astrophysics - What is the luminosity of the Milky Way galaxy?

The luminosity of the Milky Way galaxy according to this is $5\times10^{36}$ Watts, but this number suggests that there are about 10 billion stars with Solar luminosities in the Milky Way, which doesn't sound right considering that the Milky Way contains 200-400 billion stars of different luminosities.

The same goes for Andromeda, it has a luminosity similar to the Milky Way with even more stars.

I know red dwarfs are the most abundant type of stars and their luminosities are much less than the Solar luminosity, but still, that would mean that both the Milky Way and Andromeda must contain much less than 10 billion stars with masses > 1 solar mass. Does that sound correct ?

Answer

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.