Vol. Science · Astronomy · Detection

Transit
and the wobble

Two measurements, from the same orbit. If the planet crosses the star, the star dims by the area ratio δ ≈ (R_p / R_★)². A random orientation transits with probability about R_★ / a. The crossing itself lasts about (P / π) (R_★ + R_p) / a when it is central. Whether or not it crosses, the star still orbits the balance point. For a circular orbit the toward-and-away speed is K = (2πG / P)^1/3 (M_p sin i) / (M_★ + M_p)^2/3. Period and distance stay tied by the same pull, P² ∝ a³ / (M_★ + M_p), the general form of the Kepler desk. The methods, without these numbers, are on exoplanets. The star’s size and mass sit on stars. Where that orbit falls in the starlight is the habitable zone. The pull itself is gravity.

Classroom model. One star, one planet, a circular orbit, and a star that is equally bright from center to limb. No limb darkening, no eccentricity, no starspots, no atmosphere on the planet, and no second planet tugging the clock. A dip drawn here is the geometric chord, not a fitted light curve. Not a catalog and not a discovery.

Interactive · Dip and wobble

Light curve zoomed on the crossing · radial velocity is one full orbit

Light curve

Radial velocity

Light
Wobble
Transit window

Cited systems · Earth and Jupiter are the external edge-on view

Planet units · the star stays in solar units

Motion · the dip is a short slice of the orbit

R★ R☉

M★ M☉

Rp R⊕

Blank radius means it was not measured. The dip stays undrawn.

Mp M⊕

Period days

a AU

i deg

The sliders need JavaScript. The cited anchors are still below. With the fact-sheet volumetric radii, an edge-on Earth blocks 83.86 ppm of the Sun (0.008386%). The circular formula with the fact-sheet GM values and a 365.256-day period gives K = 8.95 cm/s. R★/a is 0.4650%, about 1 in 215. The central crossing lasts 13.10 h. Jupiter, same assumptions, blocks 1.010% (10,098 ppm) and wobbles the Sun by 12.46 m/s. That crossing lasts 32.56 h. R★/a is about 1 in 1,119.

Why the numbers look like that

Small stars, missed orbits, and a mass that is only a minimum.

Depth is an area. Shrink the star and the same planet covers a larger fraction of it. TRAPPIST-1 is about a tenth of the Sun’s radius, so an Earth-size planet is a dip of thousands of parts per million instead of tens. The wobble grows too, because K scales roughly as M_p / M_★^2/3 and as P^−1/3. A light star on a short orbit is tugged harder, as seen from here, than the Sun is by Earth.

Most systems never transit. R★/a for the Earth–Sun preset is 0.4650%, about 1 in 215 random orientations. Jupiter is closer to 1 in 1,119 because it is farther out. A hot Jupiter can be closer to 1 in 10, which is why the early finds were close-in giants. The probability is not the same thing as this page’s inclination. 51 Pegasi b has a measured tilt of 49.8°, so this line of sight misses the star even though a random observer would have had a fair chance.

The sine wave measures M_p sin i, not M_p. Face-on, sin i is 0 and the star’s speed along our line of sight is zero, whatever the planet weighs. Edge-on, sin i is 1 and K is the true mass. Anything between is a minimum: the true mass is larger by 1 / sin i. 51 Pegasi b was announced that way. The m sin i line on the card is the mass this K would imply if you did not already know i.

Instruments, at the precision their papers actually quote. HARPS, in its first-results report, reached a short-term precision of 0.2 m/s during one night and a long-term precision of order 1 m/s (Rupprecht, Pepe, Mayor, Queloz et al., SPIE 5492). An Earth at 9 cm/s is under that long-term floor. ESPRESSO was designed for 10 cm/s. Pepe et al. 2021 demonstrate better than 25 cm/s in a single night and 50 cm/s over months, and they call that performance compatible with a 10 cm/s instrument. At high signal-to-noise they reach 10 cm/s on M stars and 10–20 cm/s on K and G dwarfs. Kepler’s photometer was designed for 20 ppm in 6.5 hours on a 12th-magnitude star, and Caldwell et al. 2010 describe an Earth–Sun equivalent transit as an 80 ppm signal. This desk’s Earth preset, from the fact-sheet radii, is 83.86 ppm.

Literacy · the words on the card

Depth: δ = (R_p / R_★)². The fraction of the star’s disk the planet covers, for a uniform disk and a planet fully in front. Parts per million are that fraction times 10⁶. No limb darkening, so the dip is flat-bottomed when the planet is smaller than the star and fully on the disk.
R★ / a: The chance a randomly oriented circular orbit transits, if the planet is treated as a point. Including the planet’s radius gives (R_★ + R_p) / a. Both are geometry, not a statement about the inclination you set.
Impact parameter: b = (a cos i) / R_★. Zero is a central crossing. The planet misses when |b| is greater than 1 + R_p/R_★. i = 90° is edge-on. i = 0° is face-on.
Duration: First contact to fourth, for a circular orbit: (P / π) times the arcsin of (R_★ / a) √((1 + k)² − b²) / sin i, with k = R_p/R_★. A central transit is close to (P / π) (R_★ + R_p) / a. This is the geometric chord, not a fitted light curve.
K and m sin i: K is the semi-amplitude of the star’s toward-and-away speed. For e = 0 it is (2πG/P)^1/3 (M_p sin i) / (M_★ + M_p)^2/3. This desk writes that with the fact-sheet GM values, so G never has to be chosen. m sin i is M_p times sin i, in the planet’s mass unit. It is what a measured K returns when the tilt is unknown.
One orbit, two clocks: Editing the period recomputes the distance. Editing the distance recomputes the period. Either way, P = 2π √(a³ / (GM_★ + GM_p)). Around the Sun, in years and AU, that is the P² = a³ of the Kepler desk, plus the planet’s own GM, which is small for Earth and visible for Jupiter. The fact-sheet semimajor axis is quoted in the note. The card uses the two-body distance so the period and the distance cannot disagree.

Sources · fact sheets, the archive, and the papers

Cited, not invented

NASA NSSDCA — Sun Fact Sheet — GM 132,712 × 10⁶ km³/s². Volumetric mean radius 695,700 km. The same GM the vis-viva and Kepler desks use.
NASA NSSDCA — Earth Fact Sheet — GM 0.39860 × 10⁶ km³/s². Volumetric mean radius 6,371.0 km. Sidereal orbit period 365.256 days. Semimajor axis 149.598 × 10⁶ km. Orbit inclination 0.000° is to the ecliptic. This preset’s 90° is an external edge-on view.
NASA NSSDCA — Jupiter Fact Sheet — GM 126.687 × 10⁶ km³/s². Volumetric mean radius 69,911 km. Sidereal orbit period 4,332.589 days. Semimajor axis 778.479 × 10⁶ km. The sheet’s 1.304° is to the ecliptic.
IAU 2012 Resolution B2 — 1 au = 149,597,870,700 m exactly. Used when a period is turned into an AU.
Winn 2010, “Transits and Occultations,” arXiv:1001.2010 — the circular-orbit geometry on this page: a uniform-disk depth (R_p/R_★)², impact parameter b = a cos i / R_★, the R_★/a approximation and the (R_★ + R_p)/a probability, and the first-to-fourth-contact duration.
NASA Exoplanet Archive — HD 209458, Bonomo et al. 2017 column (A&A 602, A107) — M_★ = 1.119 ± 0.033 M☉, R_★ = 1.155 (+0.014, −0.016) R☉, M_p = 0.682 (+0.014, −0.015) M_Jup, R_p = 1.359 (+0.016, −0.019) R_Jup, P = 3.52474859 ± 0.00000038 days, i = 86.710 ± 0.050°, a = 0.04707 (+0.00045, −0.00047) AU, K = 84.27 (+0.69, −0.70) m/s, e < 0.0081.
Cont et al. 2026, A&A 710, A345 — 51 Pegasi b, Table 1: M_p = 0.61 (+0.06, −0.05) M_Jup, i = 49.8 (+5.8, −5.7)°, P = 4.2307966 ± 0.0000027 days, stellar K = 55.77 ± 0.15 m/s. Stellar mass 1.07 ± 0.05 M☉, stellar radius 1.19 ± 0.03 R☉, and a = 0.052 AU are taken there from Rosenthal et al. 2021. The planet does not transit at that inclination.
Mayor and Queloz 1995, Nature 378, 355 — the discovery of 51 Pegasi b. The archive lists that paper’s minimum mass as 0.47 ± 0.02 M_Jup. This desk does not load a planet radius for the preset. The archive marks the composite radius as a calculated value.
Agol et al. 2021, PSJ 2, 1 — TRAPPIST-1 e: R = 0.920 (+0.013, −0.012) R⊕, M = 0.692 ± 0.022 M⊕, P = 6.101013 ± 0.000035 days, i = 89.793 ± 0.048°, a = 0.02925 ± 0.00025 AU. Stellar radius 0.1192 ± 0.0013 R☉. Stellar mass 0.0898 ± 0.0023 M☉, adopted from Mann et al.
Mahajan, Eastman, and Kirk 2024, ApJL 963, L37 — GJ 1214 b, Table 3 central values: M_★ = 0.1820 M☉, R_★ = 0.2162 R☉, P = 1.580404531 days, R_p = 2.733 R⊕, M_p = 8.41 M⊕, a = 0.01505 AU, i = 88.980°, K = 14.38 m/s, (R_p/R_★)² = 0.013430, e = 0.0062. Kepler-22 b is not a preset here. GJ 1214 b is the small-star case with a published mass and radius.
Rupprecht, Pepe, Mayor, Queloz et al., SPIE 5492 — HARPS abstract: a short-term precision of 0.2 m/s during one night and a long-term precision of the order of 1 m/s, using the simultaneous thorium reference.
Pepe et al. 2021, A&A 645, A96 — ESPRESSO was designed for 10 cm/s. They demonstrate better than 25 cm/s in a single night and 50 cm/s over several months, limited by photon noise and stellar jitter, and call that compatible with an instrumental precision of 10 cm/s. At S/N = 430, 10 cm/s is reached for M stars, and K and G dwarfs fall between 10 and 20 cm/s.
Caldwell et al. 2010, Kepler instrument performance — the mission is designed to detect the 80 ppm signal of an Earth–Sun equivalent transit, and the photometer is designed for 20 ppm on a 12th-magnitude star in 6.5 hours. This page’s Earth depth is computed from the fact-sheet radii. It is not their 80 ppm pasted in.

Classroom picture of a transit and a circular-orbit wobble. Uniform disk, no limb darkening, no eccentricity. Earth–Sun, from the fact sheets and i = 90°, is 83.86 ppm and 8.95 cm/s. Jupiter–Sun is 1.010% and 12.46 m/s. Not a catalog and not a fitted light curve. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

Other suns

Light	Zoomed on the crossing so the dip is visible. A year-long orbit would hide an Earth-size dip in one pixel. The vertical scale is the depth, labeled in ppm or in percent.
Wobble	One full orbit. Mid-transit is the center, where the sine wave crosses zero. Positive is the star receding. The shaded band, when there is a transit, is the same window as the light curve.
No dip	If this inclination misses the star, the light curve is flat and the duration is blank. The geometric probability stays, because that is about a random observer.
No radius	51 Pegasi b loads no planet radius. Depth and duration stay blank until you type one. Typing leaves the preset.

Presets	Earth, Jupiter, HD 209458 b, 51 Pegasi b, TRAPPIST-1 e, GJ 1214 b. Earth and Jupiter are edge-on external views. The others use the cited inclination.
Units	Earth or Jupiter for the planet’s radius and mass. The star stays in solar units. Switching units keeps the same planet.
Period and a	Kepler’s third law with both masses. The last one you edit wins. The other updates.
Inclination	0° is face-on. 90° is edge-on. K scales with sin i. The transit, if there is one, requires i near 90°.
Motion	Dip sweeps the crossing. Full orbit runs the year, and the light-curve marker is only in frame while the planet is crossing. Pause holds mid-transit.

Depth	(Rp/R★)², in percent and in ppm. Computed from the radii, not copied from a paper.
R★ / a	The point-planet geometric probability, also as “1 in N”.
K	The circular formula. Compared, in the note, with a published K when the paper measured one.
m sin i	Planet mass times sin i. The minimum mass a wobble alone would give.

Light curve	The canvas takes focus. Left and Right change the inclination. Shift is a coarser step.
Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Transit
and the wobble

Small stars, missed orbits, and a mass that is only a minimum.

Cited, not invented

Exoplanets

Habitable zone

Kepler

Stars

Gravity

Solar map

Transit and wobble

The curves

Controls

Readouts

Keyboard and touch

Transitand the wobble

Small stars, missed orbits, and a mass that is only a minimum.

Cited, not invented

Exoplanets

Habitable zone

Kepler

Stars

Gravity

Solar map

Transit
and the wobble