Stars are self-gravitating balls of plasma in hydrostatic equilibrium, powered by nuclear fusion. Understanding stellar structure reveals the origin of chemical elements, the life cycle of stars, and the endpoints — white dwarfs, neutron stars, black holes.
PrerequisitesGravitation(Ch.7)⋅Nuclearphysics(Ch.NP)⋅Statisticalmechanics(Ch.S)⋅Generalrelativity (Ch. GR)
Learning Goals
Derive the hydrostatic equilibrium equation and use it to estimate central pressures and temperatures in stars.
Apply the virial theorem to compare the Kelvin-Helmholtz timescale with the nuclear burning timescale of the Sun.
Explainthemass−luminosityrelationL∝M3.5andcomputemain−sequencelifetimesasafunction of stellar mass.
Describe stellar endpoints (white dwarfs, neutron stars, black holes) and the mass thresholds that determine them.
Use the Chandrasekhar limit and the distance ladder (parallax, Cepheids, Type Ia SNe) to connect stellar physics to cosmological measurements.
AS.1 Hydrostatic Equilibrium
A star's structure is determined by the balance between gravity (inward) and pressure gradient (outward). For a spherical shell at radius r containing mass M(r):
dP/dr=−GM(r)ρ/r2(hydrostaticequilibrium)(AS.1)
dM/dr=4\pir2ρ(masscontinuity)(AS.2)
These two equations govern the pressure and density profiles. The virial theoremfor a self-gravitating system states: 2K + U_grav = 0, so E_total = −K = U_grav/2. Contracting a star releases gravitational energy, half radiated away, half heating the gas.
The Kelvin-Helmholtz timescale (gravitational contraction time):
The Sun's actual age is 4.6×10⁹ yr — 300× longer. This proved that nuclear burning, not gravity, powers the Sun (Thomson/Kelvin's classical estimate was wrong).
AS.2 Energy Transport
Heat flows outward through a star by three mechanisms:
Radiative transport: photons random-walk outward. The opacity κ (m²/kg) determines how far they travel. The radiative temperature gradient:
where a = 4σ/c is the radiation constant and L(r) is the luminosity at r.
Convection: when |dT/dr|_rad exceeds the adiabatic gradient, the gas is convectively unstable (Schwarzschild criterion). Convection is efficient — nearly adiabatic in stellar interiors. The outer ∼30% of the Sun (by radius) is convective; the core is radiative.
AS.3 Nuclear Burning in Stars
The proton-proton (pp) chain dominates in solar-mass stars (T_core ≈ 1.5×10⁷ K):
pp I: 4 p → ⁴He + 2e⁺ + 2νe + 26.7 MeV. Rate ∝ T⁴ (steep T dependence). CNO cycle dominates for M > 1.5 M☉ (T_core > 1.7×10⁷ K): ε ∝ T¹⁶ to T²⁰.
The steep temperature dependence is a thermostat: if T rises, ε rises, pressure rises, star expands, T falls — negative feedback keeps stars stable. Main sequence lifetime:
where f_H ≈ 10% of the mass is burned (core hydrogen fraction). Since L ∝ M³ to M⁴, massive stars (10 M☉) live only ∼30 Myr; low-mass stars (0.1 M☉) live > 100 Gyr.
Main sequence:SinceL∝M3.5andτ∝M/L∝M−2.5:a10M⊙starlives∼316×shorterthantheSun:1010/316≈30Myr.ConsistentwiththeageoftheOrionOBassociation.
AS.4 Stellar Evolution and Death
After main sequence, the hydrogen shell burns, and the core contracts as the envelope expands → Red Giant. Helium ignites (triple-alpha process: 3 ⁴He → ¹²C + 7.27 MeV), then carbon, oxygen in more massive stars.
Stellar endpoints by mass:
M < 8 M☉: planetary nebula + white dwarf (supported by electron degeneracy pressure). Chandrasekhar limit: M_Ch = 1.44 M☉. Above this, no stable white dwarf exists.
M > ~20 M☉: collapse to black hole. Event horizon at r_s = 2GM/c². For 10 M☉ BH: r_s ≈ 30 km.
Theorem AS.1 — Chandrasekhar Limit
White dwarfs are supported by electron degeneracy pressure. For non-relativistic electrons: P_∝ρ5/3.Themass−radiusrelation:R∝M−1/3—moremassiveWDsaresmaller.AsMincreases,electronsbecomerelativistic(Pdeg∝ρ(4/3:MCh=(5.87/\mue2)(\hbarc/G)3/2mH−2≈1.44M⊙where\mueisthemeanmolecularweightperelectron(≈2forC/O).AtM=MCh,theradiusgoes to zero — no stable configuration. Type Ia supernovae occur when a white dwarf accretes to this limit and undergoes thermonuclear explosion — they are standard candles (discovered dark energy, Nobel 2011).
AS.5 Cosmological Distance Ladder
Measuring cosmic distances requires a hierarchy of methods:
Parallax (up to ∼1 kpc, Gaia satellite): d = 1/p arcsec (parsec).Cepheids: period-luminosity relation (Leavitt 1912): log L = a log P + b, P = 1–100 days. Used to ∼100 Mpc (HST Key Project, H₀ calibration).Type Ia supernovae: standard candles (M_V ≈ −19.3), usable to z ∼ 2. Used to discover accelerating expansion (dark energy).
The Hubble constant H₀ = v_rec/d (recession velocity per distance) measures the expansion rate. Current tension: H₀ = 73.0 ± 1.0 km/s/Mpc (local, Cepheids/SNe Ia) vs H₀ = 67.4 ± 0.5 km/s/Mpc (CMB/Planck). 5σ discrepancy — possibly new physics.
Definition AS.1 — Common Traps
Luminosity and brightness are different: observed flux falls with distance squared.
Magnitude is logarithmic: smaller magnitudes are brighter.
Hydrostatic balance is dynamic equilibrium: gravity is balanced by pressure gradients, not absent.
Stellar lifetimes depend strongly on mass: massive stars burn brighter and die sooner.
Exercises — AS.1–AS.5 Astrophysics
1.
Calculate the number of pp-chain reactions per second in the Sun, the mass converted to energy per year, and the neutrino flux at Earth.