Initial Mass Function approximated by the Salpeter function

[mrp]

1. For stellar objects, M>0.08, then at the epoch of star formation: What fraction of stars are Type-II supernova progenitors (M>8)? What is the fraction of the mass of stars that are Type-II supernova progenitors (M>8)? What is the fraction of the light from stars that are Type-II supernova progenitors (M>8)?



2. Given Equations:

for M>0.5;

for 0.08<M<0.5;

The constants (I'm using "E" to represent the greek letters in the given functions) E₀ and and E₁ are chosen to match the observed local density of stars and to be continuous across M=0.5.



3. I understand what the initial mass function is but I don't understand how to use the Salpeter approximation to answer these questions. I can't find any kind of example anywhere on the internet or in my book and I don't have to a clue where to start. A similar example or any kind of help would be appreciated. Thanks

 

[mark726]

I can provide some guidance on how to approach this problem using the Salpeter approximation. The Salpeter initial mass function (IMF) is a mathematical representation of the distribution of stellar masses at the time of star formation. It is described by the equation:

ξ(M) = E0M^-α

where ξ(M) is the number of stars with mass between M and M+dM, and α is a constant typically ranging from 1.3 to 2.3. This equation can be used to answer the questions posed in the forum post.

1. To determine the fraction of stars that are Type-II supernova progenitors (M>8), we first need to integrate the IMF equation over the mass range of 8 to infinity:

∫8∞ E0M^-α dM

This integral can be solved using the power rule of integration, giving us:

E0/(α-1) * 8^(1-α)

To determine the fraction, we then divide this value by the total number of stars, which is given by the integral of the IMF over the entire mass range (0.08 to infinity):

∫0.08∞ E0M^-α dM

This integral can be solved in a similar manner, giving us:

E0/(α-1) * 0.08^(1-α)

Therefore, the fraction of stars that are Type-II supernova progenitors is:

(8^(1-α) - 0.08^(1-α)) / (8^(1-α) - 0.08^(1-α))

2. To determine the fraction of the mass of stars that are Type-II supernova progenitors, we need to integrate the IMF equation over the same mass range (8 to infinity), but this time multiplying by the mass, M:

∫8∞ E0M^-α * M dM

This integral can be solved using the power rule of integration, giving us:

E0/(α-2) * 8^(2-α)

To determine the fraction, we then divide this value by the total mass of all stars, which is given by the integral of the IMF multiplied by the mass over the entire mass range (0.08 to infinity):

∫0.08∞ E0M^-α * M dM

This integral can be solved in a similar manner, giving