Exoplanet Transits: Reading a Planet's Size from a Shadow

A shadow one part in ten thousand deep

When an exoplanet's orbit carries it directly between its star and an observer's telescope, it blocks a tiny, geometrically simple fraction of the star's light. To first approximation, the fractional dip in brightness at mid-transit is just the ratio of the two disks' areas — the planet's silhouette over the star's:

delta(F/F_star)  =  (R_planet / R_star)^2

  Earth transiting the Sun:    ~ 84 parts per million   (tiny)
  Jupiter transiting the Sun:  ~ 1.0 percent             (easily seen)

Why the light curve is not a flat-bottomed rectangle

A star is not a uniform disk — it is dimmer near its edge than at its centre, an effect called limb darkening, because a line of sight toward the limb samples cooler, higher, more slanted layers of the stellar atmosphere. As the planet crosses a limb-darkened star, it blocks progressively more light as it moves toward brighter disk centre and less as it approaches the far limb, which rounds off what would otherwise be a flat-bottomed, trapezoidal dip into the smooth, curved transit shape astronomers actually observe. The standard analytic model that includes this correctly is the Mandel-Agol formalism (2002), which computes the overlap area of a limb-darkened stellar disk and an opaque circular planet in closed form, and remains the workhorse fitting model for transit photometry today.

Shape encodes geometry: inclination and impact parameter

How deep and how long the transit lasts depends on more than just the size ratio. The impact parameter b — how far off-centre the planet's path crosses the star, in units of stellar radii — depends on orbital inclination i and sets both the transit duration and, for large enough b, whether the transit even reaches full depth (a grazing transit only partially covers the disk, even at mid-transit) or occurs at all:

b = (a / R_star) * cos(i)          impact parameter
                                    (a = orbital semi-major axis)

  b < 1 - Rp/Rs   : full transit  (planet disk entirely inside stellar disk)
  1 - Rp/Rs < b < 1 + Rp/Rs : grazing transit (partial coverage)
  b > 1 + Rp/Rs   : no transit at all

A near edge-on orbit (i close to 90 degrees, b close to 0) gives a long, deep, symmetric, well-defined transit; as inclination departs from edge-on, b grows, the transit shortens, and eventually the geometry misses the star's disk completely — which is why only a small fraction of exoplanets, those whose orbital planes happen to align with our line of sight, are detectable by the transit method at all.

Getting the planet's radius, and a hint of its atmosphere

Because the transit depth gives (Rp/Rs)² directly, and Rs is independently estimated from the star's spectral type, mass and evolutionary models, a transit measurement alone yields the planet's physical radius — no direct imaging required. Combine that with the planet's mass from radial-velocity follow-up and you get the planet's bulk density, the single most useful number for telling a rocky world from a gas-rich one. Pushed further, comparing transit depth at different wavelengths — transmission spectroscopy — reveals that the planet's apparent radius is slightly larger at wavelengths its atmosphere absorbs strongly, since starlight grazing the upper atmosphere at those wavelengths is blocked too; this is how molecules like water vapour and sodium have been detected in exoplanet atmospheres from the ground and from space telescopes like JWST.

Why period, not just one dip, matters

A single dip in brightness is not enough to confirm a planet — stellar spots, instrumental noise, and eclipsing binaries in the background can all mimic one. Confirmation normally requires multiple transits repeating with a strictly periodic cadence consistent with Kepler's third law given the star's mass, plus a consistent depth and duration each time; that periodicity, more than the shape of any single dip, is what separates a genuine planetary transit from a false positive in large surveys like Kepler and TESS.

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