How Accurate is the Initial Mass Function in Predicting Stellar Distributions?

[Physics Dad]

Homework Statement

A star cluster forms with a total mass of 10[SUP]5[/SUP]M⊙. After 3Myr, the cluster emerges from its natal cloud, making it observable for the first time. The reason we can see the cluster is that the winds of the most massive stars (M>30M⊙) have punched a hole through the cloud. How many such stars do you expect there are in this cluster, given its total mass?

Relevant Equations

M[SUB]tot[/SUB]=ξ[SUB]0[/SUB]∫MΦ(M) dM (eq 1)
N=ξ[SUB]0[/SUB]∫Φ(M) dM (eq 2)

Assumptions:

1) The minimum stellar mass in this cluster is 0.1M⊙
2) The maximum stellar mass in this cluster is 150⊙

First calculate the local stellar density constant (ξ₀) for this cluster using eq 1:

Having rearranged this equation and using the limits of the minimum and maximum stellar masses defined in the assumptions I get...

ξ₀ ≅ 16000

Next using eq 2, the newly calculated constant and using the limits of 30M⊙ and 150M⊙ (to get the number of stellar objects in this mass range) I get the answer to be ≅ 100

I just wanted to check that this is the correct approach, the maths side of things I can handle.

All feedback greatly appreciated.

 

[haruspex]

Not a topic I know anything about, but...
How do you know what the function φ(M) is?
How can ~100 stars of mass at most 150M⊙ each add up to 10⁵M⊙?

 

[Physics Dad]

Not a topic I know anything about, but...
How do you know what the function φ(M) is?
How can ~100 stars of mass at most 150M⊙ each add up to 10⁵M⊙?

Hi, thank you for your response.

The massive stars do not add up to 10⁵M⊙. They are very rare in a star cluster, it is a case of trying to work out how many of the total stars in the cluster are in this 30 - 150M⊙ category.

The Salpeter function Φ(M) has been found to be M^-2.35 consistently for most star clusters and is considered to be acceptable for such calculations.

Thanks again

 

[haruspex]

how many of the total stars in the cluster are in this 30 - 150M⊙ category.

Ha! I missed the word "such" in the question.
Then I agree with your method and answers, except..
- I get nearly 17000 for ξ₀, so just a little more for the answer
- having looked it up, the Salpeter function looks rather an overestimate for masses under 0.5⊙.