Vanadium 50 said:
As far as the bell-like distribution, well, the shape needs to be generally "belly". At one end you need very bright stars to see them, and they are rare, and at the other end you need dim stars to be very close and they are rare too.
Dim stars don't need to be rare (they aren't really) to see the rise at low ##d## (why isn't this showing up as LaTeX?), they only have to not be
way more common than the Sun (which indeed they aren't). For ##d## small enough that dust extinction is little issue (which does seem to be the case here), the distribution ##dN/dln(d)## with apparent brightness above some limit (in this case the limit determined by the list of brightest stars) is proportional to ##d^3 f(d^2)##, where ##f(L)## is the relative number of stars with intrinsic luminosity above ##L##, and ##d## is measured in units appropriate to the chosen brightness cutoff that defines the list. So as long as ##f## does not rise toward smaller ##L## faster than ##L^{-3/2}##, we will see a rising distribution at low ##d## due to the volume effect. (And if ##f(L)## did rise faster than that, the night sky would be chalk full of dim objects, which is your point, but that's why it doesn't require they be rare, just not super common-- for example, you'd need some 30 times the density of stars 1/10 the luminosity of the Sun to have this problem).
You make the point that the turn over in the plot could be due to a change in the ##f(L)## function where it does start to fall faster than ##L^{-3/2}##. That's probably right, the turnover is at a few hundred LY, but dust extinction should not be strong until several thousand. In fact, we can see that a lot of the dimmest stars that make this list are red giants that are about a hundred times more luminous than the Sun, and red giants don't spend a lot of time much brighter than that. So that's probably what we are seeing in the turnover of the distribution, the fact that to see stars that far away, a lot of the stars will need to be giants, and giants evolve faster as they get brighter, so they are rarely in those brighter stages that we need them to be in to see them at the largest distances in the plot.
The list also includes a smattering of massive stars, which are very bright while still on the main sequence, and in a region of steady-state star formation, there would be enough of them to keep the plotted distribution from turning over (their ##f(L)## would not be far from ##L^{-3/2}## as it happens). But our region of the Milky Way is not a place where there is steady-state star formation, since it's not inside a spiral arm, so a lot of the massive stars formed too long ago to still be on the main sequence, and that should also contribute to the turnover.
So yeah, the turnover is probably due to the rarity of very luminous stars. The Gaussian nature still seems like a coincidence, as the rising and falling parts are due to very different reasons and it really shouldn't look that symmetric or that Gaussian, and the statistics are too poor to expect that it would anyway.
Vanadium 50 said:
The x-axis is in magnitudes as opposed to something more reasonable like Janskies.
I believe the x axis is ln(distance), so the only place brightness comes into play is in the cutoff minimum apparent brightness that goes into the set of stars, which can be measured in any unit without consequence to the shape of the curve.
Vanadium 50 said:
Also, a normal distribution has 3 parameters and we only have 6 significant points. So I don't think this is surprising. More an "oh" than a "holy smokes!"
That's what I meant about the fact that there are really only 3 statistically well-quantified points, and the other 3 should have significant uncertainties so it's pretty coincidental they fit perfectly to
any smooth curve, let alone a nice symmetric Gaussian. I suspect that a more complete distribution (say if it included the 10,000 brightest stars) would not be Gaussian, because the rise seems to be due to an ##f(L)## function that is less steep than ##L^{-3/2}## due to red giants, and the turnover is due to the fact that we live in a region where there has not been a lot of recent star formation, so that should not even make a symmetric curve, let alone Gaussian. So it's really a remarkable coincidence that the curve looks so perfect, given the statistics.
