I How close is Earth to the closest black hole?

[JMz]

Buzz, as Dr. Jeffries has noted, this estimate is sensitive to a number of unknowns - lncluding historical supernova rates and stellar mass distributions, so getting a decent estimate based on current observational data is limited. My best guess is our current empirical data is at no better than the 1 sigma confidence level, so the relative accuracy of the one easily verifiable prediction [nearest white dwarf] is fairly impressive. Fortunately, this is great news for aspiring astrophysicists

Ahh, back to good old days, when "astrophysical accuracy" meant "within a factor of 2" (i.e., the error was in the exponent). ;-)

 

[JMz]

Purely technical question: In what language are you working? (Hence, what is the RAND function?) Matlab? XL? something else?

(I didn't see your statement that you are using XL until after I posted this.)

 

[JMz]

Hi @JMz:

I have decided that I need to redo my Monte Carlo calculations. First, I have found buggy behavior in the spread sheet package I am using on my Windows 7, so I am going to switch to a very old Excel on my very old XP PC that seems to still be a reliable tool, although more limited. Then I need to take into account that the random variable for the number of stars within a specified radius sphere is 10^R where R is a random number from an approximate Gaussian distribution with a mean of 3 and a standard deviation of 0.5, since the calculation is for a sphere containing 1000 stars with a random range between about 300 and 3000. I will post the results when I complete the new spread sheet.

RESULTS
I performed 20 runs of 100 trials each. Each trial generated a random number of stars and of BHs. This was done by generating the log based 10 of the number from an approximate Gaussian distribution with a standard deviation of 0.5, an respective means of 8 and 11. (I assumed that estimates of the 100 million BHs and 100 billion stars were independent.) For each run I calculated the expected value of the ratio of stars to BHs, as well as the corresponding value of 10^(mn+dv) and 10^(mn-sd). The results of the 20 runs showed a mean of approximately 1000 with 65% of the trials in the range 200 to 5000.

The average distance of a BH from the center of a sphere (at a random point in the sphere) with 1000 stars is approximately 40 ly. Taking into account that the size of a sphere corresponding to the star/BH ratio is proportional to the cube root of this ratio, the range of expected distance with a 65% confidence level would therefore be between
40 / 5^1/3 ~= 24 ly and 40 × 5^1/3 ~= 680 ly.

Regards,
Buzz

Typo in that last number: 40*5^(1/3) ~ 68. This all seems quite believable.

 

[Buzz Bloom]

Purely technical question: In what language are you working? (Hence, what is the RAND function?) Matlab? XL? something else?

Hi JMz:

RAND() is a function provided by Excel. It generates a (pseudo) random number with a flat distribution between zero and one.

Regards,
Buzz

 

[JMz]

Hi JMz:

RAND() is a function provided by Excel. It generates a (pseudo) random number with a flat distribution between zero and one.

Regards,
Buzz

OK. Note that there are simple methods to quickly generate exact standard Normal samples from Uniform ones: e.g., https://en.wikipedia.org/wiki/Box–Muller_transform. The polar version (Bell, Knop) needs about 20% more U's than can be used, but both methods otherwise produce ~ 1 Normal for each Uniform, rather than needing several U's for a single N.

 

[hilbert2]

Could there be some "anthropic" reason why a black hole can't exist near a planet that has life on it? If a significant amount of mass per second is spiralling into a BH, how far away from it do you have to be to not get fried by X-rays and gamma rays? I don't even know how to do an order of magnitude estimate for this, but I remember seeing a claim that a supernova explosion taking place too close would destroy everything living on Earth.

 

[graybass]

So we agree the estimated (but not known) distance to be approx. 3-400 LY?

 

[stefan r]

Could there be some "anthropic" reason why a black hole can't exist near a planet that has life on it? If a significant amount of mass per second is spiralling into a BH, how far away from it do you have to be to not get fried by X-rays and gamma rays? I don't even know how to do an order of magnitude estimate for this, but I remember seeing a claim that a supernova explosion taking place too close would destroy everything living on Earth.

No, Jupiter gives of x-rays for example. We also get lots of x-rays from sun storms. Doctors on Earth use molybdenum targets in x-ray generators because the k-alpha radiation is a higher energy. Copper targets are better because they cool faster. Copper is also cheaper and mechanically better. Copper k-alpha radiation is used in research x-ray machines. No high energy radiation is really safe but the medical/dental x-rays are much less damaging to living tissue. The atmosphere blocks most of the x-rays that are most harmful.

The level of x-rays that are life threatening and the level that is detectable has many orders of magnitude difference. So there would be plenty of sources for Chandra to find.

Particles falling in radiate energy.

This process of accretion is one of the most efficient energy-producing processes known; up to 40% of the rest mass of the accreted material can be emitted as radiation.

Someone with a better background with plasmas might be able to estimate the rate interstellar gas falls in. The density of the interstellar medium varies a lot.

 

[Buzz Bloom]

So we agree the estimated (but not known) distance to be approx. 3-400 LY?

Hi graybass:

I can't figure out where your numbers come from, but the statement I would make based on my Monte Carlo calculation with @JMz 's correction, is:
the probability is 65% that the distance to the nearest black hole is between 30 and 85 ly.

Regards,
Buzz

 

[JMz]

Hi graybass:

I can't figure out where your numbers come from, but the statement I would make based on my Monte Carlo calculation with @JMz 's correction, is:
the probability is 65% that the distance to the nearest black hole is between 30 and 85 ly.

Regards,
Buzz

Out of curiosity, Buzz, what are the median and quartile distances? These won't depend on any parameter estimates like the std.dev. They are robust to outliers (non-Normal variates).

 

[mfb]

We know 60 black holes in our galaxy today.

Gaia is expected to find a few thousand. It is basically guaranteed that many of them will be closer than the currently closest known.
As these are double systems, we might have to wait for the final data release, 2022-2024.

 

[stefan r]

... find a few thousand. ...

I 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.​

...

The arxiv article says 10⁵ black holes in the milky way. Did someone miss by x10³?

 

[mfb]

That is a good question. Both numbers are recent, and a factor 1000 is large even for astrophysical uncertainties.

Gaia will measure roughly 1% of the stars in our galaxy, measuring O(1%) of the black holes in binary systems doesn't look unreasonable.

 

[Buzz Bloom]

The arxiv article says 10⁵ black holes in the milky way. Did someone miss by x10³?

Hi stefen:

Here is a quote from the https://arxiv.org/pdf/1703.02551.pdf article.
... a galaxy like the Milky Way should host millions of ∼30M_sun black holes ...
I confess that the bulk of the article is over my head. However, Figure 2 seems to show the basis for estimating the MW BHs.
https://en.wikipedia.org/wiki/Milky_Way says the mass of the MW is ~10¹² solar masses.

As I read this chart, the black line gives the number of BHs of mass greater than 10 solar masses. The X-axis shows the mass of a galaxy. One has to extrapolate the black line since the max Y-axis value is 10⁸.

Regards,
Buzz

 

[Buzz Bloom]

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

Hi JMz:

I will have to go back and explore the Monte Carlo data. It will try to post the answer later today or tomorrow.

Regards,
Buzz

 

[Buzz Bloom]

Gaia is expected to find a few thousand.

Hi mfb:

Thanks very much for the link. The previous abstract about GAIA failed to mention anything about finding black holes.

Regards,
Buzz

 

[snorkack]

That is a good question. Both numbers are recent, and a factor 1000 is large even for astrophysical uncertainties.

The article contains a bland statement that most black holes should be binaries with a luminous partner.

Absurd on its face.

Luminous stars are short lived. A great majority of black holes should not have a luminous partner now, even if they had one at some point of evolution.

 

[mfb]

The article contains a bland statement that most black holes should be binaries with a luminous partner.

Absurd on its face.

Luminous stars are short lived. A great majority of black holes should not have a luminous partner now, even if they had one at some point of evolution.

Why do you expect binary stars to have symmetric masses?

 

[snorkack]

Why do you expect binary stars to have symmetric masses?

I did not.

On closer examination, it turns out that the article does interpret "luminous" apparently to mean "giving off any, even small amount of light", and expressly considers binaries of black hole and low mass old main sequence star - although these are hard to detect.

So: binary systems of a black hole and a massive star should commonly form, but are short-lived.
Binary systems of a black hole and a low mass main sequence star may form less often, but then last longer.

 

[stefan r]

Hi mfb:

Thanks very much for the link. The previous abstract about GAIA failed to mention anything about finding black holes.

Regards,
Buzz

I believe all the black holes GAIA will find have luminous companions. Gaia is looking at momentum and position.

Why do you expect binary stars to have symmetric masses?

There is this paper

...massive stars preferentially have massive companions...

If they came from the same cloud at the same time my first guess would be somewhat similar stars form. They have nearly identical metalicity and were probably kick started by the same shockwave. Most of the poster image open clusters are all the same color. The small dim companions do not show up as well in the poster so that is not an accurate measurement. In pictures of stellar nurseries like Orion nebula you have a group of B and O stars blowing material away and smaller stars forming in the remnants. Again that is not statistical evidence but would be a good first guess.

 

[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.)