One common argument that I’ve heard from young-earth advocates is that measurements of the sun show that it is steadily decreasing in size. If this is the case, it would have reached its present size in a hundred million years, or possibly less. Therefore, it hasn’t been burning for the several billion years that are usually claimed for it. The theory behind this is that the sun is actually burning by gravitational contraction, not by nuclear fusion. Prior to the discovery of nuclear power, this was how everyone thought the sun burned. In defense of this, they point out that nuclear burning produces neutrinos, and the amount of neutrinos that have been detected are only a third of what they should be if the sun was burning by nuclear power.
But there are major problems with this scenario. First, if the sun were really getting smaller, total solar eclipses wouldn’t have been possible until very recently. A total solar eclipse occurs when the moon moves between the sun and the earth, blocking out the sun except for its corona. If the sun used to be larger, even slightly, the moon would not have completely blocked it out, and the corona would not have been observable; in fact, prior to the 20th century, this was one of the only ways that people could observe and learn about the sun and its corona. Yet total solar eclipses have been observed throughout all of recorded human history. Obviously the sun is the same size today as it has been for as long as human beings have been observing solar eclipses.
Second, the pressure and temperature at the center of any mass the size of the sun inevitably ignites nuclear fusion. This is a basic law of physics. Nuclear power was not discovered until relatively recently, because it involves the subatomic realm that was not accessible until science had advanced to a certain level; nevertheless, it is one of the fundamental properties of matter. This argument that the sun actually burns via gravitational contraction could only be valid if nuclear energy didn’t operate. This would require us to believe that nuclear power plants don’t actually produce energy, and so are essentially a giant conspiracy.
Third, neutrinos are almost entirely produced by nuclear burning; the fact that any significant amount is observed shows that the sun is burning by nuclear power. It was certainly unusual that astronomers didn’t observe as much neutrino output as they expected, but this is not even remotely the same thing as evidence that the sun wasn’t really burning by nuclear power.
At any rate, the problem has since been solved: neutrinos come in three different “flavors” (electron, muon, and tau). Nuclear burning only produces the first of these, so scientists were only measuring one-third of the types of neutrinos that exist. But it turns out that neutrinos oscillate or convert from one flavor to another. With this, everything falls into place: the reason they were only detecting a third of the neutrinos they expected is because they were only measuring a third of the neutrinos that were being produced. As soon as they expanded their measurements to include the other two flavors, the number of neutrinos they measured perfectly matched the number of neutrinos they predicted.
Fourth, the measurements selected to argue that the sun is shrinking were cherry picked. The most extreme samples were chosen in order to invent this argument, even though they fell well within the error bars at the time. Since then, much more accurate measurements have been taken, which prove beyond any reasonable doubt that the sun is not shrinking. As with the depth of meteoritic dust on the moon, this argument was rigged by selecting the most extreme measurements and misrepresenting them as if they were the most reliable ones.
One final point: even if we ignore all of the above information for the sake of argument, this scenario would only allow us to conclude that the sun is less than a hundred million years old, not a few thousand years old, as the young-earth position requires. A hundred million years is about fifteen times less than the several billion years usually ascribed to the sun, but it’s about 15,000 times more than several thousand years. At best, it would show that both sides are wrong, but the young-earth position much more so.