Important equations
The key equations of the masterclass, collected unit by unit. Each one is derived or motivated in its unit, with worked examples.
The Astronomer's Toolkit
The small-angle formula
\[ \theta = \frac{D}{d} \quad (\theta\ \text{in radians}), \qquad\qquad \theta = 206{,}265''\times\frac{D}{d} \quad (\theta\ \text{in arcseconds}) \] An object of linear size \(D\) at a distance \(d \gg D\) has the angular size \(\theta\). \(D\) and \(d\) must be in the same unit. Any two of the three quantities give the third.
Rules for logarithms
\[ \log(ab) = \log a + \log b, \qquad \log\frac{a}{b} = \log a - \log b, \qquad \log\left(a^{n}\right) = n\log a \] The rules hold for positive \(a\) and \(b\) and for any base. They turn multiplication into addition and powers into multiplication.
Uniform circular motion
\[ v = \frac{2\pi r}{T}, \qquad\qquad a = \frac{v^{2}}{r} = \frac{4\pi^{2}r}{T^{2}}, \qquad\qquad F = \frac{mv^{2}}{r} \] An object of mass \(m\) on a circle of radius \(r\) with period \(T\) has the speed \(v\) and the centripetal acceleration \(a\), directed toward the center. \(F\) is the net force that some agent, such as a string or gravity, must supply.
Temperature, particle energy and gas pressure
\[ \tfrac{1}{2}m\langle v^{2}\rangle = \tfrac{3}{2}kT, \qquad\qquad P = nkT \] In a gas at temperature \(T\), particles of mass \(m\) have the mean kinetic energy \(\tfrac{3}{2}kT\), where \(\langle v^{2}\rangle\) is the mean of the squared speed and \(k = 1.381\times10^{-23}\unit{J\,K^{-1}}\) is the Boltzmann constant. The pressure \(P\) of an ideal gas is set by its number density \(n\) and its temperature.
Propagation of uncertainty
\[ z = x \pm y:\quad \sigma_{z}^{2} = \sigma_{x}^{2} + \sigma_{y}^{2} \qquad\qquad z = x^{a}y^{b}:\quad \left(\frac{\sigma_{z}}{z}\right)^{2} = a^{2}\left(\frac{\sigma_{x}}{x}\right)^{2} + b^{2}\left(\frac{\sigma_{y}}{y}\right)^{2} \] For sums and differences the absolute uncertainties add in squares. For products, quotients and powers the relative uncertainties \(\sigma/x\) do, each multiplied by its exponent. Both rules assume that the errors of \(x\) and \(y\) are independent and small.
The Sky and Celestial Mechanics
Altitude at culmination
\[ h_{\max} = 90^\circ - \varphi + \delta \] A star of declination \(\delta\) reaches this altitude above the southern horizon when it crosses the meridian at northern latitude \(\varphi\). A result above \(90^\circ\) means that the star passes north of the zenith.
Distance from parallax
\[ \frac{d}{1\unit{pc}} = \frac{1''}{p} \] The distance \(d\) of a star in parsecs is the reciprocal of its parallax \(p\) in arcseconds. Proxima Centauri, the nearest star, has \(p = 0.768''\) and lies at \(1/0.768 = 1.30\unit{pc}\), which is 4.25 light years.
Newton’s law of gravitation
\[ F = \frac{G M m}{r^2}, \qquad G = 6.674\times10^{-11}\unit{m^3\,kg^{-1}\,s^{-2}} \] Two bodies of masses \(M\) and \(m\) attract each other with the force \(F\) along the line that joins them. For spherical bodies \(r\) is the distance between their centers.
Kepler’s third law in Newton’s form
\[ P^2 = \frac{4\pi^2}{G(M+m)}\,a^3 \] Two bodies of masses \(M\) and \(m\) orbit each other with period \(P\); \(a\) is the semi-major axis of their relative orbit. Most masses in astronomy are measured this way.
Vis-viva equation
\[ v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right), \qquad E = -\frac{GMm}{2a} \] The speed \(v\) of a small body of mass \(m\) at the distance \(r\) from a mass \(M\), on an orbit with semi-major axis \(a\) and total energy \(E\). A circular orbit has \(a = r\) and \(v_{\mathrm{circ}} = \sqrt{GM/r}\). For \(a \to \infty\) the body just escapes: \(v_{\mathrm{esc}} = \sqrt{2GM/r} = \sqrt{2}\,v_{\mathrm{circ}}\).
Light, Radiation and Spectra
Inverse square law
\[ F = \frac{L}{4\pi d^2} \] The flux \(F\) received at distance \(d\) from a source of luminosity \(L\) that radiates equally in all directions through empty space. Twice the distance means a quarter of the flux.
Distance modulus
\[ m - M = 5\,\log_{10}\!\left(\frac{d}{10\unit{pc}}\right) \qquad\Longleftrightarrow\qquad d = 10^{(m-M+5)/5}\unit{pc} \] The difference between apparent magnitude \(m\) and absolute magnitude \(M\) depends only on the distance \(d\), as long as no dust dims the light (Section 6). Every factor of 10 in distance adds 5 magnitudes.
Wien’s displacement law
\[ \lambda_{\max}\,T = b = 2.898\times10^{-3}\unit{m\,K} \] A black body of temperature \(T\) is brightest at the wavelength \(\lambda_{\max}\).
Stefan-Boltzmann law for a star
\[ L = 4\pi R^2 \sigma T^4 \qquad\text{or}\qquad \frac{L}{\Lsun} = \left(\frac{R}{\Rsun}\right)^{2} \left(\frac{T}{\Tsun}\right)^{4} \] The luminosity \(L\) of a spherical black body of radius \(R\) and surface temperature \(T\), with \(\sigma = 5.670\times10^{-8}\unit{W\,m^{-2}\,K^{-4}}\). The second form compares the star with the Sun, \(\Tsun = 5772\unit{K}\).
Hydrogen levels and lines
\[ E_n = -\frac{13.6\unit{eV}}{n^2} , \qquad \frac{1}{\lambda} = R_{\mathrm{H}}\left(\frac{1}{n_l^{2}} - \frac{1}{n_u^{2}}\right) \] A jump between the lower level \(n_l\) and the upper level \(n_u\) emits or absorbs a photon of wavelength \(\lambda\). \(R_{\mathrm{H}} = 1.097\times10^{7}\unit{m^{-1}}\) is the Rydberg constant of hydrogen.
Doppler shift
\[ \frac{\Delta\lambda}{\lambda_0} = \frac{\lambda_{\mathrm{obs}} - \lambda_0}{\lambda_0} = \frac{v_r}{c} \] \(\lambda_0\) is the wavelength of a line at rest in the laboratory, \(\lambda_{\mathrm{obs}}\) the observed wavelength, and \(v_r\) the radial velocity, positive for a receding source. The formula holds for speeds much smaller than \(c\).
Telescopes and Observational Techniques
Light gathering power
\[ \frac{P_1}{P_2} = \left(\frac{D_1}{D_2}\right)^{2} \qquad\text{and}\qquad \Delta m = 5\log_{10}\!\left(\frac{D_1}{D_2}\right) \] The rate \(P\) at which photons from a source are collected grows with the square of the aperture diameter \(D\). Telescope 1 collects as many photons from a star \(\Delta m\) magnitudes fainter as telescope 2 collects from the brighter one.
Diffraction limit
\[ \theta_{\min} = 1.22\,\frac{\lambda}{D} \qquad\text{or}\qquad \theta_{\min} \approx 0.25''\times\frac{\lambda/\mu\mathrm{m}}{D/\mathrm{m}} \] The smallest angle \(\theta_{\min}\) (in radians in the first form, in arcseconds in the second) that a circular aperture of diameter \(D\) can resolve at wavelength \(\lambda\). No telescope does better, and many do worse because of the atmosphere or imperfect optics.
Signal to noise ratio
\[ \frac{S}{N} = \frac{R_*\,t}{\sqrt{\left(R_* + R_{\mathrm{sky}} + R_{\mathrm{dark}}\right)t + n_{\mathrm{pix}}\,\sigma_{\mathrm{r}}^{2}}} \] \(R_*\), \(R_{\mathrm{sky}}\) and \(R_{\mathrm{dark}}\) are the electrons per second from the star, the sky and the dark current inside the measuring aperture, \(t\) is the exposure time, \(n_{\mathrm{pix}}\) the number of pixels in the aperture and \(\sigma_{\mathrm{r}}\) the read noise per pixel. The relative error of the measured brightness is the inverse of \(S/N\).
Spectral resolving power
\[ R = \frac{\lambda}{\Delta\lambda} = \frac{c}{\Delta v} \] \(\Delta\lambda\) is the smallest wavelength difference that the spectrograph separates at wavelength \(\lambda\). By the Doppler formula of Unit 3 it corresponds to a velocity difference \(\Delta v\), with \(c\) the speed of light.
Resolution of an interferometer
\[ \theta_{\min} \approx \frac{\lambda}{B_{\max}} \] \(B_{\max}\) is the longest baseline of the array and \(\lambda\) the wavelength; the angle is in radians. The resolution depends on the separation of the dishes, the sensitivity on their total collecting area.
The Solar System
Radiometric age
\[ t = t_{1/2}\,\frac{\ln\left(1 + D/P\right)}{\ln 2} \] \(P\) and \(D\) are the numbers of parent and daughter atoms in the sample today, and \(t_{1/2}\) is the half-life of the parent. The clock starts when the mineral solidifies and stops exchanging atoms with its surroundings, and \(D\) counts only daughters made by decay inside the sample.
Equilibrium temperature
\[ T_{\mathrm{eq}} = 278\unit{K}\;(1-A)^{1/4}\left(\frac{a}{1\unit{au}}\right)^{-1/2} \] A rotating planet with albedo \(A\) at distance \(a\) from the Sun radiates as much power as it absorbs when its temperature is \(T_{\mathrm{eq}}\). The size of the planet does not enter. For another star, multiply by \((L/\Lsun)^{1/4}\).
Roche limit
\[ d_{\mathrm{R}} = 2.44\,R\left(\frac{\rho_M}{\rho_m}\right)^{1/3} \] A moon of density \(\rho_m\) that is held together only by its own gravity is torn apart by tides if it orbits closer than \(d_{\mathrm{R}}\) to the center of a planet of radius \(R\) and density \(\rho_M\).
Location of an orbital resonance
\[ a_{\mathrm{res}} = a_{\mathrm{pl}}\left(\frac{q}{p}\right)^{2/3} \] A small body that completes \(p\) orbits while a planet with semi-major axis \(a_{\mathrm{pl}}\) completes \(q\) orbits is in the \(p\,{:}\,q\) resonance with that planet and has the semi-major axis \(a_{\mathrm{res}}\). The same formula holds for moons and ring particles around a planet.
The Sun and the Properties of Stars
Luminosity, radius and effective temperature
\[ L = 4\pi R^2 \sigma T_{\mathrm{eff}}^4 \] A star of radius \(R\) and effective temperature \(T_{\mathrm{eff}}\) radiates the luminosity \(L\). The Stefan-Boltzmann constant is \(\sigma = 5.670\times10^{-8}\unit{W\,m^{-2}\,K^{-4}}\). Any two of the three quantities fix the third.
Sound travel time and mean density
\[ \Delta\nu \approx 135\unit{\mu Hz} \times \sqrt{\frac{\bar\rho}{\bar\rho_{\odot}}} \] The frequency spacing \(\Delta\nu\) of the acoustic tones of a star measures its mean density \(\bar\rho\), here in units of the solar value \(\bar\rho_{\odot} = 1410\unit{kg\,m^{-3}}\). The relation holds for stars that oscillate like the Sun, red giants included.
Kepler’s third law for binary stars
\[ \frac{M_1 + M_2}{\Msun} = \frac{(a/\mathrm{au})^3}{(P/\mathrm{yr})^2} \] Two stars that orbit each other with period \(P\) at a mean separation \(a\) (the semi-major axis of the relative orbit) have the total mass \(M_1 + M_2\). Both move around their common center of mass, the heavier one on the smaller orbit, with \(M_1 a_1 = M_2 a_2\) and \(a = a_1 + a_2\).
Mass-luminosity relation and main sequence lifetime
\[ \frac{L}{\Lsun} \approx \left(\frac{M}{\Msun}\right)^{3.5} \qquad\qquad t_{\mathrm{MS}} \approx 10^{10}\unit{yr} \times \left(\frac{M}{\Msun}\right)^{-2.5} \] A main sequence star of mass \(M\) has about this luminosity \(L\) and fuses hydrogen in its core for the time \(t_{\mathrm{MS}}\). Both rules are good to a factor of three between 0.5 and 20 solar masses.
The Lives and Deaths of Stars
Jeans mass
\begin{equation} M_{\mathrm{J}} = \left(\frac{5kT}{G\mu m_{\mathrm{H}}}\right)^{3/2}\left(\frac{3}{4\pi\rho}\right)^{1/2} \approx 5.4\,\Msun \left(\frac{T}{10\unit{K}}\right)^{3/2}\left(\frac{n}{10^{4}\unit{cm^{-3}}}\right)^{-1/2} \end{equation} A cloud of temperature \(T\) and density \(\rho\) (or particle number density \(n = \rho/\mu m_{\mathrm{H}}\), here with \(\mu = 2.33\)) collapses under its own gravity if its mass exceeds \(M_{\mathrm{J}}\).
Free-fall time
\begin{equation} t_{\mathrm{ff}} = \sqrt{\frac{3\pi}{32\,G\rho}} \end{equation} The collapse time depends only on the initial density \(\rho\). The dense core of the example above has \(t_{\mathrm{ff}} = 3.4\times10^{5}\unit{yr}\).
Cluster age from the turn-off mass
\begin{equation} t_{\mathrm{cluster}} \approx t_{\mathrm{MS}}(M_{\mathrm{to}}) \approx 10\unit{Gyr}\left(\frac{M_{\mathrm{to}}}{\Msun}\right)^{-2.5} \end{equation} \(M_{\mathrm{to}}\) is the mass of the stars now leaving the main sequence. The power law follows from \(t_{\mathrm{MS}} \propto M/L\) and the mass-luminosity relation \(L \propto M^{3.5}\) of Unit 6 and is good to a factor of about 1.5.
Size of the Roche lobe
\begin{equation} \frac{R_{\mathrm{L}}}{a} = \frac{0.49\,q^{2/3}}{0.6\,q^{2/3} + \ln\!\left(1+q^{1/3}\right)} \end{equation} \(R_{\mathrm{L}}\) is the radius of a sphere with the same volume as the Roche lobe of a star, \(a\) the orbital separation and \(q\) the mass of that star divided by the mass of its companion. This fit by Peter Eggleton (1983) is accurate to 1% for all \(q\). For equal masses \(R_{\mathrm{L}} = 0.38\,a\).
The Milky Way, Galaxies and the Expanding Universe
Enclosed mass from orbital speed
\[ M(<r) = \frac{v^2 r}{G} \] A body on a circular orbit of radius \(r\) with speed \(v\) weighs the mass inside its orbit. With \(G = 4.30\times10^{-6}\unit{kpc\,(km\,s^{-1})^2}\,\Msun^{-1}\) the result comes out in solar masses for \(r\) in kpc and \(v\) in km/s.
Period-luminosity relation of Cepheids (Leavitt law)
\[ M_V = -2.43\,(\log_{10}P - 1) - 4.05 \] \(M_V\) is the mean absolute visual magnitude and \(P\) the period in days. This calibration rests on Hubble Space Telescope parallaxes of ten Cepheids in the Milky Way (Benedict et al. 2007). Single stars scatter around it by about 0.2 mag.
Hubble-Lemaître law
\[ v = cz = H_0\, d \] The recession velocity \(v\) of a galaxy is proportional to its distance \(d\). The Hubble constant is \(H_0 \approx 70\unit{km\,s^{-1}\,Mpc^{-1}}\), and \(c = 2.998\times10^{5}\unit{km\,s^{-1}}\). The law holds for \(z\) below about 0.1; at larger redshift the history of the expansion enters (Unit 13).
Hubble time
\[ t_{\mathrm{H}} = \frac{1}{H_0} = 14.0\unit{Gyr} \times \frac{70\unit{km\,s^{-1}\,Mpc^{-1}}}{H_0} \] The Hubble time is the age the universe would have if it had always expanded at the present rate. The number follows from \(1\unit{Mpc} = 3.086\times10^{19}\unit{km}\), which makes \(H_0 = 2.27\times10^{-18}\unit{s^{-1}}\).
Stellar Structure and Nuclear Fusion
Hydrostatic equilibrium and mass continuity
\[ \frac{\dd P}{\dd r} = -\frac{G\,m(r)\,\rho(r)}{r^{2}}, \qquad\qquad \frac{\dd m}{\dd r} = 4\pi r^{2}\rho(r) \] \(P\) is the pressure and \(\rho\) the density at radius \(r\), and \(m(r)\) is the mass inside \(r\). Both equations hold in any star that is not collapsing or exploding.
Virial theorem
\[ 2K + U = 0, \qquad\qquad E = K + U = \tfrac{1}{2}U = -K \] \(K\) is the thermal kinetic energy of a star in hydrostatic equilibrium, \(U\) its gravitational potential energy and \(E\) the total. It holds for an ideal gas with negligible radiation pressure.
Gamow peak
\[ E_0 = \left(\frac{E_G\,(kT)^{2}}{4}\right)^{1/3}, \qquad\qquad \mathrm{rate} \propto \exp\!\left(-\frac{3E_0}{kT}\right) \] \(E_0\) is the collision energy at which most fusion reactions happen in a gas of temperature \(T\), and \(E_G\) is the Gamow energy of Equation . It applies to reactions between charged nuclei that have no resonance near \(E_0\).
Eddington luminosity
\[ L_{\mathrm{Edd}} = \frac{4\pi G M c}{\kappa} \approx 3.8\times10^{4}\,\Lsun\,\frac{M}{\Msun} \] Above this luminosity radiation pressure exceeds gravity, and a star of mass \(M\) loses its outer layers. The number holds for electron scattering in ionized gas with 70% hydrogen, \(\kappa = 0.034\unit{m^{2}\,kg^{-1}}\).
White Dwarfs, Neutron Stars and Black Holes
Electron degeneracy pressure
\begin{align*} P &= \frac{(3\pi^2)^{2/3}}{5}\,\frac{\hbar^2}{m_e}\left(\frac{\rho}{\mu_e m_u}\right)^{5/3} && \text{(slow electrons)}\\ P &= \frac{(3\pi^2)^{1/3}}{4}\,\hbar c\left(\frac{\rho}{\mu_e m_u}\right)^{4/3} && \text{(relativistic electrons)} \end{align*} The pressure \(P\) depends only on the density \(\rho\) and on \(\mu_e\). Temperature does not appear. The first form holds for \(\rho \ll 2\times10^{9}\unit{kg\,m^{-3}}\), the second far above that density.
Chandrasekhar mass
\[ M_{\mathrm{Ch}} = \frac{3.10}{(\mu_e m_u)^2}\left(\frac{\hbar c}{G}\right)^{3/2} = 1.46\,\Msun\left(\frac{2}{\mu_e}\right)^{2} \] No star supported by electron degeneracy pressure can exceed this mass. Small corrections (electric forces between the particles, general relativity) lower the value for a carbon-oxygen white dwarf to about \(1.4\,\Msun\), the number usually quoted.
Pulsar spin-down
\[ \dot E = \frac{4\pi^2 I\,\dot P}{P^3}, \qquad \tau = \frac{P}{2\dot P}, \qquad B \approx 3.2\times10^{15}\unit{T}\,\sqrt{\frac{P\,\dot P}{1\unit{s}}} \] The period \(P\) and its rate of change \(\dot P\) give the power \(\dot E\) that the rotation loses, the characteristic age \(\tau\) and the magnetic field \(B\) at the surface. The field estimate assumes a magnetic dipole with \(R = 10\unit{km}\). Pulsar papers often use gauss; \(1\unit{T} = 10^{4}\) gauss.
Schwarzschild radius
\[ r_{\mathrm{S}} = \frac{2GM}{c^2} = 2.95\unit{km}\times\frac{M}{\Msun} \] A mass \(M\) that lies entirely inside the radius \(r_{\mathrm{S}}\) is a black hole. For a non-rotating hole the sphere of radius \(r_{\mathrm{S}}\) is the event horizon.
Accretion luminosity
\[ L_{\mathrm{acc}} = \frac{GM\dot M}{R} = \eta\,\dot M c^2 , \qquad \eta = \frac{GM}{Rc^2} = \frac{r_{\mathrm{S}}}{2R} \] Gas that falls at the rate \(\dot M\) (in \(\mathrm{kg\,s^{-1}}\)) onto an object of mass \(M\) and radius \(R\) releases the power \(L_{\mathrm{acc}}\). The efficiency \(\eta\) is the fraction of the rest energy \(mc^2\) that is set free.
Exoplanets and the Search for Life
Radial velocity semi-amplitude
\[ K = \left(\frac{2\pi G}{P}\right)^{1/3} \frac{M_p \sin i}{(M_* + M_p)^{2/3}}\,\frac{1}{\sqrt{1-e^2}} \] \(K\) is half the full swing of the star’s line-of-sight velocity, \(P\) the period, \(e\) the eccentricity and \(i\) the inclination of the orbit (\(90^\circ\) is edge-on). For a circular orbit and \(M_p \ll M_*\), with \(M_{\mathrm{J}}\) the mass of Jupiter, \[ K \approx 28.4\unit{m\,s^{-1}}\;\frac{M_p \sin i}{M_{\mathrm{J}}}\left(\frac{P}{1\unit{yr}}\right)^{-1/3}\left(\frac{M_*}{\Msun}\right)^{-2/3} . \]
Transit depth and transit probability
\[ \delta = \left(\frac{R_p}{R_*}\right)^2 , \qquad p \approx \frac{R_*}{a} \] \(\delta\) is the fractional drop in brightness when a planet of radius \(R_p\) crosses a star of radius \(R_*\). \(p\) is the probability that a randomly oriented circular orbit of radius \(a\) shows transits at all.
Atmospheric signal in a transit
\[ H = \frac{kT}{\mu m_u g} , \qquad \Delta\delta \approx \frac{2nR_pH}{R_*^2} \] \(\Delta\delta\) is the extra transit depth inside an absorption band, \(g\) the gravity of the planet and \(\mu\) the mean molecular mass (2.3 for hydrogen and helium, 44 for carbon dioxide). Strong bands in a clear atmosphere reach \(n\) of 2 to 5.
Distance for a given stellar flux
\[ d = \sqrt{\frac{L/\Lsun}{S}}\unit{au} \] A planet at distance \(d\) from a star of luminosity \(L\) receives \(S\) times the flux that the Earth receives from the Sun. With \(S = 1.1\) and \(S = 0.36\) the habitable zone of the Sun runs from 0.95 to 1.67 au.
Active Galaxies and Supermassive Black Holes
Virial mass
\[ M \approx 5\,\frac{\sigma^2 R_e}{G} \] A system in equilibrium with line-of-sight velocity dispersion \(\sigma\) and half-light radius \(R_e\) has about this mass. The formula applies to elliptical galaxies, bulges, star clusters and, with galaxies as the particles, to clusters of galaxies. It fails for systems that are still merging or collapsing.
Mass from a Kepler orbit
\[ \frac{M}{\Msun} = \frac{(a/\mathrm{au})^3}{(P/\mathrm{yr})^2} \] \(M\) is the sum of the two masses, \(a\) the semi-major axis of the relative orbit and \(P\) the period. For a star around a black hole, \(M\) is the black hole mass. An orbit measured as an angle is converted with the distance \(d\): \(a\) in au equals the angle in arcseconds times \(d\) in parsecs.
The M-sigma relation
\[ M_{\mathrm{BH}} \approx 3.1\times10^{8}\,\Msun\left(\frac{\sigma}{200\unit{km\,s^{-1}}}\right)^{4.4} \] \(\sigma\) is the velocity dispersion of an elliptical galaxy or of the bulge of a disk galaxy. The numbers are the fit of Kormendy and Ho (2013), and single galaxies scatter around it by a factor of 2. Galaxies with a small, disk-like bulge, the Milky Way among them, tend to fall below the line.
Accretion luminosity and Eddington limit
\[ L = \eta\,\dot{M}c^2, \qquad L_{\mathrm{Edd}} = \frac{4\pi GMm_{\mathrm{p}}c}{\sigma_{\mathrm{T}}} = 1.26\times10^{31}\unit{W}\times\frac{M}{\Msun} \] \(\dot{M}\) is the mass accreted per unit time and \(\eta \approx 0.1\) the radiative efficiency. Above the Eddington luminosity of Unit 9 (\(m_{\mathrm{p}}\) is the proton mass, \(\sigma_{\mathrm{T}}\) the Thomson cross section of the electron) radiation pressure on ionized gas beats gravity and blows the infalling gas away. A steadily accreting black hole of mass \(M\) shines at no more than about \(L_{\mathrm{Edd}}\).
Apparent transverse speed
\[ \beta_{\mathrm{app}} = \frac{\beta\sin\theta}{1-\beta\cos\theta}, \qquad \beta_{\mathrm{app,max}} = \gamma\beta \] \(\beta = v/c\) is the true speed of the blob, \(\theta\) the angle between its motion and the line of sight, and \(\gamma = 1/\sqrt{1-\beta^2}\) the Lorentz factor. A measured \(\beta_{\mathrm{app}}\) requires \(\gamma \ge \sqrt{1+\beta_{\mathrm{app}}^2}\) and \(\theta \le 2\arctan(1/\beta_{\mathrm{app}})\).
Einstein radius
\[ \theta_{\mathrm{E}} = \sqrt{\frac{4GM}{c^2}\,\frac{D_{\mathrm{LS}}}{D_{\mathrm{L}}D_{\mathrm{S}}}} \] \(M\) is the mass of the lens inside the ring. \(D_{\mathrm{L}}\), \(D_{\mathrm{S}}\) and \(D_{\mathrm{LS}}\) are the angular diameter distances (Unit 13) to the lens, to the source and between the two, which do not simply add in an expanding universe. The giant arcs of a cluster lie near its Einstein radius.
The Expanding Universe and Relativistic Cosmology
Cosmological redshift
\[ 1 + z = \frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{emit}}} = \frac{1}{a(t_e)} \] The redshift \(z\) of a source tells by what factor the universe has expanded since the light was emitted at time \(t_e\). The relation holds for any expansion history.
Friedmann equation
\[ H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\,\rho - \frac{k c^2}{a^2} \] The expansion rate \(H\) at any time is set by the mean density \(\rho\) at that time and by the curvature constant \(k\), which has units of inverse length squared and never changes.
Critical density and density parameter
\[ \rho_c = \frac{3H^2}{8\pi G}, \qquad \Omega = \frac{\rho}{\rho_c} \] A universe with the critical density \(\rho_c\) is spatially flat. The density parameter \(\Omega\) expresses a density in units of the critical density of the same epoch; a subscript 0 marks present values.
Expansion rate at redshift \(z\)
\[ H(z) = H_0 \sqrt{\Omega_{r,0}(1+z)^4 + \Omega_{m,0}(1+z)^3 + \Omega_{k,0}(1+z)^2 + \Omega_{\Lambda,0}} \] The curvature term has \(\Omega_{k,0} = 1 - \Omega_{r,0} - \Omega_{m,0} - \Omega_{\Lambda,0}\), which is zero in a flat universe. With \(1 + z = 1/a\) the same formula gives \(H\) as a function of the scale factor.
Age and lookback time
\[ t_0 = \int_0^1 \frac{\dd a}{a\,H(a)}, \qquad t_0 - t(z) = \int_0^z \frac{\dd z'}{(1+z')\,H(z')} \] The first integral, from the Big Bang at \(a = 0\) to today, is the age of the universe. The second is the lookback time, the travel time of light from a source at redshift \(z\).
Distances in a flat universe
\[ d_p = c\int_0^z \frac{\dd z'}{H(z')}, \qquad d_L = (1+z)\,d_p, \qquad d_A = \frac{d_p}{1+z} \] The proper distance today \(d_p\) follows from the expansion history. The luminosity distance \(d_L\) turns luminosity into flux, and the angular diameter distance \(d_A\) turns size into angle.
The Hot Big Bang and the Cosmic Microwave Background
Temperature of the radiation at redshift \(z\)
\[ T(z) = T_0\,(1+z) = \frac{T_0}{a} \] \(T_0 = 2.7255\unit{K}\) is the CMB temperature today and \(a\) the scale factor (\(a = 1\) today). The relation holds after the first few minutes, once photons are no longer created or destroyed in bulk.
Time and temperature in the radiation era
\[ T \approx 1.0\times10^{10}\unit{K}\,\left(\frac{1\unit{s}}{t}\right)^{1/2}, \qquad kT \approx 0.86\unit{MeV}\,\left(\frac{1\unit{s}}{t}\right)^{1/2} \] \(t\) is the time since the Big Bang and \(k\) the Boltzmann constant. These coefficients apply between \(10^{-4}\unit{s}\) and \(1\unit{s}\). Once electrons and positrons have annihilated (\(t > 100\unit{s}\)) they become \(1.3\times10^{10}\unit{K}\) and \(1.15\unit{MeV}\).
Neutron to proton ratio in equilibrium
\[ \frac{n_n}{n_p} = \exp\!\left(-\frac{Q}{kT}\right), \qquad Q = 1.293\unit{MeV} \] \(n_n\) and \(n_p\) are the number densities of neutrons and protons. The formula applies while the weak reactions are faster than the expansion, which means \(kT > 1\unit{MeV}\).
Angular scale of the sound horizon
\[ \theta_* = \frac{r_s}{D_M}, \qquad \Delta\ell \approx \frac{\pi}{\theta_*} \] \(r_s\) is the comoving sound horizon, \(D_M\) the comoving distance to the last scattering surface in a flat universe (Unit 13), \(\theta_*\) the angle in radians and \(\Delta\ell\) the spacing of the peaks.
Departure from flatness
\[ \left|\Omega - 1\right| = \frac{|k|\,c^2}{a^2H^2} = \frac{|k|\,c^2}{\dot a^{2}} \] \(\Omega\) is the total density parameter and \(k\) the curvature constant. Deceleration lowers \(\dot a = aH\) and moves the universe away from \(\Omega = 1\). Acceleration drives it toward 1.
Dark Matter, Dark Energy and Cosmic Structure
Hydrostatic mass of a galaxy cluster
\[ M(<r) = -\frac{kT\,r}{G\,\mu m_p}\left(\frac{\dd\ln\rho}{\dd\ln r} + \frac{\dd\ln T}{\dd\ln r}\right) \] \(T\) and \(\rho\) are the temperature and density of the gas at radius \(r\), \(m_p\) is the proton mass and \(\mu\approx0.6\) the mean particle mass in units of \(m_p\). The two logarithmic slopes say how steeply density and temperature fall outward; X-ray images and spectra provide both. The formula holds for gas at rest, so it fails during a merger.
Linear growth of structure
\[ \delta \propto a = \frac{1}{1+z} \qquad \text{(matter era, } \delta \ll 1\text{)} \] While matter dominates the expansion, a small density contrast \(\delta\) grows in proportion to the scale factor \(a\). Growth nearly stops while radiation dominates (before \(z\approx3400\)) and slows again once dark energy takes over.
The BAO standard ruler
\[ \theta = \frac{r_d}{D_M(z)} , \qquad \Delta z = \frac{r_d\,H(z)}{c} \] Across the line of sight the ruler \(r_d\) appears under the angle \(\theta\) (in radians), which gives the comoving distance \(D_M\) to redshift \(z\). Along the line of sight it spans a redshift interval \(\Delta z\), which gives the expansion rate \(H(z)\) at that epoch.
Acceleration and the equation of state
\[ \frac{\ddot a}{a} = -\frac{4\pi G}{3}\,\rho\,(1+3w) , \qquad \rho \propto a^{-3(1+w)} \] A component with density \(\rho\) and pressure \(P = w\rho c^{2}\) decelerates the expansion if \(w > -1/3\) and accelerates it if \(w < -1/3\). With several components, add their contributions. A cosmological constant has \(w = -1\): its density stays the same while space expands.
Frontiers of Astrophysics
Strain
\[ h = \frac{\Delta L}{L} \] A gravitational wave of strain \(h\) changes the distance \(L\) between two freely falling objects by \(\Delta L\). The strain has no unit. The waves measured on Earth so far have \(h \approx 10^{-21}\) or less.
Chirp mass and frequency drift
\[ \mathcal{M} = \frac{(m_1m_2)^{3/5}}{(m_1+m_2)^{1/5}}, \qquad \frac{\dd f}{\dd t} = \frac{96}{5}\,\pi^{8/3}\left(\frac{G\mathcal{M}}{c^{3}}\right)^{5/3} f^{11/3} \] Here \(f\) is the gravitational wave frequency, and \(G\Msun/c^{3} = 4.93\times10^{-6}\unit{s}\). For two equal masses \(m\) the chirp mass is \(0.87\,m\). The formula holds while the two bodies are well separated.
Strain of an inspiraling binary
\[ h = \frac{4}{d_L}\left(\frac{G\mathcal{M}}{c^{2}}\right)^{5/3}\left(\frac{\pi f}{c}\right)^{2/3} \] Strain amplitude at wave frequency \(f\) for a binary of chirp mass \(\mathcal{M}\) seen face-on at luminosity distance \(d_L\) (Unit 13).
Dispersion delay
\[ \Delta t = 4.15\unit{ms}\times\mathrm{DM}\times\left[\left(\frac{\nu_1}{\mathrm{GHz}}\right)^{-2} - \left(\frac{\nu_2}{\mathrm{GHz}}\right)^{-2}\right], \qquad \mathrm{DM} = \int n_e\,\dd l \] \(\Delta t\) is the delay of the lower frequency \(\nu_1\) relative to the higher frequency \(\nu_2\). The dispersion measure DM is the electron density \(n_e\) summed along the line of sight \(l\), in \(\mathrm{pc\,cm^{-3}}\).