How We Find Other Worlds

Worlds you cannot see#

As of the mid-2020s, astronomers have confirmed more than five thousand planets orbiting other stars. Here is the strange part: almost none of them have ever been seen. A planet next to its star is a firefly next to a lighthouse — a billion times fainter, and separated on the sky by a hair's breadth. Point the finest telescope ever built at a nearby Sun-like star and the planet vanishes in the glare.

So we do not look for the planet. We watch the star, and we wait for the planet to give itself away. It does this in two ways. When a planet crosses in front of its star it blocks a sliver of light, and the star dims by a fraction of a percent, on schedule, every orbit — a shadow. And because a star and its planet both orbit their shared centre of mass, the star is never quite still; it traces a tiny circle and its light rocks between blue and red — a tug.

Two whole worlds, inferred entirely from a dip in brightness and a wobble in a spectrum. This article is where two earlier ideas on this site finally pay off: the reflex orbit is Kepler's laws applied to a star instead of a planet, and the wobble is the Doppler effect read off starlight. Two other techniques — direct imaging and gravitational microlensing — do exist and matter, but the transit and the wobble account for the overwhelming majority of the catalogue, so they are where we will stay.

The shadow: transits#

Imagine a planet whose orbit happens to be lined up edge-on to us, so that once per orbit it passes directly between us and its star. During that crossing — the transit — the planet's opaque disk covers part of the bright stellar disk, and the total light we receive drops. The star does not flicker or flare; it steps down by a fixed amount, holds there while the planet crosses, and steps back up. Plot brightness against time and you get a flat line with a shallow, flat-bottomed box carved out of it: the light curve.

Everything you want to know is written in the shape of that box. Its depth tells you how big the planet is relative to the star. Its duration and spacing tell you about the orbit. And its exquisite regularity — the same dip, repeating on a fixed clock — is what separates a planet from a passing sunspot or a fluke of the weather in the star's atmosphere.

Grab the planet size slider first. As you enlarge the planet, the box gets deeper — and only deeper; a bigger silhouette blocks more light. Read the depth straight off the light curve: that number is the planet's size, converted. Now drag the geometry slider. This is the impact parameter — how centrally the planet crosses the disk. A dead-central pass (b = 0) gives the longest, squarest transit; slide toward a grazing pass and the box narrows and softens as the planet clips only the star's edge. Notice what does not change as you adjust the geometry: the floor of a full transit sits at the same depth, because the same disk is being covered whichever chord the planet takes.

The math: what a shadow and a wobble each tell you#

From the transit, a radius. At the bottom of the dip, the planet's disk of radius $R_p$ blocks part of the stellar disk of radius $R_\star$. The fraction of light removed is just the ratio of the two areas — the transit depth:

$$\delta = \left(\frac{R_p}{R_\star}\right)^{2}$$

A Jupiter-sized planet ($R_p \approx 0.1\,R_\star$ for a Sun-like star) gives $\delta \approx 0.01$ — a 1% dip. An Earth ($R_p \approx 0.009\,R_\star$) gives $\delta \approx 8\times10^{-5}$, under a hundredth of a percent, which is why finding small planets demands a space telescope watching from above the atmosphere. Crucially, $\delta$ contains no reference to mass: a puffball and a cannonball of the same radius carve the same box. A transit measures size, full stop.

From the wobble, a mass. The star and planet orbit their common barycentre, so the star swings back and forth along our line of sight, and its spectral lines shift by the Doppler effect. The star's line-of-sight speed traces a curve whose semi-amplitude $K$ is

$$K = \left(\frac{2\pi G}{P}\right)^{1/3}\frac{m_p \sin i}{(M_\star + m_p)^{2/3}}\,\frac{1}{\sqrt{1-e^{2}}}$$

Strip that to what matters. For a circular orbit around a star much heavier than its planet, $K \propto m_p\,P^{-1/3}\,M_\star^{-2/3}$: the wobble grows with planet mass, and shrinks slowly as the orbit widens (longer $P$). The $\sin i$ is the catch — we see only the component of the star's motion along our line of sight, so a face-on orbit ($i = 0$) produces no shift at all, and radial velocity alone yields a minimum mass $m_p \sin i$. Jupiter tugs the Sun by about 12 m/s; Earth by a mere 9 cm/s — a walking pace, extracted from a star trillions of kilometres away.

The orbit, from Kepler. The transit hands you the period $P$ directly — it is the time between dips. Kepler's third law, the same relation that orders the solar system, converts that period into an orbital distance:

$$P^{2} = \frac{4\pi^{2}}{G(M_\star + m_p)}\,a^{3}\;\approx\;a^{3}\quad(\text{in years and AU for }M_\star=M_\odot)$$

And when you have both, a density. Transits give $R_p$; radial velocity gives $m_p$ (with $i$ nailed to $\approx 90°$ by the very fact that a transit occurs). Together they give the one number neither can supply alone:

$$\rho_p = \frac{m_p}{\tfrac{4}{3}\pi R_p^{3}}$$

Density is what tells a small rocky world from a puffy gas dwarf of identical size — the difference between a place with a surface and a ball of hydrogen. It is why the two methods are so much more than the sum of their parts.

The tug: radial velocity#

The radial-velocity curve is worth watching on its own, because it makes the mass-dependence tangible. Below, a star and planet orbit their shared barycentre; the star's small reflex circle produces the Doppler wobble, drawn as a velocity curve on the right.

Turn up the planet mass. The barycentre is the balance point of the star–planet seesaw, so a heavier planet drags it further from the star's centre — the star's own orbit grows, its wobble widens, and the velocity curve's amplitude $K$ climbs in step. Watch the star's colour as it goes around: it tints blue as it swings toward Earth (its lines crowd to shorter wavelengths) and red as it recedes, exactly the Doppler effect that reddens a receding galaxy, only here it flips sign twice per orbit. A bigger planet does not change the shape of that curve — still a clean sine for a circular orbit — only its height. That height, in metres per second, is the planet's mass talking.

The habitable zone, and why we found the wrong planets first#

Convert an orbital distance into a temperature and you can ask the question everyone actually cares about: could liquid water sit on the surface? A planet's equilibrium temperature scales as $T_{\text{eq}} \propto (L_\star / a^{2})^{1/4}$, so the band of orbits where water neither boils nor freezes — the habitable zone — sits at a distance that grows with the square root of the star's luminosity:

$$a_{\text{HZ}} \sim \sqrt{\frac{L_\star}{L_\odot}}\;\text{AU}$$

A star ten times as luminous as the Sun pushes its habitable zone out to roughly $\sqrt{10} \approx 3$ AU; a dim red dwarf at a hundredth of the Sun's output hugs its warm band in to about 0.1 AU, closer than Mercury. That last fact is a gift to planet-hunters: around small cool stars the habitable zone is a short orbit, which means frequent transits and a larger wobble, so the most detectable planets are also the ones most worth examining.

Which brings us to a caution baked into every catalogue. The first exoplanet found around a Sun-like star, 51 Pegasi b in 1995, was a hot Jupiter — a gas giant whipping around its star in four days. For years such planets dominated the tally, and it was tempting to conclude they were common. They are not; they are rare. They were found first because they are the loudest signal in both methods at once: a giant radius carves a deep transit, and a giant mass on a tight, fast orbit produces the largest, quickest radial-velocity swing. Small planets on wide orbits — the Earths — whisper, and we simply could not hear them yet. This is observational bias, and reading a raw exoplanet catalogue without correcting for it is like surveying a city's population by standing in the loudest nightclub.

One more misconception deserves correcting, because science-fiction art has cemented it: we do not, as a rule, photograph these planets. Direct imaging has captured a few dozen young, self-luminous giants far from their stars, and microlensing has caught others by the way a planet's gravity briefly bends a background star's light. But the number of exoplanets ever resolved as a dot of their own light is a tiny fraction of the total. The thousands we know are known by shadow and by tug — by a dimming and a Doppler shift — not by their portrait.