# Galaxies Lecture 06

Everyone acknowledges that astronomy’s magnitude system sucks. Nonetheless, it is necessary to understand it:

{\displaystyle {\begin{aligned}m_{1}-m_{2}&=-2.5\log _{10}\left({f_{1} \over f_{2}}\right)\\m-M&=5\log d-5\\m-M&=5\log d-5+A+K\\\end{aligned}}\,\!}

where ${\displaystyle m_{1}}$,${\displaystyle m_{2}}$ are two apparent magnitudes. ${\displaystyle M}$ is the standardized magnitude of a star at a fiducial distance of ${\displaystyle 10pc}$. ${\displaystyle A}$ accounts for foreground extinction from dust, and ${\displaystyle K}$ corrects for the redshift of galaxies:

${\displaystyle K=K_{0}+2.5\log(1+z)\,\!}$

${\displaystyle K}$ is important to account for. Since galaxy measurements are taken with particular waveband filters, there are all sorts brightening/dimming effects which are introduced by spectral features getting redshifted into/out of the wavebands where we are taking measurements. ${\displaystyle K}$ is also hard to remove because it requires knowledge of ${\displaystyle K_{0}}$, which means we have to know ad hoc what the spectrum of our galaxy looks like.\

### Dust

To work out ${\displaystyle A}$, we need to first measure a color excess ${\displaystyle E(B-V)}$, where ${\displaystyle E(X-Y)}$ is given by:

${\displaystyle E(X_{\lambda _{0}}-Y_{\lambda _{1}})=(m(X)-m(Y))-(m_{0}(X)-m_{0}(Y))\,\!}$

where ${\displaystyle m_{0}}$ describes the unreddened magnitude. We may then compute the extinction at ${\displaystyle X}$ as:

${\displaystyle A_{X}=(m-m_{0})_{X}\,\!}$

Since reddening asymptotes to 0 at long wavelengths, we can try to figure the ratio of the total extinction to the color excess at each waveband. Emperically, this has been measured in our neighborhood, and we find:

${\displaystyle {{A_{V} \over E(B-V)}=3.0}\,\!}$

where ${\displaystyle A_{V}}$ is the magnitude of extinction in the ${\displaystyle V}$ band (which was chosen to be the band that all color excesses are related to).\There was an interesting measurement done by Copernicus in the 1970’s which related the gas mass to dust mass along a line of sight. He found:

${\displaystyle {M_{gas} \over M_{dust}}=100\,\!}$

and this did not seem to change with the line of sight. This will be very useful to us because we can measure hydrogen columns quite well with 21 cm radiation, so we can get good figures for the amount of dust along a line of sight.\As long as we are measuring primarily starlight, we can keep relating the magnitudes of exinctions to ${\displaystyle A_{V}}$. For example:

${\displaystyle {A_{V} \over A_{K}}\sim 10\,\!}$

Just as a reference, the order of wavebands goes: U, B, V, R, I, J, K, L, M. L is about ${\displaystyle 3.5\mu m}$, and redward of M, we start getting features introduced by dust, and our system of comparing extinctions relative to ${\displaystyle A_{V}}$ breaks down.

### Eddington Luminosity

The Eddington Luminosity is the classic point at which a star’s radiation pressure balances the gravitational pull of the star on hydrogen atoms. The key constant which determines where this balance is (other than gravity), is the Thomson Cross-Section:

${\displaystyle \sigma _{T}={8\pi \over 3}\left({e^{2} \over mc^{2}}\right)^{2}=6.65\cdot 10^{-25}cm^{2}\,\!}$

We then balance the force of gravity ${\displaystyle F_{g}={d \over dt}{p}}$ with the per-photon momentum transfer:

${\displaystyle {d \over dt}{p}={1 \over c}{d \over dt}{E}={L_{*} \over c}{\sigma _{T} \over 4\pi R_{*}^{2}}\,\!}$

Now ${\displaystyle F_{g}={M_{*}m_{p} \over R_{*}^{2}}}$, so we find:

${\displaystyle {L_{*} \over M_{*}}={4\pi Gcm_{p} \over \sigma _{T}}=6.31\cdot 10^{4}{ergs \over sec\cdot g}\,\!}$

Besides being relevant for capping star-formation at O stars, this relevant in galactic centers. By the way, we made the assumption of spherically symmetric accretion to calculate the Eddington limit. If star formation does not proceed symmetrically (and we know it doesn’t), then this value can go significantly higher.\

Now returning to our discussion of mass functions, we know the total number of stars in a system is:

${\displaystyle N_{TOT}=\int _{0}^{\infty }{\Psi _{0}(M)dM}\,\!}$

where ${\displaystyle \Psi _{0}(M)}$ is the initial mass function again. The total mass in the system is then:

${\displaystyle M_{TOT}=\int _{0}^{\infty }{\Psi _{0}(M)MdM}\,\!}$

Emperically, Salpeter found ${\displaystyle \Psi _{0}(M)=N_{0}M^{-\alpha }dM}$, with a value of ${\displaystyle \alpha =2.35}$. This causes our integral to break because ${\displaystyle \alpha >1}$ means we have an infinite number of stars as we approach ${\displaystyle M\to 0}$. Similarly, ${\displaystyle \alpha >2}$ causes the total mass to diverge as ${\displaystyle M\to 0}$. This does not mean our integrals are bad, it just means we have not adequately limited the range over which ${\displaystyle \Psi _{0}(M)}$ can operate. Just as there is an upper mass limit introduced by the Eddington limit, there is a minimum mass, less than which stars do not form. Emperically, we have measured this to be about ${\displaystyle 0.08M_{\odot }}$. This causes a turnover in the power law Salpeter measured. Miller/Scudo measured:

${\displaystyle \Psi (M)\propto {\begin{cases}M^{-2.45}&M>10M_{\odot }\\M^{-3.27}&10_{\odot }>M>1M_{\odot }\\M^{-1.83}&1M_{\odot }>M\\\end{cases}}\,\!}$

On the other hand, Kroupa measured:

${\displaystyle \Psi (M)\propto {\begin{cases}M^{-4.5}&M>1M_{\odot }\\M^{-2.2}&1M_{\odot }>M>0.5M_{\odot }\\M^{-1.2}&0.5M_{\odot }>M\\\end{cases}}\,\!}$

This are obviously different results, and both of these measurements were done in our own galaxy. We would love to have a consistent number we could apply to different galaxies, but we can’t even work it out for our galaxy. Extrapolating to other galaxies is a tremendous leap of faith.

There are a couple of places we can look for directly measuring ${\displaystyle \Psi _{0}(M)}$. We look at star-forming clusters and in the infrared measure the luminosities of clumps. This has the added bonus that pre-main-sequence stars are more luminous than main-sequence stars, so we can measure out to lower masses.

One other thing about power laws: they have no inherent scale, because you’re just taking ratios. Another way of saying this is that they are self-similar. This is nice because properties of one portion of the distribution apply to the whole distribution.