---
title: "From a disk torque to a moving orbit"
description: "Derive the sign and units of migration, connect orbital energy with angular momentum, and calculate a one-AU example."
canonical_url: "https://sandboxphysics.com/stories/planet-migration/angular-momentum-and-waves/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/planet-migration/angular-momentum-and-waves/"
last_modified: "2026-10-28"
---

[Planet migration, from torque to real timing](https://sandboxphysics.com/stories/planet-migration/)

L1 · Planet–disk dynamics

# From a disk torque to a moving orbit

Sandbox Physics · 2026-10-28

A small torque applied on every orbit can matter more than a large force that averages to zero. We will derive the orbital drift before using any empirical disk formula.

## Before you begin

Circular motion, Newtonian gravity and differentiation. The next chapter supplies the torque coefficient.

## Fix the origin, signs and units

Take a solar-mass star and prograde coplanar planets. Positive angular momentum points out of the page; positive torque adds that component. The astronomical unit is the length scale and the stellar mass the mass scale. Setting the gravitational constant to unity fixes the time unit; it is not a year.

One code time unit is about 0.159 Julian years. An initially circular one-AU orbit takes about two pi code units, or one year. The scene’s clock is labeled in initial inner periods so changes in the later period do not quietly redefine elapsed time.

$$
t_0=\sqrt{\mathrm{AU}^3/(GM_\star)},\qquad \Omega(a)=\sqrt{GM_\star/a^3}
$$

Frequency has inverse-time units. A torque has units mass times length squared per time squared.

Reading: Paardekooper · REBOUND 5.2.2 and IAS15

## Differentiate the circular angular momentum

For a circular orbit the speed follows from the balance of gravity and centripetal acceleration. Multiply speed by orbital radius and planet mass to obtain angular momentum. Differentiate while holding the masses fixed.

The derivative contains half the fractional change in semimajor axis. Equating the angular-momentum derivative to the applied torque gives the drift rate. Negative torque reduces the orbital size; no inward radial drag is needed for that conclusion.

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

This derivation assumes a nearly circular orbit and a stellar-mass-dominated potential.

$$
\dot a=\frac{2\Gamma}{m a\Omega},\qquad \tau_a=\frac{a}{|\dot a|}=\frac{L}{2|\Gamma|}
$$

The drift time is positive by definition; the sign belongs to the drift rate.

Reading: Paardekooper · Ogilvie & Lubow

## Energy tells the same story

An inward-migrating circular orbit becomes more tightly bound. Differentiate the orbital energy and substitute the circular drift relation: the work rate equals angular frequency times torque. This is a useful independent check on units and signs.

A real integration has finite eccentricity and planet–planet interactions. Each planet’s two-body energy need not follow this relation instant by instant. The complete inertial energy balance must instead include the work done by the external disk force.

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

This circular limit supplies an independent check on the Cartesian integration.

Reading: Paardekooper · Cresswell & Nelson

## Separate the disk scale from its coefficient

Let the mass ratio compare planet and star, and let the disk aspect ratio compare pressure scale height with radius. The usual torque normalization is positive. It supplies the dimensional scale; the wave and corotation coefficients supply the sign.

The surface density here is mass per area, not volume density. The local frequency is evaluated at the planet’s osculating semimajor axis. This is a restricted local prescription applied to slowly evolving, low-eccentricity orbits.

$$
q=\frac{m}{M_\star},\quad h=\frac{H}{a},\quad \Gamma_0=\left(\frac{q}{h}\right)^2\Sigma a^4\Omega^2,\quad \Gamma=C\Gamma_0
$$

At a fixed disk profile the torque scale grows with planet mass squared; the circular drift rate grows with planet mass.

Reading: Paardekooper · Ogilvie & Lubow

## Work a one-AU example

Use an outer-planet mass of four Earth masses, stellar mass one solar mass, aspect ratio 0.055, surface-density scale one ten-thousandth in the adopted mass-per-square-AU units, temperature slope 0.5 and viscosity one millionth in code units. For this local example set eccentricity to zero.

The fitted total coefficient is about minus 0.623. Substitution gives the values below. They are local model calculations at one AU, not a measured migration speed or the final drift rate of a resonant pair. The public script reproduces the table.

The drift is about three millionths of an AU per year. Dividing one AU by that local rate gives about 321 thousand years. The eccentricity damping time is much shorter, about 1554 years; the complete pair nevertheless exchanges gravitational kicks as it migrates.

$$
\dot a=2C\frac{q\Sigma a^3\Omega}{h^2M_\star}
$$

For this example, convert code-time drift to AU per year using the stated time unit.

| Quantity | One-AU reference |
| --- | --- |
| Planet mass / star mass | 0.000012014 |
| Total torque / normalization | −0.623413 |
| Disk aspect ratio | 0.055 |
| Temperature implied by height (K) | 761.176 |
| Local drift (AU per year) | −0.00000311129 |
| Local migration time (years) | 321410 |
| Eccentricity damping time (years) | 1554.16 |

Reading: Paardekooper · Fendyke & Nelson

## Check the approximation before extrapolating

Doubling mass doubles the predicted local drift only while the small-planet approximation remains valid. It does not authorize applying this formula to a gap-opening giant planet. A thicker disk lowers the torque scale but also changes the diffusion response, so holding the coefficient fixed is a separate thought experiment.

The Lab displays a planar wake because this model is planar. The bright curves have enlarged thickness and equal visual weighting. Their brightness is schematic; the curves locate a physically defined wave phase.

Reading: Paardekooper · Ogilvie & Lubow

## Try the observation

### [Test signs before trusting a track](https://sandboxphysics.com/labs/planet-disk-migration/)

1. Compare the outward and saturated presets.
2. Identify the positive corotation term and negative wave term.
3. Use Gravity alone as the zero-external-torque limit.

## Exercises and explained answers

A circular planet receives the same negative torque for twice as long. Must its semimajor axis decrease by exactly twice as much?

Only in a short interval where radius, torque and frequency hardly change. Angular momentum changes linearly under constant torque, whereas semimajor axis is proportional to its square. The disk prescription generally changes the torque as well.

A calculation gives inward drift but positive disk work for a nearly circular isolated planet. What would you inspect first?

Check torque direction, the chosen coordinate orientation and the conversion between disk work and planet work. In the circular limit negative torque requires negative work. Finite eccentricity and interactions can break the instantaneous two-body relation, so compare the proper limit before declaring an error.

[Previous: Why a planet can leave its birthplace](https://sandboxphysics.com/stories/planet-migration/why-planets-move/) [Next: Why diffusion can reverse a migration torque](https://sandboxphysics.com/stories/planet-migration/diffusion-and-torque-reversals/)

## Sources and calculation scope

1. [Paardekooper, Baruteau & Kley, arXiv:1007.4964v1 (2010; MNRAS 2011)](https://arxiv.org/html/1007.4964v1)

   Sections 3–5.7: torque normalization, diffusion functions, effective gamma and the modified formula. Equations 3–7, 19, 22–23, 30–31, 40, 45–53; figures 2–3 and 6–8. Two-dimensional low-mass fit with a specified smoothing length.

2. [Cresswell & Nelson, arXiv:0811.4322v1 (2008)](https://arxiv.org/pdf/0811.4322v1)

   The eccentricity-damping and migration-time prescriptions, including the small-eccentricity wave time. We use the coplanar limit and restrict eccentricity below 0.7 aspect ratios.

3. [Fendyke & Nelson, arXiv:1310.0351v1 (2013; MNRAS 2014)](https://arxiv.org/pdf/1310.0351v1)

   The fitted exponential reduction of corotation torque with eccentricity. The exponent is negative. This controls the sign used here, despite the positive sign printed in Izidoro et al. equation 9.

4. [Ogilvie & Lubow, astro-ph/0111265v1 (2001; MNRAS 2002)](https://arxiv.org/pdf/astro-ph/0111265v1)

   Linear two-dimensional wake geometry: waves from different azimuthal orders align in phase. The scene uses only this kinematic phase curve; it does not reconstruct density or brightness.

5. [REBOUND 5.2.2 and IAS15](https://github.com/hannorein/rebound/blob/c34134716cd484ed9cdb9f7d54ef8d859bdf547d/README.md)

   IAS15 reference integration with accuracy parameter one times ten to minus ten. The optional generator pins REBOUND 5.2.2; no package code is copied into the public tutorial.