Instructor notes: Retrograde Motion: Apparent Longitude from Relative Motion

Scope note: These are future improvements for the retrograde-motion instrument. They are intentionally split into (A) “SoTA UX + pedagogy”, (B) “Model extensions that remain honest”, and (C) “Physics correctness hardening”.

A) SoTA UX + pedagogy (fun + teachable)

1. Time travel controls (past/future)

   • Add a window-start control (allow negative model days) and a “jump to…” input.
   • Add quick buttons: “Go back 10 months”, “Go forward 10 months”, “Center on nearest retrograde”.
   • Guardrail copy: keep “model day” messaging front-and-center (no calendar claims).

2. Story mode overlays (toggleable)

   • Velocity arrows for observer + target at the current time (recommended by design spec).
   • A small “sweep direction” indicator for the sign of $d\tilde{\lambda}/dt$ at the current time.
   • A “line-of-sight history” overlay (fading) for the last $N$ model days.

3. Plot UX polish

   • Add axis ticks/labels: $t$ (day) and $\tilde{\lambda}_{\mathrm{app}}$ (deg).
   • Add a “zoom to retrograde interval” button (sets window to [start-$\Delta$, end+$\Delta$]).
   • Add “hover/focus tooltip” at cursor: $t$, $\lambda_{\mathrm{app}}$, $d\tilde{\lambda}/dt$ (must work on keyboard focus, not hover-only).

4. Make it playful

   • “Compare two targets” mode: same observer, two targets on the same plot (clearly labeled; no color-only meaning).
   • “Race” mode: animate time at a user-controlled rate (must respect reduced motion).
   • “Challenge prompts” (via runtime hooks): find stationary points, estimate retrograde duration, compare inferior vs superior cases.

B) Versatility (without lying)

1. Configurable planet set

   • Add Mercury (interior extreme case) and optionally Uranus/Neptune (outer slow case) with explicit “teaching model” notes.
   • Allow “custom planet” elements entry $(a,e,\varpi,L_0)$ (with validation) for sandboxed exploration.

2. Better presets

   • Add presets for:
     • Earth $\to$ Jupiter (slow retrograde)
     • Earth $\to$ Saturn (slow retrograde)
     • Mars observer $\to$ Jupiter target (advanced)
   • Each preset should include a “what to notice” sentence.

3. Export richness (still stable)

   • Add an optional “Copy CSV (series)” action with downsampled rows: $t$, $\lambda_{\mathrm{app}}$, $\tilde{\lambda}$, $d\tilde{\lambda}/dt$, state.
   • Keep “Copy results” as the stable v1 summary payload.

C) Physics / correctness hardening (SoTA honesty)

1. Canonical element source + citations

   • Move planet elements into a dedicated dataset file in packages/physics (single source of truth) with citations (e.g. J2000 mean elements).
   • Add a “dataset provenance” note in model notes (still no calendar-date claims).

2. Numerical stability for far past/far future

   • Ensure all angle computations avoid catastrophic precision for large $t$:
     • compute $M(t)$ using modular arithmetic (track integer turns separately) so trig inputs stay bounded.
   • Add a regression test that computeSeries works for windows centered at large |t| (e.g. $t=\pm 10^6$ day) without NaNs.

3. Sharper event detection

   • Add tests that stationary refinement meets the spec tolerance (bracket width $<10^{-3}$ day) across multiple planet pairs.
   • Add “edge-case” tests: windows that start/end inside retrograde, very short windows, and targets with small eccentricity.

4. Model extensions (future scope, but physically meaningful)

   • Optional inclination + projection to ecliptic longitude (3D-to-2D) with careful pedagogy; keep the default coplanar mode for clarity.
   • Optional light-time correction (advanced; likely overkill for ASTR101, but could be a “math mode” extension later).