= Solution
\b[<Type I migration> is orbital angular-momentum exchange between an embedded low-<mass> <planet> and its disk without a deep gap.] The <planet> excites spiral <density> waves near <Lindblad resonances>; the disk's reaction torques act back on the <planet>. The outer wake usually supplies a negative torque and the inner wake a positive one, with their small imbalance often driving inward migration. <Gas> on <horseshoe orbits in a planetary disk> near corotation adds a <corotation torque> that depends on entropy and <vortensity> gradients.
For a thin disk, define $q=M_p/M_\star$, $h=H/a$ and
$$
\Gamma_0=\left(\frac qh\right)^2\Sigma_g a^4\Omega^2,\qquad\Gamma=C\Gamma_0.
$$
The coefficient $C$ is signed and depends on disk gradients, thermodynamics and diffusion; it is not a universal constant. For a circular orbit with fixed <planet> <mass>, $L_p=M_pa^2\Omega\propto a^{1/2}$. The <Type I migration timescale from disk torque> follows from $\dot L_p=\Gamma$:
$$
\dot a=\frac{2\Gamma}{M_pa\Omega},\qquad\boxed{\frac a{|\dot a|}=\frac1{2|C|}\frac{M_\star}{M_p}\frac{M_\star}{\Sigma_g a^2}h^2\Omega^{-1}.}
$$
Thus a more massive embedded core generally migrates faster within this regime, unlike a picture in which migration starts only after giant-<planet> formation.
In a simple locally isothermal smooth disk the net torque is commonly negative, potentially removing cores before they acquire <gas>. In a non-isothermal disk an unsaturated <entropy-related corotation torque> can balance or exceed the negative wave torque, giving a <planet trap> or outward migration. Viscosity and thermal diffusion help maintain the background gradients across the horseshoe region; without adequate replenishment <corotation-torque saturation> can suppress that contribution. The linear low-<mass> scaling requires a small disturbance, roughly $q\ll h^3$; larger planets, gap formation and strong disk feedback require a different treatment. Type I migration is therefore neither universally inward nor literally viscous advection of the <planet>.<Planet-Disc Interactions and Early Evolution of Planetary Systems by Clément Baruteau and others>[Sections on Lindblad and corotation torques, saturation and low-<mass> migration]{ref}
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