Free Access
Volume 567, July 2014
Article Number A121
Number of page(s) 18
Section Planets and planetary systems
Published online 24 July 2014

Online material

Appendix A: Saturation and disk evolution

thumbnail Fig. A.1

Position of a planet in the nominal synthesis calculation when it transitions from the unsaturated adiabatic migration regime into the saturated adiabatic migration regime. Colored are the planet which distance is less than 3% (blue, up facing triangles), 5% (green, right facing triangles) or 10% (red, left facing triangles) from a CZ.

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From the positions of planets at the time of saturation in the formation tracks in Fig. 7 in Sect. 3.5 one sees that most positions lie on two lines, one for the planets of the inner and one for the outer CZ. We study here the reasons for this feature.

Figure A.1 shows the positions of the planets at the transition from unsaturated to saturated adiabatic migration, the most common transition into a saturated migration regime. The positions of all planets undergoing this transition are shown in black, while the colored points show that those with a distance between their semimajor axis and the position of the CZ at that time are fewer than 10% of the semimajor axis. There are two large groups, again one of the inner and one of the outer convergence region. Almost all planets of the inner group are in the CZ when they saturate, while the outer ones are much more spread out. But the planets that are in the CZ form a line here as well. There is a third CZ inside of 0.3 AU in the most massive disks in the synthesis, which leads to a third minor group. However, this CZ evolves quickly and disappears after the first few 0.01 Myr and all associated planets end at 0.1 AU. The dozen points inside of 0.2 AU in Fig. A.1 correspond to planets in this small CZ.

This behavior results from the interaction of the following points:

One can calculate the saturation mass as a function of the orbital distance a and time t by setting s2 = 1 in Eq. (17). There, fvisc is 1, 0.55 or 0.125 in each of the model versions. (A.1)

For a fixed orbital distance and stellar mass it only depends on the disk aspect ratio h and the viscosity ν. Both quantities are decreasing with time, as the disk mass decreases, and therefore, the saturation mass also becomes smaller as the disk evolves.

While the photoevaporation rate is important for the lifetime of a disk, the constant value of α in all simulations of one synthesis results in a similar disk structure in the part of the disk where viscosity is dominant. There, the disks go through the same series of disk states (radial profile of temperature, surface density, etc.) and only the speed with which the disks go through the states is different and depends on the photoevaporation rate. Given one semimajor axis a, there is only one disk state where the inner (our outer) CZ lie at this position. Therefore also h and ν are fixed for this semimajor axis of the CZ. Therefore, the semimajor axis of the CZ corresponds to only one saturation mass. And as the CZ moves inward while the disk evolves, the saturation mass decreases. Both processes approximately follow power-laws and thus we see a line-like structure for the planets that saturate while they are in one CZ. Planets that saturate early in the disk evolution do so at a higher mass and farther out than planets that saturate in later times of disk evolution.

Finally, the spread in the outer group results from planets that saturated before they reached the CZ. These are planets in disks with high solid surface densities where the planet cores can grow fast. Compared with the inner group, the outer group also contains more planets that saturate outside of the CZ. The larger amount of solids outside of the iceline leads to higher accretion rates. The scatter is reduced when the saturation mass is increased by reducing fvisc. The higher saturation mass gives the planets more time to migrate to the CZ and to saturate there.

Appendix B: Impact of numerical parameters

We made several population syntheses calculation to test the effects of different numerical parameters and comment on the effects here.

We made calculations with the STD model and exponents b = 2.0 and b = 10.0 (nominal value b = 4.0) in the transition function between locally isothermal and adiabatic migration regime (Eq. (20)). This only has a small effect for low-mass planets, while for massive planets the final semimajor axis and mass is almost the same for different values of b. At smaller masses no clear pattern can be seen. An increase of b can lead to either more or less massive planets and to either larger or smaller distances from the star of a few per cent.

We also made calculations with b = 4.0 and a hard jump for the transition function between type I and type II migration (Eq. (23), nominal value b = 10). This only affects massive planets and gives only a change in distance from the star of a few percent, with a larger b leading to planets farther away from the star but almost no change in the final mass.

Overall, the range of the different parameters studied here only leads to minor changes in the overall distribution of planets in semimajor axis and mass.

© ESO, 2014

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