Ultimate Astrophysics Cheatsheet: Concepts, Formulas & Key Principles

Introduction: Understanding Astrophysics

Astrophysics is the branch of astronomy that applies the laws of physics to understand the properties, formation, and evolution of celestial objects and the universe as a whole. It seeks to explain the behavior of everything from subatomic particles to galactic superclusters using physical principles. This cheatsheet provides a comprehensive reference to the fundamental constants, equations, phenomena, and observational techniques that form the foundation of modern astrophysics.

Fundamental Constants & Conversions

Key Physical Constants

Unit Conversions

• Distance:

  • 1 pc = 3.26 light-years = 206,265 AU
  • 1 light-year = 9.461 × 10¹⁵ m
  • 1 kpc = 1,000 pc
  • 1 Mpc = 1,000 kpc

• Energy:

  • 1 keV = 1,000 eV = 1.602 × 10⁻¹⁶ J
  • 1 MeV = 1,000 keV
  • 1 GeV = 1,000 MeV
  • 1 erg = 10⁻⁷ J

• Time:

  • 1 year = 3.156 × 10⁷ s
  • 1 Myr = 10⁶ years
  • 1 Gyr = 10⁹ years

Core Astrophysical Concepts

Electromagnetic Radiation

• Wavelength-Frequency Relation: $\lambda\nu = c$
• Planck’s Law: $B_{\lambda}(T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/\lambda k_B T} – 1}$
• Wien’s Displacement Law: $\lambda_{max}T = 2.898 \times 10^{-3}$ m·K
• Stefan-Boltzmann Law: $L = 4\pi R^2 \sigma T^4$
• Electromagnetic Spectrum (in order of increasing wavelength):
  • Gamma rays: < 0.01 nm
  • X-rays: 0.01 – 10 nm
  • Ultraviolet: 10 – 380 nm
  • Visible light: 380 – 750 nm
  • Infrared: 750 nm – 1 mm
  • Microwave: 1 mm – 30 cm
  • Radio: > 30 cm

Newtonian Gravity

• Newton’s Law of Gravity: $F = \frac{GMm}{r^2}$
• Gravitational Potential: $\Phi = -\frac{GM}{r}$
• Escape Velocity: $v_{esc} = \sqrt{\frac{2GM}{r}} = \sqrt{2g_s R}$
• Orbital Period: $P = 2\pi\sqrt{\frac{a^3}{GM}}$
• Kepler’s Third Law: $\frac{P^2}{a^3} = \frac{4\pi^2}{G(M_1 + M_2)}$
• Virial Theorem: $2\langle T \rangle + \langle U \rangle = 0$ (time-averaged kinetic and potential energy)

Special Relativity

• Lorentz Factor: $\gamma = \frac{1}{\sqrt{1 – v^2/c^2}}$
• Time Dilation: $\Delta t = \gamma \Delta t_0$
• Length Contraction: $L = \frac{L_0}{\gamma}$
• Relativistic Mass: $m = \gamma m_0$
• Mass-Energy Equivalence: $E = mc^2$
• Relativistic Doppler Effect: $\nu_{observed} = \nu_{source}\gamma(1 – \frac{v}{c}\cos\theta)$

General Relativity

• Einstein Field Equations: $G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$
• Schwarzschild Radius: $R_s = \frac{2GM}{c^2} = 2.95 \frac{M}{M_{\odot}}$ km
• Gravitational Redshift: $\frac{\lambda_{observed}}{\lambda_{emitted}} = \frac{1}{\sqrt{1 – \frac{2GM}{rc^2}}}$
• Gravitational Lensing Angle: $\alpha = \frac{4GM}{bc^2}$ (for light with impact parameter $b$)
• Gravitational Wave Strain: $h \approx \frac{4G}{c^4}\frac{E}{r}$ (at distance $r$ from source with energy $E$)

Stellar Astrophysics

Stellar Classification and Properties

• Spectral Classification (OBAFGKM, from hottest to coolest)

  • O: > 30,000 K
  • B: 10,000 – 30,000 K
  • A: 7,500 – 10,000 K
  • F: 6,000 – 7,500 K
  • G: 5,200 – 6,000 K
  • K: 3,700 – 5,200 K
  • M: 2,400 – 3,700 K

• Mass-Luminosity Relation: $\frac{L}{L_{\odot}} \approx \left(\frac{M}{M_{\odot}}\right)^{3.5}$ (main sequence stars)

• Stellar Radius Calculation: $R = \sqrt{\frac{L}{4\pi\sigma T^4}}$

• Hertzsprung-Russell Diagram: Plot of luminosity vs. temperature/spectral class

  • Main Sequence: Hydrogen-burning stars
  • Giants/Supergiants: Expanded evolved stars
  • White Dwarfs: Stellar remnants

Stellar Structure Equations

• Hydrostatic Equilibrium: $\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2}$
• Mass Conservation: $\frac{dM(r)}{dr} = 4\pi r^2 \rho(r)$
• Energy Transport: $\frac{dT}{dr} = -\frac{3\kappa \rho L(r)}{64\pi \sigma r^2 T^3}$ (radiative) or $\frac{dT}{dr} = (1-\frac{1}{\gamma})\frac{T}{P}\frac{dP}{dr}$ (convective)
• Energy Generation: $\frac{dL(r)}{dr} = 4\pi r^2 \rho(r) \epsilon(r)$

Nuclear Fusion Processes

• Proton-Proton Chain (dominant in low-mass stars):

  • Net: $4 \text{p} \rightarrow \text{He} + 2e^+ + 2\nu_e + \text{energy (26.7 MeV)}$

• CNO Cycle (dominant in stars > 1.3 $M_{\odot}$):

  • Net: $4 \text{p} \rightarrow \text{He} + 2e^+ + 2\nu_e + \text{energy (26.7 MeV)}$
  • Carbon acts as catalyst

• Triple-Alpha Process:

  • Net: $3 \text{He} \rightarrow \text{C} + \text{energy (7.275 MeV)}$

• Energy Release Rate:

  • PP-Chain: $\epsilon_{PP} \propto \rho T^4$
  • CNO Cycle: $\epsilon_{CNO} \propto \rho T^{16}$

Stellar Evolution Timeline (1 $M_{\odot}$ star)

Stellar Endpoints

• White Dwarf:

  • Mass: < 1.44 $M_{\odot}$ (Chandrasekhar limit)
  • Composition: Carbon/Oxygen or Oxygen/Neon/Magnesium
  • Support: Electron degeneracy pressure
  • Density: ~10⁶ g/cm³
  • Size: ~Earth-sized

• Neutron Star:

  • Mass: 1.44 – ~3 $M_{\odot}$
  • Composition: Mostly neutrons
  • Support: Neutron degeneracy pressure
  • Density: ~10¹⁴ g/cm³
  • Size: ~10-20 km diameter
  • Rotation: ms to s periods
  • Magnetic Field: 10⁸ – 10¹⁵ G

• Black Hole:

  • Mass: > ~3 $M_{\odot}$
  • Defining Feature: Event horizon at $r = 2GM/c²$
  • Types: Stellar (3-100 $M_{\odot}$), Intermediate (10²-10⁵ $M_{\odot}$), Supermassive (10⁵-10¹⁰ $M_{\odot}$)

Galactic and Extragalactic Astrophysics

Milky Way Properties

• Structure: Barred spiral galaxy (SBc type)
• Diameter: ~100,000 light-years
• Mass: ~1-1.5 × 10¹² $M_{\odot}$ (including dark matter)
• Stars: ~200-400 billion
• Components:
  • Disk: Contains spiral arms, star-forming regions
  • Bulge: Older stars, central bar structure
  • Halo: Globular clusters, dark matter
  • Supermassive Black Hole (Sgr A*): ~4.3 × 10⁶ $M_{\odot}$

Galaxy Classification

• Elliptical (E0-E7): Featureless, old stellar populations
• Spiral (Sa-Sc): Disk with spiral arms, ongoing star formation
• Barred Spiral (SBa-SBc): Spiral with central bar structure
• Lenticular (S0): Disk and bulge but no spiral arms
• Irregular: No definite structure

Dark Matter Evidence

• Galaxy Rotation Curves: Flat velocity profiles at large radii
• Virial Theorem in Clusters: Insufficient visible mass for observed velocities
• Gravitational Lensing: Mass distribution exceeds visible matter
• Cosmic Microwave Background: Angular power spectrum
• Large-Scale Structure Formation: Requires non-baryonic matter

Active Galactic Nuclei (AGN)

• Common Features: Supermassive black hole, accretion disk, jets (sometimes)

• Types:

  • Seyfert Galaxies: Spiral hosts, prominent emission lines
  • Radio Galaxies: Powerful radio-emitting jets
  • Quasars: Extremely luminous, distant objects
  • Blazars: Jet oriented toward Earth

• Unified Model: Different AGN types are the same phenomenon viewed from different angles

Galaxy Evolution

• Merger Sequence: Spirals → Interacting/Peculiar → Elliptical
• Star Formation History: Peak at z~2 (cosmic noon)
• Downsizing: Massive galaxies formed stars earlier and faster
• Feedback Mechanisms:
  • Stellar feedback: Supernovae, stellar winds
  • AGN feedback: Jets, winds from central black hole

Cosmology

Fundamental Observations

• Hubble’s Law: $v = H_0 d$ (recession velocity proportional to distance)
• Cosmological Principle: Universe homogeneous and isotropic on large scales
• Cosmic Microwave Background: Thermal radiation at T = 2.7255 K
• Abundance of Light Elements: Consistent with Big Bang Nucleosynthesis
• Large-Scale Structure: Filaments, voids, clusters of galaxies

Friedmann Equations

• First Friedmann Equation: $\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho – \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$
• Second Friedmann Equation: $\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3P}{c^2}\right) + \frac{\Lambda c^2}{3}$
• Where:
  • $a(t)$ = scale factor
  • $\rho$ = energy density
  • $P$ = pressure
  • $k$ = curvature parameter (-1, 0, +1)
  • $\Lambda$ = cosmological constant

Cosmological Parameters

• Hubble Constant: $H_0 = 67.4$ km/s/Mpc (Planck) or $H_0 = 73.5$ km/s/Mpc (local measurements)
• Hubble Time: $t_H = 1/H_0 \approx 13.8$ Gyr
• Hubble Radius: $R_H = c/H_0 \approx 14.4$ Gpc
• Critical Density: $\rho_c = \frac{3H_0^2}{8\pi G} \approx 8.5 \times 10^{-27}$ kg/m³
• Density Parameters:
  • $\Omega_m \approx 0.31$ (matter)
  • $\Omega_\Lambda \approx 0.69$ (dark energy)
  • $\Omega_b \approx 0.049$ (baryonic matter)
  • $\Omega_r \approx 9 \times 10^{-5}$ (radiation)
  • $\Omega_k \approx 0$ (spatial curvature)

Cosmic Timeline

Redshift Relations

• Cosmological Redshift: $z = \frac{\lambda_{observed} – \lambda_{emitted}}{\lambda_{emitted}} = \frac{a_0}{a} – 1$
• Distance-Redshift (low z): $d_L \approx \frac{c}{H_0}z$
• Luminosity Distance: $d_L = (1+z)r$ where $r$ is comoving distance
• Angular Diameter Distance: $d_A = \frac{r}{1+z}$
• Time-Redshift Relation (matter-dominated): $t(z) \approx \frac{2}{3H_0\sqrt{\Omega_m}}(1+z)^{-3/2}$

Observational Techniques

Telescopes and Detectors

• Angular Resolution: $\theta \approx 1.22 \frac{\lambda}{D}$ (diffraction limit)

• Light-Gathering Power: $\propto D^2$

• Signal-to-Noise Ratio: $SNR \propto \sqrt{t\cdot A\cdot\epsilon/\Delta\lambda}$

  • $t$ = exposure time
  • $A$ = collecting area
  • $\epsilon$ = efficiency
  • $\Delta\lambda$ = bandpass

• Telescope Types:

  • Refractor: Uses lenses
  • Reflector: Uses mirrors
  • Catadioptric: Combines lenses and mirrors
  • Radio: Parabolic dishes or arrays
  • Interferometer: Multiple telescopes combined for higher resolution

Magnitude System

• Apparent Magnitude: $m = -2.5 \log_{10}(F) + C$
• Absolute Magnitude: $M = m – 5\log_{10}(d/10)$ where $d$ in pc
• Distance Modulus: $\mu = m – M = 5\log_{10}(d) – 5$ where $d$ in pc
• Bolometric Correction: $BC = M_{bol} – M_V$

Spectroscopy

• Spectral Line Identification: Element/molecule fingerprints
• Thermal Broadening: $\Delta\lambda \propto \sqrt{T/m}$
• Doppler Shift: $\Delta\lambda/\lambda = v/c$ (non-relativistic)
• Equivalent Width: $W = \int (1 – F_\lambda/F_c) d\lambda$
• Curve of Growth: Relates equivalent width to abundance

Multi-Messenger Astronomy

• Electromagnetic Radiation: Traditional astronomy (radio to gamma)
• Gravitational Waves: Ripples in spacetime from merging compact objects
• Neutrinos: Nearly massless particles from nuclear processes
• Cosmic Rays: High-energy particles (primarily protons)

Common Challenges and Solutions

Best Practices for Data Analysis

Statistical Methods

• Chi-squared Minimization: $\chi^2 = \sum_i \frac{(O_i – M_i)^2}{\sigma_i^2}$
• Maximum Likelihood Estimation: $\mathcal{L}(\theta | x) = \prod_i p(x_i | \theta)$
• Bayesian Inference: $p(\theta | x) \propto p(x | \theta) p(\theta)$
• Monte Carlo Methods: Numerical evaluation via random sampling

Common Data Processing Steps

1. Calibration:

   • Bias/dark subtraction
   • Flat fielding
   • Wavelength calibration (spectroscopy)
   • Flux calibration

2. Source Extraction:

   • Background determination
   • Source identification
   • Photometry/spectral extraction
   • Astrometric solution

3. Analysis Techniques:

   • Model fitting
   • Spectral line measurements
   • Time series analysis (periods, variability)
   • Population statistics

Resources for Further Learning

Online Data Archives

• NASA/IPAC Extragalactic Database (NED)
• Sloan Digital Sky Survey (SDSS)
• ESO Science Archive
• NASA Astrophysics Data System (ADS)
• SIMBAD Astronomical Database

Major Observatories

• Ground-based Optical/IR:

  • Very Large Telescope (VLT)
  • Keck Observatory
  • Gemini Observatories
  • Subaru Telescope
  • Giant Magellan Telescope (future)

• Space-based:

  • Hubble Space Telescope
  • James Webb Space Telescope
  • Chandra X-ray Observatory
  • XMM-Newton
  • Gaia

• Radio:

  • Atacama Large Millimeter Array (ALMA)
  • Very Large Array (VLA)
  • Square Kilometre Array (future)

Software Tools

• IRAF/PyRAF (image reduction)
• AstroPy (Python package)
• DS9 (image visualization)
• TOPCAT (table manipulation)
• CASA (radio astronomy)

Recommended Textbooks

• “An Introduction to Modern Astrophysics” by Carroll & Ostlie
• “Radiative Processes in Astrophysics” by Rybicki & Lightman
• “Galactic Dynamics” by Binney & Tremaine
• “Stellar Structure and Evolution” by Kippenhahn & Weigert
• “Extragalactic Astronomy and Cosmology” by Schneider

This cheatsheet provides a comprehensive but necessarily simplified overview of the vast field of astrophysics. Many formulas have additional terms or modifications for specific conditions. Advanced applications often require more detailed understanding of each topic.