Cluster Census: Sampling Stars from the Initial Mass Function

Name: ________________________________ Section: __________ Date: __________

Station: __________ Group members: ________________________________________________

Goal: Use the demo to make a claim supported by (1) at least one number/readout and (2) at least one sanity check.

Station card: Cluster Census (8 to 10 minutes) Part 1 — small cluster, five draws. Set the number of stars to about 300. Press Draw a new cluster five times and record the heaviest star each time.

Draw	1	2	3	4	5
Heaviest star ($M_\odot$)

Largest $\div$ smallest = __________

Part 2 — large cluster, five draws. Set the number of stars to 20,000 and repeat.

Draw	1	2	3	4	5
Heaviest star ($M_\odot$)

Largest $\div$ smallest = __________

Part 3 — one sentence. Nothing about the mass function changed between Part 1 and Part 2. So why did the answer become more reproducible?

“The heaviest star settled down because ____________________________________.”

Word bank + sanity checks Word bank:

• Initial mass function (IMF): the distribution of masses stars are born with. A property of a population, not of any one star.
• High-mass slope $\alpha$: how steeply the number of stars falls with mass. The canonical value is $2.3$. Steeper means fewer massive stars.
• Main-sequence turnoff: the heaviest star still shining. It falls as a cluster ages, which is how real clusters are dated.
• Remnant: what a star leaves when its main-sequence lifetime runs out — a white dwarf, neutron star or black hole, set by its initial mass.

Key relationship:

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

Sanity checks:

• Around 80 percent of the stars sit below $1\,M_\odot$. If your histogram does not fall steeply to the right, check which law is selected.
• The three tally numbers must add up to the number of stars you asked for.
• Raising the number of stars does not change the shape of the histogram, only how closely the bars follow the smooth curve.
• The turnoff can only fall as age rises. If it goes up, you changed something else.
• The bars and the curve are computed separately. They should agree closely at the left-hand end and disagree visibly in the last few bins, where there are few stars.