Exoplanet Population Synthesis with Account for Orbit Variation Due to Stellar Evolution

Abstract

In this paper, the orbital evolution of exoplanets at the late stages of stellar evolution was studied using the population synthesis method. Stellar evolution was traced from the main sequence stage to the white dwarf stage. The evolutionary tracks were calculated using the MESA package. The statistics of planets that were absorbed, ejected from the system, and surviving by the time of the transformation of parent stars into white dwarfs was calculated considering the variation in the star formation rate over the entire lifetime of the Galaxy. We considered planets around stars in the range of initial masses (1–8)${{M}_{ \odot }}$ since less massive stars do not leave the main sequence over the lifetime of the Galaxy, and more massive stars do not lead to the formation of white dwarfs. It was found that for the adopted initial distributions of planets on the a–M_pl plane, the majority (about 60%) of planets born around stars in the mass range under study are absorbed by parent stars at the giant stage. A small fraction of planets (less than a percent) are ejected out of their systems under the action of the stream of matter escaping the star. The estimated number of “runaway” planets with masses from 0.04 Earth masses to 13 Jupiter masses in the Galaxy is approximately in the 300 million range.

APPENDIX

1.1 EXAMPLES OF EVOLUTIONARY TRACKS

This section provides examples of the evolutionary tracks obtained in MESA and used in the study.

Figure 15 shows the effective temperature versus luminosity diagram for most of the tracks used in the simulation. The evolution from the MS stage to the helium burnout in the center of the star is illustrated.

Fig. 15.

Hertzsprung–Russell diagram for most of the tracks from the MS to the point of helium burnout in the center of the star.

Figure 16 shows the mass loss rates used in the modeling of the evolutionary tracks.

Fig. 16.

Post-MS mass loss rates of the evolutionary tracks used in the modeling. The dots correspond to the truncated tracks used in the modeling.

Figure 17 illustrates the evolution of the mass loss rate for a star with an initial mass of 2.3 ${{M}_{ \odot }}$; the points plotted on the curve correspond to the truncated tracks described in the study. Each two adjacent points confine a segment corresponding to the evolutionary stage at which the mass loss rate is calculated as the arithmetic mean of the loss rate values at these two points, i.e., at the beginning and at the end of the stage.

Fig. 17.

Fragment of the evolutionary track of a star with an initial mass of 2.3 ${{M}_{ \odot }}$ at the TPAGB stage. The dots correspond to the truncated tracks used in the modeling.

Figure 18 shows in detail the variation in the mass loss rate in the track for a star with an initial mass of 6 ${{M}_{ \odot }}$. Along with the model of a star with an initial mass of 7.5 ${{M}_{ \odot }}$, this model demonstrated the maximum local value of the loss rate among all tracks, approximately equal to 10^–2.2 ${{M}_{ \odot }}$/year.

Fig. 18.

Detailed variation in the mass loss rate for a model with an initial mass of 6 ${{M}_{ \odot }}$. In all three fragments, the ordinate corresponds to the mass loss rate of the star, and the abscissa corresponds to its age. The interval between each pair of points corresponds to one of the evolutionary stages of the truncated track.

Figure 19 shows the evolution of luminosity in the model with an initial mass of 6 ${{M}_{ \odot }}$. As indicated in Section 2, the TPAGB stage in the MESA models of massive stars turned out to be represented by a single increase in luminosity, and not by a series of thermal flares.

Fig. 19.

Luminosity evolution in a model with an initial mass of 6 ${{M}_{ \odot }}$ for a part of the track.

Figure 20 illustrates the evolution of the masses used in the track simulations.

Fig. 20.

Post-MS evolution of the stellar mass for most of the tracks used in the study.

Figures 21 and 22 show the evolution of the stellar radius in the models used in this study. The dots mark those radius values with which the semimajor axes of the planetary orbits are compared in the course of the evolution of the systems.

Fig. 21.

Post-MS evolution of the stellar radius in massive star models.

Fig. 22.

Post-MS evolution of the stellar radii in models with initial masses (right to left) of 1.6, 1.5, 1.4, 1.3, 1.2, and 1.1 ${{M}_{ \odot }}$.