# Galaxies Lecture 14

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

### The Sunyaev-Zeldovich Effect

We now discuss the Sunyaev-Zeldovich effect. This is the effect that a galaxy cluster has a collective mass (typically ${\displaystyle \sim 10^{12}M_{\odot }}$) which creates a potential well that heats (approximately virially) a proton/electron gas (for 10 Mpc, ${\displaystyle T\sim 10^{7}\to 10^{8}}$). This hot gas interacts with the 3K (1 mm) CMB photons, which undergo inverse Compton scattering. This adds some energy to the CMB photons, effectively creating a copy of the CMB blackbody spectrum which is shifted slightly to higher frequencies. There is a single point at which the two blackbody spectra (the fundamental and the upscattered) overlap. This should be around 218 GHz, but if the cluster as a whole is moving towards or away from us, there will be a net red or blue shift (called the kinematic Sunyaev-Zeldovic effect).

How big is the SZ effect? We can measure the temperature of a cluster with ionized plasma by measuring Bremmstrahlung. The emission measure of this radiation is given by:

${\displaystyle EM=K\int _{-R}^{R}{n_{e}^{2}dr}\,\!}$

where R is the radius of the cluster, and K is some constant which may be temperature-dependent. Absorption from the SZ effect, however, only goes as ${\displaystyle n_{e}}$, and is distance-independent:

${\displaystyle SZ=K\int {n_{e}dr}\,\!}$

This is because absorption only depends on the number of electrons along the line of site. Thus, we may see the SZ effect long after the x-ray bremmstrahlung flux has been diffused beyond detectability.

By measuring EM and SZ, we have 2 equations for ${\displaystyle n_{e},dr}$, and so if we approximate that these clusters are spherical (so that ${\displaystyle 2R=\theta d}$), we have the ability to directly determining the distance to these clusters. This allows us to measure Hubble’s constant via the SZ effect. The optical depth to the SZ effect is given by:

${\displaystyle \tau _{SZ}=\sigma _{T}\int {drn_{e}(r)}\,\!}$

and the emissivity is:

{\displaystyle {\begin{aligned}\epsilon (v)&=An_{e}^{2}T_{x}^{-{\frac {1}{2}}}e^{-{h\nu \over kT_{x}}}\\\epsilon (v)&=5.44\cdot 10^{-52}{\bar {z}}^{2}n_{e}^{2}T^{-{\frac {1}{2}}}ge^{-{h\nu \over kT}}\\\tau _{SZ}&=2\sigma _{T}r_{c}{\bar {n}}_{e}\\f(v)&={\frac {4}{3}}\pi r_{c}^{3}{\epsilon (v) \over 4\pi D^{2}}\\P&={\rho kT \over \mu m_{H}}\\{dP \over dr}&={-GM(r) \over r^{2}}\rho \\\end{aligned}}\,\!}

We did this last class, so we’ll just cut to the chase:

${\displaystyle {{Trk \over \mu m_{H}G}\left({-d\ln \rho \over d\ln r}-{d\ln T \over d\ln r}\right)=M(r)}\,\!}$

### Chemical Evolution in Galaxies

Suppose we take a box with a certain mass of gas ${\displaystyle M_{g}}$. To start, we’ll say this gas has metallicity ${\displaystyle Z=0}$. Some mass of gas will get converted into stars (${\displaystyle M_{s}}$). We’ll choose a parameter ${\displaystyle p}$ which describes the yield, or fractional increase in metallicity from an episode of star formation. We’ll call ${\displaystyle d^{\prime }M_{s}}$ the mass of new stars produced, and ${\displaystyle dM_{s}}$ the mass of stars after massive stellar evolution. We define metallicity to be ${\displaystyle Z={M_{h} \over M_{g}}}$, where ${\displaystyle M_{h}}$ is the mass of everything heavier than helium. If no mass leaves or enters the box, ${\displaystyle dM_{x}=-dM_{g}}$. Also, ${\displaystyle pdM_{s}}$ is the mass of heavy elements produced, and so ${\displaystyle dM_{h}=pdM_{s}}$. Then:

{\displaystyle {\begin{aligned}dZ&=d\left({dM_{h} \over dM_{g}}\right)\\&={(p-Z)dM_{s} \over M_{g}}-dM_{s}\\&={(p-Z)dM_{s} \over M_{g}}-{x \over M_{g}}dM_{g}\\&=-p{dM_{g} \over M_{g}}\\Z(t)&=-{p\ln(M_{g}(t) \over M_{g}(0)}\\\end{aligned}}\,\!}

We can then solve for the mass of gas and stars as a function of time:

{\displaystyle {\begin{aligned}M_{g}(t)&=M_{g}(0)e^{-Z(t) \over p}\\M_{s}(t)&=M_{t}(0)\left(1-e^{-{Z(t) \over p}}\right)\\\end{aligned}}\,\!}

Now let’s choose ${\displaystyle \alpha }$ to be the fraction of the present day metallicity. It tends to be around ${\displaystyle {\frac {1}{3}}}$.

${\displaystyle {M_{s}[Z<\alpha Z(t)] \over M_{s}[Z

Now we have from before ${\displaystyle {Z \over p}=\ln {M_{g}(t) \over M_{g}(0)}}$, so

${\displaystyle {{Z \over p}={1-\left({M_{g}(t) \over M_{g}(0)}\right)^{\alpha } \over 1-{M_{g}(t) \over M_{g}(0)}}}\,\!}$

${\displaystyle {M_{g}(t) \over M_{t}(0)}\approx 0.1}$ is something we can get directly from observations (a tenth of the total luminous mass of the galaxy is in gas). For stars:

${\displaystyle {M_{s}(\alpha Z(t)) \over M_{s}(Z(t))}={(1-0.1)^{\frac {1}{2}} \over (1-0.1)}=0.6\,\!}$

However, we measure this fraction to be much smaller. This is called the G-dwarf problem. Historically, it was thought that G-dwarfs might have been a solution to this. Now we know that it isn’t, and so we need to talk about accreting/leaky boxes.