# Statistically, what would the average distance of the closest black hole be?

The closest confirmed black hole is several thousand light years away from earth. Our galaxy has about 100 billion stars. I didn't find any reliable information on the black hole count of ratio versus stars for our galaxy. Some sources say about one in a thousand.

I would like to estimate how close a black hole would be given the data we have. If I did the calculation myself, I would use the volume of the galaxy and the number of stars in it. This would give the average number of stars per some volume, and out of that, the average number of black holes per volume. The distribution of stars would also be have to taken into account, since our part of the galaxy isn't as densely populated. Out of this the statistically approximate distance of the closest black hole could be obtained.

Is anyone equipped with enough information to give me an estimate?

Basically what is needed is the ratio of black holes to stars for our galaxy and then list of x nearest stars and their distances, where x is the ratio.

(I'm asking this because black holes are object of interest, and will have to be visited in the future. Given that our space capabilities might be limited to a distance of a couple of dozens of light years in the near future, this number is interesting.)

• Black holes are generally thought to inhabit the centre of galaxies, therefore the closest one would be at the centre of the Milky Way. – Carl Jun 22 '14 at 8:46
• @ Carl There is a supermassive black hole in the center of milky way. Other black holes are spread out in the entire galaxy with the same distribution as the regular stars, only that there is less of them. – this Jun 22 '14 at 11:20
• Yeah, sorry, I was thinking just supermassive BHs. This site states that each galaxy has approximately 100 million stellar mass black holes. hubblesite.org/explore_astronomy/black_holes/encyc_mod3_q7.html – Carl Jun 22 '14 at 22:33

Let us assume that $N$ stars have ever been born in the Milky Way galaxy, and given them masses between 0.1 and 100$M_{\odot}$. Next, assume that stars have been born with a mass distribution that approximates to the Salpeter mass function - $n(m) \propto m^{-2.3}$. Then assume that all stars with mass $m>25M_{\odot}$ end their lives as black holes.

So, if $n(m) = Am^{-2.3}$, then $$N = \int^{100}_{0.1} A m^{-2.3}\ dm$$ and thus $A=0.065N$.

The number of black holes created will be $$N_{BH} = \int^{100}_{25} Am^{-2.3}\ dm = 6.4\times10^{-4} N$$ i.e 0.06% of stars in the Galaxy become black holes. NB: The finite lifetime of the galaxy is irrelevant here because it is much longer than the lifetime of black hole progenitors.

Now, I follow the other answers by scaling to the number of stars in the solar neighbourhood, which is approximately 1000 in a sphere of 15 pc radius $\simeq 0.07$ pc$^{-3}$. I assume that as stellar lifetime scales as $M^{-2.5}$ and the Sun's lifetimes is about the age of the Galaxy, that almost all the stars ever born are still alive. Thus, the black hole density is $\simeq 4.5 \times 10^{-5}$ pc$^{-3}$ and so there is one black hole within 18 pc.

OK, so why might this number be wrong? Although the number is very insensitive to the assumed upper mass limit of stars, it is very sensitive to the assumed lower mass limit. This could be higher or lower depending on the very uncertain details of the late stellar evolution and mass-loss from massive stars. This could drive our answer up or down.

Some fraction $f$ of these black holes will merge with other black holes or will escape the Galaxy due to "kicks" from a supernova explosion or interactions with other stars in their dense, clustered birth environments (though not all black holes require a supernova explosion for their creation). We don't know what this fraction is, but it increases our answer by a factor $(1-f)^{-1/3}$.

Even if they don't escape, it is highly likely that black holes will have a much higher velocity dispersion and hence spatial dispersion above and below the Galactic plane compared with "normal" stars. This is especially true considering most black holes will be very old, since most star formation (including massive star formation) occurred early in the life of the Galaxy, and black hole progenitors die very quickly. Old stars (and black holes) have their kinematics "heated" so that their velocity and spatial dispersions increase.

I conclude that black holes will therefore be under-represented in the solar neighbourhood compared with the crude calculations above and so you should treat the 18pc as a lower limit to the expectation value, although of course it is possible (though not probable) that a closer one could exist.

• Help me understand, in this calculation, what's our estimate of the current total # of BH in our galaxy? – Fattie Jul 5 '16 at 13:54
• @JoeBlow I didn't work it out and it is not needed for the calculation. It would be 0.06% of how many stars you think have ever lived in the Galaxy. – Rob Jeffries Jul 5 '16 at 14:38
• okie dokey. Let's just say, 1 in every couple thousand for rough mnemonic. Thanks. – Fattie Jul 5 '16 at 14:42

This is only a rough estimate. But the number of supernovae in the Milky Way is approximately 2-5 / 100 years (here and here). There is approximately $3 \times 10^{11}$ stars in the Milky Way. Provided the supernovae rate did not change much through the history (which is a big if that probably doesn't work), the total number of supernovae would be $2$-$5\times 10^8$. I do not know how much supernovae will result in a black hole, but if we estimate 10% - 30%, it might not be that wrong. This would lead to $2\times 10^7$ - $1.5\times 10^8$ stellar black holes in the Milky Way. By other words, one star from 2000 - 15000 should be a black hole.

If there is 1000 stars in 50.9 ly around the Sun, with this density there would be one stellar black hole per 100 - 200 ly.

I have found a database of the nearest stars within 25 parsecs. The database contains 2608 stars, given the not very accurate estimate of 1 black hole per 1000 stars, would make 2.6 black holes within 81.5 ly( 1 parsec = 3.26 lightyears ).

Taking only the closest 1000 stars from the database, then the maximum distance is 50.9 ly, so there is on average one black hole within that distance. The average distance of all 1000 stars is 35.8 ly, and that is the average distance to that probable black hole.

A more accurate ratio would make this much more interesting. Imagine a ration of 1 to 100. Then the average distance becomes only 14.3 ly.

• I'm still hoping for a better answer. – this Jun 22 '14 at 23:43