Nice question! Normally, when something burns on Earth, it burns through combustion. Combustion requires three things – fuel, air and heat. When we pour water on top of that, it extinguishes the fire by forming a coat of water on top of the fuel.
But the sun produces energy by crushing small atoms to larger ones using its own mass (nuclear fusion)... Naively speaking, the sun should just keep supplying energy and vaporising all the water in the 'bucket' as long as it can keep crushing atoms to form bigger atoms inside.
That much water will actually collapse on itself because of its own gravity. It will get much smaller because of the pressure as it collapses, and it will get much hotter at the centre. This is because the molecules have sort of fallen toward each other. They're moving faster, and that is what heat is – molecules moving fast.
Eventually, they will be moving so fast, that the hydrogen atoms in the water, which will be slamming into each other from time to time, will begin to fuse together to form Helium. Sort of like if you slam oranges together hard enough they could smash through each other's skin and form one bigger mass. When they fuse, they release even more energy, which makes their neighbouring atoms move faster, and drives the process on. Eventually this will make the water 'ignite' into a new sun itself.
A really good question. Maybe too hard to answer for anybody but an astronomer with some special software, as nothing like this happens in nature. A star makes a lot of carbon before it makes any oxygen, and here the oxygen is supplied first.
The probable answer is 'no'. The sun involves a special type of fire that is able to 'burn' water, and so it will just get hotter, and six times brighter.
Water is 89 per cent oxygen by mass. And the sun's overall density is 1.4 times that of water. So if you have a volume of water the volume of the sun, it will have 1/1.4 = 0.71 times the mass of the sun, and this mass will be .71*.89 = 63 per cent of a solar mass of oxygen and 8 per cent of a solar mass of hydrogen. The sun itself is 0.74 solar masses of hydrogen and 0.24 solar masses of helium.
So you end up with a 1.7 solar mass star with composition 48 per cent hydrogen, 37 per cent oxygen, and 14 per cent helium (with 1 per cent heavier elements). Now, will such a star burn? Yes, but not with the type of fusion the sun uses... It will be bluish-white with more UV. Along with six times the heat input the Earth's biosphere will be fried, and oceans probably boil.
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