--- title: "Why a planet crosses its star late" description: "A companion's gravity changes a planet's transit schedule. Repeated early and late crossings can reveal mass, with an eccentricity ambiguity." canonical_url: "https://sandboxphysics.com/stories/transit-timing/late-planet/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/transit-timing/late-planet/" last_modified: "2026-10-28" --- [Transit timing, from an orbit to a mass](https://sandboxphysics.com/stories/transit-timing/) L1 · Transit timing variations # Why a planet crosses its star late Sandbox Physics · 2026-10-28 A planet can arrive at its star's face several minutes late even when the telescope clock is working perfectly. Another planet has been pulling on it. The useful clue is the pattern of early and late crossings over many orbits. ## First build an appointment book When a planet passes in front of its star, the star grows a little dimmer. Astronomers call this a transit. The depth mostly tells us how large the planet looks beside the star; the center of the dip gives a time. Repeat the observation and a schedule emerges: one crossing every twelve days, for example. An isolated planet on an unchanging orbit repeats that crossing regularly. Its speed can vary around an eccentric orbit and the schedule can still repeat exactly. The appointment book tracks the repeated return to one viewing direction. Reading: Deck et al. · Lithwick ## A companion changes the next appointment Put a second planet nearby. Both planets now pull on each other as well as on the star. Near a close alignment, the companion can pull the transiting planet forward or backward. Energy and angular momentum pass between their orbits. A small change in speed also changes the orbit on which the planet travels, so a forward pull need not translate into a simple permanent advance. The resulting transit timing variation is the difference between a measured crossing and a fitted repeating schedule. Astronomers fit the schedule to all the crossings. Changing its starting date shifts every residual; changing its average period tilts the whole series. The interesting structure survives those two adjustments. Reading: Lithwick · Deck et al. ## A slow beat can grow from quick orbits Suppose the outer planet takes a little more than twice as long to circle the star as the inner one. At each meeting the planets line up in almost the same part of the orbit. Their small gravitational nudges then have a long rhythm, much like two musical notes that are nearly in tune. The timing cycle can take hundreds of days even though each orbit takes only days or weeks. That long cycle is especially useful to detect. It is also easy to misread if the observations cover only its rising flank. A gently curved section can resemble a drifting period. Extending the baseline can reveal the turn, which is why a few carefully timed later transits can matter so much. Reading: Lithwick · Greklek-McKeon et al. ## The same slow wave can hide different masses A heavier companion usually pulls harder. The planets’ eccentricities also change the timing wave. Eccentricity describes the amount and direction by which an orbit departs from a circle. One combination of eccentricities can amplify a smaller mass, or partly cancel the effect of a larger one. A clean, large wave therefore need not give a unique mass. There can be a second clue: shorter changes associated with successive conjunctions, when the planets pass each other. These are often called chopping. For a suitable near-circular system, a resolved chopping component depends strongly on the perturbing mass and helps separate it from the eccentricity combination. Transit sampling hides some harmonics in the slow wave, so the data have to distinguish the right component. Reading: Deck & Agol · Lithwick ## Kepler-289 supplies a real test Kepler-289 has three known transiting planets. The outer pair takes roughly sixty-six and one hundred twenty-six days per orbit. A 2015 analysis by Katherine Deck and Eric Agol used its short timing component to demonstrate how chopping can improve a mass measurement. A later study by Michael Greklek-McKeon and colleagues reprocessed the Kepler photometry and added ground-based observations. The timing is not immune to the star itself. Kepler-289's spotted surface changes brightness, and a planet crossing a spot can distort the dip used to measure its center. The later study explicitly modeled the stellar variation and masked conspicuous spot-crossing features. A timing error bar carries work done on those light curves. Our accompanying reanalysis uses twenty-five actual times from the outer pair and a simpler two-planet harmonic model. Adding chopping narrows a conditional mass distribution, while substantial residual scatter remains. That scatter is a result to understand before interpreting a narrow interval as a precise measurement of a real planet. Reading: Deck & Agol · Greklek-McKeon et al. · Greklek-McKeon et al. ## Try the observation ### [Watch a companion change the schedule](https://sandboxphysics.com/labs/transit-timing-variations/) 1. Compare Near two-to-one with An isolated clock. Watch the stellar image, then compare the full timing series. 2. Increase the companion mass while holding orbital spacing fixed. Predict what can change before looking at the result. [Next: Build the orbit before building the clock](https://sandboxphysics.com/stories/transit-timing/orbits-before-clocks/) ## Sources and calculation scope 1. [Lithwick, Xie & Wu, Extracting Planet Mass and Eccentricity From TTV Data (2012)](https://arxiv.org/abs/1207.4192v2) Pinned v2, Sections 2.1–2.3, Equations 1–10 and Appendix B. The free-eccentricity vector and the near-resonant validity conditions. 2. [Deck & Agol, Measurement of Planet Masses with Synodic Chopping (2015)](https://arxiv.org/abs/1411.0004v1) Pinned v1, Sections 2–5.1, Equations 10–15 and Appendix A. The transit sampling aliases some harmonics; the PH3 application explicitly compares mass constraints with and without chopping. 3. [Deck et al., TTVFast (2014)](https://arxiv.org/abs/1403.1895v1) Pinned v1, Sections 2.1.3, 2.2–2.3 and 3. Mid-transit is a foreground minimum of astrocentric sky separation. This teaching calculation reuses the site's RK4 kernel and independently checks IAS15; it does not port the GPL TTVFast code. 4. [Greklek-McKeon et al., Kepler-289 Densities and Transit Timing (2023)](https://arxiv.org/abs/2208.00022v1) Pinned v1, Sections 2–3, Table 1 and Figures 2–4, 9–10. CC BY 4.0. The author analysis jointly fits transit shapes, then fits the measured times with a three-planet dynamical model. 5. [Greklek-McKeon et al., original Table 1 machine-readable file](https://content.cld.iop.org/journals/1538-3881/165/2/48/revision1/ajac8553t1_mrt.txt) Revision1, 25,058 bytes; SHA256 8f958e773fae5a6d17aa090c5d523e13829f6b69a2f3d51ceecbf5e2d0d56af9. Fifty-one measured rows are distinct from 415 prediction-only rows. The 2021 WIRC observation is absent from the observed column.