About initial mass function and mean mass in stellar cluster

In summary, using a Salpeter IMF with upper and lower mass limits of 0.1 and 20 M⊙ respectively, the mass point at which half the mass formed in a stellar cluster lies in more massive systems and half in less massive systems is 0.5 M⊙. Additionally, the mass point at which half the luminosity of the star cluster lies in more massive systems and half in less massive systems can be found using the equation fM = ∫xy M-7/3 M dM / ∫0.120 M-7/3 M dM.
  • #1
Cedric Chia
22
2

Homework Statement


Assuming a Salpeter IMF with upper and lower mass limits of 0.1 and 20 M⊙ respectively, calculate:

(i) the mass point at which half the mass formed in a stellar cluster lies in more massive systems and half in less massive systems.

ii) the mass point at which half the luminosity of the star cluster lies in more massive systems and half in less massive systems.

Homework Equations


to find number of stars in the range x and y: fN = ∫xy M -7/3 dM / ∫0.120 M -7/3 dM
to find mass occupied by the starts in the range x and y: fM = ∫xy M -7/3 dM / ∫0.120 M -7/3 dM

The Attempt at a Solution


Question(i) :
using the second equation,
fM = ∫xy M -7/3 dM / ∫0.120 M -7/3 dM
1/2 = ∫x20 M -7/3 dM / ∫0.120 M -7/3 dM
solving for x and we get,
x = 0.5 M⊙
 
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  • #2
Your two relevant equations look the same but the exponent should be different.
 
  • #3
mfb said:
Your two relevant equations look the same but the exponent should be different.
yess I'm really sorry I've forgot to put another M in the second equation!

correction:
to find mass occupied by the starts in the range x and y: fM = ∫xy M-7/3 M dM / ∫0.120 M-7/3 M dM
 
  • #4
Looks good.
 

1. What is the initial mass function (IMF) in a stellar cluster?

The initial mass function refers to the distribution of masses of stars in a stellar cluster at its formation. It describes the relative number of stars with different masses and is typically represented as a power-law function.

2. How does the initial mass function affect the evolution of a stellar cluster?

The initial mass function is a crucial factor in determining the evolution of a stellar cluster. It determines the number of high-mass stars, which have a shorter lifespan and can significantly affect the cluster's dynamics through their strong stellar winds and supernova explosions.

3. What is the mean mass in a stellar cluster?

The mean mass in a stellar cluster refers to the average mass of all the stars in the cluster. It is calculated by dividing the total mass of all the stars by the number of stars in the cluster.

4. How does the mean mass in a stellar cluster change over time?

The mean mass in a stellar cluster changes over time due to the evolution of the stars within the cluster. As high-mass stars die off, the mean mass decreases, while low-mass stars continue to evolve and potentially increase the mean mass.

5. What factors can influence the initial mass function and mean mass in a stellar cluster?

The initial mass function and mean mass in a stellar cluster can be influenced by various factors, such as the initial conditions of the cluster, the rate of star formation, and interactions between stars. External factors, such as the presence of a nearby galaxy, can also affect the IMF and mean mass in a stellar cluster.

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