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.
Total number of stars formed with masses between M1 and M2 is
N = ∫M2M1 ξ(M) dm
The Attempt at a Solution
In our case, N = A(∫80.1M-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?