---
title: "Planet Migration, Disk Torques and Resonance Interactive Lab | Sandbox Physics"
description: "What makes a planet leave its birthplace? A planet exchanges angular momentum with gas through spiral waves and gas that turns near its orbit."
canonical_url: "https://sandboxphysics.com/labs/planet-disk-migration/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/planet-disk-migration/"
last_modified: "2026-10-28"
---

AC15 · A disk can move a planet

# Planet Migration, Disk Torques and Resonance

**What makes a planet leave its birthplace?** A planet exchanges angular momentum with gas through spiral waves and gas that turns near its orbit. A negative net torque usually moves a nearly circular orbit inward; a positive torque can move it outward. Two converging planets may temporarily share a resonant rhythm. Their orbital angles test the timing of that rhythm; the period ratio alone leaves its phase undetermined.

Disk torques move planetary orbits. Change diffusion and compare outward drift, temporary capture and later resonant escape.

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

Model assumptions

Synthetic planar, finite-mass star–two-planet integration with an external, stationary, non-self-gravitating disk. The Paardekooper 2010v1/2011 two-dimensional diffusion torque fit uses softening length 0.4 disk scale heights; mutual three-body gravity is unsoftened. The adiabatic index is 1.4, density slope 0.5 and Prandtl number 0.5. Eccentricity corrections follow Cresswell–Nelson and Fendyke–Nelson within eccentricity below 0.7 disk aspect ratios. The calculation stops outside 0.35–3 AU, above 0.2 thermal masses, below Toomre parameter twenty, or within three mutual Hill radii. The disk is an imposed energy and angular momentum reservoir, not evolved hydrodynamics. Spiral curves illustrate the linear wake phase only; their brightness is not gas density. Planet sizes and wake thickness are enlarged. Frozen presets use REBOUND 5.2.2 IAS15; custom settings integrate the same forces with DOP853. The stage loops an actual eight-orbit section of the stated synthetic run with one shared clock. The default shows the same-parameter run after fifty thousand initial inner orbits, with a visible displacement from its birthplace; the temporary-capture preset shows a finite window at twelve thousand. No lifetime or escape mechanism is inferred from angle plots alone. The real Kepler-223 timing tutorial is separate from these synthetic planets.

## An orbit can keep time and still move

Gold: inner planet; cyan: outer planet. The orbital history covers the full run; angle details show its later half, relative to each angle at that window’s start. The stage above replays only its stated eight-orbit section, with one shared physical clock.

Where the orbit moved

$a\;[\mathrm{AU}]$

$t/P_{1,0}$

Unwrapped critical angles

$\theta_i(t)-\theta_i(t_{\rm start})\;[{}^\circ]$

$t/P_{1,0}$

$\theta_i=2\lambda_2-\lambda_1-\varpi_i$ · Frozen phase is unwrapped every half initial inner orbit before saving the coarser chart. Near-zero eccentricity makes periapse angles ill-conditioned; no capture verdict is assigned there. A bounded finite window is a temporary diagnosis, not a lifetime.

Physics tutorial

## Follow angular momentum through a disk

**Background** A planet pulls on nearby gas. The gas responds with waves and horseshoe turns, then pulls back. The unequal torque changes the orbit even while each revolution looks nearly Keplerian.

**Why it matters** A close-in planet may have formed farther out, but a present orbit alone cannot reveal its history.

Start with the essentials

- **Focus question**: When does the net torque change sign, and when do two planets actually share a resonance?
- **One-sentence intuition**: Migration follows angular momentum exchange. Resonant capture requires angle dynamics, not just a near-integer period ratio.

### Core mathematical model

#### Torque moves a nearly circular orbit

$$
L=m\sqrt{GM_\star a},\qquad \dot a=\frac{2\Gamma}{m a\Omega}
$$

The last relation assumes small eccentricity; the actual integration evolves Cartesian positions and velocities. Positive torque adds prograde angular momentum.

#### Separate scale from sign

$$
\Gamma_0=(q/h)^2\Sigma a^4\Omega^2,\qquad \Gamma=\Gamma_{\rm L}+\Gamma_{\rm C}
$$

The scale is positive. The fitted coefficients determine direction; diffusion changes the corotation term.

#### Look for bounded angles

$$
\theta_i=2\lambda_2-\lambda_1-\varpi_i
$$

Unwrap phase finely enough to retain revolutions. A bounded arc in a finite record supports only a finite-time diagnosis.

### Common difficulties

#### A two-to-one ratio is not a verdict

**Typical misconception** A ratio close to two proves capture.

**Better mental model** Track the critical angles and their turning points over a declared interval; compare against a circulating case.

#### The disk is a reservoir

**Typical misconception** The planets must conserve their own energy despite drag.

**Better mental model** Subtract integrated external work and torque when checking the full inertial three-body balance.

### Run the experiment

1. 01

   #### A temporary rhythm

   Open temporary capture and read the last half of both angle tracks.

   **What to observe:** The two angles remain bounded over that declared interval, although the planets continue migrating.

2. 02

   #### Wait longer with the same parameters

   Open Later: let go and compare its parameter fields with temporary capture.

   **What to observe:** The same parameters later show circulating angles. The record does not by itself identify the escape mechanism.

3. 03

   #### Change the diffusion

   Compare outward migration with the saturated case.

   **What to observe:** Weak diffusion suppresses the nonlinear corotation contribution; the signed net torque can reverse.

Physical conclusion

Compare wave and corotation contributions, then compare temporary capture with the later same-parameter escape.

## Related experiments and articles

From disk torque to resonant timing · Reading path

### From disk torque to resonant timing

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

1. [Article **Why a planet can leave its birthplace**](https://sandboxphysics.com/stories/planet-migration/why-planets-move/)
2. [Article **From a disk torque to a moving orbit**](https://sandboxphysics.com/stories/planet-migration/angular-momentum-and-waves/)
3. [Article **Why diffusion can reverse a migration torque**](https://sandboxphysics.com/stories/planet-migration/diffusion-and-torque-reversals/)
4. [Article **Resonant capture is an angle problem**](https://sandboxphysics.com/stories/planet-migration/resonant-capture-and-release/)
5. [Article **Calculate migration times and reconstruct Kepler-223 angles**](https://sandboxphysics.com/stories/planet-migration/kepler223-timing-tutorial/)
6. [Article **Read a migration paper by testing what its torque leaves out**](https://sandboxphysics.com/stories/planet-migration/reading-migration-models/)

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)