02 / Mathematics

My mathematical interests.

ASTRONOMY / CURRENT RESEARCH

Orbital
resonance

In this astronomy project, I use N-body simulations to study how Jupiter changes comet-like orbits.

In a 3:2 pattern, the comet completes three orbits while Jupiter completes two. I investigate how objects near this pattern can later enter 2:3 resonance.

Research question

How often do comet-like orbits move from 3:2 to 2:3 resonance with Jupiter?

An orbital sketch.

Method

Following a population of orbits

I run the gravitational simulations in REBOUND and analyse the saved orbital histories. JPL Horizons provides the planetary starting states, and I use REBOUND’s TRACE integrator for the simulation.

H1 and H2 are artificial near-3:2 starting orbits designed as stable and unstable simulation cases, respectively. The latest completed experiment uses the supplied H2 starting orbit. The ensemble contains one nominal orbit and 447 perturbed orbits. These perturbations explore nearby initial conditions.

The model includes the Sun and eight planetary system barycentres. It uses a 0.01-year integration step. The detector checks the motion every 0.5 years.

448Particles in the H2 ensemble

1 million yearsSimulated duration

14Parallel workers

Identifying resonance

Reading the orbital rhythm

I first screen orbital-period ratios, and I then examine a resonant angle over time. Repeated back-and-forth motion of this angle is called libration. Motion through a full turn is called circulation.

The current screen requires at least two oscillation cycles and 100 years of evidence. It also applies a 5% period-ratio band and a 300-degree angle-span limit. I save the angle histories to examine what happens around each recorded episode.

Latest completed run / 29 October 2026

The H2 million-year experiment

The detector recorded 117 candidate 3:2 episodes and 229 candidate 2:3 episodes. These episodes involved 171 distinct particles.

Retained at the endpoint: 111
Crossed the 1,000-AU removal boundary: 307
Collisions: 30

Four panels showing H2 retention, candidate residence durations, early circulation of the nominal angle, and the ordered records of clone 120. — The H2 ensemble retains 111 of 448 particles at one million years. The lower panels show initial circulation and one ordered candidate sequence.

Nineteen particles had a 3:2 record followed by a later 2:3 record. This is 4.24% of the starting H2 ensemble. The shortest gap between such records was 10,622.5 years.

The nominal angle circulates within 89 years, and 446 of 448 particles reach full circulation within 500 years. The 19 sequences describe ordered candidate records in this H2 sample. Estimating a transition probability for initially trapped objects will require a cohort with verified initial libration.

Saved angle histories

Examining individual episodes

Clone 389 has a recorded 3:2 episode lasting 4,213 years, and clone 19 has a recorded 2:3 episode lasting 18,454.5 years. Their saved angles show repeated oscillations over these intervals.

Six saved unwrapped angle histories for candidate 3:2 and 2:3 episodes in H2. Shading marks recorded intervals and dashed lines mark first confirmation. — Shaded regions mark recorded 3:2 and 2:3 episodes. The saved angles show repeated oscillations during these intervals.

Detector sensitivity

Checking the record boundaries

Clone 120 has two 2:3 records separated by 106 years. Its saved angle stays within a 255.01-degree range across the whole interval. The period ratio briefly crosses the detector’s band, and I am examining how these thresholds affect episode counts and durations.

Saved angle histories for clones 120 and 415 showing a ratio-gate gap and a 300-degree threshold crossing while bounded oscillations continue. — Detector boundaries can split or end records while the saved angle remains bounded.

Earlier validation / 25 October 2026

Checking a control orbit

An earlier control run uses an artificial Sun–Jupiter orbit near 3:2 resonance. Its detector remains active at 1,000 years, and its recent full-cycle estimates range from 260 to 283 years.

This control uses the earlier recent-cycle detector. It provides a reference for interpreting the angle plots.

An earlier artificial Sun–Jupiter 3:2 control showing wrapped critical angles, the period ratio, and local libration-period estimates over 1000 years. — An earlier artificial Sun–Jupiter control remains classified in 3:2 resonance at 1,000 years.

Next experiments

Testing the starting conditions

The next experiments will compare H1 and H2 with matched settings. These cases differ in their initial orbital phase. I will check initial libration over several cycles, and I will repeat selected trajectories with smaller integration steps.

Impact

Why this research matters

Jupiter can redirect small bodies across the Solar System. This project studies how orbital patterns change during scattering, and it connects individual trajectories to population statistics.

The results help identify which starting conditions and detector choices deserve closer study.

REBOUND documentation JPL Horizons

Earlier work / Yau Math Camp 2025

Knot colorability
and knot groups

I wrote a paper during Yau Math Camp. The paper focuses on knot colorability. It studies a surjective homomorphism from the knot group to the dihedral group D_2n.

I submitted the paper to the Yau Science Awards, and I also gave a talk titled Colorability and Knot Groups.

Study and competitions

Selected milestones

AIME: 13February 2026
SUMaC: Program ISelected for 2026
RP2: Ross online program

All pursuits Continue exploringLiterature

Orbitalresonance