The initial mass function (IMF) of main-sequence stars,

[bbc]

I need help to solve this problem:

The initial mass function (IMF) of main-sequence stars, n(M) follows a shape called the Salpeter function,which is described by n(M)=kM^-2.35, where k is a normalising constant and M is the stellar mass in solar masses. The IMF is similar in some sense to a luminosity function except that mass is used instead of luminosity, i.e. it gives the number density of stars of a given mass per unit stellar mass. Given that the luminosity of a star depends on mass as L=M^4 (where L and M are given in Solar units), unse the IMF to derive an expression for the mass to light ratio M/L from main-sequence stars in the mass range M1 to M2, and use your fromula to estimate the mass-to-light ratio for a 10 Gyr-old main-sequence population covering a stellar mass range from 0.3-1 solar mass.

thank you

 

[TEFLing]

M/L for an entire population equals total M over total L

Total M = integral of P(M)dM x M

Total L = integral of P(M)dM x L(M)
= integral of P(M)dM x M⁴

 

[Ken G]

It sounds like you have it to me, just note that P(M) is what they mean by n(M), and just do those integrals from M1 to M2, and take the ratio. You should get a number greater than 1, because the M/L of lower masses than 1 will be larger than it is for M=1, which in these units is 1.

 

[TEFLing]

Total M ~ integral of M^(-1.35)

Which depends critically on the lower mass cut off

Total L ~ integral of M^(+1.65)

Which depends critically upon the upper mass limit

Seems like stellar lifetime actually plays a vital role in the ML ratio... IMF is only the start of the show as it were... You would always observe a mature population

 

[Ken G]

Seems like stellar lifetime actually plays a vital role in the ML ratio... IMF is only the start of the show as it were... You would always observe a mature population

As does dark matter. So it sounds like the question in mind here is to compare what M/L is for typical main-sequence stars, to the kinds of numbers you get if you include very luminous stars, stellar lifetime issues, and dark matter.

 

[TEFLing]

If the age of the cluster is T...

And if the life time of stars of mass M is t(M)...

Then luminosity will shine forth from all the stars of mass M which formed from time zero ( present epoch ) back to -t(M) ...

Or, in terms of the cluster age, from T-t(M) to T...

For sake of consistency, if L ~ M^4 then t ~ M/L ~ M^-3

So total light from living stars is a double integral over M and t...

For stars of mass M to M+dM, why

Mass of living stars = integral of P(M)dM x M x dt' from dt'=minimum in magnitude of ( -M^-3,-T ) to zero
= P(M)dM x M x Delta t
= P(M)dM x M x M^-3

Above the maximum still living mass

And...

= P(M)dM x M x T

below that mass

If this is correct, then accounting for stellar life times imposes a strong ~M^-3 cutoff for all stars above the MS turn off mass

I think that hypothetically would increase ML ratios for the same standard Saltpeter IMF

 

[TEFLing]

If stellar life times in Gyr are

t(M) ~ 10 x (M/Msol)^-3

And if the population age is T

Then the MS turn off mass is

m = M/Msol ~ (T/10)^-1/3

From that mass on up,

Total living mass = integral of P(M)dM x M x M^-3
~ M^(-3.35) evaluated from m to infinity
~ m^(-3.35)

Total living mass = integral of P(M)dM x M x M^-3
~ M^(-3.35) evaluated from m to infinity
~ m^(-3.35)

Total living light = integral of P(M)dM x L(M) x M^-3
~ M^(-0.35) evaluated from m to infinity
~ m^(-0.35)

To lowest order of approximation, all of the light comes from the top mass bracket... And all of the mass from the low end ...

Total living mass ~= integral of P(M)dM x M x T
~ M^(-0.35) evaluated from zero to m
~ zero^(-0.35)

By zero I mean the lowest mass cut off, perhaps 0.08 Msol or something like that

So ML ~ (m/zero)^0.35 ~ (T/10)^(-0.12)

If so then all star populations have comparable ML regardless of / for any cluster age (?)

 

[Ken G]

Referring to your earlier post about the double integral over both dt and dM, if you can assume that the galaxy has been forming stars at a constant rate over the full lifetime of all the stars in the IMF, then you don't need to do integrals involving L to get M/L, you can do integrals over the total amount of light a star will ever emit in its entire lifetime, call that E(M). You can then just integrate E(M) over the IMF, like E(M)P(M)dM, and reference that to the solar result to infer M/L in units where this is 1 for the Sun. Mathematically, that would be the same as carrying out the dt integral in your double integral first, and it actually simplifies life because we don't need to worry about what L is at all, just how much mass gets fused. Assuming that is proportional to M, you end up only needing to integrate M*P(M)dM over the whole IMF, and that's L/M in solar units. Deviations from that result tell you something about the star forming history, if you know the IMF, but of course in real life there is always some ambiguity about how much is the IMF and how much is the star-forming history that could be speeding up or slowing down with time.

As for your last result, I didn't really follow-- it seems to me that the living mass of the cluster does not depend on age, because most of the mass is in very low-mass stars that have not had time to die.

 

[TEFLing]

The above equations seem to state, that older populations grow ever so slightly brighter, as they accumulate vast numbers of faint stars...

While continuing to always generate the same number of luminous but ephemeral ones...

Meanwhile the mass in stars grows...

So that ML increase

Please note , I think I forgot to multiply the mass numerator by T...

Then ML ~ T^0.88

 

[TEFLing]

@Ken G

You are correct, I'm implicitly assuming a time invariant IMF

I think I should have calculated as follows
--------------------------------------------------------
Total living mass ~= integral of P(M)dM x M x T
~ M^(-0.35) evaluated from zero to m xT
~ zero^(-0.35) xT

ML ~ (m/zero)^0.35 xT ~ (T/10)^(-0.12)xT ~ T^0.88 ~ T^7/8

I think that is much more intuitively acceptable, agreeing with observations of older populations having higher larger ML ratios

I think that the figure of merit is ML(T)... such a plot would tell a great deal about the SF history of the population

 

[Ken G]

So you are assuming a steady star formation rate, not just the M/L for a single cluster as a function of its age. It certainly does make sense that if you have a steady star formation rate, and the low-mass stars have not had time to die, you will accumulate M even after L has reached a steady state, so M/L will drop with time. Then once you have waited long enough for M to reach a steady state as well (by neglecting mass caught up in low-mass white dwarfs), your L/M will reach the steady state result I mentioned above, given by integrating M*P(M)dM. But this M is not all that meaningful in the L/M, because it does not include white dwarf mass (maybe yours does), and it does not include dark matter either.

 

[TEFLing]

I'm not sure that is correct

Simplistically, almost all of the mass accumulates at the lowest mass cutoff...

Those stars never die, not in the age of our universe...

So mass in living stars accumulates almost linearly with time, i.e. population age...

~T

But older clusters are still forming the fast living and furiously fusing big bright blue super stars... While also having accumulated an increasingly vast population of faint dim dwarfs...

Such that older clusters grow a tad bit brighter over time ~T^1/8

Such that the ratio ML doesn't quite grow linearly with time ~T^7/8

 

[TEFLing]

If a population were to stop forming stars, due to loss of gas as happened in many distant galaxies...

Then ML ratios would skyrocket

The calculation would be informative

 

[Ken G]

I'm not sure that is correct

Simplistically, almost all of the mass accumulates at the lowest mass cutoff...

Those stars never die, not in the age of our universe...

Yes, one would need to wait times longer than the current age of the universe. But in any event, it's not clear that there's any point in not counting the mass of "dead" stars, since it should still be there in the M/L. Generally, one would imagine that M/L is in a kind of steady state, because the M comes from dark matter that isn't changing, and the L comes from an equilibrium population of stars that is the IMF convolved with the lifetimes. Then deviations from the expected M/L represent either a different star-forming history, a different IMF, or a different dark matter ratio. Distinguishing those possibilities would require a lot of independent observations!

 

[TEFLing]

"dead" stars' masses don't vanish... At least not entirely...

What fraction of initial mass remains behind in relic remnant compact objects? E.g. WDs, NSs, BHs ?

That is totally true...
Never the less, those compact objects don't affect ML...
Because almost all of the mass in stars, i.e. almost all of the stellar mass, comes from the stupendously enormous and gargantuanly giganteanly huge population of low mass dwarves...

For MASS, you can ignore the few proud super stars in all their glory, much less the fractions of them remaining behind as COs...

And, for LIGHT, you can ignore COs too, since even W dwarves are ultra faint

In a word, there is always much more mass in rusty red dwarves than in any other population of star-related objects

 

[Ken G]

Yes, and more mass still in the dark matter. Since there really isn't any way to infer M without looking at all the M, including the dark matter, it's not really obvious what M/L tells us about the stellar population-- unless we assume a fixed ratio of dark matter.

 

[TEFLing]

Choosing a low mass cut off appears particularly problematic, mathematically capable of generating arbitrary ML