Conclusion

1. Stars spend most of their lives fusing hydrogen into helium in the cores, sitting on the main sequence of the Hertzsprung–Russell diagram. When hydrogen in the core is depleted, low-mass (M_MS < 3 M_☉) and intermediate-mass (M_MS = 3–8 M_☉) stars evolve into red giants and undergo helium fusion, producing carbon and oxygen in their cores. Massive stars (M_MS > 8 M_☉) undergo further fusion reactions, evolving as supergiant stars, and those with the highest masses (M_MS > 11 M_☉) can eventually develop an iron core.

2. As asymptotic giant branch (AGB) stars become very luminous, a strong stellar wind removes most of the hydrogen envelope. The underlying helium-rich star gets smaller and hotter, developing a fast, radiation-driven wind. Bipolar outflows and other asymmetries are shaped by orbiting planetary- or stellar-mass companions of the AGB star. Once the central star’s surface temperature reaches T ~ 10⁴ K, it ionises the ejected envelope, which is then seen as an expanding planetary nebula. The star’s H- and He-burning shells are extinguished, leaving a degenerate CO core with T ~ 10⁵ K, which subsequently cools and fades to produce a white dwarf.

3. Degeneracy can be described in three equivalent ways:

   • (i) when the separation of particles is less than the de Broglie wavelength for their momentum, (Equation 3).
   • (ii) when the number of particles per unit volume is higher than the number of available quantum states at their energy, given by the quantum concentration, (Equation 4).
   • (iii) when the temperature of the particles is less than the limiting value given by (Equation 5).

4. Electrons, protons and neutrons are fermions. The Pauli exclusion principle dictates that no more than one identical fermion can occupy a given quantum state. So, at most two fermions (with spins +1/2 and –1/2, respectively) can occupy the same quantum state. In a cold electron gas, the energy of the most energetic degenerate electron is called the Fermi energy. The Fermi kinetic energy is (Equation 6), and its momentum (Equation 8) is called the Fermi momentum.

5. Whereas the equation of state for an ideal gas, , depends on the number density of particles n and the temperature T, for a degenerate gas it depends only on the number density: (Equation 10) or (Equation 12), where K_NR and K_UR are constants for non-relativistic and ultra-relativistic conditions, respectively. This decoupling of pressure and temperature in a degenerate gas disables thermostatic regulation. An ideal gas responds to an increased temperature by increasing pressure and hence expanding and cooling slightly; in a degenerate gas, a thermonuclear runaway can develop.

6. The number density n of some type of particle can be written in terms of its mass fraction X, mass m and the gas density ρ, as . Hence, for electrons, (Equation 13), where Y_e is the number of electrons per nucleon. For pure hydrogen, Y_e = 1, whereas for helium-4, carbon-12 and oxygen-16 (and hence for all types of white dwarf), Y_e = 0.5.

7. White dwarfs are supported against further collapse by electron degeneracy pressure. When degenerate electrons provide the pressure support of a star, the Fermi energy, E_F, may be expressed in terms of the mass. For non-relativistic degenerate electrons, (Equation 17), so degenerate electrons become more relativistic in more-massive white dwarfs. For ultra-relativistic degenerate electrons, the core density approaches infinity as the mass increases towards the Chandrasekhar limit, ~ 1.4 M_☉.

8. For non-relativistic degenerate electrons, a white dwarf mass-radius relationship can be derived: (Equation 18). This is comparable to the radius of the Earth and implies that more-massive white dwarfs have smaller radii.

9. Different types (He, CO and ONeMg) of white dwarf (WD) are the end points of low- and intermediate-mass main-sequence (MS) stars of different masses. They fade and cool along lines of constant radius in the H–R diagram.

10. Stars with M_MS > 11 M_☉ form an iron core supported by ultra-relativistic degenerate electrons. When the Chandrasekhar limit is reached, the electrons can no longer support the star. Nuclear photodisintegration by thermal photons and electron capture by nuclear protons (neutronisation) remove energy so efficiently that they send the core into free fall. The total amount of energy lost in a few seconds is comparable to the energy previously liberated via nuclear burning over the star’s entire main-sequence lifetime!

    • (i) Photodisintegration of iron proceeds as , and then . This can break down about 75% of the iron core, followed by 50% of the helium, and thus can absorb of order ~ 10⁴⁵ J of energy.
    • (ii) Neutronisation is the conversion of nuclear protons into neutrons via electron capture: ; the neutrinos produced also carry away of order ~ 10⁴⁵ J of energy.

11. The collapse of a stellar core halts when the density reaches that of nuclear matter, and then rebounds. This sends a shock wave through the rest of the star that partially reverses the collapse, leading to the ejection of the outer layers as a supernova. Its typical luminous energy is ~ 10⁴² J, and the kinetic energy of the ejecta is ~ 10⁴⁴ J. The gravitational binding energy released (~ 5 × 10⁴⁶ J) is 10 times more than the energy required to photodisintegrate the iron core or that is lost by neutronisation; it is also 100 times more than the kinetic energy of the ejecta. Thus most of the liberated energy is probably lost in an intense burst of neutrinos from the resulting neutron star.

12. When the density of a neutron star’s core reaches ~ 4 × 10¹⁴ kg m^-3, neutrons drip from the nuclei and once the density exceeds that of normal nuclear matter, ~ 2 × 10¹⁷ kg m^-3 , nuclei merge into a dense gas of electrons, protons and neutrons. Neutron stars survive because the beta decay of free neutrons is blocked by the Pauli exclusion principle and so neutrons greatly outnumber protons, typically with n_n ~ 200 n_p. They are supported against further collapse by neutron degeneracy pressure.

13. Neutron stars have masses in the range M_NS ~ 1.2–2.2 M_☉ and radii R_NS ~ 10–15 km. Their maximum mass – the Tolman–Oppenheimer–Volkoff (TOV) limit – corresponds to the neutrons becoming ultra-relativistic, but is difficult to calculate because of neutron interactions and the need to use general relativity. They form from single stars with main-sequence masses M_MS ~ 11–16 M_☉.

14. Single stars with M_MS ~ 16–25 M_☉ may form neutron stars too or they may form low-mass (M_BH ~ 5–8 M_☉) black holes instead, without a supernova. An apparent dearth of compact objects between the most massive neutron stars and the least massive black holes is referred to as the mass gap. Stars with initial masses M_MS > 25 M_☉ may also form neutron stars if they are in a binary system, because mass transfer between the two stars can significantly influence their evolution. Otherwise, such massive stars will form black holes with masses M_BH > 10 M_☉ and Schwarzschild radii (which mark the event horizon of the black hole) of R_Sch ~ 3 km per solar mass.