Initial Mass Function Question (white dwarfs, neutron stars and black holes)

[J_M_R]

Homework Statement

Calculate the number of white dwarf, neutron star and black holes the galaxy will have after 5Gyr of evolution.

M<8(solar mass) for white dwarf

8(solar mass) < M < 40(solar mass) for neutron star

M > 40(solar mass) for black hole

Initial mass function is ξ(M) = AM^-s where s = 2.35 in the Milky Way, M is the mass in solar units and A is a normalisation constant.

Homework Equations

Total number of stars formed with masses between M1 and M2 is

N = ∫^M2_M1 ξ(M) dm

The Attempt at a Solution

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In our case, N = A(∫⁸_0.1M^-2.35dm + ∫⁴⁰₈M^-2.35dm + ∫¹⁰⁰₄₀M^-2.35dm)

∴ N = -(A/1.35) ((-22.33) + (-0.053) + (-4.88x10³))

So White Dwarf Total = 16.5 A
Neutron Star Total = 0.039 A
Black Hole Total = 3.61x10^-3A

How do I account for the 5Gyr of evolution within my answer as at the moment I believe all I have calculated is the number of stars formed between the masses specified?

 

[mfb]

I'm not sure if the questions is asking about that, but (a) you might have to determine A or (b) you should take into account existing stars and stellar remnants, or even (c) take into account that some stars live longer than 5 Gyr.

And keep in mind that the present distribution of stellar masses and the distribution of stellar masses of new stars are completely different things due to the different lifetime.