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By Root 822 0

the centre of the star. But let’s assume for the sake of this paragraph that it is directly proportional to it, i.e. f ∞ r2. In other words, f = Br, where B is a constant. Here, the key point is that we want f to be a pure number, whilst r is measured in (say) metres. That means that B must be measured in 1/metres, so that the units of distance cancel each other out. So what should we choose for B? We can’t just choose something arbitrary, like ‘1 inverse metres’, because this would be meaningless and has nothing to do with the star. Why not choose 1 inverse light years for example, and get a very different answer? The only distance we have to hand is R, the physical radius of the star, and so we are forced to use this to ensure that f will always be a pure number. This means that f depends only on r/R. You should be able to see that the same conclusion can be drawn if we started out by assuming that f ∞ r2 say.’ Which is just what he said, only longer.

This means that we can express the mass of our little cube, of size L and volume L3, sitting at a distance r from the centre of the star, as . We wrote f(a) instead of just f in order to remind us that f really only depends upon our choice of a = r/R and not on the large-scale properties of the star. The same argument can be used to say that we can write Min = g(a)M where g(a) is again only a function of a. For example, the function g(a) evaluated at a = 1/2 tells us the fraction of the star’s mass lying in a sphere of half the radius of the star itself, and that is the same for all white dwarf stars, regardless of their radius because of the argument in the previous paragraph.6 You might have noticed that we are steadily working our way through the various symbols which appear in equation (1), replacing them by dimensionless quantities (a, b, f and g) multiplied by quantities that depend only on the mass and radius of the star (the average density of the star is determined in terms of M and R because , the volume of a sphere). To complete the task, we just need to do the same for the pressure difference, which we can (by virtue of equation (4)) write as where h(a, b) is a dimensionless quantity. The fact that h(a, b) depends upon both of a and b is because the pressure difference not only depends on where the cube is (represented by a) but also on how big it is (represented by b): bigger cubes will have a larger pressure difference. The key point is that, just like f(a) and g(a), h(a, b) cannot depend upon the radius of the star.

We can make use of the expressions we just derived to rewrite equation (1):

That looks like a mess and not much like we are within one page of hitting the jackpot. The key point is to notice that this is expressing a relationship between the mass of the star and its radius – a concrete relation between the two is within touching distance (or desperate grasping distance, depending on how well you handled the mathematics). After substituting in for the average density of the star (i.e. ) this messy equation can be rearranged to read

Now λ only depends upon the dimensionless quantities a, b, f, g and h, which means that it does not depend upon the quantities that describe the star as a whole, M and R, and this means that it must take on the same value for all white dwarf stars.

If you are worrying what would happen if we were to change a and/or b (which means changing the locations and/or size of our little cube) then you have missed the power of this argument. Taken at face value, it certainly looks like changing a and b will change λ so that we will get a different answer for RM1/3. But that is impossible, because we know that RM1/3 is something that depends on the star and not on the specific properties of a little cube that we might or might not care to dream up. This means that any variation in a or b must be compensated for by corresponding changes in f, g and h.

Equation (5) says, quite specifically, that white dwarves can exist. It says that because we’ve been successfully able to balance the gravity–pressure equation (equation (1)).