Part II: Compact Objects

1. White Dwarfs

1.1 Electron Degeneracy Pressure

A white dwarf is supported against gravitational collapse by the pressure of a degenerate electron gas. At the densities found in white dwarfs the electrons are packed so tightly that the Pauli exclusion principle forces them into ever-higher momentum states, generating a pressure that is independent of temperature.

Consider a gas of free electrons at zero temperature. All momentum states up to the Fermi momentum are filled. The number density of electrons is related to the Fermi momentum by

The pressure of a fully degenerate, ideal Fermi gas is obtained by integrating the momentum flux:

Non-Relativistic Limit

When $p_F \ll m_e c$ we have $v \approx p/m_e$. The integral evaluates to

Since the mass density is $\rho = \mu_e m_p n_e$ we can write $P = K_{\rm NR}\,\rho^{5/3}$, a polytropic equation of state with index $n = 3/2$ and

Ultra-Relativistic Limit

When $p_F \gg m_e c$ the electrons become relativistic with $v \to c$. The pressure becomes

This is a polytrope of index $n = 3$. The softer dependence on density (4/3 vs 5/3) is ultimately responsible for the existence of a maximum mass.

1.2 The Chandrasekhar Mass Limit

Subrahmanyan Chandrasekhar showed in 1931 that a fully relativistic treatment of the degenerate electron gas yields a maximum mass for a white dwarf. The argument proceeds as follows.

In the non-relativistic regime the total energy of a uniform-density star of mass $M$ and radius $R$ can be estimated as

The kinetic (degeneracy) energy scales as $R^{-2}$ while gravity scales as $R^{-1}$, so there is always an equilibrium radius. However, in the ultra-relativistic limit the kinetic energy scales as $R^{-1}$ — the same as gravity. Therefore equilibrium is possible only for a unique mass; above that mass gravity wins and no equilibrium exists.

Setting the ultra-relativistic kinetic energy equal to the gravitational energy and solving for $M$gives the Chandrasekhar mass:

A more rigorous derivation uses the Lane-Emden equation for an $n=3$ polytrope. The mass of the configuration is

For $n = 3$ the exponent of $\rho_c$ vanishes, so the mass is independent of the central density: there is a unique mass that can be in hydrostatic equilibrium, which is $M_{\rm Ch}$.

1.3 Mass-Radius Relation

In the non-relativistic regime ($n = 3/2$ polytrope) the radius and mass of the white dwarf are related by

In the limit $M \ll M_{\rm Ch}$ this simplifies to $R \propto M^{-1/3}$: more massive white dwarfs are smaller. This is the opposite of main-sequence stars and is a direct consequence of the degeneracy pressure equation of state. As $M \to M_{\rm Ch}$ the radius shrinks to zero — the relativistic softening of the equation of state cannot support the star.

For a $0.6\,M_\odot$ C/O white dwarf the radius is approximately $R \approx 0.012\,R_\odot \approx 8{,}400$ km, comparable to the size of Earth.

1.4 White Dwarf Cooling

White dwarfs have no nuclear energy source; they simply radiate away their stored thermal energy. The ions in the interior form a Coulomb lattice or liquid with specific heat $C_V = 3N_{\rm ion}k_B$(Dulong-Petit limit) at high temperatures. The degenerate electrons contribute negligibly to the heat capacity but dominate the thermal conductivity, ensuring the interior is nearly isothermal.

The thin, non-degenerate surface layer acts as an insulating blanket. Mestel (1952) showed that the luminosity of a cooling white dwarf is related to its core temperature by

Equating the luminosity with the rate of thermal energy loss $L = -C_V\,dT_c/dt$ and integrating gives the cooling age:

This simple power-law cooling is modified at late times by crystallisation (latent heat release), Debye cooling (when the lattice temperature drops below the Debye temperature), and convective coupling of the envelope. Nevertheless, white dwarf luminosity functions provide one of the most reliable chronometers for the age of the Galactic disk (~8-10 Gyr).

1.5 Type Ia Supernovae

When a C/O white dwarf accretes matter from a binary companion and approaches $M_{\rm Ch}$, compressional heating ignites carbon burning under degenerate conditions. Because the pressure is independent of temperature the burning is thermally unstable: the temperature rises but the star does not expand, leading to a thermonuclear runaway that unbinds the entire star — a Type Ia supernova.

Two combustion modes are discussed in the literature:

The peak luminosity is powered by the radioactive decay chain$^{56}\text{Ni}\to{^{56}\text{Co}}\to{^{56}\text{Fe}}$ with half-lives of 6.1 and 77.3 days respectively. The tight correlation between peak luminosity and light-curve width (the Phillips relation) makes Type Ia supernovae standardisable candles, leading to the 1998 discovery of accelerating cosmic expansion (see Part VI).

An alternative progenitor channel — the double-degenerate scenario — involves the merger of two C/O white dwarfs whose combined mass exceeds the Chandrasekhar limit. Gravitational-wave inspiral (Part V) drives the merger. This channel may explain the absence of hydrogen in Type Ia spectra and the lack of a surviving companion in supernova remnants such as SNR 0509-67.5.

2. Neutron Stars

When the core of a massive star ($M \gtrsim 8\,M_\odot$) collapses at the end of silicon burning, electron capture on protons ($e^- + p \to n + \nu_e$) neutronises the matter. The resulting neutron-rich core is stabilised at nuclear density ($\rho_{\rm nuc} \approx 2.8\times10^{14}$ g cm$^{-3}$) by the strong nuclear force and neutron degeneracy pressure. The overlying envelope is ejected in a core-collapse supernova, leaving behind a neutron star with typical mass$M \sim 1.4\,M_\odot$ and radius $R \sim 10$–13 km.

2.1 The Tolman-Oppenheimer-Volkoff Equation

Neutron stars are sufficiently compact that general relativity is essential: the surface gravitational redshift is $z_s = (1 - 2GM/Rc^2)^{-1/2} - 1 \approx 0.2$–0.4. The relativistic generalisation of hydrostatic equilibrium is the Tolman-Oppenheimer-Volkoff (TOV) equation, derived from the Einstein field equations for a static, spherically symmetric perfect fluid.

Starting from the Schwarzschild interior metric

the $G^t_t$ component of the Einstein equations gives the mass continuity equation:

The $G^r_r$ component, combined with the conservation of the stress-energy tensor ($\nabla_\mu T^{\mu\nu} = 0$), yields the TOV equation:

Compared to the Newtonian equation of hydrostatic equilibrium, the TOV equation contains three general-relativistic correction factors:

All three factors are greater than unity, meaning that GR makes gravity stronger. This is why the maximum mass of a neutron star is lower than a naive Newtonian estimate would suggest, and why black holes form at a finite mass rather than at infinity.

2.2 The Equation of State of Dense Matter

To close the TOV system we need the equation of state (EOS), $P = P(\epsilon)$. This is the central unsolved problem in neutron star physics. At supra-nuclear densities ($\rho > \rho_{\rm nuc}$) the relevant degrees of freedom are uncertain: neutrons, protons, hyperons, pion/kaon condensates, or deconfined quarks could all play a role.

At the simplest level one can parametrise the EOS as a polytrope $P = K\rho^\Gamma$. A single polytrope with $\Gamma \approx 2$–3 captures the broad features of neutron star structure. More realistic approaches use piecewise polytropes, spectral representations, or full many-body calculations (Brueckner-Hartree-Fock, chiral effective field theory, lattice QCD at finite baryon density).

The Hyperon Puzzle

At densities above about $2\rho_{\rm nuc}$ the neutron Fermi energy exceeds the mass difference between neutrons and hyperons ($\Lambda, \Sigma, \Xi$). Hyperons should therefore appear, softening the EOS and reducing the maximum mass to ~1.4–1.6 solar masses — inconsistent with the observed 2 solar mass pulsars. Possible resolutions include three-body hyperon-nucleon forces, a phase transition to quark matter at slightly lower density (which stiffens the EOS), or the suppression of hyperons by other exotic phases.

2.3 Maximum Mass

The existence of a maximum mass for neutron stars was demonstrated by Oppenheimer and Volkoff (1939) using a free neutron gas EOS, which gave $M_{\rm max} \approx 0.71\,M_\odot$. Nuclear interactions stiffen the EOS considerably, raising the maximum mass.

An absolute upper bound on the maximum mass can be derived from causality alone. Rhoades and Ruffini (1974) showed that if the EOS satisfies $c_s \leq c$ above some matching density, then

Current best estimates from nuclear theory and multi-messenger observations (gravitational waves from GW170817 + electromagnetic counterpart) constrain the maximum mass to approximately 2.2–2.5 solar masses. Any compact object above this limit that is not a black hole would require exotic physics.

2.4 Internal Structure

A neutron star has a rich layered structure:

2.5 Magnetars

Magnetars are neutron stars with extraordinarily strong magnetic fields, $B \sim 10^{14}$–$10^{15}$ G at the surface (compared to $10^{12}$ G for ordinary pulsars). They manifest observationally as Soft Gamma Repeaters (SGRs) and Anomalous X-ray Pulsars (AXPs).

The magnetic energy stored in the star is

The X-ray luminosity of magnetars exceeds their spin-down luminosity, implying that the energy source is magnetic field decay rather than rotation. Giant flares (such as the 2004 December 27 event from SGR 1806-20, which released ~10⁴⁶ erg in 0.2 s) are thought to result from catastrophic magnetic reconnection or crustal fracturing driven by magnetic stresses.

The field amplification mechanism is debated. The leading hypothesis is a convective dynamo operating in the first ~10–30 seconds after the neutron star is born, when the proto-neutron star is hot, differentially rotating, and convectively unstable. If the initial spin period is short enough ($P_0 \lesssim 3$ ms) the dynamo can amplify fields to magnetar strengths.

3. Pulsars

3.1 The Lighthouse Model

Pulsars are rapidly rotating, highly magnetised neutron stars whose radio (and often X-ray and gamma-ray) emission is beamed along the magnetic axis. If the magnetic axis is misaligned with the rotation axis the beam sweeps across the sky like a lighthouse, producing periodic pulses when it crosses our line of sight.

The rotation period $P$ ranges from milliseconds to several seconds. The period is observed to increase with time ($\dot{P} > 0$) as the pulsar loses rotational energy. The angular frequency$\Omega = 2\pi/P$ and its time derivative $\dot{\Omega} = -2\pi\dot{P}/P^2$ characterise the spin evolution.

3.2 Spin-Down Luminosity

The rotational kinetic energy of the neutron star is $E_{\rm rot} = \frac{1}{2}I\Omega^2$, where $I \approx \frac{2}{5}MR^2 \approx 10^{45}$ g cm$^2$ is the moment of inertia. The rate of energy loss is

For a magnetic dipole in vacuum the radiated power is

3.3 Braking Index and Characteristic Age

The spin-down is parametrised as $\dot{\Omega} = -k\Omega^n$ where $n$ is the braking index. Pure magnetic dipole radiation gives $n = 3$. The braking index can be measured from the second derivative of the frequency:

Measured values range from $n \approx 1.4$ (Vela) to $n \approx 2.9$ (Crab), generally below 3, indicating that the spin-down is not purely dipolar. Possible contributions include a magnetospheric particle wind, magnetic field evolution, and gravitational-wave emission.

For $n = 3$ and assuming the initial period is much shorter than the current period, integration gives the characteristic age:

3.4 The P-Pdot Diagram

The $P$–$\dot{P}$ diagram is the "HR diagram" of pulsars. Lines of constant magnetic field ($B \propto \sqrt{P\dot{P}}$), constant characteristic age ($\tau_c = P/2\dot{P}$), and constant spin-down luminosity ($\dot{E} \propto \dot{P}/P^3$) are straight lines on a log-log plot.

3.5 Millisecond Pulsars (Recycled Pulsars)

Millisecond pulsars (MSPs) are old neutron stars that have been spun up ("recycled") by accretion of angular momentum from a binary companion. The accretion torque is

The equilibrium spin period (when the accretion torque balances magnetic braking) is

The accretion process also buries and dissipates the magnetic field, explaining the low fields of MSPs. The transition from accretion-powered to rotation-powered state has been directly observed in "transitional" MSPs such as PSR J1023+0038.

3.6 Pulsar Timing Arrays

MSPs are extraordinarily stable rotators with timing residuals at the ~100 ns level over decades. An array of precisely timed MSPs distributed across the sky can be used as a galactic-scale gravitational-wave detector sensitive to nanohertz frequencies (periods of years to decades).

A passing gravitational wave perturbs the metric between the pulsar and Earth, inducing correlated timing residuals across the array. The characteristic angular correlation pattern — the Hellings-Downs curve — is the smoking-gun signature of a gravitational-wave background.

In 2023 the NANOGrav, EPTA, PPTA, and CPTA collaborations reported strong evidence for a common-spectrum stochastic process with emerging Hellings-Downs correlations, consistent with a gravitational-wave background from the cosmic population of supermassive black hole binaries (see Part V for the gravitational-wave theory and Part VI for the astrophysical context).

4. Accretion Physics

4.1 The Eddington Luminosity

There is a maximum luminosity at which radiation pressure on ionised gas balances gravity. For spherical accretion of fully ionised hydrogen, equating the outward radiation force per electron ($\sigma_T L / 4\pi r^2 c$) with the inward gravitational force on the associated proton ($GM m_p / r^2$) gives

The corresponding Eddington accretion rate, assuming radiative efficiency $\eta$, is

Super-Eddington accretion is possible when the geometry is non-spherical (e.g., disk accretion with outflows along the poles) or when the opacity is reduced (e.g., by magnetic confinement of the accretion column onto a neutron star polar cap). Ultra-luminous X-ray sources (ULXs) with apparent luminosities of 10–500 times the Eddington limit are now understood to be a mix of stellar-mass compact objects with beamed or genuinely super-Eddington accretion and intermediate-mass black holes.

4.2 Bondi-Hoyle-Lyttleton Accretion

A compact object moving through a uniform medium at velocity $v$ (or equivalently, a medium flowing past a stationary object) gravitationally captures material from within the accretion radius:

The Bondi accretion rate is

Bondi accretion is relevant for isolated neutron stars accreting from the interstellar medium, for wind-fed X-ray binaries, and as a sub-grid model for AGN feedback in cosmological simulations.

4.3 Accretion Disk Physics: The Shakura-Sunyaev Model

When accreting material carries angular momentum it cannot fall directly onto the compact object but instead forms a differentially rotating disk. Viscous stresses transport angular momentum outward, allowing mass to spiral inward. Shakura and Sunyaev (1973) parametrised the unknown viscosity as

In steady state the mass accretion rate $\dot{M}$ is independent of radius. The vertically integrated angular momentum equation gives the radial velocity:

The local viscous dissipation rate per unit area is

If the disk is optically thick and radiates as a blackbody, the effective temperature is$T_{\rm eff}(r) = [Q^+/\sigma_{\rm SB}]^{1/4}$, which scales as $T \propto r^{-3/4}$far from the inner edge. The integrated spectrum is a multi-colour blackbody, peaking in the UV for AGN disks and in the X-ray for disks around stellar-mass compact objects.

The total luminosity of the disk, integrated from $r_{\rm in}$ to infinity, is

4.4 Innermost Stable Circular Orbit (ISCO)

In general relativity the effective potential for circular orbits around a Schwarzschild black hole differs from the Newtonian case. The relativistic effective potential is

The last term, absent in Newtonian gravity, causes the potential barrier to disappear at small radii. Circular orbits exist only for $r > 3r_g$ (where $r_g = GM/c^2$), and stable orbits require $r > 6r_g = 3r_s$. This defines the ISCO:

Below the ISCO matter plunges freely into the black hole without radiating further (in the simplest model). The ISCO therefore sets the inner edge of the accretion disk and determines the radiative efficiency.

4.5 X-Ray Binaries

Compact objects in binary systems can accrete from their companions, producing luminous X-ray emission. Two main classes exist:

The Roche lobe radius for the donor star in a binary with mass ratio $q = M_{\rm donor}/M_{\rm accretor}$is approximated by the Eggleton formula:

5. Black Holes (Astrophysical)

5.1 The Schwarzschild Black Hole

The simplest black hole is the Schwarzschild solution (1916) — a vacuum, spherically symmetric, static spacetime. The metric in Schwarzschild coordinates is

where the Schwarzschild radius is

The event horizon at $r = r_s$ is a coordinate singularity (removable in Eddington-Finkelstein or Kruskal-Szekeres coordinates) but a true causal boundary: no signal from within can reach an external observer. The physical singularity at $r = 0$ is a curvature singularity where the Kretschner scalar $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} = 48G^2M^2/(c^4 r^6)$ diverges.

Key properties

5.2 The Kerr Black Hole

Astrophysical black holes are expected to rotate due to angular momentum conservation during collapse or accretion. The Kerr metric (1963) describes a stationary, axisymmetric vacuum spacetime with mass $M$ and angular momentum $J$. In Boyer-Lindquist coordinates:

The horizons are located where $\Delta = 0$:

Frame Dragging and the Ergosphere

Between the event horizon and the ergosphere boundary $r_{\rm ergo} = r_g + \sqrt{r_g^2 - a^2\cos^2\theta}$lies the ergosphere, where no observer can remain stationary — spacetime itself is dragged along with the black hole's rotation. Penrose (1969) showed that this frame-dragging enables energy extraction from the black hole: a particle entering the ergosphere can split so that one fragment falls through the horizon with negative energy while the other escapes with more energy than the original particle. The rotational energy extractable by the Penrose process is up to 29% of the black hole's rest-mass energy.

The Blandford-Znajek mechanism, the magnetic analog of the Penrose process, threads the ergosphere with magnetic field lines anchored in the accretion disk. The resulting Poynting flux powers relativistic jets in AGN and microquasars, with luminosity scaling as $L_{\rm BZ} \propto a_*^2 B^2 r_g^2 c$.

5.3 ISCO Dependence on Spin

The ISCO radius for the Kerr metric depends on the spin parameter and the direction of orbital motion relative to the spin. For prograde equatorial orbits:

The spin of astrophysical black holes can thus be inferred from the thermal continuum spectrum of the accretion disk or from the profile of the relativistically broadened iron K-alpha emission line at 6.4 keV.

5.4 Hawking Radiation

Hawking (1974) showed that quantum field theory in curved spacetime predicts that black holes emit thermal radiation with temperature

The radiation is thermal with a black-body spectrum. The luminosity is

For stellar-mass and supermassive black holes the Hawking temperature is negligibly small ($T_H \sim 10^{-7}$–$10^{-14}$ K), making the radiation undetectable against the cosmic microwave background. However, for primordial black holes with $M \lesssim 10^{15}$ g the Hawking temperature exceeds $10^{11}$ K and the black hole evaporates on a timescale shorter than the age of the universe. The evaporation time is

Hawking radiation implies that black holes have an entropy (the Bekenstein-Hawking entropy) proportional to their horizon area, connecting general relativity, quantum mechanics, and thermodynamics in a profound way that remains at the frontier of theoretical physics.

5.5 Observational Evidence for Black Holes

6. Simulation: White Dwarf Mass-Radius Relation

This simulation solves the Lane-Emden equation for an $n = 3/2$ polytrope (non-relativistic electron degeneracy) and uses the polytropic scaling relations to compute the white dwarf radius as a function of mass for different compositions (parametrised by the mean molecular weight per electron$\mu_e$). Helium and carbon-oxygen white dwarfs both have $\mu_e = 2$ but differ in their detailed structure; iron white dwarfs ($\mu_e \approx 2.15$) are smaller at a given mass. Adjust the composition parameters and the number of integration points below.

7. Simulation: TOV Equation Solver

This simulation numerically integrates the Tolman-Oppenheimer-Volkoff equation with a polytropic equation of state $P = K\rho^\Gamma$ to generate the neutron star mass-radius relation. A family of stellar models is computed by sweeping the central density from $10^{14.5}$ to$10^{16}$ g cm$^{-3}$. The maximum mass along the sequence is highlighted. Adjust the polytropic index $\Gamma$ and constant $K$ to explore how the EOS stiffness affects the maximum mass and typical radii. Stiffer EOS (larger $\Gamma$ or $K$) produce larger maximum masses and radii.

Summary and Connections