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1. Homework Statement

1. Homework Statement

The Initial Mass Function (IMF) defines the distribution of stellar masses created in a star-forming event. The probability P(m)dm of forming a star with a mass between m and m + dm is given by

P(m)dm = km−α dm

where α is the exponent of the power law and k is a constant of proportionality determined by requiring that the probability integrated over all masses be equal to unity. For the Salpeter IMF, which is observed locally,

α = 2.35. If there is a collection of Ntot stars created all at once (e.g., in a cluster or an event in a galaxy), the number N(m)dm of stars with mass between m and m + dm is given by

N(m)dm = NtotP(m)dm

Note that the minimum mass of a star is about m = Mmin = 0.07 M and the maximum mass of a star is about

m = Mmax = 50 M.

SO

a)Symbolically derive an expression for the fraction of the number of stars with a mass above a mass Mref in terms of α, Mref , Mmin, and Mmax. HINT: This will require you to integrate N(m)dm. Make sure to note any conditions you need to impose to carry out the integration.

b)Stars with masses exceeding 8 M explode. Using your result for (b), work out for α = 2.35 the fraction of stars that have a mass above 8 M?

## Homework Equations

P(m)dm = km−α dm

N(m)dm = NtotP(m)dm

## The Attempt at a Solution

*note i have not taken a proper integration class, which a lot of my issues are coming from, so heres my attempt:

(Mmin//Integral S//Mmax)km^-Alpha dm

k (Mmin//Integral S//Mmax) (m^-alpha+1 dm)/-+1

(k(Mmax)^-alpha+1/-alpha+1) -k(Mmin)^-alpha+1)/-alpha+1

\frac{50^{-α+1}k-0.07^{-α+1}k}/{-α+1}

I know I need to solve for K but im not sure how to.

After i solve for K for b) I need to somehow put in 8 solar masses, but im also not sure where that goes into the equation.

Any help is greatly appreciated! Thank you.