Astrophysics Advanced

A star in hydrostatic equilibrium is described by four coupled ODEs:

The four equations of stellar structure

1. Mass continuity: dM_r/dr = 4πr²ρ
2. Hydrostatic equilibrium: dP/dr = −GM_r ρ / r²
3. Energy transport: dL_r/dr = 4πr² ρ ε (ε = nuclear + gravitational energy release per unit mass)
4. Temperature gradient: depends on transport mechanism:
   Radiative: dT/dr = −3κρL_r / (16πacr²T³)
   Adiabatic (convective): dT/dr = −(1−1/γ_ad)(T/P)(dP/dr)

The Virial theorem

2⟨K⟩ + ⟨U⟩ = 0

For a star: total energy E = −½|U_grav|. A contracting protostar heats up — the virial theorem tells us half the gravitational energy goes into heat, half radiated away.

Stars are nucleosynthesis factories, building heavy elements from hydrogen over their lifetimes.

Proton-proton chain (Sun)

Net: 4¹H → ⁴He + 2e⁺ + 2ν_e + 26.7 MeV. Three branches (ppI, ppII, ppIII) dominate at different temperatures.

CNO cycle (massive stars)

Carbon, nitrogen, oxygen act as catalysts: net result the same (4p → He) but dominant above ~15 million K (T > T_Sun).

Helium burning

Triple-alpha: ⁴He + ⁴He → ⁸Be (unstable), then ⁸Be + ⁴He → ¹²C*→ ¹²C + γ. The Hoyle state in ¹²C is crucial — its existence was predicted by Hoyle and then discovered.

s-process and r-process

Elements heavier than iron are built by neutron capture. The s(slow)-process occurs in AGB stars; the r(rapid)-process requires extreme neutron fluxes — neutron star mergers (GW170817 + kilonova confirmed this).

Supernovae are the most energetic stellar explosions and the main drivers of chemical enrichment of the interstellar medium.

Type II (core collapse) supernovae

A massive star's iron core (~1.4 M_☉) collapses in <1 second to a proto-neutron star. The collapse releases ~3 × 10⁴⁶ J, mostly as neutrinos (99%). A shock wave rebounds and ejects the envelope at ~10⁴ km/s. Observed optical light: ~10⁴³ J over weeks. The neutrino burst from SN 1987A was detected — the only observed extragalactic SN neutrinos.

Type Ia (thermonuclear) supernovae

A white dwarf accretes to near the Chandrasekhar limit (or two WDs merge) and detonates by thermonuclear runaway. Peak luminosity ~10¹° L_☉. Nearly uniform peak brightness makes them standardisable candles — used to discover dark energy (Nobel Prize 2011).

Neutron stars are the densest directly observable objects in the universe.

Structure

~1.4 M_☉ packed in R ~ 10–12 km. Crust: ordinary nuclei and electrons. Outer core: neutron-rich nuclear matter. Inner core: composition uncertain — possibly quark matter.

Equation of state

Neutron star structure follows the TOV equation (relativistic hydrostatic equilibrium). The maximum mass (TOV limit ~2–3 M_☉) depends on the nuclear equation of state — measuring massive neutron stars (up to ~2.1 M_☉ observed) constrains dense matter physics.

Pulsars

Rapidly rotating neutron stars with strong magnetic fields (~10⁸–10¹⁵ T) emit beamed radiation. Period P ≈ milliseconds to seconds. Millisecond pulsars (MSPs) are the most precise clocks in nature. Pulsar timing arrays (PTAs) are searching for nanohertz gravitational waves from SMBH binaries.

Black holes range from ~3 M_☉ (stellar mass) to ~10¹° M_☉ (supermassive), with intermediates (IMBHs) still being identified.

X-ray binaries

A stellar-mass BH in a binary system accretes from a companion star. Accretion disc temperatures ~10⁷ K emit in X-rays. Systems like Cygnus X-1 (the first BH identified) show quasi-periodic oscillations from the inner disc.

Supermassive black holes

Found in the nuclei of virtually all large galaxies. Sgr A* (4 × 10⁶ M_☉) and M87* (6.5 × 10⁹ M_☉) have been imaged by the Event Horizon Telescope. The M-σ relation (BH mass correlates with host bulge velocity dispersion) suggests BH-galaxy co-evolution.

Tidal disruption events

A star passing too close to an SMBH is torn apart by tidal forces; ~half the debris accretes, producing a bright flare across the spectrum.

Accretion — the infall of gas onto a compact object — is one of the most energy-efficient processes in astrophysics.

Accretion efficiency

Gravitational energy released: ΔE = GMΔm/r. For a neutron star: η ~ 20%; for a maximally spinning BH (Kerr): η ~ 42%. Compare to nuclear fusion: η ~ 0.7%. Quasars outshine entire galaxies powered by accretion alone.

Shakura-Sunyaev disc

The standard thin disc model: gas orbits at near-Keplerian velocities, viscosity (α-disc parametrisation) transfers angular momentum outward, and the released energy is radiated locally as a blackbody. Temperature profile: T(r) ∝ r^{−3/4}.

Relativistic jets

Many accreting objects launch collimated jets of plasma at near-c. Mechanism: possibly magnetic extraction of BH spin (Blandford-Znajek process) or magneto-centrifugal launching. AGN jets can extend to megaparsec scales.

Mass bends light — a prediction of general relativity confirmed in 1919 and now a powerful astrophysical tool.

Lens equation

A lens of mass M deflects a light ray by α = 4GM/(c²b), where b is the impact parameter. The lens equation gives the angular positions of images: θ − β = α̂ × D_LS/D_S.

Einstein ring

Perfect alignment produces an Einstein ring of radius θ_E = √(4GM/c² × D_LS/(D_L D_S)).

Strong lensing

Multiple images or arcs of background galaxies, seen around galaxy clusters and massive galaxies. Used to map dark matter distributions and discover high-redshift galaxies.

Weak lensing

Slight statistical distortion of background galaxy shapes — used to map the dark matter distribution over vast areas. Stage-IV surveys (Euclid, LSST/Rubin) will map dark matter in 3D using weak lensing tomography.

Microlensing

Magnification of a star as a foreground compact object crosses the line of sight — used to search for MACHOs and detect exoplanets.

Galaxies form from primordial density fluctuations amplified by gravity in the expanding universe.

Hierarchical structure formation

In the ΛCDM model, small structures form first and merge into larger ones. Dark matter halos collapse, gas cools and falls to the centre, forms stars. Mergers of galaxies drive morphological transformation.

Star formation rate history

The cosmic star formation rate peaked at z ~ 2–3 (about 10 billion years ago) and has declined by a factor of ~10 to the present.

Feedback mechanisms

• Stellar feedback: supernovae and stellar winds eject gas, quenching star formation in low-mass haloes.
• AGN feedback: jets and radiation from a central SMBH heat or eject gas from massive galaxies — "quenching" — explaining why the most massive ellipticals have old stellar populations.

Observations

JWST discovered galaxies at z > 10 — less than 500 million years after the Big Bang — some surprisingly massive and evolved, challenging simple formation models.