What was the scientific theory of how the sun worked around or just prior to 1905?

That the sun produced heat internally through a radioactive process.

Here’s an interesting article on the history of solar physics – How the Sun Shines by John N. Bahcall

In short, before the discovery of radioactivity, it was thought the Sun’s heat was from chemical reactions (burning) or gravity turning to heat. And based on that and the mass of the Sun, the lifetime of the Sun could be about 30 million years.

Then the discovery of radioactivity allowed for much, much energy from the same mass.
Ernest Rutherford in 1904:
“The discovery of the radio-active elements, which in their disintegration liberate enormous amounts of energy, thus increases the possible limit of the duration of life on this planet, and allows the time claimed by the geologist and biologist for the process of evolution.”

And in 1920 nuclear fusion was posited as the source of solar energy:

“Sir Arthur Eddington, the brilliant English astrophysicist. Eddington argued in his 1920 presidential address to the British Association for the Advancement of Science that Aston’s measurement of the mass difference between hydrogen and helium meant that the sun could shine by converting hydrogen atoms to helium. This burning of hydrogen into helium would (according to Einstein’s relation between mass and energy) release about 0.7% of the mass equivalent of the energy. In principle, this could allow the sun to shine for about a 100 billion years”

Bravo. Wonderful answer