The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [110]
What we’re about to find is that as the star gets more massive, the pressure exerted by the electrons will no longer be proportional to the density raised to the power ; instead, the pressure increases less quickly with density. We will do the calculation in a moment, but straight away we can see that this could have catastrophic consequences for the star. It means that when we add mass, there will be the usual increase in gravity but a smaller increase in pressure. The star’s fate hinges on just how much ‘less quickly’ the pressure varies with density when the electrons are moving fast. Clearly it is time to figure out what the pressure of a ‘relativistic’ electron gas is.
Fortunately, we do not need to wheel in the heavy machinery of Einstein’s theory because the calculation of the pressure in a gas of electrons moving close to light speed follows almost exactly the same reasoning as that we just presented for a gas of ‘slow-moving’ electrons. The key difference is that we can no longer write that the momentum p = mv, because this is not correct any more. What is correct, though, is that the force exerted by the electrons is still equal to the rate of change of their momentum. Previously, we deduced that a fleet of electrons bouncing off a mirror exerts a pressure P = 2mv × (nv). For the relativistic case, we can write the same expression, but providing that we replace mv by the momentum, p. We are also assuming that the speed of the electrons is close to the speed of light, so we can replace v with c. Finally, we still have to divide by 6 to get the pressure in the star. This means that we can write that the pressure for the relativistic gas as P = 2p × nc/6 = pnc/3. Just as before, we can now go ahead and use Heisenberg’s Uncertainty Principle to say that the typical momentum of the confined electrons is h(n/2)1/3 and so
Again we can compare this to the exact answer, which is
Finally, we can follow the same methodology as before to express the pressure in terms of the mass density within the star and derive the alternative to equation (4):
where . As promised, the pressure increases less quickly as the density increases than it does for the non-relativistic case. Specifically, the density increases with a power of rather than . The reason for this slower variation can be traced back to the fact that the electrons cannot travel faster than the speed of light. This means that the ‘flux’ factor, nv, which we used to compute the pressure saturates at nc and the gas is not capable of delivering the electrons to the mirror (or face of the cube) at a sufficient rate to maintain the ρ5/3 behaviour.
We can now explore the implications of this change because we can go through the same argument as in the non-relativistic case to derive the counterpart to equation (5):
This is a very important result because, unlike equation (5), it does not have any dependence upon the radius of the star. The equation is telling us that this kind of star, packed with light-speed electrons, can only have a very specific value of its mass. Substituting in for κ′ from the previous paragraph gives us the prediction that
This is exactly the result we advertised right at the start of this chapter for the maximum mass that a white dwarf star can possibly have. We are very close to reproducing Chandrasekhar’s result. All that remains to understand is why this special value is the maximum possible mass.
We have learnt that for white dwarf stars that are not too massive, the radius is not too small and the electrons are not too squashed. They therefore do not quantum jiggle to excess and their speeds