Since the moon is receding from earth, solar and lunar eclipses become more rare. Earth will receive the last solar eclipse in around a billion years, but at the same time the Earth will look smaller from the moon. When will the last lunar eclipse happen?
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asked Oct 27, 2022 at 23:52
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1$\begingroup$ Most likely the last lunar eclipse will not be due to the current orbital effects. The moon already moves far enough away to produce an annular eclipse, yet won't produce the last total eclipse for more thatn a billion years. The Sun will swell large enough to disrupt the Earth's orbit in about 5 billion, so that will hapen first. The Sun's size would also have an effect on total vs partial eclipses. $\endgroup$– Greg MillerCommented Oct 28, 2022 at 0:29
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1$\begingroup$ Let's just assume that the sun's lifespan is increased, when would the last lunar eclipse happen? $\endgroup$– InfinitySwordofDiamondCommented Oct 28, 2022 at 2:04
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2 Answers
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Never, unfortunately.
Ignoring the Sun, the angular momentum of the Earth-Moon system is constant. As the tides slow down the Earth’s rotation and increase the Earth-Moon distance, more and more angular momentum is transferred from the Earth’s rotation to the Moon’s orbit.
At present the angular momentum of the Earth’s rotation is $7.2\times 10^{33} kg\, m^2\, s^{-1}$ and that of the Moon’s orbit is $41\times 10^{33} kg\, m^2 \,s^{-1}$.
Let’s over-estimate the final state of the Moon by assuming that all the Earth’s angular momentum is transferred to it. That would increase the Moon’s angular momentum from $41$ to $48.2$ in these units: a factor of about $1.175$.
The radius of an orbit is proportional to the square of the angular momentum. So the radius of the Moon’s orbit will increase by less than 40%.
This reduces the width of the Earth’s shadow at lunar distance - but given how wide the shadow is relative to the Moon, we will still have lunar eclipses, and decently long ones at that.
answered Oct 28, 2022 at 7:51
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First off, the sun will completely throw off the orbits of the Earth and moon, if not obliterate them both. But, like you said in your last comment, we will expand the sun’s lifetime with some magic.
The moon moves away 3.78 centimeters a year, and is already 384,000 kilometers away (or 238,855 miles). With a bit of research, I have figured out that Earth’s umbra is 2.578 to 2.725 moon radii (2.652 average). If we multiply 384,000 km by 2.652 (because that’s how much bigger the moon has to be in order to no longer produce a total lunar eclipse), we get an answer of 1,018,368 kilometers. Subtracting our answer with the moon’s current distance gives us 634,368 km. This is how much further the moon has to move back in order for total lunar eclipses to be no longer possible (on average). Done with some simple division, if the moon moves back 3.78 cm a year, it will take approximately 16,782,222,222.2 years (16.782 billion years) for total lunar eclipses to be no longer possible on average.
