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.

Outer wave torque, normalized $-2.2$

Outer corotation, normalized $1.5$

Current period ratio $2.00$

Inner semimajor axis $0.873\,\mathrm{AU}$

Let the disk change the orbit

Outer planet mass

4

m_2\;[M_\oplus]

Disk thickness at one AU

0.055

h_1=H/r

Viscous diffusion

-6

\log_{10}(\nu/[\mathrm{AU}^2/t_0])

Temperature gradient

0.5

\beta_T=-d\ln T/d\ln r

Initial period ratio

2.01

P_{2,0}/P_{1,0}

Drag the cyan planet sideways to change its adopted mass. Frozen presets open immediately; other settings start a new physical integration.

What is drawn, what is calculated

Gold is the inner planet and cyan the outer planet. Dashed rings mark their initial semimajor axes. Distances share one scale; planet sizes and wake thickness are enlarged. Solid curves are instantaneous osculating Kepler ellipses; the integrated planets also interact. The view is planar because the force model is coplanar. Faint moving arcs show unperturbed circular gas at the same replay time; wake curves show linear wave phase. Neither supplies an evolved gas density image.

\Gamma=\Gamma_{\rm L}+\Gamma_{\rm C},\qquad \dot a\simeq \frac{2\Gamma}{m a\Omega}

Readout torques are normalized by the outer planet's local torque scale. The imposed disk can supply or remove energy and angular momentum. The default surface-density scale is one ten-thousandth of the stellar mass per square AU. Diffusivity is twice viscosity; disk density slope is one half, adiabatic index 1.4 and fit softening 0.4 scale heights. Mutual planetary gravity is unsoftened. Stop limits are stated below the Lab.

The captured preset ends at twelve thousand initial inner orbits; the later preset keeps exactly those physical parameters and ends at fifty thousand. Neither a bounded short arc nor a near-integer period ratio proves indefinite stability. Custom runs use the same force law for twelve thousand initial inner orbits, then replay the final eight; a model-boundary stop is shown explicitly.

Read the course and reconstruct real Kepler-223 angles

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.

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 misconceptionA ratio close to two proves capture.

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

The disk is a reservoir

Typical misconceptionThe planets must conserve their own energy despite drag.

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

Run the experiment

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.
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.
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.

Planet Migration, Disk Torques and Resonance

An orbit can keep time and still move

Follow angular momentum through a disk

Core mathematical model

Torque moves a nearly circular orbit

Separate scale from sign

Look for bounded angles

Common difficulties

A two-to-one ratio is not a verdict

The disk is a reservoir

Run the experiment

A temporary rhythm

Wait longer with the same parameters

Change the diffusion

Related experiments and articles

From disk torque to resonant timing