---
title: "Read the Kepler-289 mass measurement from its light curves"
description: "A figure-by-figure guide to the pinned Kepler-289 paper, from stellar spots and partial transits to dynamical masses and density assumptions."
canonical_url: "https://sandboxphysics.com/stories/transit-timing/reading-kepler289/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/transit-timing/reading-kepler289/"
last_modified: "2026-10-28"
---

[Transit timing, from an orbit to a mass](https://sandboxphysics.com/stories/transit-timing/)

L4 · Transit timing variations

# Read the Kepler-289 mass measurement from its light curves

Sandbox Physics · 2026-10-28

Open Greklek-McKeon et al., arXiv:2208.00022v1, submitted 29 July 2022 and published in AJ 165, 48 in 2023. Follow Sections 3.1–3.3 and Figures 2–4, 9–10. They show how a fluctuating stellar light curve becomes a timing dataset and then a mass distribution; the steps carry different uncertainties.

## Before you begin

The measured-time tutorial supplies the data-selection exercise. Keep the actual paper and Table 1 MRT open beside this guide. The guide walks through the pinned author version; its full photometry reduction and posterior reanalysis lie beyond this saved-time exercise.

## Figure 1 asks what the star is doing

The broad brightness modulation is much larger than the shallow planetary transits. Section 3.1 begins by masking expected transit windows, dividing the time series at long gaps and fitting a cubic spline to the remaining stellar trend. Dividing by that trend gives a flattened light curve on which transit fitting is easier.

A mask is a modeling choice. If it excludes too little, the stellar trend can absorb a transit; if it interpolates poorly across a gap, the local transit baseline can shift. The paper records the knot prescription and the initial timing catalogs used for masking. Those details matter when comparing timing differences between reductions.

There is a useful unit check in the pinned v1: the introduction describes shallow transits near 400 parts per million, while Section 3.1 says 0.4 percent. Those units differ by a factor of ten. Table 2's small planet-to-star radius ratios imply depths of several hundred parts per million before limb effects. Use the radius ratios and figures to check the scale, and consult the final paper before copying the percentage.

Reading: Greklek-McKeon et al. · Greklek-McKeon et al.

## Figure 2 shows why a transit center can move

Read the ten outer-planet Kepler transits individually before looking at the stacked profile. Several have bumps associated with a planet crossing a darker patch on the star. The initial masks are gray; the final outlier mask is red. The blue curve is the fitted common transit profile.

The ninth transit has poor instrumental quality and is excluded from the stacked profile used to constrain shape. This is not permission to discard an inconvenient timing residual after a dynamical fit. The selection occurs in the photometric stage and has a documented observational reason.

For the shallow inner planets, the authors first determine a common transit shape and then fit individual center times with that shape fixed. The errors on those fitted centers are products of this observation model. Treating every time as an instantaneous, independent clock reading would overlook the shared shape assumptions.

Reading: Greklek-McKeon et al. · Deck et al.

## Figure 3 combines two bands and incomplete nights

The first two panels are partial J-band transits observed at Palomar with WIRC in 2019 and 2021; the third is the Kepler profile. The observing window cannot cover the roughly eight-hour transit fully from the ground. Missing ingress or egress makes the inferred center depend more strongly on the common shape.

The joint fit shares geometric parameters while allowing different radius ratios and limb-darkening coefficients in the two bandpasses. It also includes band-dependent extra photometric scatter and instrumental trend terms. The reported WIRC center precision is about four minutes; the median Kepler precision is about 0.55 minutes.

Open the MRT after reading this figure. The 2019 measured center is present, but the 2021 observed field is blank even though that row has a forecast. The public extraction keeps that observed field blank and excludes its forecast from the likelihood. The 25-time tutorial probes the Kepler-only outer pair; the paper’s extended-baseline result uses larger observation and dynamical models.

Reading: Greklek-McKeon et al. · Greklek-McKeon et al.

## Figure 4 introduces a second inference

Black points are Kepler times expressed relative to a linear schedule; red points are the ground follow-up observations. Blue curves are random draws from the dynamical posterior, not extra measurements. Identify the long timing cycle and the shorter structure in the middle planet before reading any mass value.

Section 3.3 integrates three planets with TTVFast at the stated reference epoch 2455865 BJD. Orbital inclinations are fixed to a right angle and nodes to zero for the timing calculation. The original transit-shape analysis still uses its measured projected geometry; the timing approximation relies on the small mutual inclination dispersion.

The third planet is important to the scope. A two-planet harmonic teaching model can isolate a mechanism, but its residuals need not match a full three-planet dynamical fit. Our tutorial's large extra scatter makes that distinction visible in the saved output.

Reading: Greklek-McKeon et al. · Deck et al. · Deck & Agol

## Read Figure 10 as a joint distribution

Each off-diagonal panel pairs two parameters. A tilted or curved cloud means changing one can be compensated by the other. A narrow marginal histogram is only one projection of that cloud. The caption notes that periods and initial transit times were varied too, even though they are omitted from the corner plot.

The eccentricity coordinates use square root of eccentricity times the sine or cosine of periapse direction. A prior uniform in those coordinates implies a different measure from one uniform in eccentricity-vector components. The parameterization and prior have to be read together; the plotted coordinates are not merely cosmetic.

Compare this with Deck and Agol v1 Section 5.1 and Figure 7. Their old PH3 middle/outer pair shows a curved mass ridge without independently resolved chopping, then a tighter region when the appropriate component is added. Its precision belongs to that particular data version and prior choice, and must be reassessed for later reduced models.

Reading: Greklek-McKeon et al. · Deck & Agol · Lithwick

## Check the stellar scale before interpreting density

Transit depth constrains a radius ratio, while dynamics constrains a mass ratio. To form a planet's density, multiply those ratios by adopted stellar mass and radius. The stellar radius enters cubed, so a small radius error can matter a great deal.

The pinned v1 Table 2 deserves a normalization check. Its Kepler + Palomar column gives an outer mass-ratio entry of 47.19 in units of one part in a hundred thousand. Multiplying it by the caption's stellar scale of 1.08 solar masses gives about 169.7 Earth masses, whereas its printed absolute mass is 157.18. The latter is close to using a one-solar-mass scale. The Kepler-only column follows the 1.08 scale. Medians of correlated posterior products need not equal products of medians, so do not silently replace an author value; verify the final version or original chain before treating these scalar columns as interchangeable.

Our tutorial fixes its own 1.08-solar-mass conversion and states that choice. It does not tune a subset to reach either printed mass. A full density inference propagates the joint planetary and stellar posterior, carrying mass and radius correlations into the density transformation.

$$
\rho_p=\frac{3\mu_p M_\star}{4\pi k_p^3 R_\star^3},\qquad \mu_p=m_p/M_\star,\quad k_p=R_p/R_\star
$$

With mass fixed, a one-percent radius increase lowers density by approximately three percent. For broad posteriors, transform the joint samples directly.

Reading: Greklek-McKeon et al. · Greklek-McKeon et al.

## Distinguish a timing fit from photodynamics

The 2023 analysis proceeds in stages: model the flux to obtain transit times and shape parameters, then fit a dynamical timing model. A joint photodynamical analysis instead integrates the orbital system and predicts the flux at every exposure inside one likelihood. It can carry shape, timing and dynamical correlations together, at much greater computational cost.

This distinction sets a useful next question. If a timing-only fit requires large extra scatter, does the mismatch come from a missing dynamical body, an incomplete eccentricity expansion, stellar activity or a shared photometric assumption? More exact dynamics alone cannot identify which one. The original light curves and an expanded observation model become necessary.

The paper's formation discussion goes one step further again, combining masses and radii with interior and disk models. A composition or birthplace estimate inherits those additional assumptions. Keep the evidence chain visible: measured flux, inferred times, dynamical mass ratios, stellar scale, then structural interpretation.

Reading: Greklek-McKeon et al. · Deck et al. · Deck & Agol

## Compare a resonant chain with a joint flux fit

Now open Mills et al., arXiv:1612.07376v1, the published Kepler-223 study. Figure 1 first shows the measured quarterly timing shifts. The Methods passages headed Photodynamic Inputs and Photodynamic Fits then put all four interacting planets into a single flux calculation. Transit shapes and perturbations constrain a joint set of orbital, radius and mass parameters, together with stellar information.

Read the stability selections separately from the measured likelihood. A configuration can fit four years of photometry and become unstable later. The authors therefore test later evolution and report selected posterior families. Those choices add physical information; they are not extra telescope exposures. Extended Data Figure 3 estimates Laplace-angle behavior over the observing interval, while later integrations assess what follows beyond it.

This comparison gives the next project a concrete target: replace an extracted-time likelihood with one that predicts every exposure, propagate stellar and dilution uncertainty, and state how a stability condition changes the posterior. Our two-planet timing exercise and the separate disk-migration lesson supply parts of that reasoning; neither reproduces the paper’s full photodynamic posterior or establishes permanent resonance from a short interval.

Reading: Mills et al. · Deck et al.

## Try the observation

### [Follow the physical part of the evidence chain](https://sandboxphysics.com/labs/transit-timing-variations/)

1. Identify the foreground crossing in the stellar image and the matching point in the timing plot.
2. Compare the giant-companion preset with the default before reading the real-data posterior.
3. Use the source guide's checklist to separate geometric timing, fitted photometric timing and absolute mass conversion.

[Previous: Reanalyze twenty-five measured Kepler-289 transit times](https://sandboxphysics.com/stories/transit-timing/kepler289-timing-tutorial/)

## 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.