# Galaxies Lecture 09

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Getting back to what a galaxy is, we look at M100, which is a spectacular spiral galaxy. It is important to note that between the spiral arms of a galaxy, even though we don’t see much brightness, there are as many stars as in the spiral arms. The spiral arms are bright as a result of the initial luminosity function: OB stars are alive and very bright.\

We then look at M33, which has less well defined spiral arms when viewed in the continuum, but pronounced arms are visible when a technique called unsharp masking is used to process the image. This technique subtracts a “jiggled” image from the original image to undo the effects of isophotometric rings tricking our eyes away from the spirals.\

In the text, there are some spectacular pictures of rings in elliptical galaxies. This is the result of spindown caused by dynamical friction.\

### Potential Theory

So far, we’ve been discussing galaxies as collections of point-masses. In this model, we have to do N-body simulations to understand the dynamics of galaxies. However, we can make some simplifying assumptions. If we consider the gravitational influence of a region on a point, we can describe the force as:

{\displaystyle {\begin{aligned}F({\vec {x}})&=G\int {{{\vec {x}}^{\prime }-{\vec {x}} \over \left|{\vec {x}}^{\prime }-{\vec {x}}\right|^{3}}\rho ({\vec {x}}^{\prime })d^{3}{\vec {x}}^{\prime }}\\\Phi ({\vec {x}})&=-G\int {{\rho {{\vec {x}}^{\prime }} \over \left|{\vec {x}}^{\prime }-{\vec {x}}\right|}d^{3}{\vec {x}}^{\prime }}\\{\vec {\nabla }}_{k}\left({1 \over \left|{\vec {x}}^{\prime }-{\vec {x}}\right|}\right)&={{\vec {x}}^{\prime }-{\vec {x}} \over \left|{\vec {x}}^{\prime }-{\vec {x}}\right|^{3}}\\F({\vec {x}})&={\vec {\nabla }}_{x}\int {{G\rho ({\vec {x}}^{\prime }) \over \left|{\vec {x}}^{\prime }-{\vec {x}}\right|}d^{3}{\vec {x}}}\\F({\vec {x}})&=-{\vec {\nabla }}\Phi ({\vec {x}})\\\int _{V}{{\vec {\nabla }}\cdot {\vec {F}}d^{3}x}&=\int _{S}{{\vec {F}}\cdot dA}\\{\vec {\nabla }}\cdot {\vec {F}}&=-4\pi G\rho (x)\\\nabla ^{2}\Phi &=-4\pi G\rho \\\end{aligned}}\,\!}

Thus, we have defined a potential field ${\displaystyle \Phi }$ (potential energy per mass), and we can get our energy from this field as:

${\displaystyle W={\frac {1}{2}}\int {\rho (x)\Phi (x)d^{3}x}\,\!}$

Potential fields are nice because they are scalar fields. If we say that our total potential is the sum of various components:

${\displaystyle \Phi _{TOT}=\Phi _{x}+\Phi _{y}+\Phi _{z}\,\!}$

then we can determine our potential as a function of radius for circularly orbiting matter:

${\displaystyle {d\Phi \over dR}={v_{x}^{2} \over R}+{v_{y}^{2} \over R}+{v_{z}^{2} \over R}\,\!}$

We know that for a centrally concentrated object, its gravitational potential is ${\displaystyle \Phi ={GM \over R}}$. We may then define circular velocity to be:

${\displaystyle {v_{c}^{2}=R{d\Phi \over dR}}\,\!}$

where ${\displaystyle v_{c}^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}$. We can also define escape velocity to be:

${\displaystyle v_{esc}^{2}=2\left|\Phi (R)\right|\,\!}$

Note that this tells us that the units of potential are ${\displaystyle velocity^{2}}$. We go on to prove that for a spherically symmetric density function, all shells exterior to a point do not exert a force on that object (although they do create a constant, non-zero potential), and all shells interior behave as if all their mass were concentrated at the center. If we do not have a spherically symmetric density function, then we may break up the integral over ${\displaystyle \rho }$ as a function of ${\displaystyle r}$ as:

${\displaystyle {\Phi =-4\pi G\left[{1 \over r}\int _{0}^{r}{r^{2}\rho (r)dr}+\int _{r}^{\infty }{r\rho (r)dr}\right]}\,\!}$

We can verify from this equation that for an object with diameter ${\displaystyle a, that ${\displaystyle \Phi ={GM \over r}}$. A curiousity is that if ${\displaystyle \Phi }$ is constant in a sphere, then ${\displaystyle v_{c}=r{d\Phi \over dr}=\left({\frac {4}{3}}\pi G\rho \right)^{\frac {1}{2}}r}$. Then the orbital period:

${\displaystyle T={2\pi r \over v_{c}}=\left({3\pi \over G\rho }\right)^{\frac {1}{2}}\,\!}$

does not depend on radius. This happens in nature: in galaxies we find that we have a density function that turns over. Therefore, for some region in a galaxy, there is a corotating belt.\

The time it takes a particle to fall to the center from a radius ${\displaystyle r}$ is ${\displaystyle {T \over 4}=\left({3 \over 16\pi G\rho }\right)^{\frac {1}{2}}=t_{dyn}}$ and is called the dynamical time. The dynamical time is not quite the same as the free-fall time, because free-fall time assumes the mass interior to a particle is constant, whereas the dynamical time assumes constant density. The two are related: ${\displaystyle t_{ff}={t_{dyn} \over {\sqrt {2}}}}$.\

Ultimately, what we are going for here is to estimate mass distributions in spiral galaxies. We’ll assume that ${\displaystyle v_{gas}=v_{c}}$ because gas cannot support crossing orbits (the gas would then shock until it is all flowing together circularly). Thus, we can measure the circular velocity (and thus, ${\displaystyle \Phi }$), and figure this out.