zero to cosmos

Learn the universe one equation at a time.

Free astrophysics lessons that start with basic math and build, step by step, to stars, galaxies, black holes and the Big Bang.

Start at Level 0 See a sample lesson

Change a star’s temperature

Spectral classG
Peak wavelength502 nm
Brightness vs Sun1.0×
Peak wavelength from Wien’s law, \(\lambda_{\max} = b/T\). Brightness for a Sun-sized star, \(L/L_\odot = (T/T_\odot)^4\). You’ll derive both in Level 3.

Seven levels, from zero to the edge of the universe

Each level only uses ideas from the levels before it. Levels 0 to 2 need algebra and trigonometry; calculus is taught right before Level 3, when you need it.

  1. 0

    Foundations

    The toolkit: very big and very small numbers, units, logarithms and the angles you need to measure the sky.

    Math: arithmetic and basic algebra
    \(1\,\text{AU} \approx 1.496\times10^{11}\,\text{m}\)
    \(1\,\text{pc} \approx 3.26\,\text{ly}\)
    \(\theta \approx \dfrac{s}{d}\)
  2. 1

    Light and the sky

    How we measure distances to stars, why brightness fades with distance, and what light carries.

    Math: algebra and logarithms
    \(d = \dfrac{1}{p}\)
    \(m - M = 5\log_{10}\!\left(\dfrac{d}{10\,\text{pc}}\right)\)
    \(F = \dfrac{L}{4\pi d^2}\)
  3. 2

    Gravity and orbits

    Newton’s gravity, Kepler’s laws and how to weigh a planet or a star by watching something orbit it.

    Math: algebra and trigonometry
    \(F = \dfrac{GMm}{r^2}\)
    \(P^2 = \dfrac{4\pi^2 a^3}{G(M+m)}\)
    \(v_{\text{esc}} = \sqrt{\dfrac{2GM}{r}}\)
  4. 3

    Radiation physics

    Why stars have colors, how spectra reveal what stars are made of, and how we measure their motion.

    Math: introduction to calculus
    \(\lambda_{\max} T = b\)
    \(L = 4\pi R^2 \sigma T^4\)
    \(\dfrac{\Delta\lambda}{\lambda} = \dfrac{v}{c}\)
  5. 4

    Stars

    What holds a star up, how fusion powers it, and how it ends as a white dwarf, neutron star or black hole.

    Math: derivatives and simple differential equations
    \(\dfrac{dP}{dr} = -\dfrac{G m(r)\rho}{r^2}\)
    \(E = mc^2\)
    \(L \propto M^{3.5}\)
  6. 5

    Galaxies

    How galaxies spin, the evidence for dark matter, and the discovery that the universe is expanding.

    Math: calculus
    \(v(r) = \sqrt{\dfrac{G M(r)}{r}}\)
    \(v = H_0 d\)
    \(L_{\text{Edd}} = \dfrac{4\pi G M m_p c}{\sigma_T}\)
  7. 6

    Relativity and cosmology

    Space, time, black holes and the history of the whole universe, from the first second to today.

    Math: calculus, then an introduction to tensors
    \(r_s = \dfrac{2GM}{c^2}\)
    \(1 + z = \dfrac{\lambda_{\text{obs}}}{\lambda_{\text{emit}}}\)
    \(H^2 = \dfrac{8\pi G}{3}\rho - \dfrac{kc^2}{a^2}\)

Every lesson follows the same six steps

Intuition first, then the equation, then you use it on real stars and check yourself. Here’s a short piece of Lesson 1.2.

  1. 1The idea
  2. 2The equation
  3. 3Where it comes from
  4. 4Worked example
  5. 5Practice
  6. 6Try it live
Level 1 · Lesson 1.2

Measuring the distance to a star

The idea

Hold a finger at arm’s length and close one eye, then the other. Your finger jumps against the background. Earth does the same thing on a giant scale: as it moves around the Sun, nearby stars shift slightly against distant ones. The closer the star, the bigger the shift.

The equation

\[ d = \frac{1}{p} \]

Here \(d\) is the distance in parsecs and \(p\) is the parallax angle in arcseconds. One arcsecond is 1/3600 of a degree, smaller than a coin seen from 4 km away.

Worked example

Proxima Centauri, the nearest star to the Sun, has a parallax of \(p = 0.768''\).

\[ d = \frac{1}{0.768} \approx 1.30\ \text{pc} \approx 4.24\ \text{light-years} \]

Practice

Sirius has a parallax of \(0.379''\). How far away is it, in light-years?

Show the answer

\(d = 1/0.379 \approx 2.64\ \text{pc}\). Multiply by 3.26 ly/pc: about \(8.6\) light-years.