Did anyone predict the death of the Sun before science?

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The Aztecs believed that the sun was the fifth in a line of suns, and that each of the preceding four had been destroyed before its successor was created. Various natural disasters had destroyed the previous suns. This one would be destroyed as well, by an earthquake. Its destruction could be delayed by human sacrifices, but not prevented forever. See this article:

Set I. The Five Ages- War and Destruction. Duran (1971: 412) relates that on the walls of the Temple of Warriors within a sacred compound surrounding the Templo Mayor was a large emblem of a butterfly. He tells us that this emblem meant "ollin" or "motion." The root of the word "heart" also meant motion. In this age, not only were still-beating hearts offered as the life-blood of the universe but the warriors, when sacrificed, went to the sky to help the sun each day and, after four years, became butterflies which flitted from flower to flower.

This central image suggests what appears to be the primary ordering principle of this first set - an ordering principle for time. Its basic impetus is a historically sequential, temporal message of eschatology - a vision of time with an emphasis on destruction. The scenario presented in this story is one in which the original timeless unity is divided by four deities into a series of ages, each called a "sun." Each sun is destroyed, apparently completely, for in each sun the inhabitants either "disappear" or "perish." In the fourth sun even the heavens collapse. The fifth sun is begun by divine sacrifice and is also doomed to an inevitable destruction. There is no mention of a sixth age, or of any new creation in the future, nor is there ever a return to that primordial age of of cosmic creation.

So this was not 'circle of life' stuff. The sun was going to be destroyed and there was no happy ending - just putting it off.

The Aztecs are a striking example because the sun was the centre of their mythology. I don't know much about mythology, but it's not uncommon for mythologies to predict the end of the world. Even myths with cyclical notions of time often depict the world recreated and destroyed forever. The Aztecs' mythology was linear, with one beginning and one final destruction of the world. Depending on how cultures conceptualize the world, this often entails the destruction of the sun. From the book of Revelation, Chapter 21:

And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea. And I John saw the holy city, new Jerusalem, coming down from God out of heaven, prepared as a bride adorned for her husband.

...

And I saw no temple therein: for the Lord God Almighty and the Lamb are the temple of it. And the city had no need of the sun, neither of the moon, to shine in it: for the glory of God did lighten it, and the Lamb is the light thereof.

...

And there shall be no night there [and] neither light of the sun; for the Lord God giveth them light: and they shall reign for ever and ever.

Interestingly, earlier in Revelation the sun goes dark because of an earthquake - but then flares up again and burns people. I doubt it means anything, but if you want to write an alternate history in which ancient Israelites sail the Atlantic to meet the Aztecs, there's your excuse.

Read, Kay A. “The Fleeting Moment: Cosmogony, Eschatology, and Ethics in Aztec Religion and Society.” The Journal of Religious Ethics, vol. 14, no. 1, [Journal of Religious Ethics, Inc, Wiley, Blackwell Publishing Ltd], 1986, pp. 113–38, http://www.jstor.org/stable/40015027.

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Yes. Probably the first rigorous hypothesis about the source of Sun's energy is the Helmholtz-Thompson mechanism (or Helmholtz-Kelvin) of gravitational contraction which predicts something like 20 million years solar lifetime.

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In Norse mythology, the Sun, named Sól, is chased by a wolf, named Sköll or Fenrir. At the end of times (Ragnarök), Fenrir will catch Sól and eat her. This is not great news:

High then quotes stanza 45 of Völuspá. Next, High describes that the wolf will swallow the sun, then his brother will swallow the moon, and mankind will consider the occurrence as a great disaster resulting in much ruin. The stars will disappear. The earth and mountains will shake so violently that the trees will come loose from the soil, the mountains will topple, and all restraints will break, causing Fenrir to break free from his bonds.

Alternatively, Sól will have had a daughter before being eaten, and her daughter will replace her in the sky.

Vafþrúðnismál stanza 45 is then quoted. The personified sun, Sól, will have a daughter at least as beautiful as she, and this daughter will follow the same path as her mother.

In stanza 46, Odin asks what sun will come into the sky after Fenrir has consumed the sun that exists. Vafþrúðnir responds that Sól will bear a daughter before Fenrir assails her, and that after Ragnarök this daughter will continue her mother's path.

In yet another version, the Sun would just turn black.

The god Freyr fights Surtr and loses. After this, people flee their homes, and the sun becomes black while the earth sinks into the sea, the stars vanish, steam rises, and flames touch the heavens.

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