Astronomy- Initial Mass Function problem

  • #1

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.
 

Answers and Replies

  • #2
gneill
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I know I need to solve for K but im not sure how to.
Since you're looking for the fraction of stars formed in some range of mass values, it would make sense if you normalized your function over the entire mass range to be unity. Then when you looked at a smaller range, it would give some value between zero and one, i.e. the fraction of the whole.
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.
You integrated between mass values of ##M_{min}## and ##M_{max}## to give you the total for the whole possible range. If you plug in other values for those variables, keeping the same value for k, you'll get the fraction of stars between those mass values.
 

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