00:01
All right, so let's say we're given a relationship between the number of stars and the mass of stars in a cluster of a to the m to be minus 2 .35.
00:11
So we're told that the total mass of this cluster will call this m is 10 to the 6 solar masses.
00:20
And we're ranging over, we'll put this, m is in the interval of, let's see, zero.
00:30
Point 1 to 20.
00:33
So first off, we want to find the constant a.
00:36
So the number of stars within this interval we can write as like, you know, the integral from 0 .1 to 20 of a times m to the negative 2 .35 dm.
00:47
And that means the mass of all those stars in that interval is going to be the integral of the same limits of a to the m to the negative 2 .35 times m.
00:56
So a to the negative 1 .35 d .m.
01:01
So if we evaluate this integral to find the constant a, we'll find that this is like negative a over 0 .35 times m to the negative 0 .35, from 0 .1 to 20.
01:17
And when we do this, this should come out to something like 5 .395 times a.
01:25
And this needs to equal 10 to the 6th solar masses.
01:29
So that means a equals 1 .85 times 10 to the fifth, something like that.
01:37
So that's part a of our problem.
01:40
Part b, aswan, the total luminosity of the cluster, assuming a mass luminosity relationship of like l is proportional to m to the fourth.
01:55
So what we can do is look at the luminosity across all these stars, the integral of l times d .n.
02:03
Which should be equal to basically, what was it, 1 .9 times 10 to the 5th times m to the negative 2 .35 dm times m to the 4th.
02:21
So this should be m to the 2 .65 dm...