A silent death.

Intro 

The end. We will be looking at the final stage of our star's life cycle. Specifically, we will be looking at our star becoming a white dwarf.

Principles

• Hydrostatic equilibrium

Hydrostatic equilibrium refers to the perfect balance between the gravitational force and the gas pressure (du e to energy generation in the core of the star). The balance is self-regulating in the sense that if the rate of energy generation goes down and we thus have less gas pressure the gravity takes over and the star begins to contract which then increases the temperature and pressure of the stellar interior, so then it goes back to equilibrium again. We have equations describing hydrostatic equilibrium that we can use 

• 
• Degenerate pressure/matter

Degenerate matter is basically matter on steroids, it's extremely dense ferminonic matter that exerts extreme amounts of pressure due to high gravitational pressure that makes it such that quantum mechanical effects become significant, no, more like dominant. Here's how Pauli's exclusion principle leads to high pressure. The principle states that no two electrons with the same spin can occupy the same energy state in that same volume, what happens when the gravity is too high and it tries to squeeze too many electrons into small volume? They occupy higher and higher states with progressively higher speeds, some of them reaching close to the speed of light, when electrons move that fast they create a pressure capable of supporting a whole star can you believe it? Our normal gas laws and equations don't apply and we must use other equations that describe the pressure made by such degenerate matter, this however is beyond the scope of this scroll. 

How large is a white dwarf star? 

Well using principles mentioned above, the equations of both the hydrostatic equilibrium and the pressure generated by degenerate matter and a ballsy assumption (constant uniform density) we get the following equations

Here we assumed that the number of protons per nucleus is double that of the number of neutrons.  per nucleus which is a somewhat reasonable assumption. All we need to do is insert the mass of our sun which is 

However, I unfortunately have no time left.

Star life

The life cycle of a star begins with the joining of gases in a nebula due its gravity pulling it together. Their initial temperature is fairly low so they will be placed to the right of a Hertzsprung-Russel diagram, also called a HR-diagram. In such a diagram, the luminosity is often plotted against decreasing temperate. Due to the gravitational force, the molecules are attracted to each other and contract, causing the nebula to have a reduction in size, that is, to have a smaller radius. The contraction of gases causes an increase in temperature. As this happens the star moves to the left, and when temperatures are high enough the gases start to undergo nuclear fusion, and the star has settled into the main sequence. A main sequence star is a star that is undergoing nuclear fusion. Since our star has a fairly little mass, it will settle more to the right of the diagram. As the life reaches the end of its life, the star can take on two different journeys, and these depend on its mass. A low mass star is a star that has has M <8Mo (Where Mo is solar masses from here onwards). As our star is 65% the mass of our sun, and hence less than 8Mo it is considered to be a low mass star. The core of the star will contract until electron density is so high that it becomes electron degenerate (concept explained later). This creates a high pressure that combats the gravitational pull of the star, maintaining it in hydrostatic equilibrium, that is, at rest. As more hydrogen is fused, the surface of the star has an increase in temperature, but since hydrogen fusion is more efficient, the helium present goes into a deeper shell (like a layer that¡¯s closer to the core). As you get closer to the core, the temperature is higher, so helium is able to start fusing as well. The flash produced from helium fusion causes the hydrogen burning shell to expand. Due to its higher surface area, the luminosity increases. As hydrogen fusion decreases, the star contracts, and initiates hydrogen fusion once again. The same happens with helium and the cycle happens again. It is believed that this star blowing out these helium flashes is what causes the mass loss. These helium flashes blow out the outer portions of the star, into what is called a planetary nebula. This will look like a cloud of gas that is surrounding the remaining degenerate carbon/oxygen core from the star. This core has now become what is known to us as a white dwarf. A white dwarf has extremely high temperatures, but low luminosity. Hence the lower left positioning in the HR-diagram. As there is a lack of fuel, the white dwarf¡¯s temperature lowers over time.

Below is a HR-diagram of our star¡¯s journey. The steps are described after the figure.

1. The star stars fusing and outer layers are blown off, increasing its radius and hence increasing luminosity, but lowering temperature
2. The ¡®equilibrium¡¯ between hydrogen fusion (red) and helium fusion (yellow) causing the giant to shrink and expand
3. As the star ejects the outer layers, the star expands into a supergiant
4. Temperature increases due to massive helium flashes, and outer layers of planet are ejected into a gaseous planetary nebula
5. The core is left behind into a white dwarf, that is high in temperature but its small surface area causes a very low luminosity. Dwarf proceeds to loose heat over time.