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Sandbox Physics
Planet migration, from torque to real timing

L2 · Planet–disk dynamics

Why diffusion can reverse a migration torque

Sandbox Physics ·

The previous chapter left the torque coefficient unspecified. We now calculate it, keeping its empirical calibration separate from the conservation law that turns torque into drift.

Declare the disk before evaluating a formula

Use a surface density falling with the square root of radius. The temperature is a power law unless the transition preset is selected. A pressure scale height follows from the isothermal sound speed divided by orbital frequency. The physical adiabatic index is 1.4 and the Prandtl number is one half, so thermal diffusivity is twice kinematic viscosity.

For the displayed temperature conversion take mean molecular weight 2.34, solar gravitational parameter 1.3271244004193938 times ten to the twentieth in SI, and the exact astronomical unit. The default height implies about 761 kelvin at one AU. This is an adopted local gas model, not a temperature measurement of Kepler-223.

Σ∝r−p,T∝r−βT,p=0.5,h=csrΩ,ξ=βT−(γ−1)p\Sigma\propto r^{-p},\quad T\propto r^{-\beta_T},\quad p=0.5,\quad h=\frac{c_s}{r\Omega},\quad \xi=\beta_T-(\gamma-1)p
The entropy-gradient combination uses the physical adiabatic index. The effective index defined below belongs to a different part of the response.

Reading: Paardekooper · Ogilvie & Lubow

Wave and turning-gas torques have different gradient dependence

The wave coefficient is usually negative for these gradients. The barotropic horseshoe term depends on the surface-density slope. The entropy-related term depends on the temperature and density combination just defined. Both terms are normalized by the positive local torque scale.

These coefficients come from a two-dimensional low-mass fit with smoothing length 0.4 pressure scale heights. That smoothing describes the disk force calibration. It does not soften the mutual Newtonian gravity of the star and planets.

ΓLΓ0=−2.5−1.7βT+0.1pγeff\frac{\Gamma_L}{\Gamma_0}=\frac{-2.5-1.7\beta_T+0.1p}{\gamma_{\rm eff}}
This is the circular wave coefficient before the small-eccentricity correction.
Γhs,baroΓ0=1.1(1.5−p)γeff,Γhs,entΓ0=7.9ξγeff2\frac{\Gamma_{\rm hs,baro}}{\Gamma_0}=\frac{1.1(1.5-p)}{\gamma_{\rm eff}},\qquad \frac{\Gamma_{\rm hs,ent}}{\Gamma_0}=\frac{7.9\xi}{\gamma_{\rm eff}^2}
The unsaturated horseshoe terms are not the final corotation torque. Diffusion weights and a high-diffusion linear contribution remain to be applied.

Reading: Paardekooper · Ogilvie & Lubow

Compare diffusion with a horseshoe turn

The horseshoe half-width is a fraction of the orbital radius. Diffusion across that width competes with the time for gas to revisit the planet. The dimensionless saturation parameters encode this competition; large values correspond to weak diffusion across the horseshoe region.

The viscous and thermal parameters differ by a prefactor as well as by their diffusivities. Using identical definitions for both silently changes the fit. The approximate saturation function approaches unity with strong enough replenishment and decreases toward zero as diffusion becomes weak.

xs=1.1γeff−1/4q/h,pν=23a2Ωxs32πν,pχ=a2Ωxs32πχx_s=1.1\gamma_{\rm eff}^{-1/4}\sqrt{q/h},\quad p_\nu=\frac23\sqrt{\frac{a^2\Omega x_s^3}{2\pi\nu}},\quad p_\chi=\sqrt{\frac{a^2\Omega x_s^3}{2\pi\chi}}
The half-width is dimensionless. Viscosity and diffusivity both have units length squared per time.
F(p)=11+(p/1.3)2F(p)=\frac{1}{1+(p/1.3)^2}
This is equation 23’s approximation, not the exact Bessel-function expression in equation 22. The two cutoff functions also recover the linear response at extremely strong diffusion.
D(p,t0)={1625(45π4t0)3/4p3/2,p<4t045π1−925(4t045π)4/3p−8/3,p≥4t045πD(p,t_0)=\begin{cases}\frac{16}{25}(\frac{45\pi}{4t_0})^{3/4}p^{3/2},&p<\sqrt{\frac{4t_0}{45\pi}}\\1-\frac9{25}(\frac{4t_0}{45\pi})^{4/3}p^{-8/3},&p\geq\sqrt{\frac{4t_0}{45\pi}}\end{cases}
Use turnover two for the horseshoe cutoff and seven for the linear cutoff. This piecewise function is the analytic integral of the adopted streamline response, with exponent three quarters.
ΓC,b=Γhs,bFνGν+(1−Kν)Γlin,b\Gamma_{C,b}=\Gamma_{\rm hs,b}F_\nu G_\nu+(1-K_\nu)\Gamma_{\rm lin,b}
The linear barotropic coefficient is 0.7 times the density-gradient factor divided by the effective index.
ΓC,s=Γhs,sFνFχGνGχ+(1−Kν)(1−Kχ)Γlin,s\Gamma_{C,s}=\Gamma_{\rm hs,s}F_\nu F_\chi\sqrt{G_\nu G_\chi}+\sqrt{(1-K_\nu)(1-K_\chi)}\Gamma_{\rm lin,s}
The entropy-related linear coefficient is the entropy slope times the factor below. Both thermal and viscous diffusion enter the entropy horseshoe response.
Γlin,bΓ0=0.7(1.5−p)γeff,Γlin,sΓ0=(2.2−1.4/γeff)ξγeff\frac{\Gamma_{\rm lin,b}}{\Gamma_0}=\frac{0.7(1.5-p)}{\gamma_{\rm eff}},\qquad \frac{\Gamma_{\rm lin,s}}{\Gamma_0}=\frac{(2.2-1.4/\gamma_{\rm eff})\xi}{\gamma_{\rm eff}}
The two subscripts distinguish the barotropic and entropy-related terms; the same positive torque normalization applies to both.

Reading: Paardekooper · Ogilvie & Lubow

Do not confuse two adiabatic indices

Thermal diffusion changes the wave’s effective compressibility. The paper’s effective index tends to the physical adiabatic value when heat diffusion is weak, and to unity in the rapid-diffusion limit. The implementation uses an algebraically rationalized expression to avoid subtracting nearly equal numbers in the weak-diffusion limit.

The entropy slope still uses the physical index. The effective index belongs in the torque coefficients and horseshoe width. This distinction survives even though the two values happen to be close for some settings.

Q=2χ3h3a2Ω,γeff⟶{γ,Q→01,Q→∞Q=\frac{2\chi}{3h^3a^2\Omega},\qquad \gamma_{\rm eff}\longrightarrow\begin{cases}\gamma,&Q\to0\\1,&Q\to\infty\end{cases}
This dimensionless parameter describes thermal diffusion. Gravitational stability uses the separate Toomre parameter, defined from disk density, sound speed and orbital frequency.
γeff=2γQγQ+122(γ2Q2+1)2−16Q2(γ−1)+2γ2Q2−2\gamma_{\rm eff}=\frac{2\gamma Q}{\gamma Q+\tfrac12\sqrt{2\sqrt{(\gamma^2Q^2+1)^2-16Q^2(\gamma-1)}+2\gamma^2Q^2-2}}
This is the paper’s effective-index expression. The script evaluates an algebraically equivalent rationalized form near zero diffusion to retain its limit.

Reading: Paardekooper · Ogilvie & Lubow

Evaluate a controlled sign change

Keep mass four Earth masses, radius one AU, thickness 0.055, density slope one half and eccentricity zero. Change only temperature slope and viscosity. The public standard-library script evaluates all diffusion weights and produces this table.

At the steeper gradient, the intermediate-diffusion case is positive, whereas weak diffusion suppresses the nonlinear corotation contribution and restores a negative total. Very strong diffusion approaches a weaker linear response and is negative in this example. The table is local; a migrating pair samples changing radii and eccentricities.

Temperature slopeViscosity (code units)Total torque / scale
0.50.000000001−2.35148
0.50.000001−0.623413
0.50.0001−1.80401
1.20.000000001−3.20134
1.20.0000010.717460
1.20.0001−1.96087

Reading: Paardekooper · Fendyke & Nelson

A zero-torque line is a hypothesis about stopping

The transition preset changes the temperature slope smoothly with radius. A zero of the local total torque can act as a restoring location if the torque points inward just outside and outward just inside. Two interacting planets need not both settle at that single-planet zero.

The prescription omits migration-dependent dynamical torques, three-dimensional buoyancy, luminosity feedback and evolving gas. Calling every static zero a permanent trap would exceed the model. The frontier guide shows a radiation-hydrodynamic example where the missing terms matter.

Reading: Paardekooper · Ziampras

Try the observation

Find the torque reversal

  1. Choose Move outward.
  2. Choose Corotation fades and compare the viscosity while retaining the steeper temperature gradient.
  3. Read the two signed contributions, then compare the semimajor-axis histories.

Exercises and explained answers

Keep the total torque coefficient fixed and double the disk surface density. What changes?

Torque, circular drift and eccentricity damping rates double. The viscous and thermal saturation parameters do not depend directly on density in this prescribed disk. The gravitational-stability check does change, so this scaling cannot be extended to an arbitrarily massive disk.

A code uses the effective index inside the entropy slope. How would you detect the mistake?

Choose rapid heat diffusion so the effective index approaches unity while the physical index remains 1.4. For density slope one half, the entropy slope must still subtract 0.2 from the temperature slope. A sign-change grid is more revealing than a single default value.

Sources and calculation scope

  1. 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.

  2. 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.

  3. 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.

  4. 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.

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