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Finally, we are ready to feed our equation for the quantum pressure into the key equation (1), which is worth repeating here:

But this is not as easy as it sounds because we need to know the difference in the pressures at the upper and lower faces of the cube. We could re-write equation (1) entirely in terms of the density within the star, which is itself something that varies from place to place inside the star (it must be otherwise there would be no pressure difference across the cube) and then we could try to solve the equation to determine how the density varies with distance from the star’s centre. To do this is to solve a differential equation and we want to avoid that level of mathematics. Instead, we are going to be more resourceful and think harder (and calculate less) in order to exploit equation (1) to deduce a relationship between the mass and the radius of a white dwarf star.

Obviously the size of our little cube and its location within the star are completely arbitrary, and none of the conclusions we are going to draw about the star as a whole can depend upon the details of the cube. Let’s start by doing something that might seem pointless. We are quite entitled to express the location and size of the cube in terms of the size of the star. If R is the radius of the star, then we can write the distance of the cube from the centre of the star as r = aR, where a is simply a dimensionless number between 0 and 1. By dimensionless, we mean that it is a pure number and carries no units. If a = 1, the cube is at the surface of the star and if a = ½ it is halfway out from the centre. Similarly, we can write the size of the cube in terms of the radius of the star. If L is the length of a side of the cube, then we can write L = bR where, again, b is a pure number, which will be very small if we want the cube to be small relative to the star. There is absolutely nothing deep about this and, at this stage, it should seem so obvious as to appear pointless. The only noteworthy point is that R is the natural distance to use because there are no other distances relevant to a white dwarf star that could have provided any sensible alternatives.

Likewise, we can continue our strange obsession and express the density of the star at the position of the cube in terms of the average density of the star, i.e. we can write where f is, once again, a pure number and ρ is the average density of the star. As we have already pointed out, the density of the cube depends on its position inside the star – if it is closer to the centre, it will be more dense. Given that the average density does not depend on the position of the cube, then f must do so, i.e. f depends on the distance r, which obviously means it depends on the product aR. Now, here is the key piece of information that underpins the rest of our calculation: f is a pure number and R is not a pure number (because it is measuring a distance). This fact implies that f can only depend upon a and not on R at all. This is a very important result, because it is telling us that the density profile of a white dwarf star is ‘scale invariant’. This means the density varies with radius in the same way no matter what the radius of the star is. For example, the density at a point ¾ of the way out from the centre of the star will be the same fraction of the mean density in every white dwarf star, regardless of the star’s size. There are two ways of appreciating this crucial result and we thought we’d present them both. One of us explained it thus: ‘That’s because any dimensionless function of r (which is what f is) can only be dimensionless if it is a function of a dimensionless variable, and the only dimensionless variable we have is r/R = a, because R is the only quantity which carries the dimensions of distance that we have at our disposal.’

The other author feels that the following is clearer: ‘f can in general depend in a complicated way on r, the distance of the little cube from