# Galaxies Lecture 07

• Morphology
• The Hubble Tuning Fork Classification (this is not a precise classification scheme).
• Ellipticals (E0-E5), notated En, where ${\displaystyle n=10(1-{b \over a})}$ is a measure of eccentricity. Dwarf Ellipticals (dE) are less luminous and hard to observe outside the Local Group. Dwarf Spheroidals (dSph) have very low stellar densities.
• Lenticulars (S0, SB0), which are between ellipticals and spirals. Lenticular galaxies have a disk with a very large central bulge. For normal-type lenticulars, they are enumerated (S0-S3) according to increasing dust absorption. For barred lenticulars, they are enumerated according to the prominence of the bar.
• Normal-Type Spirals (Sa-Sc). Subtypes are ordered according to: bulge-to-disk light ratios, tightness with which spiral arms are wound, and the degree to which spiral arms are resolved into stars and HII regions.
• Barred Spirals (SBa-SBc) are subtyped the same way as normal-type galaxies, but also by rings and lenses, noting the difference between inner rings and outer rings, etc.
• Irregulars. Some examples are the Magellenic Clouds. Type I Irregulars have bright knots of O and B stars. Type II have smooth images and exhibit dust lanes. Some subtyped irregular galaxies are Starburst galaxies (which have jets, tails, and rings, and are formed by the merger of two disk galaxies), and AGN.
• Surface Brightness Measurements
• The specific intensity of galaxies, as we showed in 201, does not depend on distance, so you’d think we can just measure the surface brightness of galaxies directly. However, there are 5 corrections we must make to this:
• Air Glow (the luminosity of our atmosphere)
• Zodiacal Light (reflected sunlight off of dust)
• Faint/Unresolved Stars
• Extra Galactic Light
• Seeing (the displacement of photons travelling through the atmosphere).
• de Vaucouleur’s ${\displaystyle R^{\frac {1}{4}}}$ Law
${\displaystyle m_{B}=x-yR^{\frac {1}{4}}\,\!}$

Where ${\displaystyle m_{B}}$ is the magnitude of brightness in the B band. This gives us the intensity of a galaxy as a function of radius:

${\displaystyle I(R)=Ie^{-7.67\left[\left({R \over R_{e}}\right)^{\frac {1}{4}}\right]}\,\!}$

where ${\displaystyle R_{e}}$ is half-light radius. However, ${\displaystyle I(R)=\int _{-\infty }^{\infty }{j(r)dz}}$, which is to say that our observation of the surface brightness of a galaxy is actually the the integral of the emissivity of the galaxy along the z axis. Taking these we get the Modified Hubble Law:

${\displaystyle I(R)={I_{0} \over 1+\left({R \over r_{0}}\right)^{2}}\leftrightarrow j(r)={j_{0} \over \left[1+\left({r \over r_{0}}\right)^{2}\right]^{\frac {3}{2}}}\,\!}$

Note that ${\displaystyle R}$ is the actual radius of a galaxy, and ${\displaystyle r}$ is the radius at which we are viewing a galaxy. Using these equations, we may derive the total luminosity of a galaxy as a function of radius:

{\displaystyle {\begin{aligned}L(R)&=2\pi \int _{0}^{R}{I(R)R^{\prime }dR^{\prime }}=2\pi \int {{I_{0}R^{\prime } \over 1+\left({R \over r_{0}}\right)^{2}}dR^{\prime }}\\&=I_{-}\pi \ln \left(1+\left({R \over r_{0}}\right)^{2}\right)r_{0}^{2}{\bigg |}_{0}^{R}\\\end{aligned}}\,\!}

This diverges as ${\displaystyle R\to \infty }$. To prevent this from happening, we have to model the emissivity as:

${\displaystyle j(r)={3-\gamma \over 4\pi }{La \over r^{\gamma }(r+a)}\,\!}$

It is important to note that for a radially symmetric galaxy, if we observe it to be centrally concentrated in surface brightness, it must be even more highly concentrated volumetrically.

• The Galaxy Luminosity Function ${\displaystyle \Phi (L)}$
• How many galaxies of each (luminosity) type are there in a representative volume of the universe? To answer this question we discuss the Galaxy Luminosity Function:
${\displaystyle \Phi (L)={dn \over dL}={\# \over V\cdot L}\,\!}$
• Observations are taken to solve the equation:
${\displaystyle B-M_{B}=5\log d-5+A_{B}+K\,\!}$

However, these observations are subject to several biases:

• The Malmquist (Volume) Bias
• ${\displaystyle d}$ is poorly determinted for nearby galaxies, where peculiar velocities outweigh Hubble velocities.
• Galaxies are not uniformly distributed in space.
• Accounting for these biases as best we can, we find that for low luminosities, the number of galaxies goes as a power law, and then steepens to an exponential falloff at high luminosities:
${\displaystyle \Phi (L)={\phi _{*} \over L_{*}}e^{-{L \over L_{*}}}\left({L \over L_{*}}\right)^{\alpha }\,\!}$

where ${\displaystyle \phi _{*}}$ is a normalization factor for the mean density and is of order 1, and ${\displaystyle \alpha \approx -1}$. This is called the Schechter Function.

• We could try to calculate the luminosity density of the universe:
{\displaystyle {\begin{aligned}L_{tot}&=\int _{0}^{\infty }{{dn \over dL}LdL}\\&=\phi _{*}\int _{0}^{\infty }{\left({L \over L_{*}}\right)^{\alpha +1}e^{-{L \over L_{*}}}dL}&=\phi _{*}L_{X}\Gamma (\alpha +2)\\\end{aligned}}\,\!}

Where ${\displaystyle L_{*}\sim 10^{10}L_{\odot }}$. Note that the luminosity of the Milky Way is about ${\displaystyle 10^{10}L_{\odot }}$

• We then may attempt to compute the # density of Galaxies:
{\displaystyle {\begin{aligned}n&=\int _{0}^{\infty }{{dn \over dL}dL}\\&=\phi _{*}\underbrace {\Gamma (\alpha +1)} _{divergent}\\\end{aligned}}\,\!}

To prevent divergence, we can’t integrate from ${\displaystyle L=0}$.

• Global Correlations of Elliptical Galaxies
• We try to classify ellipticals by shape: (eccentricity, "boxiness", velocity dispersions).
• Also, by "nonshape" characteristics: (luminosity ${\displaystyle L_{e}}$, radius ${\displaystyle R_{e}}$, Surface-Brightness ${\displaystyle I_{e}}$, B-V, ${\displaystyle \sigma }$).
• Surface-Brightness is related to the effective radius ${\displaystyle R_{e}}$:
${\displaystyle R_{e}\propto \left\langle I_{e}\right\rangle ^{-0.83}\,\!}$

Then using ${\displaystyle L_{e}=\pi R_{e}^{2}\left\langle I_{e}\right\rangle }$, we have:

${\displaystyle L_{e}\propto I_{e}^{-0.66}\,\!}$

This tells us that larger galaxies have less average surface brightness, and that more luminous galaxies also have less average surface brightness.

• The Faber-Jackson relationship says that
${\displaystyle L_{e}\propto \sigma _{0}^{4}\,\!}$

where ${\displaystyle \sigma _{0}}$ is the central velocity dispersion. This tells us that larger galaxies have higher velocity dispersions.

• The color-magnitude relation was the observation that more luminous galaxies have stronger absorption lines, and that more luminous galaxies are redder. Since reddening is related to metallicity and age of a galaxy (fewer blue stars), this has important implications for galaxy evolution.
• Relating the three quantities ${\displaystyle \log(R_{e}),\left\langle I_{e}\right\rangle ,}$ and ${\displaystyle \sigma }$, we have 2 independent variables. This is called the Fundamental Plane Relation. In the space of these 3 variables, the normal vector of this plane is ${\displaystyle (-0.65,0.22,0.86)}$. An edge of the plane is given by:
${\displaystyle \log(R_{e})=0.36\left({\left\langle I_{e}\right\rangle \over \mu _{B}}\right)+1.4\log \sigma \,\!}$

We need to come up with a model of galaxy formation which accounts for the flatness of this distribution.

• Finally, we have the Dn-sigma relation, where ${\displaystyle D_{n}}$ is the diameter at which the intensity equals ${\displaystyle 20.75\mu _{B}}$. Using the ${\displaystyle R^{\frac {1}{4}}}$ law and the fundamental plane relation, we find that
${\displaystyle {\sigma _{0}^{\frac {4}{3}} \over D_{n}}\propto const\,\!}$

Measurements of the Virgo Cluster have determined that:

${\displaystyle {D_{n} \over kpc}=2.05\left({\sigma \over 100{km \over s}}\right)^{1.33}\,\!}$

This relation is good to about a factor of 2 because of inherent scatter in this relation for different elliptical galaxies.