The mass–radius relation for polytropes is introduced and analyzed. The Newtonian Lane–Emden equation is derived analytically. Known analytic solutions are discussed. The famous Chandrasekhar mass is obtained as a solution of the Lane–Emden equation. Corrections to the equation of state for white dwarfs are pointed out by estimating the Coulomb corrections for a lattice of nuclei immersed in a sea of electrons. The different layers of a typical white dwarf and the typical sizes are worked out in detail. Thermal effects for the mass relation are examined. Finally, astrophysical observations of white dwarfs are shown and discussed in terms of the overall composition of white dwarfs and the resulting mass–radius relation.

The detection of gravitational waves from the merger of black holes and the merger of two neutron stars are discussed. The linearized theory of general relativity is introduced. The concept of the gauge invariance is put forward and the transverse-traceless gauge for gravitational waves is presented. Einstein‘s famous quadrupole formula for gravitational waves is developed. The principle of detecting gravitational waves is outlined. As applications, the emission of gravitational waves from a nonvanishing ellipticity of rotating neutron stars is derived. The chirp mass is introduced and the emission of gravitational waves from compact binary systems is obtained. The formula for the tidal deformability and the Love number is put forward and discussed with regard to the recent measurement of a neutron star merger by the LIGO–Virgo scientific collaboration.

The rich history of neutron stars, observationally as well as theoretically, is sketched. The different layers of a neutron star are discussed step-by-step. The outer crust is described as a lattice of nuclei with a sea of electrons. The nuclear shell model and the phenomenon of magic numbers are introduced. The sequence of nuclei in the outer crust of a neutron star is shown explicitly. Features of the inner crust, as nuclear superfluidity and superconductivity, and the connection to pulsar glitches and cooling of neutron stars are described. The possible appearance of geometric structures in the inner crust, the pasta phases, are discussed. The concept of nuclear matter is established. The present-day knowledge of the neutron matter equation of state is put in place. The mass–radius of neutron stars are conceptionally investigated with a detailed discussion of the stability of neutron star configurations. The possible presence of exotic matter in the inner core of neutron stars is considered. The experimental data on hypernuclei is summarized and the paradigm of the so-called hyperon puzzle for neutron stars is analyzed.

After a short description of the historical discovery of pulsars, the different possible interpretations for pulsars are discussed. The different observational classes of pulsars are summarized. The dipole model for pulsars is introduced to motivate the definition of the characteristic age and the magnetic field of pulsars. The notion of the braking index of pulsars is confronted with the dipole model and the emission of gravitational waves from pulsars. The pulsar diagram is discussed in terms of the evolution of pulsars and the recycling mechanism for binary pulsars. The model of the aligned rotator introduces the magnetosphere of pulsars and a more sophisticated approach for the pulsar emission mechanism. The details of extracting the neutron star masses from pulsar timing is outlined. Observational data from the Hulse–Taylor pulsar and the double pulsar are confronted with predictions from general relativity in strong fields.

The theory of quantum chromodynamics (QCD) is introduced. Features of QCD as the nontrivial vacuum due to quark and gluon condensate and asymptotic freedom at high-energy scales are discussed. The concept of perturbative QCD and the running of the coupling constant is established. The equation of state of QCD at high temperatures from lattice QCD is reviewed and confronted with perturbative QCD calculations. The QCD equation of state at high baryon density is discussed. Properties of selfbound stars are developed where the equation of state has a nonvanishing pressure at a nonvanishing energy density. The mass–radius relation of pure quark stars is examined and compared to the limits from causality.

This chapter introduces compact objects, white dwarfs, neutron stars, and black holes. The properties of compact objects are summarized as the typical radius, mass, and compactness. These compact objects are the final end point in stellar evolution. The stellar evolution in terms of the mass of the star is outlined, focusing on the burning stages and the final collapse of the star, either as a white dwarf or in a core-collapse supernova. Historical notes are given for the discovery of white dwarfs and neutron stars.

The thermodynamics potentials for describing matter at nonzero temperatures and densities or chemical potentials are summarized. Emphasis is put on the thermodynamically correct description within the canonical and grand canonical ensemble for dense matter. The notion of chemical equilibrium is introduced for several conserved quantities and used to describe matter in β-equilibrium where charge and baryon number are conserved. The limit for nonrelativistic and relativistic particles is worked out in detail. The concept of an equation of state is introduced and applied to free Fermi gases. The pressure integral is solved analytically and the nonrelativistic and relativistic limits for the equation of state are delineated. Finally, the properties of polytropes are discussed and connected to the limiting cases of the equation of state of a free Fermi gas.