AC19 · A planet misses its appointment

Transit Timing Variations and Planet Masses

What makes a planet cross its star late? A nearby planet pulls a transiting planet ahead of or behind a steady orbital clock. Repeated close alignments can build a slow timing cycle, while shorter kicks carry extra mass information. Eccentric orbits can imitate part of that signal, so timing uncertainties and the pattern across many transits determine how well a mass can be measured.

A companion changes transit times through gravity. Drag its orbit and compare the resulting stellar crossings with a fitted clock.

Clock offset · positive is late $14.9\,\mathrm{min}$

Stellar flux $1.00$

Inner timing range $29.5\,\mathrm{min}$

Recorded inner transits $199$

Move the companion

Companion mass

20

m_2\;[M_\oplus]

Initial period ratio

2.04

P_{2,0}/P_{1,0}

Inner eccentricity

0

e_{1,0}

Drag the cyan planet radially to change its initial orbit. Gold is the inner planet; the image shows the same synthetic stellar crossing.

Clock, scale and model

The stellar image is an enlarged sky plane viewed along the orbital horizontal direction. The orbital overview uses a separate distance scale and enlarged bodies; its faint curves mark the initial Kepler guides. The radial brightness follows the same limb-darkening law as the flux calculation; colored rims identify the planet silhouettes. The vertical mark locates the stellar center. The stage loops one actual crossing from a two-hundred-orbit synthetic integration.

(O-C)_n=t_n-(T_0+nP)

Geometric foreground crossings and limb-darkened flux come from the same finite-mass orbit. Light travel time, activity, relativity and tides are omitted. Custom settings start a complete asynchronous integration; the last valid record remains visible until replacement. A close-approach stop is reported.

Course and actual Kepler-289 timing tutorial

Model assumptions

Synthetic coplanar, edge-on star–two-planet Newtonian integration with finite masses and unsoftened mutual gravity. Numerical units are stellar mass, initial inner semimajor axis and gravitational constant equal to one; physical labels adopt a one-solar-mass star and twelve-day initial inner period. The inner planet has about ten Earth masses. For a nonzero companion mass, both planet silhouettes have radius nine percent of the star; sizes are enlarged in the orbital overview. At zero companion mass, the numerical reference particle is omitted from displayed flux, the scene and the timing chart. The stellar radius is 0.045 initial inner semimajor axes. Geometric mid-transits use foreground minimum projected separation; the same orbit supplies limb-darkened flux, including overlapping silhouettes. Light travel time, stellar activity, tides and relativity are omitted. Each record spans two hundred initial inner orbits; the stage loops one stated crossing window from that record. Fixed presets and custom calculations use the same RK4 kernel with at least 960 steps per initial period, refined at pericenter. A candidate step is rejected below 0.12 initial inner semimajor axes of planet–planet separation or 0.15 of star–planet separation; the last valid state is retained. The real 25-time Kepler-289 tutorial is a separate approximate inference with measured errors, jointly varied ephemerides and explicit priors; its posterior is conditional on this reduced model, while the published analysis includes all three planets.

Each point starts with a crossing

Synthetic Newtonian record · two hundred initial inner orbits.

Early and late over the full record

O-C\;[\mathrm{min}]

t/P_{1,0}

The selected stellar crossing

F/F_\star

t-T_{\rm calc}\;[\mathrm{h}]

Every event is fitted against its planet's own linear clock. The gold line marks the replayed event. Flux uses the same foreground positions, including overlapping silhouettes; the orbital baseline and the magnified crossing window have different time spans.

Start with the essentials

Focus question: Which timing pattern comes from a companion, and how much of it could eccentricity imitate?
One-sentence intuition: Calculate each crossing from the orbit before subtracting a fitted ephemeris. A slow timing amplitude mixes mass and free eccentricity; an independently resolved synodic component can help separate them.

Core mathematical model

Subtract a clock

(O-C)_n=t_n-(T_0+nP)

Time and period share the same day unit. The epoch number retains gaps in an observed series.

Two distinct beats

P_{\rm syn}=\frac1{|P_1^{-1}-P_2^{-1}|},\qquad P_{\rm sup}=\frac1{|2/P_2-1/P_1|}

The first is the conjunction interval. The second is the near-two-to-one timing beat, which can exceed the observation span.

Locate the crossing

\boldsymbol r_\perp\cdot\boldsymbol v_\perp=0,\quad r_x>0

Relative positions are planet minus star; the observer lies along positive x. Select a minimum and check the finite stellar disk.

Common difficulties

Depth is not mass

Typical misconceptionA deeper transit directly weighs a planet.

Better mental modelDepth is mainly a projected area ratio. Dynamical mass information comes from the gravitational timing pattern.

A large slow wave can remain ambiguous

Typical misconceptionA well-measured timing amplitude uniquely fixes a mass.

Better mental modelThe slow term also depends on the free eccentricity vector. Resolve another harmonic or retain a broad mass interval.

Run the experiment

01
Compare two clocks
Watch the near-two-to-one preset, then open An isolated clock.
What to observe: The companion makes timing vary. The isolated residual is limited by numerical error.
02
Move the companion
Drag the cyan orbit handle radially, then change the companion mass.
What to observe: Spacing changes the beat; mass changes the gravitational response. Both can affect the finite-baseline pattern.
03
Wait for the beat
Compare A longer beat with the default and inspect the full two-hundred-orbit timing series.
What to observe: A short observed arc can resemble a drifting period. The fitted clock removes an offset and slope; mass and eccentricity still need separate constraints.