Image: NASA, ESA, S. Beckwith (STScI), and The Hubble Heritage Team (STScI/AURA)Ten equations every astronomy student should know
The mass of the Sun, the temperature of a star and the distance to a galaxy each come out of one short formula and one measurement. The ten equations below cover most of a first course in astronomy. Each is shown with real numbers, because an equation you have never put numbers into is one you do not yet know. Solar values are taken from the NASA Sun fact sheet.
Distance and brightness
1. Parallax distance
\[ d = \frac{1}{p} \]A nearby star shifts against the background as the Earth orbits the Sun. With the parallax angle \(p\) in arcseconds, the distance \(d\) comes out in parsecs (1 pc = 3.26 light years). Proxima Centauri has \(p = 0.768\) arcseconds, so \(d = 1.30\) pc. All other distances in astronomy are calibrated on this one, as the article on the distance ladder explains.
2. The inverse square law
\[ F = \frac{L}{4\pi d^2} \]A source of luminosity \(L\) (in watts) spreads its light over a sphere of radius \(d\), and \(F\) is the flux that arrives per square meter. For the Sun, \(L = 3.83 \times 10^{26}\) W and \(d = 1.496 \times 10^{11}\) m give \(F = 1{,}361\) W per square meter at the Earth. At Jupiter, 5.2 times farther out, sunlight is 27 times weaker.
3. The distance modulus
\[ m - M = 5 \log_{10}\!\left(\frac{d}{10\ \mathrm{pc}}\right) \]This is the inverse square law in the units astronomers use. The apparent magnitude \(m\) says how bright a star looks, and the absolute magnitude \(M\) how bright it would look from 10 parsecs. Smaller numbers mean brighter. Sirius has \(m = -1.46\) and lies at 2.64 pc, so \(M = -1.46 - 5\log_{10}(0.264) = +1.43\). The Sun's absolute magnitude is +4.83, which makes Sirius about 23 times more luminous in visible light.
Light and temperature
4. Wien's displacement law
\[ \lambda_{\max}\, T = 2.898 \times 10^{-3}\ \mathrm{m\,K} \]A hot body radiates most strongly at a wavelength \(\lambda_{\max}\) that is inversely proportional to its temperature \(T\). The Sun's surface at 5,772 K peaks at 502 nanometers, in green light. The cosmic microwave background at 2.725 K peaks at 1.06 mm. Read the other way round, the color of a star is a thermometer.
5. The Stefan-Boltzmann law
\[ L = 4\pi R^2 \sigma T^4 \]The luminosity of a star depends on its surface area and on the fourth power of its surface temperature, with \(\sigma = 5.670 \times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}\). Put in the Sun's radius, \(6.957 \times 10^{8}\) m, and 5,772 K, and you recover \(3.83 \times 10^{26}\) W. The law is most useful as a ratio. A red giant with 100 times the Sun's radius and half its temperature has \(100^2 \times (1/2)^4 = 625\) solar luminosities. Astronomers use it to find the radii of stars that are far too distant to show a disk.
6. The Doppler shift
\[ \frac{\Delta\lambda}{\lambda} = \frac{v}{c} \]A source moving along the line of sight at speed \(v\), much smaller than the speed of light \(c\), has its spectral lines shifted by \(\Delta\lambda\): to the red if it recedes, to the blue if it approaches. The red hydrogen line at 656.28 nm shifts by 0.066 nm for a star receding at 30 km/s. The first planet found around a Sun-like star, 51 Pegasi b in 1995, makes its star wobble at about 56 m/s. That is a shift of roughly one part in five million, and measuring it won a Nobel Prize.
Gravity and orbits
7. Kepler's third law in Newton's form
\[ P^2 = \frac{4\pi^2 a^3}{G\,(M_1 + M_2)} \]Two bodies with masses \(M_1\) and \(M_2\) orbit each other with period \(P\) at a separation \(a\) (the semi-major axis), and \(G = 6.674 \times 10^{-11}\ \mathrm{m^{3}\,kg^{-1}\,s^{-2}}\). This equation is how nearly every mass in astronomy is measured. The Earth's orbit, with \(a = 1.496 \times 10^{11}\) m and \(P = 3.156 \times 10^{7}\) s, gives \(1.99 \times 10^{30}\) kg for the Sun. In units of years, astronomical units and solar masses it shrinks to \(M = a^3/P^2\). The star S2 orbits the center of the Milky Way with \(a \approx 1{,}000\) au and \(P = 16\) years, which puts about 4 million solar masses inside its orbit.
8. The Schwarzschild radius
\[ r_s = \frac{2GM}{c^2} \]The escape speed from distance \(r\) of a mass \(M\) is \(v = \sqrt{2GM/r}\), which is 11.2 km/s at the surface of the Earth. Set \(v = c\) and solve for \(r\), and you obtain the radius of a black hole's event horizon. The Newtonian shortcut happens to give the same formula as general relativity. The result is 2.95 km per solar mass: about 12.7 million km for the black hole in the center of our galaxy. More on what this radius means is in what a black hole is and what it is not.
Telescopes and the expanding universe
9. The diffraction limit
\[ \theta = 1.22\,\frac{\lambda}{D} \]A telescope with an aperture of diameter \(D\) cannot separate details closer than the angle \(\theta\) (in radians) when it observes at wavelength \(\lambda\). Multiply by 206,265 to convert to arcseconds. Hubble, with 2.4 m at 550 nm, reaches 0.058 arcseconds. The James Webb Space Telescope has a 6.5 m mirror but works at longer wavelengths, and at 2 micrometers it reaches 0.077 arcseconds. To resolve the shadow of a black hole, the Event Horizon Telescope linked radio dishes across the Earth, for an effective diameter of some 10,000 km at a wavelength of 1.3 mm. That yields about 30 millionths of an arcsecond.
10. The Hubble-Lemaître law
\[ v = H_0\, d \]Galaxies recede at a speed proportional to their distance. With a Hubble constant \(H_0\) of 70 km/s per megaparsec, a galaxy at 100 megaparsecs recedes at 7,000 km/s, and its lines are redshifted by \(z = v/c = 0.023\). Measure a redshift and you have a distance. The inverse \(1/H_0\) is 14 billion years, close to the age of the universe. The exact value of \(H_0\) is disputed between 67 and 73, a story told in the Hubble tension explained for students.
How to use them
Check units before numbers. If the left side is a length and the right side comes out in seconds, no calculator will help. Keep everything in SI units until the end, or use the solar-unit shortcuts and stay in them.
Work with ratios where you can. Most questions in astronomy ask how a star compares with the Sun, and then the constants cancel. The red giant in equation 5 needed neither \(\sigma\) nor \(\pi\).
The equations also combine. Equations 2 and 5 together give the temperature of a planet at a given distance from its star. Equations 6 and 7 together are how the masses of exoplanets and of binary stars are measured: the Doppler shift gives the orbital speed, and Kepler's law turns speed and period into mass. An eleventh equation, \(E = mc^2\), explains why stars shine at all. The Sun's luminosity divided by \(c^2\) is 4.3 million tonnes per second, the mass it converts into light.
In the A&A Masterclass these equations are derived where they are first needed. Unit 1 covers the mathematics, Unit 2 parallax and Kepler's laws, Unit 3 light, magnitudes and the radiation laws, and Unit 4 the diffraction limit. The Schwarzschild radius appears in Unit 10 and the Hubble-Lemaître law in Units 8 and 13. The curriculum shows the order.
