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

In summary, the conversation discusses the calculation of the local stellar density constant (ξ0) for a star cluster, using equations 1 and 2 and the limits of the minimum and maximum stellar masses defined in the assumptions. The result is approximately 16000 for ξ0 and approximately 100 for the number of stars in the 30-150M⊙ category. The Salpeter function Φ(M) is used for these calculations, although it may overestimate for masses under 0.5⊙.
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.

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⊙?

haruspex said:
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

how many of the total stars in the cluster are in this 30 - 150M⊙ category.
Ha! I missed the word "such" in the question.
- 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⊙.

## 1. What is the Initial Mass Function (IMF) Problem?

The IMF Problem refers to the discrepancy between the observed distribution of stellar masses in a given star-forming region and the predicted distribution based on theoretical models. In other words, the IMF Problem is the challenge of understanding how stars of different masses are formed and distributed within a galaxy.

## 2. Why is the IMF Problem important to study?

The IMF is a fundamental property of star formation and is crucial for understanding the evolution and structure of galaxies. It also has implications for other areas of astrophysics, such as the formation of planetary systems and the chemical enrichment of the universe.

## 3. What are some proposed solutions to the IMF Problem?

There are several proposed solutions to the IMF Problem, including variations in the physical conditions of star-forming regions, the role of magnetic fields in shaping the IMF, and the influence of feedback processes from massive stars on the formation of lower mass stars.

## 4. How do scientists study the IMF?

Scientists study the IMF by observing star-forming regions and measuring the masses of the stars within them. They also use theoretical models and simulations to understand the physical processes involved in star formation and how they may affect the IMF.

## 5. What are some current challenges in studying the IMF?

Some current challenges in studying the IMF include the difficulty of accurately measuring the masses of stars, the complex interplay between various physical processes in star-forming regions, and the need for more comprehensive observations and simulations to fully understand the IMF on a galactic scale.

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