Astrophysics & Cosmology — formula sheet
Every boxed result from the 18 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Equation of Stellar Hydrostatic Equilibrium
At every radius the outward push from the pressure decreasing with height (\(dP/dr<0\)) exactly cancels the inward pull of the weight of a unit-volume fluid element, \(g\rho = Gm\rho/r^2\). The mass equation says the enclosed mass grows by one thin shell's worth per step outward. Two coupled first-order ODEs in the two unknowns \(P(r)\) and \(m(r)\); a third relation (an equation of state, \(P=P(\rho,\dots)\)) is needed to close the system.
Virial Theorem for Bound Systems
On average a bound self-gravitating system carries exactly twice as much kinetic energy (in magnitude) as its total energy, and half as much as its (negative) potential energy. It is virialised: the mean kinetic energy is pinned by the depth of the gravitational well. Because \(\langle E\rangle=-\langle T\rangle<0\), losing energy makes \(\langle T\rangle\) rise — the negative-specific-heat behaviour of gravity.
The Lane-Emden Equation for Polytropes
All the dimensional structure of a polytrope collapses onto one universal function \( \theta(\xi) \) depending only on the index \( n \). The physical density is \( \rho_c\theta^n \), the physical radius is \( \alpha\xi \), and the star's surface is the first zero \( \theta(\xi_1)=0 \), giving radius \( R=\alpha\xi_1 \). Two numbers per index — \( \xi_1 \) and \( \xi_1^2|\theta'(\xi_1)| \) — set the mass and radius of every polytrope of that index.
The Chandrasekhar Mass Limit
The maximum mass is built entirely from constants: the quantum-relativistic combination \((\hbar c/G)^{3/2}\), which is a mass (the Chandrasekhar mass is a few times this "gravitational mass scale"), divided by the square of the mass per electron. It does not depend on radius or central density — the defining feature of an \(n=3\) polytrope — so as a white dwarf's mass rises toward \(M_{\mathrm{Ch}}\) its radius shrinks toward zero. Composition enters only through \(\mu_e\): carbon–oxygen and helium dwarfs (\(\mu_e=2\)) give \(1.44\,M_\odot\); heavier compositions give less. The scale is set by gravity being weak enough that a macroscopic number \(\sim(\hbar c/Gm_H^2)^{3/2}\approx 2\times10^{57}\) of nucleons is needed before it wins.
Tolman-Oppenheimer-Volkoff Equation
The left side is the pressure gradient needed to hold the star up against gravity; the right side is that gravitational pull. Three factors distinguish it from Newton's \(-Gm\rho/r^2\), each making gravity stronger: (i) \(\rho\to\rho+P/c^2\) — pressure contributes inertia and weight; (ii) \(m\to m+4\pi r^3P/c^2\) — pressure integrated over the interior acts as an additional gravitating source; (iii) the denominator \((1-2Gm/rc^2)^{-1}>1\) — spacetime curvature amplifies the field. Integrating inward from the surface (\(P(R)=0\)) with a chosen central density fixes the star's radius \(R\) and total mass \(M=m(R)\); scanning central densities traces the mass–radius curve.
The Saha Ionization Equation
The degree of ionization rises with temperature both through the Boltzmann factor \(e^{-\chi_i/k_B T}\) (more particles clear the ionization barrier) and through the phase-space factor \(T^{3/2}\) (a hotter electron gas has more accessible translational states, so ionized states are entropically favoured). Crucially, \(n_e\) sits on the left: raising the electron density at fixed \(T\) drives the equilibrium back toward the neutral stage, because recombination becomes more likely. Ionization is therefore a competition between temperature and density, not temperature alone.
The FRW Metric from Homogeneity and Isotropy
Homogeneity and isotropy leave exactly two pieces of freedom in the entire metric: a single positive function of time, the scale factor \( a(t) \), which stretches all comoving separations by the same factor; and one constant \( k \) whose sign selects a closed (\( k>0 \), 3-sphere), flat (\( k=0 \), Euclidean), or open (\( k<0 \), hyperbolic) spatial geometry. Comoving coordinates \( (r,\theta,\varphi) \) are frozen onto the matter; physical separations grow as \( a(t) \) grows. All the dynamics is deferred to \( a(t) \), which symmetry does not fix — the Einstein equations do.
The Friedmann Equations from Einstein's Equations
The first equation says the square of the expansion rate \(H=\dot a/a\) is set by the total energy density, reduced by positive spatial curvature and boosted by the cosmological constant — geometry balances against content. The second says pressure gravitates: a fluid with \(\rho+3p/c^{2}>0\) decelerates the expansion, while a sufficiently negative pressure (dark energy, \(w<-1/3\)) or a positive \(\Lambda\) drives acceleration. The third is local energy conservation carried by the expansion: as space stretches, density dilutes at a rate set by \(H\) and the equation of state, so \(\rho\propto a^{-3(1+w)}\) for \(p=w\rho c^{2}\).
Cosmological Redshift from the Scale Factor
The factor by which a spectral line is shifted to longer wavelength equals the factor by which the Universe has expanded between emission and reception. A galaxy at \(z=1\) emitted its light when the Universe was half its present size (\(a(t_e)=a(t_0)/2\)); the surface of last scattering at \(z\approx1100\) dates from when everything was about \(1100\) times more compressed. Wavelength is dragged along with the expanding metric, not shifted by motion through space.
Comoving, Luminosity, and Angular-Diameter Distances
One integral over the expansion history sets the comoving distance; curvature bends it into the transverse comoving distance \(D_M\); and the two observable distances then follow by dividing or multiplying by \((1+z)\). The angular-diameter distance is smaller (nearby objects look big), the luminosity distance larger (distant candles look faint), and their ratio is locked to \((1+z)^2\) by geometry alone.
Particle Horizon, Horizon and Flatness Problems
In a hot big bang the age of the universe is finite and light has only had time to cross a finite comoving distance, so the sky is tiled by many patches that have never exchanged a signal — yet the CMB shows them at one temperature to \(1\) part in \(10^5\) (horizon problem). The same exponent makes \(|\Omega-1|\) grow with \(a\): to land near \(\Omega_0\approx1\) today the universe must have begun with \(|\Omega-1|\) tuned to \(\sim10^{-60}\) at the Planck epoch (flatness problem).
Slow-Roll Inflation from a Scalar Field
When the potential is flat enough that both \(\varepsilon\) and \(|\eta|\) are small, the inflaton crawls down \(V(\phi)\) at a terminal velocity fixed by the slope, the energy density stays nearly constant, and the scale factor grows exponentially. Inflation persists until \(\varepsilon\to1\); the accumulated growth is \(e^N\) with \(N\) the field-space integral of \(V/V'\). Requiring the observable universe to have been in causal contact needs \(N\gtrsim 50\)–\(60\), which flat potentials supply easily.
CMB Blackbody Spectrum and T ∝ 1/a
Free adiabatic expansion is a similarity transformation on a Planck spectrum: it slides the whole curve to lower frequencies and lower amplitude while keeping the dimensionless shape variable \(x=h\nu/k_BT\) fixed. The spectrum stays a blackbody; only the single label \(T\) changes, and it changes as \(1/a\). Because \(1+z=a_0/a\), a measured redshift is a direct measurement of past CMB temperature.
Recombination and Photon Decoupling
The free-electron fraction is set by a competition between the huge phase-space/dilution factor \((m_ec^2/k_BT)^{3/2}/\eta\sim 10^{15}\), which favours ionization, and the Boltzmann suppression \(e^{-B/k_BT}\), which favours neutrality. Neutrality wins only once \(k_BT\) has fallen to about \(B/42\approx 0.32\ \text{eV}\) — a temperature roughly forty times smaller than the binding energy, entirely because photons outnumber baryons by \(\sim 1.6\times10^{9}\). Half-ionization defines recombination at \(z\approx 1380\); the plasma becomes transparent slightly later, at \(z_{\rm dec}\approx 1090\), once \(x_e\) has fallen to \(\sim 10^{-2}\).
Primordial Helium from Big Bang Nucleosynthesis
About a quarter of the baryonic mass of the universe emerges from the first minutes as \({}^4\mathrm{He}\) and three-quarters as hydrogen. The number is set almost entirely by the neutron fraction surviving to nucleosynthesis, which is itself pinned by the competition between the weak interaction and cosmic expansion near \(1\ \mathrm{MeV}\).
The Jeans Instability Criterion
A perturbation behaves like a sound wave shifted downward by gravity. Short wavelengths (\(k>k_J\), \(\lambda<\lambda_J\)) keep \(\omega^2>0\) and merely oscillate: pressure crosses the region faster than gravity can collapse it. Long wavelengths (\(k<k_J\)) have \(\omega^2<0\), so one root grows as \(e^{|\omega|t}\) — gravitational collapse. The Jeans mass \(M_J\propto c_s^3\rho_0^{-1/2}\propto T^{3/2}\rho_0^{-1/2}\) is the least mass that collapses at given \(T\) and \(\rho_0\).
Linear Growth of Density Perturbations
The density contrast behaves like a damped, anti-restoring oscillator. The middle term \( 2H\dot\delta \) is Hubble friction: expansion continually drains the peculiar motion and slows collapse, which is why gravitational growth is a power law in time rather than the exponential runaway of a static medium. The bracket sets the competition between self-gravity, which wants to grow \( \delta \), and pressure, which wants to oscillate it; the crossover is the Jeans scale. In the matter era gravity wins and the dominant solution grows exactly in step with the scale factor. In the radiation era the matter term is overwhelmed by the radiation-driven Hubble rate, so matter perturbations merely limp along logarithmically until matter–radiation equality releases them.
Flat Rotation Curves and Dark Matter Halos
A flat rotation curve is not a statement that gravity is weak far out — it is a statement that mass keeps accumulating. Every time you double the radius you double the enclosed mass, so the invisible halo must extend well beyond the luminous disc and its density must decline as the inverse square of radius. The luminous matter alone would give \( M(r)\to\text{const} \) and \( v_c \propto r^{-1/2} \); the excess is the dark-matter halo.