p1: interactions near the hill sphere

The Hill sphere is the region around a body where its gravity dominates over the larger one it orbits.

In this project, we will investigate what happens if a test particle approaches the Hill sphere.


An Example of the Simulation Setup

  1. Intialize the Sun and the Earth on a circular orbit at
  2. Initialize a test particle on a circular orbit at
  3. Run the simulation until the Hill sphere encounter, explore this 3-body interaction as function of and time .
(c) Tian Yi

p1: interactions near the hill sphere

The Hill sphere is the region around a body where its gravity dominates over the larger one it orbits.

In this project, we will investigate what happens if a test particle approaches the Hill sphere.


An Example of the Simulation Setup

  1. Intialize the Sun and the Earth on a circular orbit at
  2. Initialize a test particle on a circular orbit at
  3. Run the simulation until the Hill sphere encounter, explore this 3-body interaction as function of and time .
(c) Chris Ormel

p2: disintegrating exoplanet

Planets that come to close to the star may vaporize their refractory elements. The vapor can fill the Roche lobe and escape the system through L1 or L2. This is because the particles are subject to a radiative force away from the star, with the equation of motion as:


The Goal of This Project

  • Investigate the fate of particles escaping through L1 and L2.
  • Treat as a fixed parameter, consider how diiferent would change the final orbit of the planets.
(c) NASA

p3&p4: settling and particles orbit decay


Settling

Particles will settle to the midplane in a disk due to the component of the star gravity.

Gas drag opposes the vertical acceleration with the equation of motion of:

In this project, investigate how settling behavior changes with or .

vp,φ
Fdrag
orbital decay
(c) Tian Yi

p3&p4: settling and particles orbit decay


Particles Orbital Decay

  • The gas drag also opposes the Keplerian metion and leads to a orbital decay with:
  • In this project, you could first investigate the orbital decay process with a fixed or .
  • Next, you could switch to a fixed particle size and allow to depend on position.
vp,φ
Fdrag
orbital decay
(c) Tian Yi

p5: the earth-moon problem

The Moon originally was much closer to Earth and then migrated outwards due to tidal effects. In this project, you should:

  • include the tidal interactions between the Earth and Moon,
  • analyze the rate of orbital expansion and the corresponding spin-down.


Parameters Needed

  • Love number, moment of inertia, and spin vector for the Earth and the Moon.
  • Tidal parameter for the Earth and the Moon.
(c) Answers in Genesis

p5: the earth-moon problem

The Moon originally was much closer to Earth and then migrated outwards due to tidal effects. In this project, you should:

  • include the tidal interactions between the Earth and Moon,
  • analyze the rate of orbital expansion and the corresponding spin-down.


Note

  • You need to use rebound and reboundx for this project. Note that reboundx is not supported in Windows.
  • There is a reboundx tutorial about this. You can build your project on this example.
(c) Answers in Genesis

p6: pebble accretion or escape

Pebble refers to millimeter- to centimeter-sized solid particles in protoplanetary disks.

The pebble drift inwards due to radial drift. Therefore, it will either be accreted by the protoplanet or pass by it.


Simulation Setup

  • Initialze a star and a planet on a circular orbit.
  • Intialize pebbles with fixed .
  • Add the drag force and the gravitational drag, investigate the fraction of particles that will be accreted.
(c) Ormel & Klahr (2010); Lambrechts & Johansen (2012); Ormel (2017)

p6: pebble accretion or escape

Pebble refers to millimeter- to centimeter-sized solid particles in protoplanetary disks.

The pebble drift inwards due to radial drift. Therefore, it will either be accreted by the protoplanet or pass by it.


Simulation Setup

  • Initialze a star and a planet on a circular orbit.
  • Intialize pebbles with fixed .
  • Add the drag force and the gravitational drag, investigate the fraction of particles that will be accreted.
(c) Liu & Ormel (2018)

p7: Orbits in the Alpha Centauri system

The Centauri system is the closest star system to Earth which inspired the setting of Liu Cixin's The Three-Body Problem.

In the system, Cen A and Cen B are a binary on an eccentric orbit, while Cen C (Proxima Centauri) orbits the binary at a much remote distance of ∼10,000 au.


The Goal of This Project

  • Replicate the dynamical configuration of the system.
  • Insert test particles and investigate stable orbits.
  • Explore the suitability of life for these hypothetical planets.
(c) universemagazine.com
(c) Comicpop Library

p8: Resonance trapping

Planets in disk would experience disk damping force which damps the semi-major axis and the eccentricity of the planet:

Such force would lead to (inward) migration of the planets. If there are multiple planets in the system, resonance may form.


The Goal of This Project

  • Intialize a inner Jupiter and an outer Earth
  • Let the outer planet migration under disk damping force. Investigate the resonance trapping criteria.
(c) Tian Yi

p9: Kozai-Lidov oscillations

the Kozai-Lidov effect describes the phenomenon that the orbit of a binary system perturbed by a distant third body.

The distant third body leads to an oscillation between its eccentricity and inclination:


Example Simulation Setup

  • Intialize a binary and a test particle orbiting one of the star on a circular orbit with an inclination.
  • Reproduce Kozai-lidov oscillations of the test particles.
(c) www.planetary.org
(c) Naoz (2016)

p10: Orbit crossing

Orbit crossing refer to the status where the apoapsis and periapsis of neighboring body start to become equal.

When protoplanets experience orbit crossing, there orbit will become dynamically unstable, potentially leading to planet merging or other violent events.


Example Simulation Setup

  • Consider at least 10 Mars-sized protoplanets around 1 au at a close mutual distance.
  • Integrate the system in time until the point of the first orbit crossing.
  • You could use rebound with whfast algorithm to speed up your integration.
(c) Chamber & Wetherill (1998)

p11&p12: Strring and Scattering of Planetesmials

Planetesimals refer to kilometer-sized rocky or icy bodies, serving as building blocks for planets.


Stirring

  • Place the planetesimals initially on near-circular orbits with random spacing.
  • Plot the rms-values of the eccentricity and inclination as function of time.


Scattering

  • Make one of the planetesimals 1000x more massive and place it at the center of the belt, so it scatters other planetesimals.
(c) Carter & Steward (2020)

p13: The solar system back in time

We understand the current configuration of the Solar System in detail. Therefore, a reliable approach to studying its history involves backward integration of planetary orbits through numerical simulations.


Possible Investigation Point

  • Plot eccentricities and inclination of the planets evolve during the Solar System history.
  • Identify Milankovitch cycles, which refers to climate patterns over long period due to the variation in the oribit of the Earth.
eccentricity
time
(c) Ito & Tanikawa 2002