How We Find Other Worlds
We have never seen most exoplanets. We found them by watching stars blink and wobble.
Finding planets you cannot see
Junior level — plain language, no maths
There are thousands of planets circling other stars. Almost none of them have ever been photographed, and most never will be - they are far too faint and far too close to a star that outshines them a billion times over. So how do we know they are there?
Two tricks, and both work on the star rather than the planet.
The first is the blink. If a planet's orbit happens to be edge-on from our point of view, then once every orbit it slides across the face of its star and blocks a sliver of the light. The star dims - by about 1% for a big planet, and by a hundredth of that for an Earth. Measure the dip, and you have measured the planet's size.
The second is the wobble. A planet does not really orbit its star; the two orbit each other, around a shared balance point. So the star swings in a small circle, and its light shifts slightly towards blue as it comes at us and towards red as it moves away. From the size of that shift you get the planet's mass.
Put the two together - size from the blink, mass from the wobble - and you can work out what the planet is made of, without ever seeing it.
Things worth knowing
- A Jupiter-sized planet dims its star by about 1%. An Earth-sized one dims it by 0.008% - like spotting a moth crossing a searchlight from a thousand kilometres away.
- A transit only works if the orbit is edge-on to us, which is why we find far more planets than we would if we could see every system.
- The blink gives the size, the wobble gives the mass, and together they give density - the difference between a ball of rock and a ball of gas.
Transit depth, radial velocity and detection bias
Student level — the core equations
The transit method measures a ratio of areas. A planet crossing its star blocks a fraction \((R_p/R_\star)^2\) of the light, so the depth of the dip gives the planet's radius directly once the star is characterised. A Jupiter across a Sun-like star gives about 1%; an Earth gives 84 parts per million, which is why Kepler needed a space telescope and photometric precision of a few tens of ppm.
Radial velocity measures the star's reflex motion. The semi-amplitude is \(K \approx 28.4\ \text{m/s}\,(M_p/M_J)(a/\text{AU})^{-1/2}\) for a solar-mass star, so Jupiter at 5 AU moves the Sun by 12.5 m/s and Earth at 1 AU by just 9 cm/s. Current spectrographs reach around 30 cm/s, which is why Earth analogues around Sun-like stars remain out of reach by radial velocity alone.
Both methods are strongly biased towards big planets on short orbits: transits need an edge-on alignment whose probability is roughly \(R_\star/a\), and short periods mean more repeat events in a given survey. That bias is why the first exoplanets found were hot Jupiters, and why the true population - dominated by small planets - only emerged once Kepler stared at one field for four years.
Combine the two and you get bulk density. A 1.5 Earth-radius planet at 5 Earth masses is rock; the same radius at 1.5 masses has a thick envelope of hydrogen. The observed radius valley near 1.8 Earth radii, a gap in the size distribution, is thought to mark where stellar irradiation strips those envelopes away.
Key Formulas
| Transit depth | \(\delta = \left(\dfrac{R_p}{R_\star}\right)^{2}\) | |
|---|---|---|
| Radial velocity | \(K \approx 28.4\ \tfrac{\text{m}}{\text{s}}\ \dfrac{M_p/M_J}{\sqrt{a/\text{AU}}}\) | |
| Period | \(P^{2} = \dfrac{a^{3}}{M_\star}\) | years, AU, solar masses |
| Transit probability | \(p \approx R_\star/a\) | |
Things worth knowing
- Transit depth equals (Rp/R★)². Jupiter across the Sun gives 1%, Earth gives 84 ppm.
- Earth moves the Sun at 9 cm/s. The best spectrographs today reach about 30 cm/s, so that signal is still just out of reach.
- The chance that a random orbit is edge-on enough to transit is roughly R★/a - about 0.5% for an Earth-like orbit, which is why surveys must watch hundreds of thousands of stars.
Occurrence rates, the radius valley and what comes next
Scholar level — full mathematical depth
Kepler turned exoplanets from a curiosity into statistics. After correcting for geometric and detection completeness, planets between 1 and 4 Earth radii with periods under 100 days orbit a large fraction of Sun-like stars - a class with no Solar System analogue at all. The occurrence rate of Earth-size planets in the conservative habitable zone of Sun-like stars, \(\eta_\oplus\), remains poorly pinned down, with published values spanning roughly 2% to 50% depending on the completeness treatment.
The radius valley at \(1.8\,R_\oplus\) is the sharpest structural feature in the distribution. Photoevaporation and core-powered mass loss both predict it: a rocky core either keeps a percent-level hydrogen envelope, which doubles its radius, or loses it entirely. Which mechanism dominates is still argued, and the valley's dependence on stellar mass and orbital period is the discriminant currently being measured.
Transit spectroscopy is where the field is going. During transit a thin annulus of the planet's atmosphere is backlit, imprinting absorption features on the stellar spectrum with amplitude of order \(2 R_p H / R_\star^2\) where \(H\) is the scale height - parts per million, but JWST reaches it. Detections of CO\(_2\), SO\(_2\) and photochemical products in hot giant atmospheres are now routine; the same technique on temperate rocky planets around M dwarfs is the current frontier and remains at the edge of feasibility.
The systematic worry is stellar contamination: unocculted starspots and faculae imprint signals that mimic planetary absorption, and disentangling them is now a limiting factor rather than photon noise.
Key Formulas
| Transmission signal | \(\Delta\delta \approx \dfrac{2 R_p H}{R_\star^{2}}\) | H = scale height |
|---|---|---|
| Scale height | \(H = \dfrac{k T}{\mu g}\) | |
| Radius valley | \(R_p \approx 1.8\,R_\oplus\) | |
| Occurrence | \(\eta_\oplus \sim 0.02 - 0.5\) | still unsettled |
Things worth knowing
- Kepler statistics imply small planets (1-4 R⊕) are the most common class in the galaxy - a category the Solar System entirely lacks.
- The radius valley near 1.8 R⊕ is thought to separate stripped rocky cores from those that kept a percent-level hydrogen envelope.
- Transmission spectroscopy signals scale as 2RpH/R★², a few tens of ppm for a hot Jupiter - and stellar spots, not photon noise, are now often the limiting systematic.