Elementary Astronomy - Initial Mass Function

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Homework Statement

Estimate the number of low and high mass stars that you expect to find in our Milky Way Galaxy now. Use the following (very approximate but about right numbers) for your estimate:

The Star Formation Rate in our galaxy is approximately 1 Solar Mass per year and virtually all of this goes into making =~ 0.5 Solar Mass stars. The Initial Mass Function, Xi, describes the number of stars at each mass and is something like:

$$\xi$$ = $$\xi$$₀ M^-2.3

where M is the mass of the stars in Solar Masses. In lecture it was shown that the lifetime of stars is given by

L = 10¹⁰y / M^2.5

If the galaxy is 10 billion years old, how many 0.5 and 50 Solar Mass stars do you expect to find in the galaxy.

Homework Equations

$$\xi$$ = $$\xi$$₀ M^-2.3
L = 10¹⁰y / M^2.5

The Attempt at a Solution

My attempt lies around trying to understand exactly what $$\xi$$₀ is. Although we haven't learned it in class, I know that the IMF is meant to describe a distribution of initial solar masses in stars in the galaxy. Obviously, in this case we're not meant to do it in the standard fashion.

I note that when $$\xi$$₀ is 1 then
$$\xi$$ = 5 @ M = 0.5 and
$$\xi$$ = 0.02 @ M = 50

Which tells me that the function is working to some extent but without some explanation as to what $$\xi$$₀ is I can barely comprehend what I am meant to do with this question. I hope I'm not missing something completely obvious, but this question is confusing.

 

[Acidic]

Update:

I consulted my professor as I was handing in this assignment (Without this question completed) And he explained to me that $$\xi$$₀ is a constant, that can be determined from $$\xi$$ = 2 when M = 0.5. And from there the rest of the variables can be determined.