Modern Physics · Advanced Topics

High-Energy Astrophysics & Compact Objects

Neutron stars, black holes, gamma-ray bursts, and cosmic rays are laboratories for extreme physics — densities beyond nuclear, magnetic fields 10¹⁵ G, relativistic jets, and particle energies 10²⁰ eV. These objects probe GR, nuclear physics, plasma physics, and particle physics simultaneously.

PrerequisitesGeneralrelativity(Ch.GR)⋅Nuclearphysics(Ch.Nuc)⋅Astrophysics(Ch.Astro)⋅StatiGeneral relativity (Ch. GR) \cdot Nuclear physics (Ch. Nuc) \cdot Astrophysics (Ch. Astro) \cdot Statistical mechanics (Ch. SM)
Learning Goals
  • Apply the TOVequationtoneutronstarsandexplainhowthenuclearequationofstatedeterminesMmTOV equation to neutron stars and explain how the nuclear equation of state determines M_{m}
  • Compute the surface magnetic field, characteristic age, and spindown luminosity of a pulsar from P and Ṗ.
  • Derive the Eddington luminosity and compute the Eddington accretion rate for a compact object.
  • Describe the Kerr metric, the ergosphere, and the Penrose process for extracting black hole rotational energy.
  • Derive the GZK cutoff energy and estimate the maximum distance from which UHECR can reach Earth.

HEA.1 Neutron Stars

Neutron stars (NS): stellar remnants with M ~ 1.4 M_☉ compressed to R ~ 12 km. Central density ρ ~ 5–10 ρ_nuc (ρ_nuc = 2.7×10¹⁴ g/cm³). Supported by neutron degeneracy pressure (Fermi pressure of neutrons + nuclear interactions).

Tolman-Oppenheimer-Volkoff (TOV) equation: the GR analog of hydrostatic equilibrium:

dP/dr=−(G/r2)(ρ+P/c2)(M+4\pir3P/c2)/(1−2GM/(rc2))(TOV)dP/dr = -(G/r^{2})(\rho + P/c^{2})(M + 4\pir^{3}P/c^{2})/(1 - 2GM/(rc^{2})) \qquad (TOV)(HEA.1)

The factors (ρ + P/c²) and (1 − 2GM/(rc²)) are GR corrections — absent in Newtonian gravity. Maximum NS mass (Oppenheimer-Volkoff limit): M_max depends on the nuclear equation of state (EOS). Observed: J0348+0432, M = 2.01 M_☉; PSR J0952-0607, M ≈ 2.35 M_☉ — constrains EOS. Minimum mass for a black hole (after NS collapse): M > M_max ~ 2.3 M_☉.

Definition HEA.1 — Pulsar
ArapidlyrotatingNSwithastrongmagneticfieldB 108–1015G.Rotationfrequency:upA rapidly rotating NS with a strong magnetic field B ~ 10^{8}–10^{15} G. Rotation frequency: up tof=716Hz(PSRJ1748−2446ad).Lighthousemodel:amisalignedrotatingmagneticdipoleto f = 716 Hz (PSR J1748-2446ad). Lighthouse model: a misaligned rotating magnetic dipoleradiates and sweeps a beam across Earth. Pulsar period derivative Ṗ gives the spindown luminosity: L_/P3(I=NSmomentofinertia 1045g\cdotcm2).Surfacemagneticfield:B=3.2×1019(PP˙)GauP^{3} (I = NS moment of inertia ~ 10^{45} g\cdotcm^{2}). Surface magnetic field: B = 3.2\times10^{19} \sqrt(PṖ) Gaus.Pulsarsarethemostaccurateclocksinnature(stability10−14—rivalingatomicclocks. Pulsars are the most accurate clocks in nature (stability 10^{-14} — rivaling atomic clocks at long basescales).

HEA.2 Accretion and X-ray Binaries

Accretion onto compact objects converts gravitational potential energy to radiation. Efficiency η = ΔE/Mc²:

L=ηM˙c2ηNS≈GM/(Rc2)≈0.2,ηBH≈0.06–0.42(dependsonspin)L = \eta Ṁ c^{2} \qquad \eta_NS \approx GM/(Rc^{2}) \approx 0.2, \qquad \eta_BH \approx 0.06–0.42 (depends on spin)(HEA.2)

For a 1 M_☉ NS (R = 10 km): η ≈ 0.2 — 20% mass-to-energy conversion. Compare: nuclear fusion η_nuc ≈ 0.007 (0.7%). Accretion is the most efficient known energy source (after matter-antimatter annihilation η = 1).

Eddington luminosity: maximum luminosity before radiation pressure exceeds gravity (for electron scattering opacity): L_Edd = 4πGMm_p c/σ_T = 1.26×10³⁸ erg/s × (M/M_☉). Above L_Edd: super-Eddington accretion (jets, winds). ULX sources (ultra-luminous X-ray): L > L_Edd for M = 1 M_☉ — either super-Eddington accretion or intermediate-mass BHs.

Example HEA.1 — Crab Pulsar Energy Budget

TheCrabpulsar(PSRB0531+21):P=33.1ms,P˙=4.22×10−13s/s.CalculateBsurface,ageThe Crab pulsar (PSR B0531+21): P = 33.1 ms, Ṗ = 4.22\times10^{-13} s/s. Calculate B_{surface}, age, and spindown luminosity. Compare to the Crab Nebula luminosity.

Magnetic field:B=3.2×1019(PP˙)G=3.2×1019(33.1×10−3×4.22×10−13)G=3.2×1019(1.40×10−13)G=3B = 3.2\times10^{19} \sqrt(PṖ) G = 3.2\times10^{19} \sqrt(33.1\times10^{-3} \times 4.22\times10^{-13}) G = 3.2\times10^{19} \sqrt(1.40\times10^{-13}) G = 3.2×1019×1.18×10−7G≈3.8×1012G.Magnetarrange:B 1014–1015G(1000×stronger).TheC2\times10^{19} \times 1.18\times10^{-7} G \approx 3.8\times10^{12} G. Magnetar range: B ~ 10^{14}–10^{15} G (1000\times stronger). The Cab is a normal young pulsar.
Characteristic age:τc=P/(2P˙)=33.1×10−3/(2×4.22×10−13)s=3.92×1010s≈1240yr.Actualage:SN1054→\tau_c = P/(2Ṗ) = 33.1\times10^{-3}/(2 \times 4.22\times10^{-13}) s = 3.92\times10^{10} s \approx 1240 yr. Actual age: SN1054 \to age=971yr(asof2025).Goodagreement(characteristicageoverestimatesslightlyifinage = 971 yr (as of 2025). Good agreement (characteristic age overestimates slightly if intial spin was much faster).
Spindown luminosity:Lsd=4π2IP˙/P3=4π2×1045gcm2×4.22×10−13/(33.1×10−3)3=4π2×4.22×1032/3.63×1L_{sd} = 4\pi^{2}I Ṗ/P^{3} = 4\pi^{2} \times 10^{45} g cm^{2} \times 4.22\times10^{-13} / (33.1\times10^{-3})^{3} = 4\pi^{2} \times 4.22\times10^{32} / 3.63\times10−5erg/s≈4.5×1038erg/s=1.2×105L⊙.Thisistherotationalkineticenergybeingradi0^{-5} erg/s \approx 4.5\times10^{38} erg/s = 1.2\times10^{5} L_\odot. This is the rotational kinetic energy being radiated.
Nebula luminosity:CrabNebula:LNeb≈1.3×1038erg/s(synchrotron+optical+X−ray).Lsd/LNeb≈3.5—thCrab Nebula: L_{Neb} \approx 1.3\times10^{38} erg/s (synchrotron + optical + X-ray). L_{sd}/L_{Neb} \approx 3.5 — thepulsarwindcarriesabout3×theradiatednebulaluminosity(restgoesintoacceleratinge pulsar wind carries about 3\times the radiated nebula luminosity (rest goes into accelerating the remnant). The Crab pulsar is the engine of the Crab Nebula — confirmed by the pulsar-powered nebula (PWN) model. Discovery (1968): first identified pulsar inside a supernova remnant, establishing NS as SN remnants.

HEA.3 Black Hole Physics

Schwarzschild BH: ds² = −(1−r_s/r)c²dt² + dr²/(1−r_s/r) + r²dΩ². Event horizon at r = r_s = 2GM/c². Last stable circular orbit (ISCO): r_ISCO = 3r_s = 6GM/c² (Schwarzschild). Efficiency of accretion: η = 1 − √(1−2/3) × ... = 1 − √(8/9) ≈ 5.7%.

Kerr BH (rotating): ergosphere at r = r_s (equator), ISCO shrinks. Penrose process: extract energy from BH by splitting particle in ergosphere — one fragment falls in, the other escapes with more energy than the original. Maximum extractable rotational energy: 29% of M_BH c². Blandford-Znajek mechanism: magnetic field threading a spinning BH drives relativistic jets.

Supermassive BHs: M = 10⁶–10¹⁰ M_☉ in galactic nuclei (AGN, quasars). M87* image (Event Horizon Telescope, 2019): M = 6.5×10⁹ M_☉, r_s = 19 billion km, shadow diameter ≈ 40 μas — resolved at radio wavelengths. Sgr A* (Milky Way center): M = 4.15×10⁶ M_☉, image released 2022.

HEA.4 Gamma-Ray Bursts and Cosmic Rays

Gamma-ray bursts (GRBs): brightest electromagnetic events in the universe. E ~ 10⁵¹–10⁵³ erg (isotropic equivalent), duration 0.1–1000 s. Short GRBs (< 2 s): binary NS/NS or NS/BH mergers — confirmed by GW170817. Long GRBs (> 2 s): collapsar model — rapidly rotating massive star core collapse, forming a BH + accretion disk + relativistic jet (Lorentz factor Γ ~ 300). Internal shocks: variability in jet → shocks → γ-ray emission (prompt). External shock (afterglow): jet decelerates in ISM → X-ray/optical/radio.

Fireball model: opacity problem solved if jet is ultra-relativistic (Γ > 100): comoving photon energy E' = E/Γ drops below pair-production threshold. Compactness parameter: ℓ = (σ_T L)/(4πR²m_ec³) — requires Γ ≥ 100 to give ℓ < 1 (transparent).

Ultra-high-energy cosmic rays(UHECR): E > 10¹⁸ eV (EeV). GZK cutoff (Greisen-Zatsepin-Kuzmin, 1966): protons above 5×10¹⁹ eV interact with CMB photons: p + γ_CMB → Δ⁺ → n + π⁺ — lose energy over ~50 Mpc. Sources: AGN, magnetars, NS mergers (uncertain). Air shower experiments: Auger Observatory (Argentina), Telescope Array (Utah). Maximum energy from acceleration: E_max = ZeBR (Hillas criterion, B = field, R = size).

Definition HEA.2 — Common Traps
  • Nonthermal spectra are common: high-energy sources often need synchrotron or inverse-Compton models.
  • Compactness sets timescales: rapid variability implies small emitting regions.
  • Accretion efficiency can exceed fusion: gravity near compact objects is an enormous energy source.
  • Jets are relativistic: beaming can make luminosities appear much larger along the line of sight.
Exercises — HEA.1–HEA.4 High-Energy Astrophysics
1.
DerivetheEddingtonluminosityfora10M⊙blackhole.WhataccretionratedoesthisimpDerive the Eddington luminosity for a 10 M_\odot black hole. What accretion rate does this imply?Howlongwouldittaketogrowa109M⊙quasarstartingfrom1M⊙ly? How long would it take to grow a 10^{9} M_\odot quasar starting from 1 M_\odot?
W
Straightforward
2.Discuss the Tolman-Oppenheimer-Volkoff equation and the neutron star maximum mass. What is the Buchdahl limit and how do GW observations constrain the nuclear equation of state?
Intermediate
3.
Derive the GZK cutoff energy for cosmic ray protons interacting with CMB photons. What is the "GZK horizon" — the maximum distance from which UHECR can reach Earth?
eV
Intermediate
4.Explain pulsar timing arrays as gravitational wave detectors. What is the Hellings-Downs correlation? Describe the NANOGrav 2023 detection of the gravitational wave background.
Challenging
Key Takeaways
  • TOVequation:GRhydrostaticequilibrium.NSmasslimit 2−3M⊙(EOSdependent).TidaldTOV equation: GR hydrostatic equilibrium. NS mass limit ~ 2-3 M_\odot (EOS dependent). Tidal deformability constrains EOS.
  • Pulsars:rotatingmagnetizedNS.B=3.2×1019(PP˙)G.Lsd=4π2IP˙/P3.MostaccuratenaturPulsars: rotating magnetized NS. B = 3.2\times10^{19}\sqrt(PṖ) G. L_{sd} = 4\pi^{2}IṖ/P^{3}. Most accurate natural clocks.
  • Accretionefficiency:η 6−42Accretion efficiency: \eta ~ 6-42% for BH (vs 0.7% for fusion). L_{Edd} = 1.26\times10^{38}(M/M_\odot) erg/s.
  • Kerr BH: ergosphere, Penrose process (29% rotational energy extractable), Blandford-Znajek jet mechanism.
  • GRBs:short=NSmergers(GW170817confirmed).Long=collapsars,Γ 300,E 1052erg.IGRBs: short = NS mergers (GW170817 confirmed). Long = collapsars, Γ ~ 300, E ~ 10^{52} erg. Internal shock model.
  • GZKcutoff:EGZK 5×1019eV.UHECRfrom>1020eVmustoriginatewithin 50Mpc(GZKhGZK cutoff: E_{GZK} ~ 5\times10^{19} eV. UHECR from >10^{20} eV must originate within ~50 Mpc (GZK horizon).