The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [100]
But what happens if the star is not massive enough to initiate helium fusion? For stars less than about half the mass of our Sun, this is the case, and for them something very dramatic happens. The star heats up as it contracts, but, before the core reaches 100 million degrees, something else halts the collapse. That something is the pressure exerted by electrons due to the fact that they are in the grip of the Pauli Exclusion Principle. As we have learnt, the Pauli principle is crucial to understanding how atoms remain stable, and it underpins the properties of matter. Here is another string to its bow: it explains the existence of compact stars that survive despite the fact that they no longer burn up any nuclear fuel. How does this work?
As the star gets squashed, so the electrons within it get confined to a smaller volume. We can think of an electron in the star in terms of its momentum p and hence its associated de Broglie wavelength, h/p. In particular, the particle can only ever be described by a wave packet that is at least as big as its associated wavelength.1 This means that, when the star is dense enough, the electrons must be overlapping each other, i.e. we cannot imagine them as being described by isolated wave packets. This in turn means that quantum mechanical effects, and the Pauli principle in particular, are important in describing the electrons. Specifically, they are being squashed together to the point where two electrons are attempting to occupy the same region of space, and we know from the Pauli principle that they resist this. In a dying star, therefore, the electrons avoid each other and this provides a rigidity that resists any further gravitational collapse.
This is the fate of the lightest stars, but what of stars like our Sun? We left them a couple of paragraphs ago, burning helium into carbon and oxygen. What happens when they run out of helium? They too must then start to collapse under their own gravity, which means they will have their electrons squashed together. And, just as for the lighter stars, the Pauli principle can eventually kick in and halt the collapse. But, for the most massive of stars, even the Pauli Exclusion Principle has its limits. As the star collapses and the electrons get squashed closer together, so the core heats up and the electrons move faster. For heavy enough stars, the electrons will eventually be moving so fast that they approach the speed of light, and that is when something new happens. When they close in on light-speed, the pressure the electrons are able to exert to resist gravity is reduced to such an extent that they aren’t up to the job. They simply cannot beat gravity any more and halt the collapse. Our task in this chapter is to calculate when this happens, and we’ve already given away the punchline. For stars with masses greater than 1.4 times the mass of the Sun, the electrons lose and gravity wins.
That completes the overview that will provide the basis for our calculation. We can now go ahead and forget all about nuclear fusion, because stars that are burning are not where our interest lies. Rather, we are keen to understand what happens inside dead stars. We want to see just how the quantum pressure from the squashed electrons balances the force of gravity, and how that pressure becomes diminished if the electrons are moving too fast. The heart of our study is therefore a balancing game: gravity versus quantum pressure. If we can make them balance we have a white dwarf star, but if gravity wins we have catastrophe.
Although not relevant for our calculation, we can’t leave things on such a cliff-hanger. As a massive star implodes, two further options remain open to it. If it is not too heavy then it will keep squashing the protons