Skip to content

model.md

Latest commit

 

History

History
106 lines (78 loc) · 5 KB

model.md

File metadata and controls

106 lines (78 loc) · 5 KB
Raw

Model - Solar System Models

This document describes the model (the underlying physics, math, and behavior) for the simulation, in terms appropriate for an educator. It is the companion to implementation-notes.md, which targets developers.

Overview

This sim ports the NAAP Solar System Models lab and has two screens that look at the same historical question — "why do planets sometimes appear to move backward against the stars?" — from two different models.

  • Ptolemaic System reconstructs the Earth-centered (geocentric) explanation: each planet rides a small circle (the epicycle) whose center travels around a larger circle (the deferent) centered near, but not exactly on, the Earth. Presets cover Venus, Mars, Jupiter, and Saturn (Mercury has no preset). Students can store/recall parameter sets and watch a zodiac longitude trail.
  • Planetary Configurations shows the Sun-centered (heliocentric) explanation: planet 1 is always the observer and planet 2 is always the target, both on circular orbits. Retrograde motion is the geometric effect of one body periodically overtaking the other. A timeline schedules synodic events with RUN, PAUSE, or STOP actions at each alignment.

Students can compare the two models directly: the same retrograde behavior an observer sees from Earth falls out of very different underlying geometries.

Quantities and units

Quantity Symbol Units Range
Epicycle size (Ptolemaic) R_e deferent radii 0 – 0.75
Eccentricity (Ptolemaic) ecc deferent radii 0 – 0.2
Motion rate (Ptolemaic) — °/day 0.01 – 4.5
Apogee angle (Ptolemaic) — degrees 0 – 360
Elapsed time (Ptolemaic) t days 0 – ∞
Path trail duration — years 0.3 – 10
Animation rate (Ptolemaic) — days per second 1 – 500 (default 100)
Semimajor axis (Configurations) a₁, a₂ AU 0.25 – 10
Orbital period (Configurations) P years derived: a^1.5
Elongation — degrees, signed (− = East, + = West) −180 – 180
Elapsed time (Configurations) t years 0 – ∞
Animation rate (Configurations) — × inner angular speed 0 – 6

Governing equations

Ptolemaic System — deferent, epicycle, and equant

The deferent has a fixed radius (normalized to 1) but is offset from Earth by the eccentricity, along the apogee direction. The equant sits twice as far from Earth as the deferent's center, on the same line. Motion around the deferent is uniform as seen from the equant, not from Earth.

Which angle drives the deferent depends on whether the planet is superior (Mars, Jupiter, Saturn) or inferior (Mercury, Venus):

  • Superior planets: deferent driven by the planet's orbital angle (anomaly); epicycle locked to the Sun's angle.
  • Inferior planets: deferent follows the Sun's angle; epicycle follows the planet's orbital angle.

The Sun moves at fixed rate ≈ 2π/365.25 rad/day on a circle of radius 2.25 deferent units (diagram scale). Ecliptic longitude is atan2(y, x) of geocentric planet position — retrograde appears as non-monotonic longitude in the zodiac strip.

Planetary Configurations — Kepler orbits and synodic events

Both bodies move on circular orbits around the Sun:

period = semimajorAxis^1.5    (AU and years)

Elongation measures the target's apparent position relative to the Sun from the observer's viewpoint.

Synodic period (time between repeats of the same configuration):

T_syn = 1 / (1/P_inner − 1/P_outer)

Event names depend on which orbit is inner (a₁ < a₂ means planet 1 is inner observer):

Inner observer (a₁ < a₂) Outer observer (a₁ > a₂)
1. Opposition 1. Inferior conjunction
2. Eastern quadrature 2. Greatest elongation (W)
3. Conjunction 3. Superior conjunction
4. Western quadrature 4. Greatest elongation (E)

Within each synodic cycle, event times are [0, t_Q, T_syn/2, T_syn − t_Q] where t_Q = acos(a_inner/a_outer) / ω_syn. The schedule also depends on starting epoch angles via a cycle offset — events are not functions of semimajor axes alone.

Playback at events: RUN (pass through), PAUSE (hold + countdown), STOP (hold indefinitely).

Defaults: a₁ = 1 AU, a₂ = 2.4 AU.

Simplifications and assumptions

  • All orbits are circular — deliberate NAAP simplification.
  • Configurations: exactly two bodies + fixed Sun; no mutual perturbations.
  • Ptolemaic distances are diagram-normalized, not astronomical scale.
  • Presets on Configurations set semimajor axes to named planets' AU values.

References

  • NAAP Solar System Models lab: ../Baseline/Astronomy/astroUNL/naap/ssm/modeling.html, ../Baseline/Astronomy/astroUNL/naap/ssm/naap_ssm_sg.pdf (student guide).
  • Original Flash simulators: Ptolemaic System Simulator (ptolemaic023) and Planetary Configurations Simulator (configurationsSimulator044).