Cluster Census: Sampling a Star Cluster from the Initial Mass Function

The mass function

$$ \xi(m) \propto m^{-\alpha} $$

• For $m \gtrsim 0.5\,M_\odot$ the canonical slope is $\alpha = 2.3$.
• Steeper $\alpha$ means fewer massive stars for the same total number.
• Maschberger (2013) replaces the broken power law with one smooth formula that turns over at low mass. Kroupa (2001) keeps the break at $0.5\,M_\odot$.
• The smooth curve on the histogram is the law itself, computed analytically. The bars are what you actually drew. They are not the same computation.

Small numbers at the top

• Most stars are small. Around 80 percent of a cluster sits below $1\,M_\odot$, which is why the histogram falls away so steeply to the right.
• So the massive end is only a handful of draws. Press Draw a new cluster a few times with a few hundred stars and watch the heaviest star jump around by a factor of several.
• Turn the number of stars up and the same readout settles down. Nothing about the law changed; there are just more draws from its tail.

Age carves the diagram from the top

• A star's main-sequence lifetime falls steeply with mass, so the heaviest stars leave first and the HR diagram loses its top-left corner.
• The heaviest star still shining is the turnoff. It is how a cluster's age is read off in practice.
• Lifetimes here come from Hurley, Pols & Tout (2000), not from the common shortcut $t_{\rm MS} \approx 10\,{\rm Gyr}\,(M/M_\odot)^{-2.5}$, which is off by more than a factor of thirty at $100\,M_\odot$.

What this model does not do

• The giant branch is schematic. Two of its three timescales are real: the main-sequence lifetime and the time to the base of the giant branch both come from Hurley et al. (2000). The shape of the track is textbook rather than computed, so positions are good to a factor of a few. Read the branch for its shape and its turnoff, not for a star's exact radius.
• The model has a domain, and the two ends differ. The Tout et al. (1996) fits are published for $0.1$ to $100\,M_\odot$. Above the ceiling they extrapolate cleanly, so stars heavier than $100\,M_\odot$ are plotted with a ring and counted as extrapolated. Below $0.1\,M_\odot$ they are not used at all, so the draw starts there rather than at the true hydrogen-burning limit of $0.08\,M_\odot$. That missing sliver is narrow in mass but holds roughly a tenth of the stars, because the mass function is so steep. Drawing them and then leaving them off the HR diagram taught nothing; clamping them up to $0.1\,M_\odot$ would have invented a spike that is not in the mass function.
• No white dwarfs, and no horizontal branch. Once a star burns through its giant phase it leaves the tally as a remnant and is drawn grey. The cooling track a real cluster shows at bottom left is not modelled.
• No dynamics. Positions are drawn from a static profile. Nothing orbits, nothing sinks, and there is no mass segregation.
• Dot size is legibility, not radius. It scales as the cube root of mass so one heavy star cannot swamp the frame.