Estimating number of Neutron Stars in a Star Cluster

[QuantumCrash]

Homework Statement

An open cluster is observed to contain 1000 stars. Clusters contain many more low mass
stars compared to high mass stars, reflecting the star formation process. This depen-
dence of the number of stars formed in a given mass range is expressed in the so-called
initial mass function, which turns out to be the same for different clusters.

The standard initial mass function leads to 940 stars with masses less then
8 M⊙ , 53 between 8M⊙ and 40 M⊙ and 7 stars with a mass larger than > 40 M⊙ . We will
assume that no further star formation will occur in this cluster during its evolution.

How many neutron stars would you roughly expect to find in the cluster?

Homework Equations

Neutron star is 2-3M⊙ .
Possibly time/Mass/Luminosity relations.

The Attempt at a Solution

I am somewhat stuck here and need a prod in the right direction. The closest thing I can think of is assume an even/normal distribution, and use the fractional mass range (1M⊙ ) of the neutron star over the number of stars under 8M⊙ , i.e. 940/8. However, this seems a bit silly and unsatisfactory really. I also considered looking at the initial mass function as the question initiall states , but a brief wiki suggests that I am not suppose to know that yet, since I have never seen the equation version of it before.

 

[phyzguy]

Only stars in a certain mass range end up as neutron stars. Too little mass, and it ends up a white dwarf, too much and it collapses directly to a black hole. A little research should help you uncover what that mass range is.

 

[QuantumCrash]

I am aware that stars 5-8 M⊙ up to 25-40M⊙ are likely to produce neutron stars upon reaching the end of their life-cycle. However, I am having trouble seeing how this helps me in estimating the number of neutron stars currently in the star cluster.

 

[Steely Dan]

The initial mass function tells you the distribution of stars as they enter the main sequence (which happens at the same time for all of them, more or less). But being the initial mass function, it doesn't account for stars that have already evolved off the main sequence (e.g. neutron stars). You have to infer that a certain number of stars were formed in the correct mass range to form neutron stars. That is what the IMF tells you; it is not an account of stellar masses now, but at the time the cluster was formed.

 

[QuantumCrash]

Ah, I see, I understood the initial mass function as the star distribution that would result after stars stopped being produced as the question mentioned, ie that would be the current distribution , including the neutron stars.

I suppose the key I should have noticed was in another sub question, where it mentioned that the end of the main sequence was 6 M⊙. I did not see it a big deal for stars of larger masses in the cluster, because I was operating under the assumption that such that different stars would be formed at different times, and did not connect the dots.

Hence, would the simple answer of 53(or ~50) be a reasonable answer to a rough estimate to the expected number of neutron stars in the cluster?

On a related note, some searching lead me to the fact that the age of nebulae are on the order of ~10000 years, would be it a decent approximation to assume that all the stars in a particular star cluster are created around the same time, considering the age of clusters being discussed are typically in the order of 10^10 years.

 

[Steely Dan]

Hence, would the simple answer of 53(or ~50) be a reasonable answer to a rough estimate to the expected number of neutron stars in the cluster?

It wouldn't be a bad answer, but many of the stars in the 8-40 mass range will form black holes (probably the ones starting above ~20). So for that you'll have to make some sort of estimate about how many are in the lower half of the range.

would be it a decent approximation to assume that all the stars in a particular star cluster are created around the same time, considering the age of clusters being discussed are typically in the order of 10^10 years.

Yes, that's exactly the idea. The time it takes for all the cloud fragments to collapse is very short compared to the average lifetime of the cluster. Now, they specify it's an open cluster, which means it's fairly young, but still, it's likely that we're past the point where the most massive stars have died, for the average open cluster. On the other hand, the average lifetime of an open cluster isn't that long either (generally less than $10^9$ years, so you definitely can only make an estimate here.

 

[QuantumCrash]

What would be the best way to take that into account? If I take into account that possibility, would I not need to take into account the number of stars in the 940 group, since quite a few stars from that range would form neutron stars too. In this case it would be in the 6-8M⊙ range, since it was given that the main sequence at ends at a star with 6M⊙. Though again, at this point, making the assumption that the stars in each group are evenly distributed is a bit iffy, or perhaps it's possible to create some sort of normalised distribution based on the ranges given?

 

[Steely Dan]

What would be the best way to take that into account? If I take into account that possibility, would I not need to take into account the number of stars in the 940 group, since quite a few stars from that range would form neutron stars too.

The generally accepted rule is that stars with $\leq 8 M_\odot$ form white dwarfs, and stars with greater masses form neutron stars.

In this case it would be in the 6-8M⊙ range, since it was given that the main sequence at ends at a star with 6M⊙.

Not sure what you mean here but this is not standard terminology. The "main sequence" refers to the part of a star's lifetime where it principally burns hydrogen in the core. All stars necessarily do this at some point in their lifetime; that lifetime is just very short for massive stars.

 

[QuantumCrash]

6M⊙ is in reference to the fact that in another subquestion, which I mentioned earlier, the main sequence ends a star with aforementioned mass. My lecture notes mentioned the ranges are:
0.8M⊙<M<5-8M⊙ produces a white dwarf
5-8M⊙<M<25-40M⊙ produces a neutron star
>25-40M⊙ produces black holes.

which I suppose is an ambiguous and unhelpful range.

I would say stars in the 5-8M⊙ range can produce either dwarf stars or neutron stars. Since it mentioned that the main sequence in this particular star cluster ended at 6M⊙, I am making the assumption that all the stars bigger than this mass have died (granted, they could well be giants), which is why I stated 6-8M⊙, since stars of lower mass than that would still be in the main sequence.

 

[Steely Dan]

6M⊙ is in reference to the fact that in another subquestion, which I mentioned earlier, the main sequence ends a star with aforementioned mass. My lecture notes mentioned the ranges are:
0.8M⊙<M<5-8M⊙ produces a white dwarf
5-8M⊙<M<25-40M⊙ produces a neutron star
>25-40M⊙ produces black holes.

which I suppose is an ambiguous and unhelpful range.

Astrophysics is not yet an exact science. There are many areas in which we have to give ambiguous answers because that's all we have. But as a general rule, almost all stars below 8 solar masses on the main sequence do not form neutron stars, even if some can under specific circumstances (e.g. binary systems, etc.)

I would say stars in the 5-8M⊙ range can produce either dwarf stars or neutron stars. Since it mentioned that the main sequence in this particular star cluster ended at 6M⊙, I am making the assumption that all the stars bigger than this mass have died (granted, they could well be giants), which is why I stated 6-8M⊙, since stars of lower mass than that would still be in the main sequence.

Well, the giant phase of the lifetime is pretty short, astronomically speaking, so it's a safe guess that if the main sequence ends at 6 solar masses, that cluster has formed basically all the neutron stars it is going to form.

 

[QuantumCrash]

I see, thank you, that rule is new to me. However, what would be a good method for estimating the number of stars that turn into neutron stars instead of black holes in that range of 53? Surely it's not just dividing by half, or is it? I can think of several more 'fancy' steps, like (1/2)*total ( 8-20 that will surely turn into neutron stars), and (1/2)*(1/2)*(total) representing the rest of the stars that may or may not turn into black holes.

 

[Steely Dan]

I don't have a particularly good answer for you, because there's really no way to know based on the information given. Surely the answer is greater than zero and less than 53, but beyond that there's no way to justify an estimate based solely on the information given. One way to make such an estimate is to look around for initial mass functions of similar clusters that have actually been measured with finer resolution in mass, or use a common analytical relationship (e.g. the Salpeter power law relationship).