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I think there is a misprint in an article. I will include the link if you don't mind and cut paste the sentence that I think is a misprint. Here is the link https://www.abc.net.au/science/articles/2012/04/17/3478276.htm#:~:text=The%20power%20output%20of%20the,can%20it%20be%20so%20low%3F

...and here is the sentence in the article"

" On average, any given hydrogen atom will run into another hydrogen atom only once every five billion years."

Overall I think it is an interesting article but maybe I am not grasping what the author has intended with this sentence.

By the way I wasn't specifically looking for this but ran across it by accident while trying to find the rate that fusion occurs in the sun's core and compare that to man made fission or fusion events I haven't finished researching that and then I ran into this and became puzzled...

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    $\begingroup$ this seems to be a popular phrase , found it here also zhihu.com/question/62023350/answer/1009119177 with google . I think the clue is on "given", if the plasma of hydrogen is dilute enough. This question/answer is relevant, physics.stackexchange.com/q/73970 $\endgroup$
    – anna v
    Commented Jul 24, 2022 at 20:06
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    $\begingroup$ If it means "fuse with another hydrogen nucleus", think of this fact: Then Sun is 4.6 billion years old. But most of the hydrogen nuclei in it are still hydrogen atoms, after this many years. So the probability a "random" hydrogen nucleus in the Sun will merge with another hydrogen nucleus within a period of 4.6 billion years is less than 50%. $\endgroup$
    – Jeppe Stig Nielsen
    Commented Jul 24, 2022 at 20:20
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    $\begingroup$ @JeppeStigNielsen I don't know if it's a good argument. If one divides total helium atoms fused per sun lifetime, i.e. $((0.25 \times 1.9885 \times 10^{30}~kg) / (2m_p+2m_n)) / (4.6~\text{billion years})$ (assuming main abundance of ${}^4He$ isotope, then one gets that Hydrogen conversion to Helium frequency is $10^{38}~\text{atoms}/\text{second}$. As for me, this number is huge and it shows that extrapolating "fusion probability" from a sun lifetime is not very appropriate thing, because low fusion rates are compensated with big particle numbers and "enormous" lifetime of sun. $\endgroup$
    – Agnius Vasiliauskas
    Commented Jul 24, 2022 at 21:09
  • $\begingroup$ FWIW, the ABC used to run an internet forum for fans of Dr Karl. We could attend to little mistakes that he made on his radio show, and answer follow-up questions from listeners. Sadly, due to budget cuts, they closed that forum about a decade ago. $\endgroup$
    – PM 2Ring
    Commented Jul 25, 2022 at 13:38

2 Answers 2

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This statement is indeed misleading.

For a given hydrogen atom, the frequency of collisions in the center of the Sun is enormously high, some $10^{17} s^{-1}$ or thereabout. So it will "run into and collide with other protons" all the time.

In contrast, the Sun's livetime is of order $10^{10}$ years, so it will burn through half its hydrogen inventory over a time of order $5\times 10^9$ years, by means of nuclear fusion. Thus, it will take a given hydrogen nucleus billions of years to "run into another proton and fuse with it".

You can cross-check that: On average that's $10^{17}*10^7*10^9=10^{33}$ collisions before the Coloumb barrier (a few MeV) is overcome. The $10^7$ Kelvin at the center of the Sun correspond to 1keV. Particles follow a Boltzmann distribution, which at high energies goes like $\sim exp(-1/T)$. You just need to integrate this distribution (still just an exponential...) to find that tail where the integral is $10^{-33}$, which for a characteristic temperature of 1keV, should be about 1MeV.

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  • $\begingroup$ As I said here: physics.stackexchange.com/a/540199/123208 The main bottleneck in the proton-proton chain isn't the fusion of two protons to form a diproton, it's the conversion of the diproton to a deuteron. [...] It's estimated that (in the solar core) the probability of a diproton converting to a deuteron is in the order of $10^{-26}$. $\endgroup$
    – PM 2Ring
    Commented Jul 25, 2022 at 13:14
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If “run into” means takes part in a fusion reaction then this is correct as an order of magnitude approximation. Wikipedia says:

... each proton (on average) takes around 9 billion years to fuse with one another using the PP chain

As noted in comments, if the average lifetime of a lone proton were very much shorter than this then the Sun would have already used up most of its hydrogen. However, we know that the proportion of hydrogen in the core of the Sun is still around $40\%$ and the Sun is only about half way through its main sequence life span.

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