What is the distribution of Population I and II stars in the Milky Way galaxy? 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) ?
 A: 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 however some evidence for initial mass function variations in our Galactic populations - Li et al. (2023) suggest that lower metallicity populations contain fewer low-mass stars, perhaps by a factor of two. Other mass functions are available, but using them is a bit more complicated and won't change the result beyond this and 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.
