Earth aphelion would add about 1.7% to the umbra length.

But!  Oover half a billion years, the Sun will have expanded a bit (4-5% radius), reducing the umbra in proportion.

Then of course, there will be changes in the eccentricity of both Moon's & Earth's orbits over that time scale…

On Mon, Aug 28, 2017 at 11:44 AM, Dan Corrin (via tml list) <xxxxxx@simplelists.com> wrote:
According to sources the moon's umbra is 380,000 km, the moon orbits at 384,400 km (362,600 to 405,400) while the radius of the earth is 6378 km.
So when the moon moves far enough away that the umbra doesn't touch the earth, there will be no total eclipses.
On average then the umbra is 1978km more than the distance to the surface of the earth, but if the moon is in Perigree or Apogee when the eclipse occurs it can be ten times this distance, or already too small to touch the earth.
Also the size of the umbra presumably differs slightly as the Earth's orbit is an ellipse as well.
In any case at 3.8cm per year, the 1978km would take some 52Myr, and in the best case 10x that or ~520Myr


On Monday, August 28, 2017 7:58 AM, Phil Pugliese (via tml list) <xxxxxx@simplelists.com> wrote:



Can anyone calculate how long it will be before the moon moves so far away from the earth that only annular eclipses will be possible?

Just curious,

TIA,

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