Fundamentals of Astrophysics Definitive Study Notes

Fundamentals of Stellar Properties

1. Introduction to Observational and Physical Properties

  • Direct Observational Properties (Point-Source Data):

    • Position on the Sky: Corrected for Earth's motion, measured on a celestial sphere. Current accuracies: $\sim 0.01$ arcsec (ground) to milliarcseconds (space).

    • Apparent Brightness (Flux $F$): Logarithmic scale called magnitude ($m$). $\Delta m = 5$ represents a factor 100 decrease in $F$.

    • Color or Spectrum: Measures flux variation over wavelength $F_{\lambda}$. Spectral resolution $\lambda/\Delta\lambda$ allows detection of absorption lines representing discrete atomic transitions.

  • Inferred Physical Properties: By combining observations with physical principles, we can determine:

    1. Distance

    2. Luminosity ($L$)

    3. Temperature ($T$)

    4. Size (Radius $R$)

    5. Elemental Composition ($X, Y, Z$ flags for Hydrogen, Helium, and Metals)

    6. Velocity ($V_{radial}$ and $V_{transverse}$)

    7. Mass ($M$) and Surface Gravity ($g$)

    8. Age

    9. Rotation (Period $P$ or $V_{rot}$)

2. Astronomical Distances

  • Angular Size ($\alpha$): For distance $d$ and size $s$:

    • $\tan(\alpha/2) = s / 2d$

    • Small angle approximation (radians): $\alpha \approx s/d$

    • Distance to the Sun: $d = 1\,au \approx 1.5 \times 10^8\,km$. Angular diameter $\alpha_{\odot} \approx 0.5^{\circ} = 30\,arcmin = 1800\,arcsec$.

  • Trigonometric Parallax: The apparent shift of a nearby star against background stars due to Earth's 1 au orbital radius.

    • $d = s / \alpha$ where $\alpha$ is the parallax angle in radians.

    • Distance in Parsecs ($pc$): d[pc]=1p[arcsec]d[pc] = \frac{1}{p[arcsec]}

    • $1\,pc = 206,265\,au \approx 3.26\,ly \approx 3 \times 10^{16}\,m$.

  • Solid Angle ($\Omega$): Measured in steradians ($sr$) or square degrees.

    • $\Omega \approx \pi \alpha^2$ (for circular patch with angular radius $\alpha$).

    • Full sky $= 4\pi\,sr \approx 41,253\,degree^2$.

3. Stellar Luminosity and Magnitude

  • Inverse-Square Law for Flux ($F$):

    • F=L4πd2F = \frac{L}{4\pi d^2}

  • Surface Brightness (Intensity $I$): Flux per solid angle. Independent of distance.

    • I≈FΩ=L4π2R2=F∗πI \approx \frac{F}{\Omega} = \frac{L}{4\pi^2 R^2} = \frac{F_*}{\pi}

  • The Magnitude System:

    • Apparent Magnitude ($m$): Difference between two stars: m2−m1=2.5log⁡(F1F2)m_2 - m_1 = 2.5 \log\left(\frac{F_1}{F_2}\right)

    • Absolute Magnitude ($M$): Apparent magnitude if the star were at $d = 10\,pc$.

    • Distance Modulus: m−M=5log⁡(d10 pc)m - M = 5 \log\left(\frac{d}{10\,pc}\right)

    • Solar Benchmarks: $L_{\odot} \approx 4 \times 10^{26}\,W$ and $M_{\odot} \approx +4.8$.

4. Surface Temperature and Blackbody Radiation

  • Nature of Light: $\lambda \nu = c$. Photon energy $E = h\nu$.

  • Planck Blackbody Law ($B_{\lambda}(T)$):

    • Bλ(T)=2hc2/λ5ehc/λkT−1B_{\lambda}(T) = \frac{2hc^2/\lambda^5}{e^{hc/\lambda kT} - 1}

  • Wien's Displacement Law: The peak wavelength $\lambda_{max}$ is inversely proportional to $T$.

    • λmax≈500 nmT/T⊙\lambda_{max} \approx \frac{500\,nm}{T/T_{\odot}}

  • Stefan-Boltzmann Law: Surface flux $F_*$ and Bolometric Intensity $B(T)$.

    • F∗=σsbT4F_* = \sigma_{sb} T^4 where $\sigma_{sb} \approx 5.67 \times 10^{-8}\,W/m^2/K^4$.

    • Luminosity Scaling: LL⊙=(TT⊙)4(RR⊙)2\frac{L}{L_{\odot}} = \left(\frac{T}{T_{\odot}}\right)^4 \left(\frac{R}{R_{\odot}}\right)^2

  • Color Temperature: Inferred from the color index $(B - V) = m_B - m_V$. Negative values indicate higher temperatures.

5. Composition, Spectra, and the H-R Diagram

  • Spectral Lines: Absorption lines occur in cooler outer layers (photosphere) via transitions between discrete atomic energy levels.

  • Solar Composition: Mass fractions $X \approx 0.72$ (H), $Y \approx 0.26$ (He), $Z \approx 0.02$ (metals).

  • Spectral Classification: Based on temperature ($OBAFGKM$). O is hottest ($50,000\,K$), M is coolest ($3,500\,K$). Sun is $G2V$.

  • Hertzsprung-Russell (H-R) Diagram: Plots Luminosity vs. Temperature.

    • Main Sequence (MS): Long-lived phase of hydrogen fusion.

    • Giants/Supergiants: Evolved stars with high $L$, low $T$.

    • White Dwarfs: Hot, tiny remant cores.

6. Gravity, Mass, and Orbits

  • Surface Gravity ($g$): g=GMR2g = \frac{GM}{R^2}

    • For the Sun, $g_{\odot} \approx 27 g_e$. $10^4$ higher for white dwarfs; $10^{10}$ higher for neutron stars.

  • Velocities:

    • Escape Speed: Vesc=2GMRV_{esc} = \sqrt{\frac{2GM}{R}}

    • Circular Orbital Speed: Vorb=GMrV_{orb} = \sqrt{\frac{GM}{r}}

  • Virial Theorem: For bound systems, Total Energy $E = U/2 = -T$ (Kinetic $T$ is half the magnitude of Potential $U$).

7. Stellar Ages and Lifetimes

  • Kelvin-Helmholtz (Gravitational) Timescale: Time to radiate total gravitational binding energy.

    • $t_{KH} \approx \frac{3 G M_{\odot}^2}{10 R_{\odot} L_{\odot}} \approx 30\,Myr$.

  • Nuclear (H-fusion) Timescale: Efficiency $\epsilon_{nuc} \approx 0.007$. Core fraction $f \approx 0.1$.

    • tnuc=ϵnucfMc2L≈10 GyrM/M⊙L/L⊙t_{nuc} = \frac{\epsilon_{nuc} f M c^2}{L} \approx 10\,Gyr \frac{M/M_{\odot}}{L/L_{\odot}}

  • Mass-Lifetime Scaling: Given $L \sim M^3$, then $t_{ms} \approx 10\,Gyr (M_{\odot}/M)^2$.

  • Cluster Age: Determined by the Main Sequence Turnoff point luminosity $L_{to}$.

8. Space Motion and Binary Systems

  • Doppler Shift: Δλλo=Vrc\frac{\Delta\lambda}{\lambda_o} = \frac{V_r}{c}

  • Proper Motion ($\mu$): Angular drift in arcsec/yr. Transverse velocity:

    • Vt=4.7μp km/sV_t = 4.7 \frac{\mu}{p}\,km/s

  • Kepler's Third Law (Binary Mass):

    • M1+M2=a3[au]P2[yr]M⊙M_1 + M_2 = \frac{a^3[au]}{P^2[yr]} M_{\odot}

    • Spectroscopic Binaries: $M \sim V^3 P$. Max Doppler shifts give orbital speeds.

    • Eclipsing Binaries: Contact times $(t_2 - t_1)$ reveal stellar radii.

9. Observational Methods and Instrumentation

  • Telescopes:

    • Aperture ($D$): Light gathering area $A \propto D^2$. $m_{lim} \approx 7.5 + 5 \log(D[cm])$.

    • Angular Resolution: Diffraction limit $\alpha = 1.22 \lambda / D$.

  • Space Emissions:

    • UV/X-ray/Gamma-ray blocked by atmosphere; require space telescopes.

    • Infrared: Probes cool dust and star-forming regions.

  • Polarimetry: Stokes parameters $IQUV$. Circulary polarized light ($V$) scales with line-of-sight magnetic field strength $V \sim B_{los}$.

Fundamentals of Stellar Structure and Evolution

10. Hydrostatic Equilibrium

  • Pressure Balance: Outward gas pressure stops gravitational collapse.

    • dPdr=−ρ(r)g(r)\frac{dP}{dr} = -\rho(r) g(r)

  • Ideal Gas Pressure ($P$): $P = n k T = \rho k T / \mu_{bar}$, where $\mu_{bar}$ is mean molecular weight ($\approx 0.6 m_p$ for Sun).

  • Interior Virial Temperature ($T_{int}$): Derived from $H \approx R$.

    • T_{int} \approx \frac{GM ̄\mu}{kR} \approx 14 \times 10^6 \, K \frac{M/M_{\odot}}{R/R_{\odot}}

11. Energy Transport

  • Radiative Diffusion: Photons undergo a random walk. Central optical depth $\tau_c \approx 10^{11}$. Total diffusion time $t_{diff} \approx 7000\,yr$.

  • The $L \sim M^3$ Scaling: Arises from combining hydrostatic equilibrium and radiative diffusion equations without needing nuclear specifics.

  • Convection: Occurs when the radiative temperature gradient $dT/dr_{rad}$ exceeds the adiabatic gradient $dT/dr_{ad}$.

    • Caused by high opacity $\kappa$ (hydrogen recombination) or steep temperature dependence of nuclear burning (CNO cycle).

    • Hayashi Track: Fully convective proto-stars evolve at nearly constant $T$ with decreasing $L$.

12. Nuclear Fusion Requirements

  • Temperature for H-fusion: Must overcome electrostatic repulsion. Tunneling occurs when thermal de Broglie wavelength $\lambda \sim b$ (repulsion separation).

    • Characteristic speed: $V_{th,nuc} = 2e^2/h \approx 690\,km/s$.

    • Core temperature: $T_{nuc} \approx 17\,MK$.

  • Limits:

    • Lower Mass Limit: Brown Dwarfs ($M < 0.08 M_{\odot}$) fail because electron degeneracy stops core contraction before $T_{nuc}$ is reached.

    • Upper Mass Limit: Eddington Limit. Radiative force $g_{rad} = ̄̄\kappa F / c$ exceeds gravity $g$. $M_{max} \approx 200 M_{\odot}$.

13. Low-Mass Star Evolution ($M < 8 M_{\odot}$)

  • Red Giant Branch (RGB): Core H exhaustion $\rightarrow$ core contraction $\rightarrow$ H-shell burning $\rightarrow$ stellar expansion.

  • Helium Flash: Triggered in degenerate cores of stars $M < 2 M_{\odot}$ once $120 \, MK$ is reached.

  • Horizontal Branch (HB): Stable core He-burning.

  • Asymptotic Giant Branch (AGB): Helim-shell burning. Leads to massive pulsation.

  • End State: Planetary Nebula (envelope ejection) followed by a White Dwarf remnant.

    • White Dwarfs: Supported by electron degeneracy pressure. $R_{wd} \approx 0.01 R_{\odot}$. $R$ decreases as $M$ increases.

    • Chandrasekhar Limit: $M < 1.4 M_{\odot}$.

14. High-Mass Star Evolution ($M > 8 M_{\odot}$)

  • Multiple Shell Burning: Synthesis up to Iron (Fe). Beyond Iron, fusion is endothermic.

  • Core-Collapse Supernova: Iron core collapses, protons + electrons $\rightarrow$ neutrons. Shock wave ejects envelope at $0.1c$.

  • Remnants:

    • $8 M_{\odot} < M < 30 M_{\odot} \rightarrow$ Neutron Star (radius $\approx 10\,km$).

    • $M > 30 M_{\odot} \rightarrow$ Black Hole.

  • Gravitational Waves: Detected from binary mergers (LIGO). Convert mass deficit to wave energy.

Interstellar Medium (ISM) and Formation

15. The ISM and Components

  • Mean density: $n \approx 1\,cm^{-3}$.

  • Phases (Roughly Isobaric, $nT \approx 10^3\,K/cm^3$):

    • Cold: $10-100\,K$, Giant Molecular Clouds (GMCs). Site of star formation.

    • Warm: $10^4\,K$, HII regions (photoionized hydrogen).

    • Hot: $10^6\,K$, shock-heated by supernovae.

  • Interstellar Dust: Causes reddening/extinction. $\kappa_d(v) \propto v^{\beta}$ ($\beta \sim 1-2$).

16. Star and Planet Formation

  • Jeans Criterion: Cloud collapses if mass exceeds MJ≈92M⊙T3/2n1/2M_J \approx 92 M_{\odot} \frac{T^{3/2}}{n^{1/2}}

  • Protostellar Disks: Form due to conservation of angular momentum. Disk radius $r_d \approx 2\beta_{eq} R$.

  • Nebular Model: Solid grains grow into planetesimals. Ice Line separates inner rocky planets from outer gas/ice giants.

  • Equlibrium Temperature $T(d)$: T(d)≈290 K1 audT(d) \approx 290\,K \sqrt{\frac{1\,au}{d}}

Galaxies and Cosmology

17. The Milky Way and Galaxies

  • Galaxy Components: Disk (Pop I stars), Halo and Bulge (Pop II stars, Globular Clusters).

  • Rotation Curves: Flat curves ($V \approx constant$) imply mass $M(R) \propto R$. Evidence for Dark Matter.

  • Supermassive Black Holes (SMBH): Present in most nuclei. Milky Way central SMBH $M \approx 4 \times 10^6 M_{\odot}$.

18. Expansion and Quasars

  • Hubble Law: $V = H_o d$. Recession speed proportional to distance.

  • Quasars (QSOs): High-$L$ active galactic nuclei powered by accretion onto SMBHs. Efficiency $\epsilon \sim 0.1$.

  • Tully-Fisher Relation: $L_{gal} \propto V_{rot}^4$. Used as a standard candle for distances.

19. Cosmology and the Big Bang

  • Critical Density ($\rho_{co}$): Balance point for expansion vs. gravity. $\sim 10^{-29}\,g/cm^3$.

  • Parameters: $\Omega_m$ (matter fraction), $\Omega_{\Lambda}$ (dark energy fraction). Ι Γ-measured flat universe: $\Omega_m \approx 0.3, \Omega_{\Lambda} \approx 0.7$.

  • CMB: Radiation from the recombination era ($T \approx 3000\,K, z \approx 1100$). Currently $2.728\,K$.

  • Early Universe Eras:

    • Nucleosynthesis: First 3 minutes ($T=10^9\,K$). Forms 25% Helium.

    • Particle Era: Photon energy created matter/antimatter pairs.

    • Inflation: Exponential expansion solving the Flatness and Horizon problems.

  • Large Scale Structure: The "Cosmic Web" formed by gravitational amplification of initial quantum fluctuations.