# Galaxies Lecture 05

### The Initial Mass Function

When we look at galaxies, we get information about their brightness and spectrum. From this, we would like to make inferences about their evolution. To do this, we need information about the mass of the galaxy. Since most of the visible mass of a galaxy is in the form of stars, it is important to have a solid understanding of stars to make inferences about observations of galaxies.

We measure the masses of stars using binary star systems and applying Kepler’s 3rd law. Ultimately, we would like to have a stellar mass function ${\displaystyle \Phi (M)}$ that gives us the number of stars of a given mass per unit volume. This function depends on time because of the effects of stellar evolution. The initial mass function ${\displaystyle \Psi (M)}$ has these effects removed. A current topic of research is how/whether ${\displaystyle \Psi (M)}$ varies over the history of the universe.

The mass of a star depends on the magnitude of a Jean’s instability in a region. Since Jean’s instabilities can have masses lower than that required for H-He fusion, we have, in addition to main sequence stars, brown dwarfs such as L stars (${\displaystyle T=1500-2000K}$) which contain Lithium, and T stars (${\displaystyle T<1000K}$) which have methane. These stars are sustained by deuterium fusion.

We can measure ${\displaystyle \Phi (M_{V})}$, the general luminosity function, where ${\displaystyle M_{V}}$ is the magnitude of light in the visible band. The simplest way to do this is to take a picture with a CCD, measure the parallax of each of the stars in the picture, and then plot ${\displaystyle \Phi (M_{V})}$ vs. ${\displaystyle N}$. However, there are two biases which make this an inaccurate measurement. First, there is the Malmquist Bias, which states that since there is a limit to the sensitivity of any instrument, we can see brighter stars out further than dim stars. Thus, any picture we take will be sampling different volumes for different kinds of stars. This effect needs to be removed to determine ${\displaystyle \Phi (M_{V})}$, which is supposed to be a per volume measurement.

We can do this by doing proper-motion surveys to determine which are the closer stars and determining the ratios of star types for a given distance. The second bias is the Lutz-Kelker Bias, which states that when we take a picture of the sky, we are sampling a solid angle on the sky. If we have some uncertainty in the aperature of our telescope, we will have an error in our calculated solid angle. This error will have a greater effect on measurements of far-away stars than on close-by stars. In proper-motion surveys, this will tend to cause us to oversample distant stars, and will introduce a second bias as we try to correct the first bias. There are additional problems in measuring ${\displaystyle \Phi (M_{V})}$. For example, we don’t even know ${\displaystyle M_{V}}$ because of extinction due to interstellar dust.

We would then like to work from ${\displaystyle \Phi (M_{V})}$ to ${\displaystyle \Phi (M)}$. To do this, we use theoretical stellar models which are informed by observations of binary stars. We restrict our calculations to stars on the main sequence in order to get a direct mapping of luminosity to mass. This is stellar structure theory.

Important general trends to know about the solar vicinity are: most of the light (luminosity) is from O and B stars, most stars are M stars, and most of the mass is in K and M stars.

To work from ${\displaystyle \Phi (M)}$ to ${\displaystyle \Psi (M)}$, we need to use stellar evolutionary theory (the fact that big stars die young). The first work of this type was done by Salpeter in 1954. This isn’t as simple as dividing each star type by it’s lifetime because ${\displaystyle \Phi (M)}$ has sampled multiple generations of stars, and so it includes the dependence of star formation rate on star size. This is especially problematic when observing far-away galaxies because we have no clue about how star formation rates vary between galaxies and as a function of time since the big bang.