Astrophysics
The Physics of Stars, Galaxies, and the Cosmos
New to Astrophysics?
Check out our beginner-friendly course designed for undergraduate students! Learn about stars, galaxies, black holes, and cosmology with accessible explanations, real-world examples, and links to advanced topics when you're ready.
Start with Astrophysics for Beginners âCourse Overview
This graduate-level astrophysics course spans the physical processes governing celestial objects from individual stars to the large-scale structure of the universe. Beginning with the equations of stellar structure and nuclear burning, we progress through compact-object physics, galactic dynamics, and cosmological structure formation, concluding with the frontiers of exoplanet science, astrobiology, and multi-messenger astronomy.
4
Course Parts
16
Topics
6
Key Equations
4
Reference Texts
Prerequisites
Classical Mechanics
Newtonian dynamics, orbital mechanics, Lagrangian and Hamiltonian formulations
Electrodynamics
Maxwell's equations, electromagnetic radiation, and radiative processes
Thermodynamics
Equations of state, blackbody radiation, and thermal equilibrium
Nuclear Physics
Nuclear reactions, binding energies, and nucleosynthesis processes
Special Relativity
Lorentz transformations, relativistic kinematics, and mass-energy equivalence
Mathematics
Differential equations, vector calculus, and tensor analysis
Key Equations of Astrophysics
The foundational equations that govern stellar structure, stability, luminosity limits, mass thresholds, gravitational collapse, and stellar populations.
1. Hydrostatic Equilibrium
A star remains stable when the inward pull of gravity at every shell is exactly balanced by the outward pressure gradient. Consider a thin shell at radius $r$ with thickness $dr$, density $\rho(r)$, and enclosed mass $M(r)$. The gravitational force per unit volume pulling inward is $G M(r) \rho / r^2$, while the net pressure force is $-dP/dr$. Setting these equal gives:
$$\boxed{\frac{dP}{dr} = -\frac{G\,M(r)\,\rho(r)}{r^2}}$$
This is the most fundamental equation of stellar structure. It couples to the mass continuity equation $dM/dr = 4\pi r^2 \rho$ and determines how pressure, density, and temperature vary from the stellar core to the photosphere.
2. Virial Theorem
For a self-gravitating system in quasi-static equilibrium, there is a deep connection between the total gravitational potential energy $E_{\text{grav}}$ and the total thermal (kinetic) energy $E_{\text{th}}$. Integrating the hydrostatic equation over the entire star and using the ideal gas relation, one obtains:
$$\boxed{2\,E_{\text{th}} + E_{\text{grav}} = 0}$$
This means the total energy is $E_{\text{tot}} = E_{\text{th}} + E_{\text{grav}} = -E_{\text{th}} = \tfrac{1}{2}E_{\text{grav}} < 0$, so the system is bound. Counter-intuitively, as a star radiates and loses energy, it contracts and heats up â a hallmark of negative heat capacity in self-gravitating systems.
3. Eddington Luminosity
There is a maximum luminosity at which radiation pressure on electrons (via Thomson scattering) exactly balances gravitational attraction on protons. For a fully ionised hydrogen envelope of mass $M$, equating the radiation force per electron $\sigma_T L / (4\pi r^2 c)$ to the gravitational force per proton $G M m_p / r^2$ yields:
$$\boxed{L_{\text{Edd}} = \frac{4\pi\,G\,M\,m_p\,c}{\sigma_T} \approx 1.26 \times 10^{38}\left(\frac{M}{M_\odot}\right)\;\text{erg\,s}^{-1}}$$
Stars exceeding $L_{\text{Edd}}$ drive powerful winds that strip their outer envelopes. This limit governs the maximum masses of main-sequence stars and the peak luminosities of accreting compact objects such as X-ray binaries and active galactic nuclei.
4. Chandrasekhar Mass
White dwarfs are supported against gravity by electron degeneracy pressure. As mass increases, the electrons become relativistic and the pressure support weakens. Chandrasekhar showed that there is an upper mass limit beyond which no cold, degenerate electron gas can sustain hydrostatic equilibrium. For a composition with mean molecular weight per electron $\mu_e$:
$$\boxed{M_{\text{Ch}} = \frac{5.83}{\mu_e^2}\;M_\odot \approx 1.44\;M_\odot}$$
For a carbon-oxygen white dwarf, $\mu_e = 2$, giving the canonical value of $\approx 1.46\,M_\odot$ (the exact figure depends on corrections for Coulomb interactions and general relativity). Exceeding this limit triggers either a Type Ia supernova (thermonuclear) or gravitational collapse to a neutron star.
5. Jeans Mass
A gas cloud collapses under its own gravity only when gravitational energy overcomes thermal pressure. The critical mass scale is set by comparing the free-fall time to the sound-crossing time across the cloud. For a uniform cloud of temperature $T$, number density $n$, and mean molecular weight $\mu$:
$$\boxed{M_J = \left(\frac{5\,k_B\,T}{G\,\mu\,m_H}\right)^{3/2}\left(\frac{3}{4\pi\,\rho}\right)^{1/2}}$$
Clouds with $M > M_J$ are gravitationally unstable and fragment to form stars. In giant molecular clouds ($T \sim 10\;\text{K}$, $n \sim 10^3\;\text{cm}^{-3}$), the Jeans mass is roughly $\sim 10\,M_\odot$, setting the characteristic scale for star formation.
6. Salpeter Initial Mass Function
Edwin Salpeter (1955) empirically determined that the number of stars formed per unit mass interval follows a power law. The initial mass function (IMF) describes how newly formed stellar populations are distributed in mass:
$$\boxed{\xi(M) = \frac{dN}{dM} \propto M^{-2.35}}$$
The steep negative exponent means low-mass stars vastly outnumber high-mass stars. For every $20\,M_\odot$ O-star formed, there are roughly a thousand solar-mass stars. The Salpeter IMF remains a cornerstone of stellar population synthesis, chemical evolution models, and galaxy spectral energy distribution fitting, though modern refinements (Kroupa, Chabrier) flatten the slope below $\sim 0.5\,M_\odot$.
Course Contents
Part I: Stellar Physics
Stellar structure, nuclear reactions, evolution, and the HR diagram
Part II: Compact Objects
White dwarfs, neutron stars, supernovae, and gamma-ray bursts
Part III: Galaxies & Cosmology
Galaxy formation, AGN, dark matter halos, and large-scale structure
Part IV: Frontiers
Exoplanets, astrobiology, multi-messenger astronomy, and observational techniques
References
Carroll & Ostlie
An Introduction to Modern Astrophysics
The standard comprehensive undergraduate-to-graduate text covering stellar, galactic, and cosmological astrophysics with thorough derivations and worked problems.
Kippenhahn, Weigert & Weiss
Stellar Structure and Evolution
The definitive graduate reference for the physics of stellar interiors, nuclear burning, and evolutionary tracks from pre-main-sequence to compact remnants.
Shapiro & Teukolsky
Black Holes, White Dwarfs, and Neutron Stars
Authoritative treatment of compact object physics including equations of state, accretion, gravitational radiation, and relativistic stellar structure.
Binney & Tremaine
Galactic Dynamics
The standard reference for gravitational dynamics of stellar systems, covering orbits, equilibria, dark matter halos, disk stability, and galaxy interactions.
Related Courses
Cosmology
Big Bang, inflation, dark energy, and the cosmic microwave background
General Relativity
Spacetime curvature, Einstein field equations, and gravitational physics
Nuclear Physics
Nuclear reactions, binding energies, and nucleosynthesis processes
Gravitational Waves
LIGO, compact binary mergers, and gravitational wave astronomy
Video Lectures: Astrophysics
The Life and Death of Stars - Astrophysics Lecture
Galaxies, Dark Matter, and the Large-Scale Structure of the Universe
Mapping Cosmic Magnetism in the Space Between Stars â Susan E. Clark (Stanford)
Astrophysics in the Prize Record
Astrophysics is honoured by both the Crafoord Astronomy prize (Royal Swedish Academy) and a long line of Nobel Prizes \u2014 exoplanets (2019), gravitational waves (2017), cosmic-ray physics (1936, 2002), and pulsar timing (1974, 1993).
Crafoord Astronomy â
Royal Swedish Academy honours in astronomy and astrophysics.
Nobel Physics â
The benchmark global physics prize since 1901 â 35 laureate lectures (2013â2025).
Dirac Medal of ICTP â
ICTP Trieste annual award for theoretical physics, since 1985 â laureate lectures + ICTP Distinguished Conversations.
Max Planck Medal â
DPGâs highest honour for theoretical physics, since 1929 â laureate lectures plus the Lise Meitner Lecture series.