White Dwarfs, Neutron Stars and Black Holes
When fusion ends, a star is held up by quantum mechanics or by nothing at all. The unit derives degeneracy pressure, the Chandrasekhar limit and the spin-down of pulsars, then turns to black holes: the Schwarzschild radius, time dilation near the horizon, and the ways in which accretion and orbital motion give dark objects away.
1 Degenerate matter
Sirius B, the nearest white dwarf, holds \(1.02\,\Msun\) in a ball slightly smaller than the Earth. Its mean density is \(2.7\times10^{9}\unit{kg\,m^{-3}}\), so a teaspoon of it would weigh 13 metric tons. The surface is at about \(25\,000\unit{K}\), yet Sirius B would keep its size if it cooled to absolute zero. Heat is irrelevant to the pressure that supports it.
This unit is about what is left when fusion has stopped for good (Unit 7). Stars born with less than about \(8\,\Msun\) leave a white dwarf, heavier ones a neutron star or a black hole.
1.1 Pressure without temperature
Two rules of quantum mechanics produce a pressure that exists even at zero temperature. The Pauli exclusion principle forbids two electrons from occupying the same quantum state. The uncertainty principle says that a particle confined to a region of size \(\Delta x\) has a momentum of at least \(\Delta p \approx \hbar/\Delta x\), where \(\hbar = 1.055\times10^{-34}\unit{J\,s}\) is the reduced Planck constant.
Squeeze a gas until it contains \(n_e\) electrons per cubic meter. Exclusion gives each electron a private cell of side \(n_e^{-1/3}\), and the uncertainty principle then forces a momentum \(p \approx \hbar\,n_e^{1/3}\) on it. The denser the gas, the faster its electrons move, whatever the temperature. Pressure is momentum delivered per unit area and time, roughly \(P \approx n_e\,p\,v\) for particles of speed \(v\). With \(v = p/m_e\) this gives \(P \approx \hbar^2 n_e^{5/3}/m_e\). At very high density \(p\) exceeds \(m_e c\), the speed saturates at \(c\), and the law becomes \(P \approx \hbar c\,n_e^{4/3}\).
This degeneracy pressure comes from the electrons, while the nuclei carry the mass. With \(\mu_e\) nucleons per electron and the atomic mass unit \(m_u = 1.66\times10^{-27}\unit{kg}\), the electron density is \(n_e = \rho/(\mu_e m_u)\). Helium, carbon and oxygen all have \(\mu_e = 2\). A full count of the quantum states supplies the numerical factors in the box below.
At the center of the Sun the degeneracy pressure of the electrons is about one sixth of their thermal pressure. Inside a white dwarf at \(10^{7}\unit{K}\) it is roughly a hundred times larger, and the gas is called degenerate. It has no thermostat: heat it and the pressure barely changes, so fusion, once ignited, runs away (the helium flash of Unit 7).