AC15 · Planet–disk dynamics
Planet migration, from torque to real timing
A planet changes the gas around it; that response can change the orbit. Start with angular momentum, follow diffusion and resonant angles, then work with real measured transit timing.
Why a planet can leave its birthplace
Gas exchanges angular momentum with a planet. Follow the waves, the turning gas and the difference between a temporary resonance and a permanent home.
02 · L1From a disk torque to a moving orbit
Derive the sign and units of migration, connect orbital energy with angular momentum, and calculate a one-AU example.
03 · L2Why diffusion can reverse a migration torque
Build the low-mass torque prescription from disk gradients, horseshoe turns and two diffusion clocks; calculate its sign changes.
04 · L3Resonant capture is an angle problem
Integrate the finite-mass system, account for external disk work, and distinguish temporary angle libration from later circulation.
05 · L3Calculate migration times and reconstruct Kepler-223 angles
Reproduce a 775-cell torque map and a real 44-row timing diagnostic with standalone Python and explicit model, time and uncertainty conventions.
06 · L4Read a migration paper by testing what its torque leaves out
A pinned reading route through saturation, resonant-chain populations and three-dimensional radiative buoyancy; compare assumptions before conclusions.
Sources and calculation scope
- Paardekooper, Baruteau & Kley, arXiv:1007.4964v1 (2010; MNRAS 2011)
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.
- Cresswell & Nelson, arXiv:0811.4322v1 (2008)
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.
- Fendyke & Nelson, arXiv:1310.0351v1 (2013; MNRAS 2014)
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.
- Ogilvie & Lubow, astro-ph/0111265v1 (2001; MNRAS 2002)
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.
- Mills et al., arXiv:1612.07376v1, Nature 533, 509 (2016)
Methods on timing and Laplace-angle reconstruction, Extended Data Table 1 (PDF page 30) and Figure 3 (page 35). The tutorial freezes all 44 quarterly timing measurements and their original linear ephemerides.
- Izidoro et al., arXiv:1703.03634v2 (24 May 2017)
Sections 2.2–2.3, 5.2, 6.1 and 8; figures 7 and 14. This population study links migration, chains and later instability. Our two-planet illustration is not its population reproduction.
- Ziampras, Nelson & Paardekooper, arXiv:2406.08555v1 (12 June 2024)
Sections 2.2–2.3, 3, 4.2–4.4 and 5; figures 3–7; appendix A. Global three-dimensional radiation hydrodynamics tests buoyancy-driven dynamical torques absent from our static two-dimensional fit.
- REBOUND 5.2.2 and IAS15
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.