Randomly generating plausible star types for a synthetic galaxy?

This has actually been an area of intense research for decades now. Astronomers are quite interested in the distribution of stellar masses in a variety of different galaxies and clusters. The precise mix you're going to get of course depends on the environment you choose; galaxies with higher metal contents will produce stars with higher metallicities. In general, however, you can take your pick of different initial mass functions, or IMFs. An IMF is a function that tells you what fraction of stars have a mass between $m$ and $dm$, where $dm$ is some step size.

As an example, take the famous Salpeter IMF - one of the first created. The Salpeter IMF has the form $$\phi(m)dm=\phi_0m^{-2.35}dm$$ Therefore, if you have a given stellar population, the fraction of stars with a mass between $m_1$ and $m_2$ is $$f(m_1,m_2)=\int_{m_1}^{m_2}\phi(m)dm=\int_{m_1}^{m_2}\phi_0m^{-2.35}dm$$ Here, $\phi_0$ is a normalization constant.

The Salpeter IMF is a simple one, and it's great as a toy model. More realistic IMFs, like the Kroupa IMF (which is quite commonly used), are piecewise functions. They're also power laws, but the exponent is different over several different mass ranges.

If you want to play around, I've put some code on GitHub to let you play around with a couple IMFs (and generate stellar populations on your own, with some randomness thrown in). Here's a comparison of how stellar masses are distributed, according to the two IMFs I discussed above. The $y$-axis shows the fraction of stars $f( with a mass less than $m$:

$200M_{\odot}$ is kind of an arbitrary upper limit, but in reality any population you synthesize will contain only a tiny, tiny number of stars with masses above $100M_{\odot}$. $0.08M_{\odot}$ is a realistic lower choice - it's the boundary between stars and brown dwarfs.

How do these stack up against L.Dutch's numbers? Using some stellar mass range tables, I found that the Salpeter IMF predicted 93.1% M stars, 3.03% K stars, and 0.926% G stars; The Kroupa IMF predicted 68.1% M stars, 13.6% K stars, and 4.2% G stars. The Kroupa model appears to be more realistic - as you might expect; after all, Salpeter came up with his IMF in 1955!

A couple things to bear in mind:

• The IMF only tells you the initial distribution of a population of stars. Stars of different masses age at different rates, so over time, as massive stars die, the population will skew towards lower masses. You arguably need to evolve your population over time - computationally expensive, even for basic toy approximations.
• This model assumes only one burst of star formation; in reality, most galaxies are continuously forming stars at different rates.
• As I said at the beginning, mileage may vary depending on your galactic environment. Using an IMF based on conditions in the Milky Way should give you a Milky Way-like population.

posted almost 5 years ago

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