Capture into mean motion resonance

Abstract

Orbital or mean motion resonance (MMR) occurs when the orbital period ratio between two planets is close to a ratio of small integers. MMR can protect planets from collisions and affects the overall final architecture of the system. Observations of exoplanets have shown that the survival rate of MMR-chains is low, and most systems are near and just wide of an exact MMR. The most common chains are the 2:1 and 3:2 first order mean motion resonances. Theoretical studies have shown that it is easy to capture planets into MMR when considering convergent migration, a natural consequence of planet-disk interactions that leads to MMR capture. However, there are discrepancies when it comes to the survival rate. The first step is to better understand capture into MMR. In this project, we aim to investigate which and how, orbital initial conditions and migration parameters affect capture into MMR. Having a better view of the full parameter space and its effects, one can better constrain theoretical models. Which together with the already known aspects of resonance instabilities could be enough to correctly reproduce observations in the future.

In order to study each parameter, I conducted N-body integrations with Rebound on a planar three body system, with varied initial conditions of orbital, planetary and migration parameters. Firstly, in a controlled setting of constant damping timescales, then using a realistic prescription of planetary migration, Type I.

During this project I have seen that when varying both planetary mass and damping timescale of the semi-major axis, there is a dependency on mass for capture into MMR. These results confirmed that smaller planetary mass usually results in MMR with smaller separation. Moreover, the damping timescale can alter this; longer damping timescale leads to capture into the first encountered MMR. Whereas planets with short damping timescale cross multiple MMRs before settling into a tighter spaced MMR or becoming unstable. For the Type I migration setting, I find that capture into MMR has no dependency in planetary mass for larger planetary mass and initial surface density. For the largest masses there is no capture into MMR. However, capture into MMR depends on the initial surface density; larger values correspond to faster migration rate and capture into MMR with smaller separation, and more systems in the chaotic zone of overlapping resonances.

Furthermore, the numerical results agree with the previously derived analytical prescription, which determines if a capture is possible. The analytical criterion gives a slightly steeper dependency on planetary mass, than what numerical results here show for both constant and Type I migration.

Popular Abstract

Astronomy, the study of the most fascinating aspects of our existence that goes back thousands of years, has constantly provided clues to the big questions such as where everything started from and our place in the universe. Over time Astronomy has evolved tremendously through countless contributions from different scientists. For example Tycho Brahe, Galileo Galilei and Johanes Kepler were the first ones to conduct and document scientific and detailed observations of the solar system. The latter formulated the famous three laws of planetary motion, paving the path for Newton's gravitational theory. Just to name a few pioniers of dynamics, a branch of astronomy that is important whenever one wants to investigate the evolution of a system whether it is planets, as studied in this project, or any other astronomical object.

Today we know that our solar system, however special, is only one of many planetary systems. Statistical studies of observations have shown that every Sun-like star in our galaxy is likely to host at least one planet of different compositions and sizes, some with perhaps good conditions for life. Planets are formed in a rather flat disk of leftover material from the formation of the star. Through different processes this material is assembled into planets, and the main force involved in further evolution is gravity. It is gravitational attraction from the star that bounds the planets in the elliptical orbits that they move along, also known as Keplerian motion. A period is the time it takes to complete one turn around the star on such an orbit. Beside the star, a planet feels the gravity from the other planets in the system. Moreover, it is gravitational interactions between a planet and the gas in the disk that make a planet migrate. A natural consequence of such interactions and vital for the phenomena of mean motion resonance that this project investigates.

Mean motion resonance (MMR) occurs when the orbital period ratio of two planets is a ratio of small integers. For example, the 2:1 MMR means the planet closer to the star finishes two periods while the planet further away finishes one. Which always moves around the star at a slower rate, because the gravitational attraction from the star weakens with distance.

Since planetary migration is a natural consequence of planet-disk interactions, theory leads us to believe that it is easy to capture planets into MMR. Theoretical studies produce higher percentage of systems in MMR then observations do. Out of the many planets detected around other stars, only roughly 5\% are in MMR. This discrepency and the low survival rate is yet to be fully understood. This project will attempt to shed some light on the matter, by looking into what can affect MMR capture using numerical simulations of orbital evolution. For example, using constant evolution timescales we see that smaller mass planets are captured into MMR with smaller separation. Such a configuration can be less stable and may not survive. When using Type I, a realistic migration prescription, I show that for high initial surface density of the disk gas migration is fast, and the planets are again captured into a MMR with smaller separation. Although, capture into MMR does in this case not depend on planetary mass when the mass is large, contradicting the previously derived analytical prescription used to determine when capture occurs.

• author: Kajtazi, Kaltrina
• supervisor: Antoine Petit
• organization: Lund Observatory - Undergoing reorganization
• course: ASTM31 20212
• year: 2021
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• keywords: MMR, Mean motion resonance, Planetary dynamics, First order MMR, Numerical integration with N-body
• publication/series: Lund Observatory Examensarbeten
• report number: 2021-EXA186
• language: English
• id: 9067076
• date added to LUP: 2021-10-20 09:28:37
• date last changed: 2021-10-20 09:28:37