The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [103]
Figure 12.3. A tyre deforming slightly as it supports the weight of a car.
This is an easy equation to solve: P = (14,700/400) N/cm2 = 36.75 N/cm2. A pressure of 36.75 Newtons per square centimetre is probably not a very familiar way of stating a tyre pressure, but we can convert it into the more familiar ‘bar’. 1 bar is standard air pressure, and is equal to 101,000 Newtons per square metre. There are 10,000 square centimetres in a square metre, so 101,000 Newtons per square metre is equivalent to 10.1 Newtons per square centimetre. Our desired tyre pressure is therefore 36.75/10.1 = 3.6 bar (or 52 psi – you can work that one out for yourself). We can also use our equation to deduce that, if the tyre pressure decreases by 50% to 1.8 bar, then we’ll double the area of tyre in contact with the ground, which makes for a flatter tyre. After that refresher course on pressure we are ready to return to the little cube of star matter illustrated in Figure 12.2.
If the bottom face of the cube is closer to the centre of the star then the pressure on it should be a little bit bigger than the pressure pressing on the top face. That pressure difference gives rise to a force on the cube that wants to push the cube away from the centre of the star (‘up’ in the figure) and that is just what we want, because the cube will, at the same time, be pulled towards the centre of the star by gravity (‘down’ in the figure). If we could work out how to balance those two forces then we’d have developed some understanding of the star. But that is easier said than done because, although step 1 will allow us to work out how much the cube is pushed out by the electron pressure, we still have to figure out by how much gravity pulls in the opposite direction. By the way, we do not need to worry about the pressure pushing against the sides of our cube because the sides are equidistant from the centre of the star, so the pressure on the left side will balance the pressure on the right side and that ensures the cube does not move to the left or right.
To work out the force of gravity on the cube we need to make use of Newton’s law of gravity, which tells us that every single piece of matter within the star pulls on our little cube by an amount that decreases in strength the farther the piece is from our cube. So more distant pieces pull less than closer ones. To deal with the fact that the gravitational pull on our cube is different for different pieces of star matter, depending on their distance away, looks like a tricky problem but we can see how to do it, in principle at least – we should chop the star into lots of pieces and then work out the force on the cube for each and every such piece. Fortunately, we do not need to imagine chopping the star up because we can exploit a very beautiful result. Gauss’ law (named after the legendary German mathematician Carl Friedrich Gauss) informs us that: (a) we can totally ignore the gravity from all the pieces sitting further out from the centre of the star than our little cube; (b) the net gravitational effect of all of the pieces that sit closer to the centre is exactly as if all of those pieces were squashed together at the exact centre of the star. Using Gauss’ law in conjunction with Newton’s law of gravity we can say that the cube experiences a force that pulls it towards the centre of the star and that force is equal to
where Min is the mass of the star lying within a sphere whose radius reaches only as far out as the cube, Mcube is the mass of the cube and r is the distance of the cube from the star’s centre (and G is Newton’s constant). For example, if the cube sits on the surface of the star then Min is the total mass of the star. For all other locations, Min is smaller than that.
We’re now making progress because to balance the forces on the cube (which we remind you means that the cube doesn’t move and that means the star is not going to explode or collapse2) we require that
where Pbottom and Ptop are the pressures of the electron gas at the upper and lower faces of the cube