# Galaxies Lecture 19

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### Dark Matter Density Profiles

${\displaystyle {\partial f \over \partial t}+\sum _{i=1}^{3}{\left(v_{i}{\partial f \over \partial x_{i}}-{\partial \phi \over \partial x_{i}}{\partial f \over \partial v_{i}}\right)}={df \over dt}=0\,\!}$

which says that the change in the distribution of velocities in a volume which is comoving with the fluid (${\displaystyle {df \over dt}}$), is related to the change in distribution of a non-comoving element (${\displaystyle {\partial f \over \partial t}}$), plus a factor which represents a flux over the boundary of the comoving volume (both in position and velocity space). This equation is not equal to 0 when collisions become important.

We now begin taking moments of this equation.

{\displaystyle {\begin{aligned}0&=\int _{0}^{\infty }{{\partial f \over \partial t}d^{3}{\bar {v}}}+\sum \int _{0}^{\infty }{v_{i}{\partial f_{i} \over \partial x_{i}}d^{3}{\bar {v}}}-\sum {\partial \phi \over \partial x_{i}}\int _{0}^{\infty }{{\partial f \over \partial v_{i}}d^{3}{\bar {v}}}\\&={\partial \int \over \partial t}_{0}^{\infty }{fd^{3}{\bar {v}}}+{\partial \over \partial x_{i}}\int _{0}^{\infty }{fv_{i}d^{3}{\bar {v}}}\\\end{aligned}}\,\!}

where the last term of the first equation was eliminated using the divergence theorem: ${\displaystyle \int _{V}{{\vec {\nabla }}fd^{3}{\bar {x}}}=\int _{S}{fd^{2}{\bar {s}}}=0}$. Now ${\displaystyle n\equiv \int _{0}^{\infty }{fd^{3}{\bar {v}}}}$ is the number density of stars, and ${\displaystyle \left\langle v_{i}\right\rangle ={\bar {v}}_{i}={\int {fv_{i}d^{3}v} \over \int {fd^{3}v}}}$, so we have our first Jean’s equation:

${\displaystyle {{\partial n \over \partial t}+{\partial (n{\bar {v}}_{i}) \over \partial x_{i}}=0}\,\!}$

This equation is the star analog for the continuity equation in fluids:

${\displaystyle {\partial \rho \over \partial t}+{\partial \rho {\bar {v}}_{i} \over \partial x_{i}}\,\!}$

For stars, instead of conserving mass, we conserve number. Next, we use:

${\displaystyle 0={\partial \over \partial t}\int _{0}^{\infty }{fv_{j}d^{3}{\bar {v}}}+\sum {\partial \over \partial x_{i}}\int _{0}^{\infty }{v_{i}v_{j}fd^{3}{\bar {v}}}-\underbrace {\sum {\partial \phi \over \partial x_{i}}\int {v_{j}{\partial f \over \partial v_{i}}d^{3}{\bar {x}}}} _{integrate\ by\ parts}\,\!}$

Physically, the last term is ${\displaystyle {\overline {v_{i}v_{j}}}}$, and represents the correlation between velocities in two different directions. Instead of going through the derivation, we will jump straight to the second Jean’s equation:

${\displaystyle {n{\partial {\bar {v}}_{j} \over \partial t}+n{\bar {v}}_{i}{\partial v_{j} \over \partial x_{i}}=-n{\partial \phi \over \partial x_{i}}-{\partial (n\sigma _{ij}^{2}) \over \partial x_{i}}}\,\!}$

where ${\displaystyle \sigma _{ij}^{2}\equiv {\overline {v_{i}v_{j}}}-{\bar {v}}_{i}{\bar {v}}_{j}}$. This equation is the star-equivalent of the fluid momentum equation:

${\displaystyle {\partial {\bar {v}} \over \partial t}+({\bar {v}}\cdot \nabla ){\bar {v}}=-\nabla \phi -{1 \over \rho }\nabla p\,\!}$

In fluids, we have the an energy conservation equation. In the stellar dynamics case, this equation is not useful because the only way to change the energy of a galaxy is to hit it with another galaxy. The one exception to this is in globular clusters, where stars can get evaporated out of the group.

Now, unfortunately, we need to talk about these equations in cylindrical coordinates. The Jeans conservation equation becomes:

${\displaystyle {\vec {\nabla }}\cdot {\bar {F}}{1 \over R}{\partial \over \partial R}\left(RF_{R}+{1 \over R}{\partial F_{\phi } \over \partial \phi }+{\partial F_{z} \over \partial z}\right)\,\!}$

where ${\displaystyle {\bar {F}}_{i}=nv_{i}}$

${\displaystyle {{\partial {\bar {v}}_{\phi } \over \partial \phi }=0}\,\!}$
${\displaystyle {\partial n \over \partial t}+{1 \over R}{\partial (Tn{\bar {v}}_{R}) \over \partial R}+{\partial n{\bar {v}}_{z} \over \partial z}=0\,\!}$

The second equation becomes:

${\displaystyle {\partial n{\bar {v}}_{z} \over \partial t}+{\partial (n{\overline {v_{R}v_{z}}}) \over \partial R}+{\partial (n{\bar {v}}_{z}^{2}) \over \partial z}+{n{\overline {v_{R}v_{z}}} \over R}+n{\partial \phi \over \partial z}=0\,\!}$

For a plane-parallel sheet in steady state, we know that ${\displaystyle {\partial \over \partial t}=0}$, ${\displaystyle {\partial \over \partial R}=0}$, and ${\displaystyle {\bar {v}}_{R}=0}$, so in our second equation, that leaves us with:

${\displaystyle {{\partial (n{\bar {v}}_{z}^{2}) \over \partial z}+n{\partial \phi \over \partial z}=0}\,\!}$

Which is an equation which agrees nicely with observations of disk galaxies. Combining this with Poisson’s equation:

${\displaystyle {\partial ^{2}\Phi \over \partial z^{2}}=4\pi G\rho _{TOT}\,\!}$

we get

${\displaystyle {1 \over n}{\partial (n{\bar {v}}_{z}^{2}) \over \partial z}=-{\partial \Phi \over \partial z}\equiv -g_{z}\,\!}$

which is an acceleration in the z direction. Thus:

${\displaystyle {{d \over dz}\left[{1 \over n}{d(n{\bar {v}}_{z}^{2}) \over dz}\right]=-4\pi G\rho _{TOT}}\,\!}$

Thus, we can measure the velocity and number density in the z direction and we can get an estimate of the total mass density in the midplane of a galaxy. Typically, this is done using K giants because they are old (and therefore, more representative of the total population). We make the assumption here that all stars have the same dynamical temperature (${\displaystyle {\bar {v}}_{z}^{2}}$). It turns out that this assumption breaks down, as it was discovered that there are “old thin disk” and “thick disk” stellar populations.

If we integrate the above equation, we can get an equation for the total density along a column ${\displaystyle \Sigma _{TOT}=\int _{0}^{\infty }{\rho (z)dz}}$. This determination was used to show there was no excess dark matter associated with the disk compared to the halo.

Graphing ${\displaystyle g(z)}$, we find it grows linearly with ${\displaystyle z}$ up to 300 pc, with ${\displaystyle g(z)_{300}=5\cdot 10^{-9}cms^{-2}}$. Because ${\displaystyle g(z)}$ grows linearly, we can infer that ${\displaystyle \rho (z)}$ is relatively constant near the midplane. We measure ${\displaystyle {\rho _{0}=0.1M_{\odot }pc^{-2}}}$ and ${\displaystyle \Sigma _{disk}(R_{0})=48\pm 9M_{\odot }pc^{-2}}$