Keep the full gravitational problem
The star and two finite-mass planets move in an inertial frame. Each gravitational pair obeys the unsoftened inverse-square law. The initial inner orbit has semimajor axis one AU, the outer planet is twice as massive, and both start with eccentricity 0.001. Initial periapse longitudes are 0.2 and 1.2 radians; mean anomalies are zero and two radians.
A finite planetary mass enters the two-body frequency. The outer initial semimajor axis includes the finite-mass correction below. After constructing the heliocentric elements, all bodies are shifted to the barycentric position and velocity origin.
Reading: REBOUND 5.2.2 and IAS15 · Izidoro et al.
Write down the imposed disk acceleration
At each time, calculate a planet’s heliocentric osculating elements, then its local wave and corotation torques. The migration acceleration is parallel to the relative velocity and has a coefficient chosen to supply the desired torque about the star. A separate radial term damps eccentricity.
The radial damping term has no torque about the stellar origin. Its relative work is nonpositive. For an eccentric orbit, the velocity-parallel term is not pure semimajor-axis migration: it also affects energy and eccentricity. This is an explicitly prescribed force model, not hydrodynamics.
Reading: Cresswell & Nelson · Izidoro et al.
Reduce the corotation term with the correct sign
Eccentric motion weakens the coherent horseshoe response. We multiply the circular corotation result by the negative exponential below, following Fendyke and Nelson. Izidoro et al. equation 9 prints a positive exponent despite describing attenuation; this implementation uses the original attenuation law.
The wave term also receives the Cresswell–Nelson eccentricity correction. The Lab stops at eccentricity greater than 0.7 local aspect ratios, well before the large-eccentricity sign changes of extrapolated formulae. It also stops outside 0.35–3 AU, above 0.2 thermal masses, below Toomre parameter twenty, or inside three mutual Hill radii.
Reading: Fendyke & Nelson · Izidoro et al. · Cresswell & Nelson
Audit the energy and angular momentum reservoir
The static disk is external to the three-body system. Its force is applied to the planets; no equal-and-opposite acceleration is secretly applied to the star. The system may therefore change energy, angular momentum and linear momentum.
For the energy audit, use every body’s inertial velocity, including the star, and all three gravitational pair potentials. Subtract the time-integrated disk work. The angular-momentum audit similarly uses inertial positions and the external torque. Using heliocentric torque alone in this full-system audit omits the moving-origin term.
Reading: REBOUND 5.2.2 and IAS15 · Cresswell & Nelson
Track the phase of the kicks
Mean longitude advances around an orbit; periapse longitude locates its long axis. Near the two-to-one resonance, combine them into two critical angles. Period ratio alone discards the phase information that decides whether repeated conjunctions reinforce a coherent pattern.
Unwrap each angle at fine physical time intervals before discarding samples for a compact plot. Otherwise an angle that circles an integer number of times between saved frames can look fixed. Frozen records track phase every half initial inner orbit; the default and later cases are separately checked at finer cadence. Where very small eccentricity makes periapse ill-conditioned, the chart switches to wrapped samples and gives no capture verdict.
Reading: Mills et al. · Izidoro et al.
Read the saved example without opening the Lab
For the default disk and four-Earth-mass outer planet, the later half of the twelve-thousand-period record has critical-angle ranges about 17.5 and 18.6 degrees wide. Both show repeated turns within those ranges. We call this temporary capture over that declared window.
The same initial state and physical parameters integrated to fifty thousand periods no longer keep those angles bounded. In its later half they each sweep about twenty-seven full turns. This is evidence for release in this model, not a verified explanation of the release mechanism. Identifying a specific overstability or resonance-overlap mechanism would require further controlled experiments.
| End time (initial inner periods) | Inner axis (AU) | Outer axis (AU) | Period ratio |
|---|---|---|---|
| 12000 | 0.971254 | 1.542251 | 2.000930 |
| 50000 | 0.873217 | 1.386660 | 2.001106 |
Reading: REBOUND 5.2.2 and IAS15 · Izidoro et al.
Separate numerical agreement from physical validity
Frozen presets use REBOUND 5.2.2 IAS15 with accuracy parameter one times ten to minus ten. Custom browser runs use DOP853 with relative tolerance two times ten to minus eleven and absolute component scale one times ten to minus fourteen. An independent short-trajectory comparison checks both Cartesian phase and orbital sizes.
The public package states actual tolerances and convergence results. A numerically accurate solution of this imposed two-dimensional torque model can still miss real disk physics. The synthetic pair and the measured timing diagnostic retain their separate initial conditions and scientific roles.
Reading: REBOUND 5.2.2 and IAS15 · Ziampras