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Because the moon distance from earth grows, how long will it take, that it is no longer possible to see a total eclipse but only circular eclipses? And does it make a difference if you are on the mount everest or on sea level?

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    $\begingroup$ About 600 million years. $\endgroup$ – PM 2Ring Jan 19 at 12:50
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    $\begingroup$ This is not a duplicate. That whole thing is about the definition of transits, and the answer is 10 Tyr or 1E+13 years. The answer her 8E+08 years, a factor of 10,000 longer. @PM2Ring has also independently mentioned 6E+08 years. That answer does not answer this question, not even close! $\endgroup$ – uhoh Jan 23 at 10:00
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    $\begingroup$ @CarlWitthoft these are very different questions. By closing, you send people to the wrong answer to this question. $\endgroup$ – uhoh Jan 23 at 10:01
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    $\begingroup$ @uhoh I think the questions are similar in principle. Grimaldi made a reasonable but arbitrary decision about what the angular diameter of the Moon needs to be so that its transit of the Sun doesn't qualify as an eclipse. But the 10 trillion years he calculates isn't really valid. The Sun will be a white dwarf long before then, and the Earth-Moon system will have suffered major disruption during the Sun's red giant phase, and will most likely get swallowed at some stage. $\endgroup$ – PM 2Ring Jan 23 at 15:03
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    $\begingroup$ @Carl If there's any dupe-closing to be done, I'm inclined to go in the other direction, closing coblr's old question as a dupe of this new one. $\endgroup$ – PM 2Ring Jan 23 at 15:06
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I will take a stab at it. Assuming

  • the last eclipse happens when the moon is at perigee and the earth is at aphelion.
  • the moon's perigee is increasing by 4 cm per year,
  • the earth's aphelion does not change with time,
  • the sun and the moon are perfect spheres,
  • the radius of the sun and moon don't change, and
  • a total eclipse occurs when $r_{\mathrm{sun}}/d_{\mathrm{sun}} < r_{\mathrm{moon}}/d_{\mathrm{moon}}$ where $r$ is radius and $d$ is distance,

I get the following results:

  • The last total eclipse at sea level will occur in 721,587,917 years at sea level (obviously too many significant digits).
  • The last total eclipse at mount Everest will occur in 721,807,917 years.
  • The difference is 220,000 years = (height_of_Everest) * 25000 years/km.

Mathematica Source Code:

dSun = 152097000; rSun = 695508; dMoonNow = 357347; rMoon = 1737; 
rEarth = 6371; hEverest = 8 + 8/10;
sol2 = Solve[ rMoon/(d + t 4/100/1000) == rSun/(dSun - rEarth), t][[1]];
Print[ t /. sol2 /. d -> dMoonNow - rEarth  // Round];
Print[ t /. sol2 /. d -> dMoonNow - rEarth - hEverest // Round];
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    $\begingroup$ Over that time scale and to that level of precision, will the Sun's diameter change as it evolves? $\endgroup$ – Steve Linton Jan 19 at 19:08
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    $\begingroup$ I think so! (I added the assumption that it stays constant.) $\endgroup$ – irchans Jan 19 at 19:32
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    $\begingroup$ The Sun will get larger by 2-3% on that timescale. So this answer would be an upper limit. $\endgroup$ – Rob Jeffries Jan 20 at 9:13
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    $\begingroup$ But since the apparent size of the Moon shrinks by 8% in the same time, I would guestimate that the overestimate of the time is only by about 25%. $\endgroup$ – Rob Jeffries Jan 20 at 9:17

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