---
title: "Transit Timing Variations and Planet Masses Interactive Lab | Sandbox Physics"
description: "What makes a planet cross its star late? A nearby planet pulls a transiting planet ahead of or behind a steady orbital clock."
canonical_url: "https://sandboxphysics.com/labs/transit-timing-variations/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/transit-timing-variations/"
last_modified: "2026-10-28"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Read a planet's changing appointment

**Background** A companion exchanges orbital energy and angular momentum with a planet. The next stellar crossing then occurs a little ahead of or behind a clock fitted to all crossings.

**Why it matters** A transit depth gives a size ratio; gravity in the timing adds information about mass.

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 misconception** A deeper transit directly weighs a planet.

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

#### A large slow wave can remain ambiguous

**Typical misconception** A well-measured timing amplitude uniquely fixes a mass.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

A timing residual is a difference from an adopted clock. Its uncertainty and sampling are part of any mass inference.

## Related experiments and articles

When a planet arrives late · Reading path

### When a planet arrives late

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a planet crosses its star late**](https://sandboxphysics.com/stories/transit-timing/late-planet/)
2. [Article **Build the orbit before building the clock**](https://sandboxphysics.com/stories/transit-timing/orbits-before-clocks/)
3. [Article **Turn stellar crossings into timing residuals**](https://sandboxphysics.com/stories/transit-timing/crossings-and-residuals/)
4. [Article **Why a timing wave can leave the mass uncertain**](https://sandboxphysics.com/stories/transit-timing/mass-and-eccentricity/)
5. [Article **Reanalyze twenty-five measured Kepler-289 transit times**](https://sandboxphysics.com/stories/transit-timing/kepler289-timing-tutorial/)
6. [Article **Read the Kepler-289 mass measurement from its light curves**](https://sandboxphysics.com/stories/transit-timing/reading-kepler289/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)