How close is Earth to the closest black hole?

[Buzz Bloom]

Is it astronomically known what the closest distance is between Earth and a black hole? (I was not able to locate an answer to this question searching the Internet.) If not, the article

https://www.sciencenews.org/article/we-share-milky-way-100-million-black-holes​

says

The Milky Way teems with black holes — about 100 million of them.​

Based on this estimate, and taking into account the relative density of matter based the distance from the center of the Milky Way, what would be a reasonably accurate estimate of the distance between Earth and the closest black hole?

 

[JCMacaw]

A quick Google search I found - " The closest black hole we know of is V616 Monocerotis, also known as V616 Mon. It's located about 3,000 light years away, and has between 9-13 times the mass of the Sun. We know it's there because it's located in a binary system with a star with about half the mass of the Sun.

Also a Wikipedia article: A0620-00

 

[Buzz Bloom]

A quick Google search I found

Hi JCMacaw:

Thank you very much for your post. I confess that my searching skills could definitely use some improvement. Would you please post the text you searched on?

Regards,
Buzz

 

[Drakkith]

I quickly found it using "closest black hole" in my google search.

 

[Buzz Bloom]

Hi @JCMacaw and @Drakkith:

Thank you both for your posts. I have been thinking about this question since I first came across the 100 million black holes paper I cited in my post #1.

Wikipedia gives the radius of the Milky way as 50-90 kly.

https://en.wikipedia.org/wiki/Milky_Way​

Assuming a conservative order of magnitude estimate of 100 kly, the area of a circular disk of this radius is approximately

3 × 10¹⁰ ly².​

This gives a black hole density of

1 black hole per 300 ly².​

If I am calculating this correctly, assuming a more-or-less uniform population density of black holes, this means that the average distance from an arbitrary point to its nearest black hole would be approximately

7 ly.​

Therefore the distance of 3300 ly from Earth to the cited "nearest" black hole, 1A 0620-00, is about 470 times the approximated average distance from an arbitrary point to its nearest black hole. This suggests that there should be a closer black hole to Earth than 1A 0620-0. One plausible explanation for this discrepancy is the a black hole that is not in a binary system is so difficult to find that astronomers have not yet found many. However, the article

https://www.nasa.gov/centers/goddard/news/topstory/2008/smallest_blackhole.html​

describes a method (QPO) for finding black holes and estimating its mass that (if I understand it correctly) does not require a binary system. However, I suppose that if this method requires a lot more work than looking for binary systems with a black hole, that ratio of discovered black holes not in a binary system to those that are might be very small. Apparently the QPO method does not give a distance estimate to the black hole.

Another plausible explanation is that the distribution of black holes is very far from uniform. I found what seems to be a useful source for exploring the distribution of stars

http://www.astro.caltech.edu/~george/ay20/Ay20-Lec16x.pdf ,​

but I need some time to study it.

Comments on the above would be much appreciated.

Regards,
Buzz

 

[Grinkle]

Comments on the above would be much appreciated

I tried to find a good reference to the number of high confidence actual black holes have been mapped in the galaxy, and I didn't find an easy answer, but I think the number is around a dozen or so, and there is some judgement involved in whether even some of these objects are definitely all black holes.

https://www.quora.com/How-many-black-holes-have-been-located-in-the-Milky-WayIf the above page is more or less accurate, I don't think its significant that we haven't found any closer than 3300 LY to Earth given how few we are able to suspect we observe at all.

 

[jim mcnamara]

Here is a recent estimate:
https://academic.oup.com/mnras/article-abstract/473/1/1186/4060726 (abstract)

Counting black holes: The cosmic stellar remnant population and implications for LIGO

Oliver D. Elbert James S. Bullock Manoj Kaplinghat https://doi.org/10.1093/mnras/stx1959

 

[ohwilleke]

Hi @JCMacaw and @Drakkith:

Thank you both for your posts. I have been thinking about this question since I first came across the 100 million black holes paper I cited in my post #1.

Wikipedia gives the radius of the Milky way as 50-90 kly.

https://en.wikipedia.org/wiki/Milky_Way​

Assuming a conservative order of magnitude estimate of 100 kly, the area of a circular disk of this radius is approximately

3 × 10¹⁰ ly².​

This gives a black hole density of

1 black hole per 300 ly².​

If I am calculating this correctly, assuming a more-or-less uniform population density of black holes, this means that the average distance from an arbitrary point to its nearest black hole would be approximately

7 ly.​

Therefore the distance of 3300 ly from Earth to the cited "nearest" black hole, 1A 0620-00, is about 470 times the approximated average distance from an arbitrary point to its nearest black hole. This suggests that there should be a closer black hole to Earth than 1A 0620-0. One plausible explanation for this discrepancy is the a black hole that is not in a binary system is so difficult to find that astronomers have not yet found many. However, the article

https://www.nasa.gov/centers/goddard/news/topstory/2008/smallest_blackhole.html​

describes a method (QPO) for finding black holes and estimating its mass that (if I understand it correctly) does not require a binary system. However, I suppose that if this method requires a lot more work than looking for binary systems with a black hole, that ratio of discovered black holes not in a binary system to those that are might be very small. Apparently the QPO method does not give a distance estimate to the black hole.

Another plausible explanation is that the distribution of black holes is very far from uniform. I found what seems to be a useful source for exploring the distribution of stars

http://www.astro.caltech.edu/~george/ay20/Ay20-Lec16x.pdf ,​

but I need some time to study it.

Comments on the above would be much appreciated.

Regards,
Buzz

I think that a better estimate as a starting point would be that black hole population density is proportionate to mass density, although even this would be an underestimate because where matter density is higher stellar collisions are likely to be more frequent, while while matter density is lower, stellar collisions are less common and further, more generally, the amount of matter to feed black holes is less abundant so massive objects should grow to the critical mass threshold less often.

So, black holes should really be present at a right higher than a frequency proportionate to total mass density in high mass density parts of the galaxy and should be present at a rate less than the frequency proportionate to total mass density in low mass density parts of the galaxy.

Since mass density is relatively low at the fringes of the galaxy where we live, we would expect the number of black holes per square light year of the plane of the Milky Way to be much lower in our vicinity than near the core of the Milky Way.

Also, while black holes in binary systems are particularly easy to find, occulting other stars, and gravitational lensing effects, should make black holes that are reasonably close to Earth discernible with existing technologies.

Another point to consider is that Earth midway along a spiral arm of the galaxy. So, black holes are much more likely to be present further out or further in along the river of stars that make up that arm, than they are to be very far out in directions at right angles to that arm. So, we are really mostly looking for nearby black holes in two directions that make up a pretty modest share of the total celestial sphere around the solar system. We don't have to look in every direction to have a pretty reliable gauge of what might be out there.

So, while it is certainly possible that there is a black hole closer to Earth than the closest one located so far, (and 1 kpc is a very long way considering that we are about 8 kpc from the galactic center), the closer you get to Earth the less likely it is that astronomers could miss it.

If I were to give a gut guess using back of napkin level precision inputs, I'd suspect that the closest black hole is probably at least on the order of hundreds of light years away.

 

[Buzz Bloom]

If I were to give a gut guess using back of napkin level precision inputs, I'd suspect that the closest black hole is probably at least on the order of hundreds of light years away.

Hi ohwilleke:

Thank you for your post.

Since the nearest currently found black hole to Earth, which is in a binary system, 1A 0620-00, is roughly 3000 ly away, and based on your rough estimate that a non-binary system black hole might, as a reasonable rough approximation, be say 300 ly away, then is it plausible to estimate that in the region of the Milky Way near the Earth, the population density of non-binary black holes is about 1000 times greater than the population of binary system black holes? This conclusion seems to contradict the generally accepted concepts that (1) there are definitely more sun sized or larger stars in binary systems than not in binary systems, and (2) that black holes are the end states of sufficiently large stars.

https://www.space.com/1995-astronomers-wrong-stars-single.html

"If you go out and look at the all visible stars in the night sky and ask, 'How many of those are binary?' the answer is, 'Most of them,'" said study author Charles Lada of the Harvard-Smithsonian Center for Astrophysics (CfA). "The assumption was that because most bright stars were binaries, all stars would tend to be binaries."​

(Underlining is mine to highlight the point I want to discuss.) The article then says

The catch, however, is that most stars in the Milky Way are not bright stars like our Sun, but dim, low-mass stars called red dwarfs.​

However, red dwarfs are much to small to ever become black holes.

I am still planning to study the article I cited in post #5 about the distribution of matter. Perhaps that will show that the apparently anomaly I discussed above vanishes when a proper analysis of density is taken into account.

Regards,
Buzz

 

[stefan r]

Hi
...estimate of 100 kly, the area of a circular disk of this radius is approximately
[INDENT]3 × 10[SUP]10[/SUP] ly[SUP]2[/SUP].[/INDENT]
This gives a black hole density of
[INDENT]1 black hole per 300 ly[SUP]2[/SUP].[/INDENT]
If I am calculating this correctly, assuming a more-or-less uniform population density of black holes, this means that the average distance from an arbitrary point to its nearest black hole would be approximately
[INDENT]7 ly.[/INDENT]

Your calculation assumes disc was 1 light year thick. The [URL='https://en.wikipedia.org/wiki/Thin_disk']thin disc[/URL] is closer to 1000 light years.

[QUOTE="Buzz Bloom, post: 5943206, member: 547865"]...This conclusion seems to contradict the generally accepted concepts that (1) there are definitely more sun sized or larger stars in binary systems than not in binary systems, and (2) that black holes are the end states of sufficiently large stars.
[/QUOTE]

You could say that black holes form in super novas and in black hole mergers. Either would decrease the number of black holes in binaries. The mergers were binary now they are not. Supernova remnants can lurch in the supernova. Large stars also shed a lot of mass. The binaries usually formed in the same cloud. I would expect a large star to have a large companion. A lot of the mass lost by a giant star can end up in the companion star which makes the companion age faster.

 

[JMz]

A much simpler estimate (given that the original question didn't ask only about known BH's): From the 100M estimate for the total number in the Milky Way, we can say that this is roughly 1/1000 of the total number of stars [or a little less]. So the distance to the closest one should be ~ 10 [or a little more] times the distance to the closest star, Alpha Centauri, or 10*4.3 LY = 43 LY.

A little more careful estimate, based on a few more stars (just so we don't stumble if Alpha Centauri is much farther or closer than average): There are 10 stars within 10.3 LY as it happens, so a typical distance among stars in our neighborhood, from a typical one to its closest neighbor, is 10.3/10^(1/3) = 4.8 LY. So a typical distance to a BH would be ~ 48 LY [or, again, a little more]. (Conclusion: Alpha Cen is not much farther or closer than average.)

If you wanted to use an even larger sample, over a larger region, there are ~ 2000 stars within 50 LY. So you'll get a similar distance. (Try it yourself.)

 

[stefan r]

Is it astronomically known what the closest distance is between Earth and a black hole? (I was not able to locate an answer to this question searching the Internet.) If not, the article

https://www.sciencenews.org/article/we-share-milky-way-100-million-black-holes​

says

The Milky Way teems with black holes — about 100 million of them.​

Based on this estimate, and taking into account the relative density of matter based the distance from the center of the Milky Way, what would be a reasonably accurate estimate of the distance between Earth and the closest black hole?

Here is the free arxive article.

we can say that this is roughly 1/1000 of the total number of stars [or a little less]. ..

I do not think you can make that comparison. Most of the black holes are coming from population II and population III. We do not see population III stars anymore. The Milky Way's population II stars tend to be spread out in the bulge or halo. The disc formed several billion years after the big bang. Population I stars tend to be smaller which means they live longer. Population I stars have a harder time forming large black holes. We see a lot of population I stars in the disc of spiral galaxies. If we take the total number of stars that ever existed the frequency of large blue stars increases. The average location of all stars that ever existed would be much further from the disc's plain than the average location of stars we can see today.

 

[JMz]

Not much disagreement with your statements -- and I wouldn't feel an extra factor of 3 was unreasonable. However, unless the estimate of 100M BHs given in the OP depends sensitively on all of those specifics, then it seems like overkill to analyze their progenitors so closely.

I didn't interpret the OP as a question asking for much precision. Specifically, it mentioned basing the estimate on density, not on history, so the disk gets more attention than the halo. So I gave a back-of-the-envelope answer that Buzz Bloom could do without any special modeling.

 

[Buzz Bloom]

Your calculation assumes disc was 1 light year thick. The thin disc is closer to 1000 light years.

Hi stefan:

My intent was to ignore the thickness, since it is much smaller than the radius, and I visualized a projection of the position of all stars (including black holes) onto the central plane of the Milky Way. The per kly² calculation was the result of this projection.

Regards,
Buzz

 

[ohwilleke]

Here is the free arxive article.
I do not think you can make that comparison. Most of the black holes are coming from population II and population III. We do not see population III stars anymore. The Milky Way's population II stars tend to be spread out in the bulge or halo. The disc formed several billion years after the big bang. Population I stars tend to be smaller which means they live longer. Population I stars have a harder time forming large black holes. We see a lot of population I stars in the disc of spiral galaxies. If we take the total number of stars that ever existed the frequency of large blue stars increases. The average location of all stars that ever existed would be much further from the disc's plain than the average location of stars we can see today.

You aren't wrong, although as a first order estimate, the black hole per star figure is a pretty decent place to start. The point someone made on binary starts being more likely to have black holes is also relevant. Given those considerations though, I still think an order of magnitude estimate for the nearest undetected BH being in the hundreds of light years isn't a bad one.

 

[stefan r]

...estimate that in the region of the Milky Way near the Earth, the population density of non-binary black holes is about 1000 times greater than the population of binary system black holes? This conclusion seems to contradict the generally accepted concepts that (1) there are definitely more sun sized or larger stars in binary systems than not in binary systems, and (2) that black holes are the end states of sufficiently large stars...

... The point someone made on binary starts being more likely to have black holes is also relevant.

The paper had this to say:

Here we have introduced a new parameter that we refer to as the “binary black hole efficiency”: ≡ f_b*×f_m1/m2×f_surv×f_t<1. This dimensionless quantity parameterizes our ignorance of merg-ing black holes from massive stars. The value
f_{b*[/SUB is the massive star binary fraction (fb*∼0.5: e.g. Sana et al., 2012; Kobul-nicky & Fryer, 2007; Pfalzner & Olczak, 2007) and fm1/m2 is the fraction of massive binary systems with mass ratios near unity. Current models predict fm1/m2 ~0.1 for m1/m2 = 0.9(Sana et al.,2012). The fraction of those massive star binaries that survive as black hole pairs after stellar evolution isfsurv (Belczynski et al., 2016a; Lamberts et al., 2016). Finally,ft represents the fraction of binary black holes with orbital configurations that make them available to merge before the present day (ft<1).}

So about half of massive stars are in binary fractions. At least that is what the same authors who said there were 100 million black holes around the Milky Way believe.

They also use unit like density per cubic gigaparsec.

Hi stefan:

My intent was to ignore the thickness, since it is much smaller than the radius, and I visualized a projection of the position of all stars (including black holes) onto the central plane of the Milky Way. The per kly² calculation was the result of this projection.

Regards,
Buzz

I have no problem with projections. Just have to be careful because statements based on projections can be taken out of context: Unless I did a geometry error the galaxies of the coma cluster are closer to us than Andromeda and Triangulum galaxies. Also Earth is flat but parts of it flip upside down twice a day. Other parts of Earth follow ellipses in 24 hour cycles and the center of the ellipse is not the center of Earth's gravity except for the equator.

 

[JMz]

You aren't wrong, although as a first order estimate, the black hole per star figure is a pretty decent place to start. The point someone made on binary starts being more likely to have black holes is also relevant. Given those considerations though, I still think an order of magnitude estimate for the nearest undetected BH being in the hundreds of light years isn't a bad one.

If it's 100s of LY (say, 400 to be concrete), then the BH density is 1000x lower in our neighborhood than the galactic average. That seems like a large enough discrepancy that there should be a very clear, primary reason. (Circumstances in which lots of 2nd-order effects all push an estimated result up, but each only by a little, are uncommon in practice.) What would that primary reason be?

 

[Drakkith]

If it's 100s of LY (say, 400 to be concrete), then the BH density is 1000x lower in our neighborhood than the galactic average. That seems like a large enough discrepancy that there should be a very clear, primary reason. (Circumstances in which lots of 2nd-order effects all push an estimated result up, but each only by a little, are uncommon in practice.) What would that primary reason be?

I don't know of the "primary" reason, if there is one, but I'd venture a guess and say that the lower density of stellar objects in this area of the galaxy (compared to other regions like the core and globular clusters) and the difficulty in finding black holes are probably two main reasons.

 

[JMz]

I don't know of the "primary" reason, if there is one, but I'd venture a guess and say that the lower density of stellar objects in this area of the galaxy (compared to other regions like the core and globular clusters) and the difficulty in finding black holes are probably two main reasons.

Both of those certainly affect the proximity of known BH's. But the OP asked for an estimate to the nearest, not only a value for the nearest known one. And the local star density affects the proximity of stars in just the same way as the proximity of BHs (apart from "2nd-order" effects), so I believe the estimate still stands.

 

[Drakkith]

Both of those certainly affect the proximity of known BH's.

Those factors should affect the density (and thus proximity) of unknown black holes as well if I'm not mistaken.

 

[JMz]

Those factors should affect the density (and thus proximity) of unknown black holes as well if I'm not mistaken.

Not sure how the lower density of stars here (than in the galactic center, say) should affect the relative density of BH's -- relative to stars, that is. My back-of-the-envelope simply assumed that that ratio in our neighborhood is about equal to the galactic average. A simple statistic that we could estimate right from the OP, given just the total number of stars in the MW.

 

[Drakkith]

Not sure how the lower density of stars here (than in the galactic center, say) should affect the relative density of BH's -- relative to stars, that is. My back-of-the-envelope simply assumed that that ratio in our neighborhood is about equal to the galactic average. A simple statistic that we could estimate right from the OP, given just the total number of stars in the MW.

My apologies, I didn't realize you were talking about the black-hole-to-star ratio.

 

[JMz]

My apologies, I didn't realize you were talking about the black-hole-to-star ratio.

Ah! Yes, just a tactic to get a simple (fairly accurate?) distance estimate.

 

[Chronos]

Considering the nearest known neutron star [RX J1856.5-3854] is about 400 ly distant and that neutron stars are probably more common than black holes, it's no surprise the nearest known black hole suspect is a few thousand light years off. Both star types are difficult to detect, which makes their actual abundance uncertain, A black hole is potentially easier to detect because a] they can really stand out in a crowd with proper feeding. b] they are vulnerable to exposure via gravitational lensing in photo surveys.

 

[Tom Kunich]

Is it astronomically known what the closest distance is between Earth and a black hole? (I was not able to locate an answer to this question searching the Internet.) If not, the article

https://www.sciencenews.org/article/we-share-milky-way-100-million-black-holes​

says

The Milky Way teems with black holes — about 100 million of them.​

Based on this estimate, and taking into account the relative density of matter based the distance from the center of the Milky Way, what would be a reasonably accurate estimate of the distance between Earth and the closest black hole?

Because black holes require density of matter the presence of black holes would be exponentially higher towards the center of a galaxy.

 

[stefan r]

Because black holes require density of matter the presence of black holes would be exponentially higher towards the center of a galaxy.

A black hole requires a massive star. Larger stars are more likely to form in a cloud of hydrogen and helium. Increased metalicity decreases the likelihood of a black hole forming.

 

[JMz]

Because black holes require density of matter the presence of black holes would be exponentially higher towards the center of a galaxy.

That is a shaky argument to make. (And by "exponentially", I assume you mean ... "much".) BHs don't depend on the density of stars within a few LY, something which is indeed much larger in the core. They depend on matter within a few million miles, which is a very local phenomenon: local to a gas cloud, local to a "solar" system, etc. Those environments are more common in the core, in roughly direct proportion to the stars themselves.

 

[gary350]

No one has visited a black hole to see for sure it is really there. Black hole is a theory.

 

[Drakkith]

No one has visited a black hole to see for sure it is really there. Black hole is a theory.

Black holes are predictions of the General Theory of Relativity. They're not really a theory in and of themselves. Not anymore than planets or stars are theories. In any case, we don't really need to visit a black hole to reasonably conclude that they exist. We have substantial evidence that black holes exist backed up by solid theoretical justification for them. For example, observations of the supermassive black hole at the center of our galaxy show the presence of roughly 4 million solar masses worth of material contained within a volume of space only 45 au in radius (a little bit beyond the semi-major axis of Pluto's orbit) and possibly less. The best explanation of how such an enormous amount of mass is contained within such a small region of space is that there is a black hole there.

And that's only one piece of evidence.

 

[JMz]

No one has visited a black hole to see for sure it is really there. Black hole is a theory.

"Theory" is not a word that should be bandied about lightly. Theory is the word we use for the most well established of scientific concepts, the ones for which the evidence is overwhelming. (The very same concept used to be called a law. Just a change of vocabulary in the scientific community.)

Electrons, alpha Centauri, and the center of the Earth are also places we've never been. That doesn't mean they are mere speculations. But the evidence for all of them is overwhelming, and much of it is based on deep scientific understanding, not only direct observation. So also with BH's and General Relativity.

 

[Buzz Bloom]

The following is the clearest description of the anomaly I have been able to assemble regarding the number of black holes known near the earth. (Calculations and sources are presented later.)

Within 5000 ly of Earth there are 600 million stars.
http://www.atlasoftheuniverse.com/5000lys.html

The number of stars per black holes in the Milky Way is ~1800. Therefore assuming the ratio of stars to black holes is more-or-less a constant in smaller regions of the Milky Way, there should be about 240,000 binary system black holes within 5000 ly of Earth. But there is only one known black hole (and it is in a binary system) within 5000 ly of Earth.

CALCULATIONS
https://en.wikipedia.org/wiki/Milky_Way
Total mass of stars : 100 billion sun-mass units

https://en.wikipedia.org/wiki/Stellar_classification
TYPE MASS FRACTION
O >=16 ~0.00003%
B 2.1 - 16 0.13%
A 1.4 - 1.8 0.6%
F 1.15 - 1.4 3%
G 0.8 - 1.4 7.6%
K 0.45 - 0.8 12.1%
M 0.48 - 0.45 76.45%
(My apologies for the chart being hard to read. I do not know how to format it better here.)
-> Average star mass ~0.56 sun-mass units
-> number of stars in Milky Way ~180 billion

https://www.sciencenews.org/article/we-share-milky-way-100-million-black-holes

The Milky Way teems with black holes — about 100 million of them.​

-> Number of stars per BH ~1800

http://www.atlasoftheuniverse.com/5000lys.html
600 million stars within 5000 ly -> ~340,000 BHs
Assuming 50% of BHs are in binary systems -> 170,000 binary system BHs within 5000 ly.
However, there is only one known black hole within 5000 ly.

The following is the closest known binary system black hole to Earth at 3300 ly.
https://en.wikipedia.org/wiki/A0620-00

The second closest known binary system black hole seems to be the following at 6100 ly.
https://en.wikipedia.org/wiki/Cygnus_X-1

 

[snorkack]

Both star types are difficult to detect, which makes their actual abundance uncertain, A black hole is potentially easier to detect because a] they can really stand out in a crowd with proper feeding. b] they are vulnerable to exposure via gravitational lensing in photo surveys.

1) Neutron star and black hole both have gravity - neutron stars should also show up in gravitational lensing.
2) Both can emit, if properly fed matter AND angular momentum, to form an accretion disc. The difference is that a black hole has no hair but an event horizon. A neutron star can be a pulsar - black hole cannot. The energy left over from accretion disc must still be emitted by neutron star - in black hole, it just sinks to event horizon.

 

[JMz]

1) Neutron star and black hole both have gravity - neutron stars should also show up in gravitational lensing.
2) Both can emit, if properly fed matter AND angular momentum, to form an accretion disc. The difference is that a black hole has no hair but an event horizon. A neutron star can be a pulsar - black hole cannot. The energy left over from accretion disc must still be emitted by neutron star - in black hole, it just sinks to event horizon.

Indeed, those both affect the detection of BHs, and therefore of the closest known BH, though not of the estimated distance to the very closest.

 

[JMz]

The following is the clearest description of the anomaly I have been able to assemble regarding the number of black holes known near the earth. (Calculations and sources are presented later.)

Within 5000 ly of Earth there are 600 million stars.
http://www.atlasoftheuniverse.com/5000lys.html

The number of stars per black holes in the Milky Way is ~1800. Therefore assuming the ratio of stars to black holes is more-or-less a constant in smaller regions of the Milky Way, there should be about 240,000 binary system black holes within 5000 ly of Earth. But there is only one known black hole (and it is in a binary system) within 5000 ly of Earth.

CALCULATIONS
https://en.wikipedia.org/wiki/Milky_Way
Total mass of stars : 100 billion sun-mass units

https://en.wikipedia.org/wiki/Stellar_classification
TYPE MASS FRACTION
O >=16 ~0.00003%
B 2.1 - 16 0.13%
A 1.4 - 1.8 0.6%
F 1.15 - 1.4 3%
G 0.8 - 1.4 7.6%
K 0.45 - 0.8 12.1%
M 0.48 - 0.45 76.45%
(My apologies for the chart being hard to read. I do not know how to format it better here.)
-> Average star mass ~0.56 sun-mass units
-> number of stars in Milky Way ~180 billion

https://www.sciencenews.org/article/we-share-milky-way-100-million-black-holes

The Milky Way teems with black holes — about 100 million of them.​

-> Number of stars per BH ~1800

http://www.atlasoftheuniverse.com/5000lys.html
600 million stars within 5000 ly -> ~340,000 BHs
Assuming 50% of BHs are in binary systems -> 170,000 binary system BHs within 5000 ly.
However, there is only one known black hole within 5000 ly.

The following is the closest known binary system black hole to Earth at 3300 ly.
https://en.wikipedia.org/wiki/A0620-00

The second closest known binary system black hole seems to be the following at 6100 ly.
https://en.wikipedia.org/wiki/Cygnus_X-1

Yup. Note that your assumption of the more-or-less-constant ratio leads directly to the estimate of a few dozen LY. (I used 1000, instead of 1800, so the cube root is particularly easy. ;-)

 

[snorkack]

Analyze the statistics a bit:

Almost all M and K dwarfs ever formed are still around - the world has not lasted long enough for any of them to burn out. The only possible fates have been stellar collision or ejection from Milky Way, both of them rare.
Oldest G dwarfs are now burning out.
And for F, A, B and O dwarfs, the dwarfs presently existing are a small fraction of stars that have existed and burnt out.
Furthermore, the star formation rate of Milky Way has not been constant - young Milky Way at some time held more young stars at one time than now.
Now, what becomes or these short lived dwarfs?
G, F and A dwarfs become white dwarfs.
Above a certain mass, stars commonly become neutron stars.
What becomes of them?
Neutron stars can be destroyed by merger.
Also many pulsars have high peculiar velocities.
It appears that the formation of neutron stars often (not always) gives them high peculiar velocities.
Which fraction of pulsars in Milky Way are bound thereto? Which fraction of neutron stars formed in Milky Way is left in Milky Way?

What is the minimum mass of star to form a black hole rather than neutron star?
Does formation of black holes tend to give holes initial peculiar velocities, like the formation of pulsar does? How does the distribution of black hole initial peculiar velocities compare against the distribution of pulsar initial peculiar velocities?
Which fraction of black holes formed in Milky Way are left in Milky Way?