# SATURATED TORQUE FORMULA FOR PLANETARY MIGRATION IN VISCOUS DISKS WITH THERMAL DIFFUSION: RECIPE FOR PROTOPLANET POPULATION SYNTHESIS

## Abstract

We provide torque formulae for low-mass planets undergoing type I migration in gaseous disks. These torque formulae put special emphasis on the horseshoe drag, which is prone to saturation: the asymptotic value reached by the horseshoe drag depends on a balance between coorbital dynamics (which tends to cancel out or saturate the torque) and diffusive processes (which tend to restore the unperturbed disk profiles, thereby desaturating the torque). We entertain the question of this asymptotic value and derive torque formulae that give the total torque as a function of the disk's viscosity and thermal diffusivity. The horseshoe drag features two components: one that scales with the vortensity gradient and another that scales with the entropy gradient and constitutes the most promising candidate for halting inward type I migration. Our analysis, which is complemented by numerical simulations, recovers characteristics already noted by numericists, namely, that the viscous timescale across the horseshoe region must be shorter than the libration time in order to avoid saturation and that, provided this condition is satisfied, the entropy-related part of the horseshoe drag remains large if the thermal timescale is shorter than the libration time. Side results include a study of the Lindblad torque as amore »

- Authors:

- Instituto de Ciencias Fisicas, Universidad Nacional Autonoma de Mexico (UNAM), Apdo. Postal 48-3, 62251-Cuernavaca, Morelos (Mexico)
- Laboratoire AIM, CEA/DSM, CNRS, Universite Paris Diderot, Irfu/Service d'Astrophysique, Bat. 709, CEA/Saclay, 91191 Gif-sur-Yvette (France)

- Publication Date:

- OSTI Identifier:
- 21471249

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Astrophysical Journal; Journal Volume: 723; Journal Issue: 2; Other Information: DOI: 10.1088/0004-637X/723/2/1393

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; COMPUTERIZED SIMULATION; ENTROPY; HYDRODYNAMICS; MATHEMATICAL MODELS; NUMERICAL SOLUTION; PLANETS; PROTOPLANETS; THERMAL DIFFUSION; THERMAL DIFFUSIVITY; TORQUE; DIFFUSION; FLUID MECHANICS; MATHEMATICAL SOLUTIONS; MECHANICS; PHYSICAL PROPERTIES; SIMULATION; THERMODYNAMIC PROPERTIES

### Citation Formats

```
Masset, F. S., and Casoli, J., E-mail: masset@fis.unam.m, E-mail: jules.casoli@cea.f, E-mail: masset@fis.unam.m.
```*SATURATED TORQUE FORMULA FOR PLANETARY MIGRATION IN VISCOUS DISKS WITH THERMAL DIFFUSION: RECIPE FOR PROTOPLANET POPULATION SYNTHESIS*. United States: N. p., 2010.
Web. doi:10.1088/0004-637X/723/2/1393.

```
Masset, F. S., & Casoli, J., E-mail: masset@fis.unam.m, E-mail: jules.casoli@cea.f, E-mail: masset@fis.unam.m.
```*SATURATED TORQUE FORMULA FOR PLANETARY MIGRATION IN VISCOUS DISKS WITH THERMAL DIFFUSION: RECIPE FOR PROTOPLANET POPULATION SYNTHESIS*. United States. doi:10.1088/0004-637X/723/2/1393.

```
Masset, F. S., and Casoli, J., E-mail: masset@fis.unam.m, E-mail: jules.casoli@cea.f, E-mail: masset@fis.unam.m. Wed .
"SATURATED TORQUE FORMULA FOR PLANETARY MIGRATION IN VISCOUS DISKS WITH THERMAL DIFFUSION: RECIPE FOR PROTOPLANET POPULATION SYNTHESIS". United States. doi:10.1088/0004-637X/723/2/1393.
```

```
@article{osti_21471249,
```

title = {SATURATED TORQUE FORMULA FOR PLANETARY MIGRATION IN VISCOUS DISKS WITH THERMAL DIFFUSION: RECIPE FOR PROTOPLANET POPULATION SYNTHESIS},

author = {Masset, F. S. and Casoli, J., E-mail: masset@fis.unam.m, E-mail: jules.casoli@cea.f, E-mail: masset@fis.unam.m},

abstractNote = {We provide torque formulae for low-mass planets undergoing type I migration in gaseous disks. These torque formulae put special emphasis on the horseshoe drag, which is prone to saturation: the asymptotic value reached by the horseshoe drag depends on a balance between coorbital dynamics (which tends to cancel out or saturate the torque) and diffusive processes (which tend to restore the unperturbed disk profiles, thereby desaturating the torque). We entertain the question of this asymptotic value and derive torque formulae that give the total torque as a function of the disk's viscosity and thermal diffusivity. The horseshoe drag features two components: one that scales with the vortensity gradient and another that scales with the entropy gradient and constitutes the most promising candidate for halting inward type I migration. Our analysis, which is complemented by numerical simulations, recovers characteristics already noted by numericists, namely, that the viscous timescale across the horseshoe region must be shorter than the libration time in order to avoid saturation and that, provided this condition is satisfied, the entropy-related part of the horseshoe drag remains large if the thermal timescale is shorter than the libration time. Side results include a study of the Lindblad torque as a function of thermal diffusivity and a contribution to the corotation torque arising from vortensity viscously created at the contact discontinuities that appear at the horseshoe separatrices. For the convenience of the reader mostly interested in the torque formulae, Section 8 is self-contained.},

doi = {10.1088/0004-637X/723/2/1393},

journal = {Astrophysical Journal},

number = 2,

volume = 723,

place = {United States},

year = {Wed Nov 10 00:00:00 EST 2010},

month = {Wed Nov 10 00:00:00 EST 2010}

}