The Initial Mass Function

The Initial Mass Function describes the mass distribution of a population of stars based on their theoretical initial masses (the masses they had at "birth"). The IMF is an empirically derived function; ie, it has been found to be true in numerous actual sets of stars, and was derived from that data, rather than being theoretically predicted. The IMF has this form:



where N is the number of stars, M is the mass of a star, and A is a proportionality constant.

Suppose we have discovered a newly formed globular cluster with a total mass of a million solar masses, and the masses of the stars it contains range from .1 solar masses to 20 solar masses. What can we find out about the cluster given this information?

First, we ought to find A. The first step is to integrate, so we have N as a function of M. (And we'll be playing a little fast and loose with the rules of differential equations, so don't hate me, mathematicians.)



from one tenth of a solar mass to 20 solar masses. Don't get too hung up on the limits of integration, because we'll be integrating again right away. This will give us the total mass of the system, which, fortunately, we know:



Substituting in the values for the variables (and rounding a bit, and approximating), we get



Cool. Now, suppose we divide the stars into categories: massive (between 8 and 20 solar masses), intermediate (between 1 and 8 solar masses) and low-mass (between .1 and 1 solar masses).What is the fraction of total stars in the cluster from each of these categories?

First, we can find the total number in each category. Now that we have the value of A, we can use the first integral above (where we found N as a function of M), except changing the limits of integration to be one of the sets of boundaries for a mass category (eg, 8 masses to 20 masses for massive stars). Solving these integrals gives around 3540 massive stars, 77,600 intermediate stars, and 1,767,000 low-mass stars. This makes massive stars about .2 % of the cluster by number, intermediate stars about 4.2 %, and low-mass stars 95.6 %.

One can also find the total mass contained in each of these categories in this particular cluster by using similar limits on the second integral. This gives about 75,000 solar masses; 280,000 solar masses; and 650,000 solar masses, respectively.

I will continue discussing the Initial Mass Function in a second post, covering more of this problem.

Worked out with Mee and David.