Lecture 6. Death of Low Mass Stars

We saw last lecture that the mass of a main sequence star determines its temperature and luminosity. High mass means high gravity so for the star to be stable (hydrostatic equilibrium) requires that the thermal pressure pushing outwards which balances gravity has to go up as well, so the temperature (and density) goes up. And the nuclear reaction rate is INCREDIBLY sensitive to temperature - a little bit more temperature means that the particle velocity is a little bit higher, so MANY more protons get within the range of the strong nuclear force which can overcome the electrical repulsion. Massive stars are MUCH MUCH more luminous. But their lifetime on the main sequence is determined by how much hydrogen fuel they have available (mass), and the rate at which they spend it (luminosity). Massive stars with lots of fuel resources squander them in a short lived blaze of glory, while very low mass stars eke out the small amount of fuel they have to glow dimly over a main sequence lifetime much longer than that of the Sun. When we look at the HR diagram of different clusters of stars then we see the main sequence turnoff at different points - so we can estimate ages of clusters!

But whether its a long or short lifetime, all stars eventually come to the end of it. When all of the hydrogen in the core has been fused to helium then there is no more energy available to the core. This marks the end of the main sequence lifetime, and from here on it the the star is in its last gasp, trying but failing to find a stable energy source to combat gravity. Most of a star's life is spent on the main sequence, but this last 10 per cent of its luminous lifetime can be spectacular!

The weight of the layers above still pushes down on the core, only now there are no fusion reactions to heat the core to hold it up against gravity. So gravity takes over and the core shrinks. The layers outside the core collapse too, the ones closer to the center collapse quicker than the ones near the surface. As the layers collapse, the gas compresses and heats up as there is gravitational energy released. This holds up the core for a little while, but to get more gravitational energy it has to contract still further ...

But can the core contract indefinitely ? How close can we squash material together before it protests ? With classical ideas of electrons and protons/nuclei as being almost point-like particles, then plainly we can squash them together until they 'almost touch' before odd things start to happen. But as we've said before, electrons are not like this - they can behave a bit more like a wave. And a wave doesn't have a well defined position, it's spread out. So it might object at much larger spacings than we'd expect from point particles. And now we hit another odd property of matter - suppose you have two water or sound (or light!) waves of the same energy (amplitude and wavelength) in the same place. They would add up together to make you a wave thats twice as big. But the electron waves ARE NOT LIKE THIS. They have a Greta Garbo sense of 'I want to be alone'. You can't get two electrons with the same energy at the same place. This is NOT due to them being of the same charge (we'll see later on that neutrons do this too!!). And they can't get away with having very very slightly different energies - we saw in lecture 3 that electrons in atoms have energies which are quantized - they can only take certain values ! We looked at some justification for that by thinking the permitted energy levels were ones where the wavelength of the electron fit perfectly into the radius of the 'orbit'. So here it can either have a wavelength which fits into the little space, or it can have half this, (so two wavelengths fit into the space) or 1/3 of this (so three wavelengths fit into the space).... From what we know about light waves we then expect shorter wavelengths to be associated with more energy. So the more electrons we try and put in the box, the higher the energy they have to have. But earlier when we looked at pressure we saw that normal thermal pressure is caused by the motion of nuclei and electrons - the faster they move (i.e. the more energy they have) then the more pressure they exert. But here we are saying that the electrons have energy even if they are NOT HOT - they are moving just because they are confined to a small space. And if they are moving then they exert a pressure - this is called electron degeneracy pressure. It is a purely quantum mechanical effect - ultimate core collapse of a star can be held off by this odd property that electrons want to be alone!

For very low mass stars, masses less than 0.4x that of the Sun, then this is what happens. These stars transport energy by convection so they are all mixed inside so the hydrogen runs out in the whole star. The core shrinks, and shrinks, and shrinks, becoming degenerate before the core temperature rises high enough to do anything more entertaining! So it can't contract any more, so can't get any more gravitational energy. This is a white dwarf made up of helium. It fades and cools becomming a black dwarf.

Stars from 0.4-4 solar masses are much more complex. These carry the energy in the interior mostly by radiation so the core is made up of helium but the outer layers are still mostly hydrogen and they are not mixed together during the main sequence life of the star. So when the core starts contracting, and heating up because of gravity then eventually the layer just outside the core gets hot and dense enough for fusion to start. So there are 2 sources of energy - the gravitational shrinking and H fusion in a shell around the core, and this core and shell are much more luminous than the original main sequence core! So the outer layers of the star (which hadn't contracted by very much so still feel much the same gravity) suddenly have MUCH more luminosity going through them. So the thermal pressure is much bigger than the gravity, so the outer layers expand tremendously (our sun will be bigger than the orbit or mercury and Venus, and possibly also of Earth), and so cool down.... This is a red giant. Plotting this on an HR diagram we'd see the star go from the main sequence to much higher luminosity and much lower temperatures - ie it moves up and to the right on an HR diagram.

But the core keeps on contracting, and the shell burning around the core keeps on going, adding helium ash to the core. So gravity in the core gets STRONGER as the mass of the core increases AND its radius is decreasing. So more electrons get squashed into a smaller and smaller space, so to shorter and shorter wavelengths. Thus we get higher and higher electron velocities and so an increase in pressure (as required for the core to be stable as we increased gravity!). This means that higher masses can only be held up by squashing the core into a smaller and smaller volume. Obviously this can't continue indefinitely!

But first what happens is that the temperature in the degenerate core goes high enough (over 100 million K) that it can start to fuse Helium to carbon (triple alpha process). You'd have thought that the core would fuse 2 helium nuclei to make Berylium, but the isotope of Be which you'd make from 2 helium nuclei would have 4 protons and 4 neutrons and that happens to be highly unstable and falls apart very very fast! So you have to stick another helium on before it does so, and go to an extremely stable isotope of carbon (carbon 12 with 6 protons and 6 neutrons). So then we have Hydorgen fusing to helium in a shell and He fusing to Carbon in the core. But the core is soon all converted to Carbon (very very quickly for stars less than about 3x the mass of the Sun - its called the helium flash though it doesn't result in any flash of light from the surface as the luminosity has to struggle out). So then it has a carbon core contracting by gravity, and helium fusing to carbon in a shell and hydrogen fusing to helium in an outer shell. But the additional energy released from the core means that it expands so it cools, so the luminosity drops! So the outer layers of the star

So now the C core keeps contracting, so the He and H fusing shells get hotter, so the luminosity goes up so the outer layers expand and cool .... it goes back up to being a red giant again, and is even bigger this time than last time. So the outer layers are a long long way from the core, so the gravity holding them onto the star is quite weak. And the shell burning isn't really very stable - if there is a bit more luminosity than normal then the outer layers get pushed out and become detached from the star. So we have material illuminated by the hot core - this is a planetary nebula. The exposed core can't keep on contracting as it hits the limit where degeneracy pressure won't let it go any smaller. So the core starts off as a hot carbon white dwarf, and then cools down to a black dwarf.

For stars with mass ~3-6x then the scenario is much the same except that the gravity is higher so the core temperature on the main sequence is higher. The helium core temperature is higher, so that the core starts fusing He-C BEFORE it becomes degenerate. So the helium burns rather more steadily, giving rise to a phase where there is helium core fusion and H shell fusion. And then once the He core has all fused to C then its as above