---
title: "Transit timing, from an orbit to a mass"
description: "Derive transit times from interacting orbits and analyze 25 measured Kepler-289 times with mass and eccentricity uncertainty."
canonical_url: "https://sandboxphysics.com/stories/transit-timing/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/transit-timing/"
last_modified: "2026-10-28"
---

AC19 · Transit timing variations

# Transit timing, from an orbit to a mass

Sandbox Physics · 2026-10-28

A companion changes the moment a planet crosses its star. Follow the orbit, learn to measure the delay, then ask how much mass the timing really determines.

[01 · L1 Why a planet crosses its star late A companion's gravity changes a planet's transit schedule. Repeated early and late crossings can reveal mass, with an eccentricity ambiguity.](https://sandboxphysics.com/stories/transit-timing/late-planet/) [02 · L1 Build the orbit before building the clock Derive finite-mass orbital motion, choose barycentric units, and recover the isolated planet's exactly repeating transit schedule.](https://sandboxphysics.com/stories/transit-timing/orbits-before-clocks/) [03 · L2 Turn stellar crossings into timing residuals Locate a geometric transit, derive the fitted linear ephemeris, and separate conjunction time from the long near-resonant beat.](https://sandboxphysics.com/stories/transit-timing/crossings-and-residuals/) [04 · L3 Why a timing wave can leave the mass uncertain Derive the mass–free-eccentricity degeneracy, understand transit aliasing, and construct a normalized likelihood with explicit priors.](https://sandboxphysics.com/stories/transit-timing/mass-and-eccentricity/) [05 · L3 Reanalyze twenty-five measured Kepler-289 transit times Select actual times from the author table, fit mass and free eccentricity jointly, and inspect chopping, residual scatter and prior sensitivity.](https://sandboxphysics.com/stories/transit-timing/kepler289-timing-tutorial/) [06 · L4 Read the Kepler-289 mass measurement from its light curves A figure-by-figure guide to the pinned Kepler-289 paper, from stellar spots and partial transits to dynamical masses and density assumptions.](https://sandboxphysics.com/stories/transit-timing/reading-kepler289/)

## Sources and calculation scope

1. [Mills et al., A resonant chain in Kepler-223 (Nature 2016)](https://arxiv.org/abs/1612.07376v1)

   Pinned v1, Figure 1 and Extended Data Figure 3; Methods passages Photodynamic Inputs and Photodynamic Fits, PDF pages 19–23. Joint four-planet flux inference and subsequent stability selections are distinct operations.

2. [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.

3. [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.

4. [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.

5. [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.

6. [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.

7. [Schmitt et al., Planet Hunters VII and PH3 (2014)](https://arxiv.org/abs/1410.8114v1)

   Pinned v1, the transit/dynamical analysis and Table 5. PH3c is today's Kepler-289d, while PH3d is Kepler-289c. The older forecast table is not used as an independent measurement list.