How close is Earth to the closest black hole?

[Buzz Bloom]

Hi @JMz:

I have looked over my spreadsheet and my source articles, and I have decided I need to make some revisions, primarily because some of the more recently found references seem to be more reliable. When I complete this, I will report the means and standard deviations as well as quartile numbers with standard deviations from multiple Monte Carlo runs.

I have listed below the various variable values I will use in the next version, and the sources for the data, as well as a few comments.

https://www.sciencenews.org/article/we-share-milky-way-100-million-black-holes
It is difficult to judge from this abstract any range and confidence limits.
However, I guess that it is plausible to assume that the 1 sigma base 10 logarithmic stdev is 0.5

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

The total mass of all the stars in the Milky Way is estimated to be between
4.6×10¹⁰ M☉ and 6.43×10¹⁰ M☉.
Average: 5.515 10¹⁰ M☉.
Stdev = 1.3 10¹⁰ M☉.
Geometric average: 5.44 10¹⁰ M☉.

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

Average mass of a MW star is 0.54 M☉.
Estimated number of stars: 10 +/- 2.4 x 10¹⁰ M☉ or 10^{11 +/- log₁₀(2.4)}
It is not clear whether I should consider this to be a Gaussian distribution, or the logarithm to be a Gaussian distribution.
Estimated number of stars per BH = 1000 - Stdev to be calculated.

http://www.atlasoftheuniverse.com/12lys.html​

33 stars within 12.5 ly (bigger than brown dwarfs)
1000 stars at this density would require a sphere of radius (1000/33)^(1/3) * 12.5 = 39 ly.
Average distance from center for a random point is 39 * 3/4 ~= 29 ly.

Regards,
Buzz

 

[JMz]

Hi @JMz:

I have looked over my spreadsheet and my source articles, and I have decided I need to make some revisions, primarily because some of the more recently found references seem to be more reliable. When I complete this, I will report the means and standard deviations as well as quartile numbers with standard deviations from multiple Monte Carlo runs.

I have listed below the various variable values I will use in the next version, and the sources for the data, as well as a few comments.

https://www.sciencenews.org/article/we-share-milky-way-100-million-black-holes
It is difficult to judge from this abstract any range and confidence limits.
However, I guess that it is plausible to assume that the 1 sigma base 10 logarithmic stdev is 0.5

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

The total mass of all the stars in the Milky Way is estimated to be between
4.6×10¹⁰ M☉ and 6.43×10¹⁰ M☉.
Average: 5.515 10¹⁰ M☉.
Stdev = 1.3 10¹⁰ M☉.
Geometric average: 5.44 10¹⁰ M☉.

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

Average mass of a MW star is 0.54 M☉.
Estimated number of stars: 10 +/- 2.4 x 10¹⁰ M☉ or 10^{11 +/- log₁₀(2.4)}
It is not clear whether I should consider this to be a Gaussian distribution, or the logarithm to be a Gaussian distribution.
Estimated number of stars per BH = 1000 - Stdev to be calculated.

http://www.atlasoftheuniverse.com/12lys.html​

33 stars within 12.5 ly (bigger than brown dwarfs)
1000 stars at this density would require a sphere of radius (1000/33)^(1/3) * 12.5 = 39 ly.
Average distance from center for a random point is 39 * 3/4 ~= 29 ly.

Regards,
Buzz

This is turning into a substantial project for you, isn't it? :-)
FWIW, log-Normal will give virtually the same answers as Normal, because the SD is not large compared to the mean. (And if log-N isn't close enough to right, then the right answer is almost dependent on observational and model uncertainties in the published research, so a Normal will likely be even further from the mark.)

 

[Buzz Bloom]

FWIW, log-Normal will give virtually the same answers as Normal, because the SD is not large compared to the mean.

Hi JMz:

I will accept your advice and calculate the variables as log normal.

Regards,
Buzz

 

[Buzz Bloom]

Out of curiosity, Buzz, what are the median and quartile distances?

Hi JMz:

I completed the revised spreadsheet, and I used the Box-Muller method for generating Gaussian random numbers.

Input values

33 stars with 12.5 ly - assumed this was a good approximation of star population density in the Earth region.

http://www.atlasoftheuniverse.com/12lys.html
​

N_BH = random number of BHs in MW is a log Gaussian distribution.
log₁₀ N_BH = 8 +/- 0.5

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

The total mass of all the stars in the Milky Way is estimated to be between
Mstars[/SUB = 4.6×10^10 M☉ and 6.43×10^10 M☉.
M_stars random number of the mass of stars (stellar mass units) in MW is a log Gaussian distribution.

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

Average mass of a MW star is 0.54 M☉.
https://en.wikipedia.org/wiki/Stellar_classification

N_stars random number of stars in MW is a log Gaussian distribution.
N_stars = M_stars / 0.54
log₁₀ N_stars = 10^(11 +/-0.11) M☉.
ρ = N_stars / N_BH
ρ is also a log Gaussian distribution.
log₁₀ ρ = 3 +/- √(0.5² + 0.11²) = 3 +/- 0.512.

Monte Carlo

I generated 25 runs of 100 random numbers for ρ. I sorted each run of 100 to get percentile values.
I calculated the m=average, and std=standard deviations for each percentile from the 25 runs.
I will report below these m and std values for the following percentiles:

16, 25, 50, 75, 84.​

The 16th and 84th percentile values would correspond to to m-std and m+std for a Gaussian distribution rather than a log Gaussian distribution.

R = 12.5 × (ρ/33)^1/3
I also calculated values m, m-std and m+srd corresponding to 10³, 10^2.488, and 10^3.512 values of ρ.

For each of these eight ρ values, I calculate the corresponding radius R of a sphere containing the corresponding number of stars, and the average distance D to a point in the sphere from the center.
D = 3 R / 4

Regards,
Buzz

 

[JMz]

The only question that comes to mind (beyond previously mentioned stellar-evolution & galaxy-evolution questions that might influence our expectations of BH density vs. position in the MW) is: You cite 0.54 M_☉ as the MW mean, but does that closely match the mean for the stars in the local neighborhood, 12 LY? I am just wondering if there is a selection effect. (Obviously, we have known about all the bright nearby stars since antiquity, but not the fainter ones.)

My guess is, Yes, over that distance, we know them all. But if there were a selection effect, it would favor brighter stars, for which there would be far fewer in the MW as a whole, so the BH/star ratio would need to be adjusted upward. But even that is surely just a minor adjustment: Even a large adjustment will only change the distance estimate by a cube root.

 

[Buzz Bloom]

You cite 0.54 M☉ as the MW mean, but does that closely match the mean for the stars in the local neighborhood, 12 LY?

Hi JMz:

That's a good question, but I will need a break for a while before I can explore it.

The list of the 33 stars with their characteristics is in
http://www.atlasoftheuniverse.com/12lys.html .
The data I use to arrive 0.54 is in
https://en.wikipedia.org/wiki/Stellar_classification .

If you have the time you might get the characteristics for each of 33 stars, and look up the mass for each type, and calculate their average mass.

Regards,
Buzz

 

[JMz]

Hi JMz:

That's a good question, but I will need a break for a while before I can explore it.

The list of the 33 stars with their characteristics is in
http://www.atlasoftheuniverse.com/12lys.html .
The data I use to arrive 0.54 is in
https://en.wikipedia.org/wiki/Stellar_classification .

If you have the time you might get the characteristics for each of 33 stars, and look up the mass for each type, and calculate their average mass.

Regards,
Buzz

A quick check: Another link on the first page you listed is http://www.atlasoftheuniverse.com/nearstar.html, which gives spectral class and lists them in order of distance. The ones closer than 12 LY (and even the rest within 20 LY) are overwhelmingly class M or sometimes K. These are all low-mass stars (no red giants to complicate the relationship to spectral class). So I do believe that they are typical of the average mass in the MW, about 1/2 solar.

 

[QuasBlackWorm MIT]

But I really hope Proxima Centauri implodes oh itself and becomes a black hole soon
That should make my field of science much more interesting

 

[mfb]

It won't. Proxima Centauri will live for many billion years and then become a white dwarf.

 

[JMz]

It won't. Proxima Centauri will live for many billion years and then become a white dwarf.

In fact, this suggests a time scale that greatly understates Proxima's lifetime. It is estimated that it will last 5 to 10 trillion years. (And no black hole, ever.)