The Deaths of Resonant Planetary Systems

Abstract

White dwarf stars represent the last evolutionary stage of most stars in the Universe. As many of half of these stars are known to have metals in their atmospheres, that must have entered the atmosphere recently. Some also have observable debris discs, hinting that remnant exoplanets provide an important puzzle piece. However, we do not know the full extent of their role, as it is very hard to observe planets around these faint stars, so we do not know what kinds of planets exist. The properties of exoplanets strongly affect how much they can contribute to WD pollution. During stellar evolution, planets may collide with each other, become engulfed by the star, or be ejected from the system. Many theoretical studies focus on seeing what kinds of exoplanets survive their host star's evolution into a WD. The aims of this project are to investigate whether exoplanets in the 2:1 mean motion resonance are more or less likely to survive stellar evolution, as well as to see how the resonance itself is affected, as planets in resonance have not been studied before in this context.

I use the N-body code REBOUND and the related framework REBOUNDx to simulate planets in resonance undergoing stellar evolution. I use three observed resonant planetary systems as templates for the simulated planetary systems, and I use a theoretical model for a 3 Msol star to simulate stellar evolution. I develop a linear interpolation method for REBOUNDx that is used to interpolate the stellar mass and radius throughout the simulations, as the cubic spline method included in REBOUNDx behaves severely inaccurately for my data set, and I update the stellar mass and radius at every integration time step of the simulation runs. The simulations in the set with more massive planets experience instability and planet loss both before and after the WD phase begins, over a range of times. This is in agreement with previous works, showing that more massive planets are more likely to experience instability before, during, and after stellar evolution. Also in agreement with previous works, most of the instabilities occur soon after the start of the WD stage.

The two simulation sets with less massive planets do not experience instability. They all remain in resonance, but the resonant behaviour is changed as a result of stellar evolution. The change is consistent with stellar mass loss occurring rapidly compared to the orbital time scale. We also see that the linearly interpolated stellar parameters affect the statistical properties (especially eccentricity and resonant argument) of post-AGB resonant systems, since piecewise linear functions occasionally have an undefined second derivative, even when parameters are being updated at every integration time step. These properties are also affected by which model for stellar evolution is used, and especially how the mass loss at the end of the AGB is modelled.

The system architectures simulated here could not explain the full amount of observed pollution and other mechanisms are needed. The role of exoplanets in resonance in WD pollution is still not well-understood, and the work done here sets up for future studies to this end.

Popular Abstract

Since the dawn of time, humans have wondered if we are alone in the universe. We still do not know that, but we do know that there are other planets than the ones in our Solar System. It is very hard, as they are so far away from us, but we now know of almost 6000 so-called exoplanets, planets around stars other than the Sun. Most exoplanets are detected from seeing them affect the light of their star.

We have not found many exoplanets around the type of stars called white dwarf stars, or WDs for short. WDs are old stars that used to be similar to the Sun, but have since "died" after going through a dramatic process in which they temporarily grow to hundreds of times bigger than they were, as well as lose much, if not most of their mass. A white dwarf is a very small, but very dense star, only as big as the Earth, but it weighs as much as half the Sun. It is also not very bright, which is one reason it is hard to detect planets around WDs.

Since WD stars are so massive and dense, if anything other than hydrogen or helium enters their atmospheres, it sinks to the bottom fast. Still, we observe that as many as half of all WD atmospheres contain rocky material. This material must therefore have fallen onto the WD recently, or we would not be able to see it - it would have sunk. Our best guess for what is causing the material to fall onto the WD is that something is causing asteroids in orbit around the WD to change course, and that they end up so near the WD that they are ripped apart by its gravity, and the small pieces eventually rain onto the WD.

Exoplanets are able to deflect asteroids toward the WD, thanks to their gravitational force. Scientists have computed how good they are at it, and they find that planets with lower masses are significantly better at deflecting asteroids toward the star, as heavier planets end up throwing the asteroids out of the system instead. Properties like how circular or elliptical the planet orbit is also affect what impact the planet has on asteroids.

Since we are not able to observe many planets around WDs, we do not know if planets with the right mass and other helpful properties are common or not. If they are very common, exoplanets may be the leading cause of material in WD atmospheres. If they are not, then we have to look for other explanations. In order to replace observations, scientists run simulations of star systems where the star evolves, with planets that have various properties, and then they see which planets survive and what properties they have. One such property that has never been tested before is mean motion resonance. Two planets are in a 2:1 (pronounced "two-to-one") mean motion resonance if one planet makes two orbits around the star in the same amount of time as the other one just makes one orbit.

In this project, I run three sets of simulations with planets in the 2:1 resonance, and in all three sets the planetary masses are different, but all sets have the same star, which evolves into a WD during the simulation. The aim is to see how planetary systems in resonance are affected by stellar evolution compared to non-resonant systems. This will then help us better understand how capable planets around WDs are at throwing material toward the WD atmosphere.

• author: Anderholm Hansson, Cas
• supervisor: Alexander Mustill
• organization: Astrophysics; Department of Physics
• course: ASTM32 20251
• year: 2025
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• report number: 2025-EXA250
• language: English
• id: 9212788
• date added to LUP: 2025-09-19 09:50:49
• date last changed: 2025-09-19 09:50:49