\nabla ^{2}\phi _{1}&=4\pi G\rho _{1}\\\end{aligned}}\,\!}

Taking ${\displaystyle {\frac {\partial }{\partial t}}{}}$ of the first equation and combining it with ${\displaystyle \nabla }$ of the second equation, we have:

${\displaystyle \underbrace {{\partial ^{2}\rho _{1} \over \partial t^{2}}-c^{2}\nabla ^{2}\phi _{1}} _{wave\ equation}-4\pi G\rho _{1}\rho _{0}=0\,\!}$

For the wave equation portion, we’ll assume a solution ${\displaystyle \rho =Ce^{i({\vec {k}}\cdot {\vec {x}}-\omega t)}}$, giving us a dispersion relation:

${\displaystyle {\omega ^{2}-c^{2}k^{2}-4\pi G\rho _{0}}\,\!}$

For large wavenumber ${\displaystyle (k\gg {4\pi G\rho _{0} \over c^{2}})}$, we have ${\displaystyle \omega ^{2}=c^{2}k^{2}}$. As wavenumbers decrease, we arrive at an instability when ${\displaystyle k_{J}^{2}={4\pi G\rho _{0} \over c^{2}}}$. This is the Jeans Wavenumber. As a wavelength this reads:

${\displaystyle {\lambda _{J}^{2}={\pi c^{2} \over G\rho _{0}}}\,\!}$

This equation is simply a relation between the sound crossing time ${\displaystyle ({c \over \lambda })}$ and the free-fall time ${\displaystyle {\sqrt {G\rho _{0}}}}$. Let us remind ourselves that this relation is for collapsing gas. The Jeans criterion for instability in star systems is:

${\displaystyle \lambda _{J}=\left({G\rho _{0} \over \pi \sigma _{*}^{2}}\right)\,\!}$

where ${\displaystyle \sigma _{*}}$ is the velocity dispersion of stars.\

We may go on to ask what the Jeans mass ${\displaystyle M_{J}}$ at recombination might be:

${\displaystyle M_{J}={\frac {4}{3}}\pi \rho ({\frac {1}{2}}\lambda _{J})^{3}={\frac {1}{6}}\pi \rho _{0}\left({\pi c^{2} \over G\rho _{0}}\right)^{\frac {3}{2}}\,\!}$

At recombination, ${\displaystyle \rho _{0}\sim 10^{-23}}$. The temperature over this time period goes from ${\displaystyle T\sim 10^{4}\to 10^{2}K}$. Then using ${\displaystyle {\frac {1}{2}}mc^{2}={\frac {3}{2}}kT}$, and assuming ${\displaystyle T\sim 1000K}$, (and possibly remembering that gas at 1000K has a velocity dispersion of ${\displaystyle 3{km \over s}}$) we have:

${\displaystyle c^{2}={3kT \over 1.4m_{H}}=10\left({km \over s}\right)^{2}\,\!}$

This gives us a Jeans mass of ${\displaystyle M_{J}\sim 3\cdot 10^{6}M_{\odot }}$. This is about the mass of a dwarf galaxy of a globular cluster, suggesting that these objects were the first to form in our universe.\

We can apply this work on Jeans criteria to stars and star formation. Let’s try to figure out how hot the gas that forms a ${\displaystyle 100M_{\odot }}$ star can be, given observed densities of ${\displaystyle n=10^{3}cm^{-3}}$:

${\displaystyle M_{J}={\pi \over 6}\rho _{0}\left({3\pi kT \over G\rho _{0}1.4m_{H_{2}}}\right)^{\frac {3}{2}}=100M_{\odot }\,\!}$

This gives us a temperature of ${\displaystyle T\sim 32K}$. Note that we use the mass of ${\displaystyle M_{H_{2}}}$ because almost all of the hydrogen in these cool clouds is molecular. Observationally, we find temperatures around ${\displaystyle 10K}$ and densities around ${\displaystyle n_{0}\sim 10^{5}}$, giving us ${\displaystyle M_{J}\sim 6.4M_{\odot }}$. Naively, we’d assume this should be the mass of most of the stars we see. We know, however, that most of the mass of stars is in ${\displaystyle {\frac {1}{2}}M_{\odot }}$ stars. To explain this, we need for star formation to proceed inefficiently, with much of the mass being ejected during collapse.\

For rotating disks, we have a Jeans equation:

${\displaystyle \omega ^{2}=\kappa ^{2}-2\pi G\Sigma \left|k\right|+k^{2}c^{2}\,\!}$

where ${\displaystyle \kappa =R{d\Omega ^{2} \over dR}+4\Omega ^{2}}$ is the epicyclic frequency. For gas in this system, we have the criterion for instability being ${\displaystyle Q<1}$, where Q is:

${\displaystyle Q\equiv {c\kappa \over \pi G\Sigma _{g}}\,\!}$

where ${\displaystyle \Sigma _{g}}$ is the surface density of gas. For stars we have:

${\displaystyle Q\equiv {\sigma \kappa \over 3.36G\Sigma _{*}}\,\!}$

For the Milky Way, we find ${\displaystyle \Sigma _{*}\sim 50{M_{\odot } \over pc^{2}}}$, ${\displaystyle \sigma \sim 30km\ s^{-1}}$, and ${\displaystyle \kappa \sim 36km\ s^{-1}kpc^{-1}}$, giving us ${\displaystyle Q=1.7}$, which suggests that the galaxy is overall stable, but may become unstable in spiral arms. This work is recent by a guy named Toomre.

### Ram Pressure

The idea here is that hot, static, x-ray emitting gas in the galaxy clusters imparts pressure on the galaxies moving around inside. The ram pressure that the galaxy imparts on the gas is ${\displaystyle \rho v^{2}}$, where ${\displaystyle v}$ is the velocity of the galaxy relative to the gas. This ram pressure can cause stripping of gas off of the galaxy if it exceeds the hydrostatic pressure of gas in the galaxy:

{\displaystyle {\begin{aligned}P_{hydro}&=\alpha G\Sigma _{*}\Sigma _{g}\\\rho _{IGM}&>{\alpha G\Sigma _{*}\Sigma _{g} \over v^{2}}\end{aligned}}\,\!}

Using that ${\displaystyle \Sigma _{*}\sim 50{M_{\odot } \over pc^{2}}}$, and ${\displaystyle \Sigma _{g}\sim 7{M_{\odot } \over pc^{2}}}$ we find that this effect is present in the Milky Way.