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What's approximatively the mass of the interstellar neighborhood of the Sun ?
(for example the neighborhood of radius 50 light-years).

Remark: The mass of the observable matter should be completed by the mass of the "dark matter".

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You don't have to guesstimate to come up with the answer.

What you do is look at the dynamics of stars with respect to the Galactic plane - in particular, the velocity dispersions of stars with known distances from the plane, combined with a reasonable assessment of where the Sun is with respect to the plane (close), yields an almost model-independent assessment of the mass density in the disk of our Galaxy in the vicinity of the Sun.

This work was done using the Hipparcos cataogue by Creze et al. (1998). They found that the local mass-density (all forms of matter) in the Galactic disk was $0.076 \pm 0.015 M_{\odot}/pc^3$.

Thus within 50 light years (15.3 pc), because the Galactic disk has an exponential scale height much bigger than this (about 100-200 pc), we can assume a constant density and derive a total mass in the solar neighborhood of $1100\pm 200\ M_{\odot}$.

This mass is almost compatible with the total mass estimated to be in stars, white dwarfs, neutron stars, brown dwarfs and the interstellar medium gas. For instance, according to Chabrier (2001), the total mass density is almost accounted for in the form of 60% main sequence stars plus white dwarfs and neutrons stars, about 5% brown dwarfs and 30% gas, and so the contribution of dark matter to the local disk density is probably very small. Dark matter is not distributed in the same way as normal matter; it did not collapse to a disk, because it is dissipationless and probably exists in a spherical halo. So this consistency between the local baryonic matter density and the total mass density is not an argument against the dark matter hypothesis.

EDIT: I found a few estimates of the local dark matter density that use the kinematics of local stars as a constraint. The range of densities quoted is from 0.008 to 0.02 $M_{\odot}$/pc$^3$ (Bovey & Tremaine 2012; Garbari et al. 2012) i.e. small compared with the contribution from stars and gas.

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  • $\begingroup$ In the first two paras it may be worth qualifying "mass density" as "local mass density (within 125 pc radius)"? Not that this affects your analysis. $\endgroup$ – steveOw Mar 7 '15 at 15:18
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Very roughly: $3.5 \times 10^{33}kg$, or 1800 solar masses.

Here's how I came by that number, it is a very rough approximation.

The major mass components of the galaxy are stars, the interstellar medium, and dark matter.

According to the HYG Database there are approximately 1000 stars within 50 light years of the Earth. The average mass of a star is 0.2 solar masses (thanks to Rob Jeffries in the comments), where a solar mass is about $M_\odot \approx 2 \times 10^{30} kg$. This gives us $4\times 10^{32} kg$ of stars nearby.

The interstellar medium (ISM) is primarily atomic hydrogen, and has an average density of 1 proton per cubic centimetre, although it can vary widely in different parts of the galaxy. A proton weighs $1.6 \times 10^{-27} kg$, so a 50 light year sphere of "average" ISM will weigh about $7 \times 10^{32}kg$. We can do a little bit better than that though. Our solar system lives in the local fluff which is a cloud in the local bubble. The local fluff has a radius of about 15 ly, and a density of 0.3 atoms per cubic centimetre. The local bubble is about 150 ly across, and has a density of only 0.05 atoms per cubic centimetre. Using these figures instead we get an approximate ISM mass of $4 \times 10^{31} kg$.

This is an order of magnitude smaller than the contribution of stars, and our estimate for the stars could easily be off by more than 10%, so let's err on the high side and say the total mass of stars + ISM is $5 \times 10^{32}kg$.

We don't know how much dark matter exactly is in the galaxy, but if its similar to the cosmological average then there is roughly six times as much dark matter as baryonic. If this holds true locally, then there is about $3\times 10^{33}kg$ of dark matter nearby.

So, a rough estimate says there is about $3.5 \times 10^{33}kg$ of mass within 50 light years of us. This is equivalent to 1800 solar masses, or $2 \times 10^{60}$ protons!

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    $\begingroup$ The average star is not 1 solar mass. $\endgroup$ – Rob Jeffries Mar 2 '15 at 7:13
  • $\begingroup$ Cool! I should have researched that better, do you know approximately what the average stellar mass is locally? Perhaps I should add more emphasis to very rough. $\endgroup$ – Geoff Ryan Mar 2 '15 at 7:25
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    $\begingroup$ About 0.2-0.25 solar masses, so only a factor of 5 out... I should add that if you've read the wikipedia page correctly, it is hopelessly wrong about the number of stars there are far more than 200 within50 light years. Possibly your two errors cancel. $\endgroup$ – Rob Jeffries Mar 2 '15 at 7:27
  • $\begingroup$ I'll modify the answer, thanks! While you're at it, any idea what the local dark matter concentration is? I looked around a bit, but there seems to be some debate as to whether or not there's local enhancement in the plane of the disk, so I left it as (approximately) just the cosmological ratio. $\endgroup$ – Geoff Ryan Mar 2 '15 at 7:31
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    $\begingroup$ -1 for adding in dark matter. Your assumption is invalid. Using the results from Nesti, et al., "The dark matter halo of the Milky Way, AD 2013," Journal of Cosmology and Astroparticle Physics 2013.07 (2013): 016, there is about 200 or so solar masses of dark matter within 50 ly of the Sun. This makes dark matter a small contributor to the total answer; your estimate has dark matter as the main contributor. The reason dark matter is a small contributor is that dark matter has a radial density distribution. $\endgroup$ – David Hammen Mar 2 '15 at 13:32

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