I have been trying to find out the distribution of Population I and II stars in the Milky Way. The distribution I mean is the percentage of each population to the total stars in the galaxy. So in other words, if the Milky Way contains 200 billion stars, how many of these have formed more than 10 billion years ago (Pop II stars) and how many have formed less than 10 billion years ago (Pop I stars) ?
[Physics] the distribution of Population I and II stars in the Milky Way galaxy
astrophysicsgalaxiesmilky-waystars
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It turns out that it is the distribution of birth stellar masses and most importantly, the lifetimes of stars as a function of mass that are responsible for your result.
Let's fix the number of stars at 200 billion. Then let's assume they follow the "Salpeter birth mass function" so that $n(M) \propto M^{-2.3}$ (where $M$ is in solar masses) for $M>0.1$ to much larger masses. There are more complicated mass function known now - Kroupa multiple power laws, Chabrier lognormal, which say there are fewer low mass stars than predicted by Salpeter, but they don't change the gist of the argument. Using the total number of stars in the Galaxy, we equate to the integral of $N(M)$ to get the constant of proportionality: thus $$n(M) = 1.3\times10^{10} M^{-2.3}.$$
Now let's assume most stars are on the main sequence and that the luminosity scales roughly as $L = M^{3.5}$ ($L$ is also in solar units), thus $dL/dM = 3.5 M^{2.5}$.
We now say $n(L) = n(M)\times dM/dL$ and obtain $$ n(L) = 3.7\times10^{9} M^{-4.8} = 3.7\times10^{9} L^{-1.37}.$$
The total luminosity of a collection of star between two luminosity intervals is $$ L_{\rm galaxy} = \int^{L_2}_{L_1} n(L) L \ dL = 5.9\times 10^{9} \left[L^{0.63} \right]^{L_{2}}_{L_1}$$ This equation shows that although there are far more low-mass stars than high mass stars in the Galaxy, it is the higher mass stars that dominate the luminosity.
If we take $L_1=0.1^{3.5}$ we can ask what is the upper limit $L_2$ that gives $L_{\rm galaxy} = 1.3\times 10^{10} L_{\odot}$ ($=5\times10^{36}$ W)?
The answer is only $3.5L_{\odot}$. But we see many stars in the Galaxy that are way brighter than this, so surely the Galaxy ought to be much brighter?
The flaw in the above chain of reasoning is that the Salpeter mass function represents the birth mass function, and not the present-day mass function. Most of the stars present in the Galaxy were born about 10-12 billion years ago. The lifetime of a star on the main sequence is roughly $10^{10} M/L = 10^{10} M^{-2.5}$ years. So most of the high mass stars in the calculation I did above have vanished long ago, so the mass function effectively begins to be truncated above about $0.9M_{\odot}$. But that also then means that because the luminosity is dominated by the most luminous stars, the luminosity of the galaxy is effectively the number of $\sim 1M_{\odot}$ stars times a solar luminosity.
My Salpeter mass function above coincidentally does give that there are $\sim 10^{10}$ star with $M>1M_{\odot}$ in the Galaxy. However you should think of this as there have been $\sim 10^{10}$ stars with $M>1 M_{\odot}$ born in our Galaxy. A large fraction of these are not around today, and that is actually the lesson one learns from the integrated luminosity number you quote!
EDIT: A postscript on some of the assumptions made. The Galaxy is much more complicated than this. "Most of the stars present in the Galaxy were born 10-12 billion years go". This is probably not quite correct, depending on where you look. The bulge of the Galaxy contains about 50 billion stars and was created in the first billion years or so. The halo also formed early and quickly, but probably only contains a few percent of the stellar mass. The moderately metal-poor thick disk contains perhaps another 10-20% and was formed in the first few billion years. The rest (50%) of the mass is in the disk and was formed quasi-continuously over abut 8-10 billion years. (Source - Wyse (2009)). None of this detail alters the main argument, but lowers the fraction of $>1M_{\odot}$ stars that have been born but already died.
A second point though is assuming that the luminosity of the Galaxy is dominated by main sequence stars. This is only true at ultraviolet and blue wavelengths. At red and infrared wavelengths evolved red giants are dominant. The way this alters the argument is that some fraction of the "dead" massive stars are actually red giants which typically survive for only a few percent of their main sequence lifetime, but are orders of magnitude more luminous during this period. This means the contribution of of the typical low-mass main sequence stars that dominate the stellar numbers is even less significant than the calculation above suggests.
Ironically, it's actually harder to measure the mass of the Milky Way than that of other galaxies. You'd think that with it being RIGHT THERE it would be easy, but alas. Most of the difficulty comes from (1) the galaxy spans a huge part of the sky, so it takes an extremely long time to observe any particular feature in detail across the whole thing (say mapping the strength of an emission line, for instance), and (2) it's hard to get an overall picture of the galaxy because parts of it get in the way of seeing other parts - there's a lot of dust in the galactic disk that obscures our view of the more distant parts of the disk, and the disk is where most of the stars are.
Stellar mass is actually the easiest mass to measure in astronomy, because you can see it much more directly than other mass components. All that needs to be done is measure the intrinsic (rather than apparent) luminosity of a galaxy, assume a "mass-to-light ratio" and multiply to get the stellar mass. Mass to light ratios are on the order of $$\Upsilon\sim1{\rm M}_\odot/{\rm L}_\odot$$ So a galaxy with a luminosity a billion times solar has a stellar mass of about a billion solar masses. More accurate estimates get complicated quickly, as you need to account for the initial distribution of stars in the stellar population(s) involved (the initial mass function: IMF), the age of the populations, dust extinction, etc. etc.
Gas mass is not too bad either. Depending on the phase of the gas - whether it's ionized, molecular or atomic (neutral) hydrogen it may be possible to measure line emission. Neutral hydrogen shows up in the radio at 21cm from the hyperfine transition (spin flip). Most of the gas mass is in neutral hydrogen. Depending on conditions, the Lyman or Balmer series lines may be visible (the first Balmer line is called ${\rm H}\alpha$ in astronomy jargon, it's a common one to observe). Molecular hydrogen - the stuff that stars are made from directly, think Pillars of Creation, is tougher to measure as it has no strong emission lines. What's usually done is to measure emission from other molecular species - ${\rm CO}$ is a common one - and assume something about what fraction of the gas mass that species makes up.
Dark matter mass is inferred from things like galactic rotation curves or gravitational lensing, which both probe the total mass of the system. When we get a total mass from one of these tracers, we always seem to come up about an order of magnitude short (I'm using "always" very loosely here). This, coupled with cosmological observations that seem to imply there is a lot of matter ("dust" in cosmology jargon) that is not "baryonic", but is rather something else that outguns baryons a little less than 10:1 in mass.
As to the Milky Way, there are a number (about 10 that I know of) of ways you can try to measure the mass. I've co-authored a paper which uses several methods. One fairly well known measurement of the total (not just stellar) mass of the MW and M31 is this one, which is more than a factor of 2 bigger than the one you quote. Other sources are more in line with your number... the uncertainty is still rather large. Here's another paper that does the total mass with a different methodology (and get about $1.26\times10^{12}{\rm M}_\odot$), and also models the stellar mass, finding about $6.43\times10^{10}{\rm M}_\odot$, which is about the same ballpark as most estimates for the Milky Way.
If you're adventurous and want to get your hands dirty, stellar mass estimates for at least several hundred thousand galaxies from the SDSS are readily available. These are based on the luminosity of the galaxies, more or less as I've described above. Total mass estimates also exist, but I can't recall where they're easily obtained right now, and they're more uncertain.
Jerry Schirmer mentioned black holes in the comments, so I may as well add a note. The MW black hole is thought to be about $10^{6}{\rm M}_\odot$, so less than one part in ten thousand of the stellar mass, and perhaps a millionth of the total mass. This is more or less typical, though some particularly large black holes get up to perhaps a hundredth of the mass of their galaxy, at most. SMBH's are not thought to be the dominant mass component in any known galaxy (though of course they do dominate in the very central regions).
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This is difficult to answer in an unarguable way because the old bimodal classification of population I and II is more nuanced these days - e.g. thin disk, thick disk, bulge population etc. However, if you define population II as meaning those stars that were born in the first billion years of our Galaxy's evolution, then the following rough calculation gives an idea of the proportions.
Assume that all star are born according to the Salpeter mass function $n(M) = A M^{-2.3}$, where $M$ is in solar units and $A$ is some constant. Assume that the minimum mass is 0.1 and the maximum is 100.[There is no strong evidence for initial mass function variations in our Galactic populations. Other mass functions are available, but using them is a bit more complicated and won't change the result beyond other uncertainties I'll mention.]
Assume that the star formation rate, $\Phi(t)$ has been uniform and began approximately 12 billion years ago . This is more difficult to justify. It is quite likely that the star formation rate was a lot higher at the beginning of the Galaxy's evolution - I'll discuss this assumption at the end. The star formation rate is $\Phi(t)=C$ in units of stars per year. Assume that we are only looking at main sequence stars and that stars spend a negligible fraction of their lives off the main sequence (again, not quite right, but it will do here). Assume the main sequence lifetime is given by $10^{10} M^{-2.5}$ years, where $M$ is in solar units. Ignore white dwarfs.
The number of stars per unit mass that have been formed up to time $t$ $$ N(M) = \int^{t}_0 C n(M)\ dt = CAM^{-2.3}t $$ But if a star was born at $t$, then it will have lived and died if $t < 1.2\times10^{10} - 10^{10}M^{-2.5}$. For a uniform star forming rate, the fraction of stars of mass $M$ that are still alive at time $t$, $f(t) = (5/6)M^{-2.5}$.
So if the Galaxy is 12 billion years old, only stars with $M<0.93$ that were born right at the beginning are still alive. In addition, all pop II stars with $M>0.96M_{\odot}$ have died. These two limits are so close that we will assume there are a negligible number of stars between these masses.
For Pop I and Pop II stars with $M<0.93$, the ratio of Pop II/Pop I stars is just the ratio of their formation timescales, because all that have been born are still alive - i.e. $N_{II}/N_{I} = 10^{9}/1.1\times10^{10} = 0.09$ and the total number of stars is $$N(<0.93) = 1.2\times10^{10}CA \int_{0.1}^{0.93} M^{-2.3}\ dM = 1.74\times10^{11}CA$$
Now you might have thought that this number was an upper limit, because surely you have to add to the population I number, all the stars with $M>0.96$ that were born in the past and have not yet died. Well it turns out that number is small. For pop I stars with $M>0.96$: $${N_{I}} = \int_{0.96}^{100} 10^{10} CAM^{-2.3}\frac{5}{6}M^{-2.5}\ dM = 3.1\times10^{9}CA $$ Even ignoring stellar lifetimes, the number of stars with $M>1$ is $7.7\times10^{9} CA$.
The final result is then that $N_{II}/N_{I} \sim 0.09$. To be more exact, it will be $$N_{II}/N_{I} \simeq \frac{\int_{\tau_{II}} \Phi_{II}(t)\ dt}{\int_{\tau_{I}} \Phi_{I}(t) \ dt},$$ where $\tau_{I}$ and $\tau_{II}$ represent the periods over which the types of star were born and $\Phi(t)$ is the star formation rate at that time.
How sensitive is the calculation to variations in $\Phi$? It is likely that the star formation rate was actually much higher in the early Galaxy. Well, if the star formation rate was higher, then more high-mass stars would have been produced early on and these would have more rapidly enriched the ISM. Once the ISM is rich in metals then metal-poor stars cannot form. So there will be a trade-off (though probably not an exact one). $\Phi$ could have been bigger, but then $\tau_{II}$ would be smaller.
It is very difficult to accurately ascertain this number observationally, because the spatial distributions of the two populations is very different and we cannot directly tell the ages of stars by looking at them. Population II stars are more spherically distributed around the Galaxy, whilst population I stars are concentrated in the Galactic plane, where the Sun is. However, the bulge stars, whilst metal-rich are also probably very old - so do you include those? There is also an intermediate "thick disk" population with intermediate metallicities that probably formed over 2-3 billion years early on.
Thus when we look around us, the stars nearby are hugely dominated by Pop I stars by about 200:1 (here my definition is that Pop II stars are metal-poor; we cannot tell the age of a star by just looking at it!). Extrapolation of this using estimates of the density distribution of metal-poor stars suggests that halo population II contributes only a few percent of the stellar mass of the Galaxy. In turn, this suggests the formation epoch of population II stars lasted much less than 1 billion years. I'm trying to pin this number down a bit better, but the interpretation is confused by what is classified as Pop II, what metallicity cut-off is used, and also by the possibility that our Galaxy halo might include populations due to a number of merger and accretion events, not all of which are metal-poor. Finally there is the question of the bulge. Roughly 20-25% of the stellar mass is here and it probably formed rapidly (about billion years) at the beginning of the Galaxy. For the reasons I discussed above, such a period of intense star formation means that the ISM was enriched and most bulge stars have high metallicity.