ASTR 474 Problem Set 03: N-Body Conservation and Close Encounters

Objectives and instructions

Practice computational reasoning, quantitative analysis, validation, and scientific communication for lecture 04.

Use SI or explicitly stated astronomical units consistently; show conversions and significant figures.
Cite all data, software, and adapted code. Distinguish supplied measured data from synthetic records.
Include runnable code and record environment, command, and seed where used.
Collaboration on concepts is permitted; each submitted derivation, code, and interpretation must be the student's own and contributors must be named.

Revisions and resubmissions are encouraged to improve mastery and grade under the course policy.

Problem 3.1

From the three-body conservation table, calculate the ratio of energy errors between the coarsest and finest timesteps. Does angular momentum show the same trend? Explain with the fixed-step symplectic method.

Deliver: derivation or code, units/conventions, a reproducible result, uncertainty or validation evidence, and a short physical interpretation.

Problem 3.2

Write the equations for the three-body accelerations and identify the smallest dynamical timescale that would control a step-size choice in a close encounter.

Deliver: derivation or code, units/conventions, a reproducible result, uncertainty or validation evidence, and a short physical interpretation.

Problem 3.3

Given a close-encounter interval shorter than the chosen WHFast step, recommend an alternative REBOUND integrator and define two validation criteria before trusting the result.

Deliver: derivation or code, units/conventions, a reproducible result, uncertainty or validation evidence, and a short physical interpretation.

Reproducibility check

Use this generated reference result as one consistency check, not as a substitute for showing reasoning: Across 12 years, the three-body WHFast relative energy error falls from 8.56e-12 at dt=5.707 d to 5.33e-13 at dt=1.427 d; angular-momentum and scaled momentum errors are tabulated separately.

No textbook exercise is adopted for this course; problems are original and cite the primary software/data sources in the reference log.

Submission

Submit a readable PDF report, source code, environment/version record, and any generated tables/figures. The assignment is complete only when another student can rerun the command and identify the inputs and validation test.