The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [109]
The argument that we just presented can be improved upon because it is possible to calculate exactly what the value of λ should be, but to do that we would need to solve a second-order differential equation in the density, and that is a mathematical bridge too far for this book. Remember, λ is a pure number: it simply ‘is what it is’ and we can, with a little higher-level maths, compute it. The fact that we did not actually work it out here should not detract at all from our achievements: we have proven that white dwarf stars can exist and we have managed to make a prediction relating their mass and radius. After calculating λ (which can be done on a home computer), and after substituting in the values for κ and G, the prediction is that
which is equal to 1.1 × 1017 kg1/3m for cores of pure helium, carbon or oxygen (Z/A = 1/2). For iron cores, Z/A = 26/56 and the 1.1 reduces slightly to 1.0. We trawled the academic literature and collected together the data on the masses and radii of sixteen white dwarf stars sprinkled about the Milky Way, our galactic backyard. For each we computed the value of RM1/3 and the result is that astronomical observations reveal RM1/3 ≈ 0.9 × 1017 kg1/3m. The agreement between the observations and theory is thrilling – we have succeeded in using the Pauli Exclusion Principle, the Heisenberg Uncertainty Principle and Newton’s law of gravity to predict the mass–radius relationship of white dwarf stars.
There is, of course, some uncertainty on these numbers (the theory value of 1.0 or 1.1 and the observational number equal to 0.9). A proper scientific analysis would now start talking about just how likely it is that the theory and experiment are in agreement, but for our purposes that level of analysis is unnecessary because the agreement is already staggeringly good. It is quite fantastic that we have managed to figure all this out to an accuracy of something like 10%, and is compelling evidence that we have a decent understanding of stars and of quantum mechanics.
Professional physicists and astronomers would not leave things here. They would be keen to test the theoretical understanding in as much detail as possible, and to do that means improving on the description we presented in this chapter. In particular, an improved analysis would take into account that the temperature of the star does play some role in its structure. Furthermore, the sea of electrons is swarming around in the presence of positively charged atomic nuclei and, in our calculation, we totally ignored the interactions between the electrons and the nuclei (and between electrons and electrons). We neglected these things because we claimed that they would produce fairly small corrections to our simpler treatment. That claim is supported by more detailed calculations and it is why our simple treatment agrees so well with the data.
We have obviously learnt an awful lot already: we have established that the electron pressure is capable of supporting a white dwarf star and we have managed to predict with some precision how the radius of the star changes if we add or remove mass from the star. Unlike ‘ordinary’ stars that are eagerly burning fuel, notice that white dwarf stars have the feature that adding mass to a star makes it smaller. This happens because the extra stuff we add goes into increasing the star’s gravity, and that makes it contract. Taken at face value the relationship expressed in equation (5) seems to imply that we would need to add an infinite amount of mass before the star shrinks to no size at all. But this isn’t what happens. The important thing,