AC19 · Transit timing variations
Transit timing, from an orbit to a mass
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.
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.
02 · L1Build the orbit before building the clock
Derive finite-mass orbital motion, choose barycentric units, and recover the isolated planet's exactly repeating transit schedule.
03 · L2Turn stellar crossings into timing residuals
Locate a geometric transit, derive the fitted linear ephemeris, and separate conjunction time from the long near-resonant beat.
04 · L3Why 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.
05 · L3Reanalyze 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.
06 · L4Read 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.
Sources and calculation scope
- Mills et al., A resonant chain in Kepler-223 (Nature 2016)
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.
- Lithwick, Xie & Wu, Extracting Planet Mass and Eccentricity From TTV Data (2012)
Pinned v2, Sections 2.1–2.3, Equations 1–10 and Appendix B. The free-eccentricity vector and the near-resonant validity conditions.
- Deck & Agol, Measurement of Planet Masses with Synodic Chopping (2015)
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.
- Deck et al., TTVFast (2014)
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.
- Greklek-McKeon et al., Kepler-289 Densities and Transit Timing (2023)
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.
- Greklek-McKeon et al., original Table 1 machine-readable file
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.
- Schmitt et al., Planet Hunters VII and PH3 (2014)
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.