Instructor notes: Angular Size: The Sky’s Ruler

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• Instructor hub: /demos/_instructor/
• Student demo: /play/angular-size/
• This demo: Model · Activities · Assessment · Backlog

This guide is instructor-facing Student demo: /play/angular-size/
UI markup: apps/demos/src/demos/angular-size/index.html
Demo logic: apps/demos/src/demos/angular-size/main.ts
Physics model: packages/physics/src/angularSizeModel.ts

Where to go next

• Model + math + assumptions: apps/site/src/content/instructor/angular-size/model.md
• In-class activities: apps/site/src/content/instructor/angular-size/activities.md
• Assessment bank: apps/site/src/content/instructor/angular-size/assessment.md
• Future enhancements: apps/site/src/content/instructor/angular-size/backlog.md

New: Station Mode + Help

• In the student demo, click Station Mode to build a data table you can print or copy as CSV.
• Click Help / Keys (or press ?) for shortcuts; press g to open Station Mode.

Why this demo exists

Why This Matters Angular size is one of the most transferable “astronomy math” ideas: what you see depends on physical size and distance. This single relationship explains why the Moon and Sun look similar in size, why planets look like points to the naked eye, and why telescopes have a resolution limit.

Learning goals (ASTR 101)

Students should be able to:

• Predict how angular size changes when distance or physical size changes (inverse proportional reasoning).
• Read and compare angular sizes in degrees, arcminutes, and arcseconds.
• Explain the coincidence: Sun and Moon have similar angular diameters, enabling total solar eclipses (with the eclipse demo as the follow-up).

10–15 minute live-teach script (projector)

1. Start with Sun preset. Ask: “What does $0.5^\circ$ look like?” Connect to the “thumb at arm’s length” reference if you use it.

2. Switch to Moon (Today). Ask: “Does the Moon’s angular size feel ‘about the same’ as the Sun?” Point out the similarity.

3. Use Moon time mode: Orbit. Slide through orbit angle and show the angular size range changing slightly (perigee vs apogee).

4. Switch to Moon time mode: Recession time. Ask: “If the Moon slowly moves away, what happens to its angular size?” Tie to the eclipse consequence: total eclipses cannot last forever.

5. Switch to everyday objects (quarter/thumb/ISS). Ask: “Which unit makes the most sense here: degrees, arcminutes, or arcseconds?”

Suggested connections to the other demos

• Eclipse geometry: total solar eclipses require alignment and comparable angular sizes of Sun and Moon.
• Moon phases: phases change over the synodic month; angular size does not define phase, but it matters for eclipse appearance.
• Seasons: a cautionary parallel — “what we see” can mislead unless we connect it to a model.

Navigation

• Instructor hub: /demos/_instructor/
• Back to this demo guide: Guide
• Student demo: /play/angular-size/
• This demo: Model · Activities · Assessment · Backlog

Links Student demo: /play/angular-size/
Main guide: apps/site/src/content/instructor/angular-size/index.md
Model deep dive: apps/site/src/content/instructor/angular-size/model.md

MW Quick Exploration (3–5 min, pairs)

TPS: The Sun–Moon coincidence Think (30 s): “Which looks larger in the sky: the Sun or the Moon?”

Pair (60 s): Decide whether you expect their angular sizes to differ by a lot, a little, or be about the same.

Share (1–2 min): Use the demo:

1. Select Sun preset → read the angular size.
2. Select Moon (Today) preset → read the angular size.
3. Ask: “What does this imply about total solar eclipses?”

Debrief script: “It’s a coincidence: the Sun is about $400\times$ larger but also about $400\times$ farther, so the ratios nearly cancel.”

MW Short Investigation (8–12 min, pairs/triads)

Investigation: Scaling without a calculator Task: Pick one object (e.g., the Moon preset). Then do three controlled changes and predict first:

1. Double the distance $d$ (keep diameter fixed).
2. Halve the distance.
3. Double the diameter $D$ (keep distance fixed).

For each, record the angular size and write one proportional-reasoning sentence like: “When $d$ doubles, $\theta$ roughly halves.”

Expected pattern: $\theta \propto D/d$ for small angles.

Friday Astro Lab (20–30+ min, groups of 3–4)

Astro Lab: When are total solar eclipses possible? Deliverable: A short written claim with evidence + a “future Moon” graph.

Tip: Use Station Mode to build a data table you can print or copy as CSV.

Part 1 (evidence table): Record angular sizes for:

• Sun
• Moon (today)
• Moon at perigee-like and apogee-like settings (orbit mode)

Part 2 (recession): Switch to Moon “recession time” mode. Choose two future times (e.g., +500 Myr and +1000 Myr). Record Moon angular size at each.

Prompt: “At what point does the Moon become too small to ever fully cover the Sun?”

Reasoning cue: totality requires $\theta_{\text{Moon}} \gtrsim \theta_{\text{Sun}}$ (plus alignment from the eclipse demo).

ASTR 201 extension (optional): Compare the exact formula $\theta = 2\arctan(D/2d)$ to the small-angle approximation $\theta\approx D/d$ and estimate the fractional error for the Moon.

Station version (for the Cosmic Playground capstone rotation)

Station card: Angular Size (6–8 minutes) Demo setup: Compare Sun and Moon (Today); then toggle Moon orbit mode.
Tip: Click Station Mode to add rows and print/copy your table.

Your station artifact (fill in):

1. Control(s): diameter $D$, distance $d$

2. Observable(s): angular diameter $\theta$ (deg/arcmin/arcsec)

3. Governing relationship: write this equation in words:

   $$\theta = 2\arctan!\left(\frac{D}{2d}\right)$$

4. Sanity check: what happens to $\theta$ if $d$ doubles?

5. Connection sentence: “This matters for eclipses because…”

Word bank + sanity checks Word bank:

• Angular size $\theta$ (degrees/arcmin/arcsec): how big an object looks on the sky (an angle).
• Physical diameter $D$ (km in this demo): the object’s actual size.
• Distance $d$ (km in this demo): how far the object is from the observer.
• Small-angle idea: larger $D$ → larger $\theta$; larger $d$ → smaller $\theta$.
• Unit ladder: $1^\circ = 60,\mathrm{arcmin}$ and $1,\mathrm{arcmin} = 60,\mathrm{arcsec}$.

Sanity checks:

• If $d$ doubles, $\theta$ should get about half as big (for small angles).
• The Sun and Moon have similar angular sizes today, which is why total solar eclipses are possible sometimes.
• Perigee vs apogee: the Moon’s angular size is slightly larger at perigee than at apogee.

Navigation

• Instructor hub: /demos/_instructor/
• Back to this demo guide: Guide
• Student demo: /play/angular-size/
• This demo: Model · Activities · Assessment · Backlog

How to use this bank Each item includes a demo setup so you can reproduce the scenario live, plus distractors tied to common misconceptions.

Clicker questions

Clicker 1: Distance scaling Prompt: If an object moves twice as far away but stays the same physical size, its angular size will…

A. double
B. halve
C. stay the same
D. become zero instantly

Correct: B.

Reasoning: For small angles, $\theta \propto D/d$.

Demo setup: Pick any preset, then double the distance slider and watch the angular size update.

Clicker 2: Pick the right unit Prompt: Which unit is usually most appropriate for the angular size of the full Moon?

A. arcseconds
B. arcminutes
C. degrees
D. radians

Correct: C (about half a degree).

Demo setup: Select Moon (Today) and read the unit that the demo displays.

Clicker 3: Why total eclipses are possible Prompt: Why can the Moon sometimes completely cover the Sun during a solar eclipse?

A. The Moon is physically larger than the Sun
B. The Moon is closer, so it can have a similar angular size to the Sun
C. The Sun turns off during eclipses
D. Earth’s shadow blocks the Sun

Correct: B.

Demo setup: Compare Sun and Moon (Today) presets and show their angular sizes are similar.

Short answer

Short answer 1: Read the equation like a sentence Prompt (2–3 sentences): The demo uses $\theta = 2\arctan(D/2d)$. Explain what this equation is saying in words, including what happens when you increase $D$ or increase $d$.

Answer key (core idea): Angular size depends on the ratio of physical diameter to distance. Increasing $D$ increases $\theta$; increasing $d$ decreases $\theta$. For small angles, it behaves like $\theta \approx D/d$ (in radians).

Short answer 2: Quick mental math (small-angle) Prompt: A 2 m tall person stands 20 m away. Use $\theta \approx D/d$ to estimate the angular size in radians and degrees.

Answer key: $\theta \approx 2/20 = 0.1$ rad. In degrees, $0.1\times(180/\pi)\approx 5.7^\circ$.

Sanity check: At 20 m, a person looks “several degrees” tall — consistent with everyday experience.

Exit ticket (1 minute)

Exit ticket: Totality condition (conceptual) Prompt: Write the condition for a total solar eclipse in angular-size language (no calculations required).

Expected: The Moon’s angular diameter must be at least as large as the Sun’s angular diameter ($\theta_{\text{Moon}} \gtrsim \theta_{\text{Sun}}$), and the alignment must be right (from the eclipse geometry demo).

Navigation

• Instructor hub: /demos/_instructor/
• Back to this demo guide: Guide
• Student demo: /play/angular-size/
• This demo: Model · Activities · Assessment · Backlog

Links Student demo: /play/angular-size/
Model code: packages/physics/src/angularSizeModel.ts
UI/visualization code: apps/demos/src/demos/angular-size/main.ts

What the demo is modeling (big picture)

This demo models angular diameter (how large something appears on the sky) from first principles: physical size and distance.

It’s intentionally multi-scale:

• Nearby objects (cm–km; meters to km away)
• Solar system (km objects; $10^5$–$10^9$ km away)
• Galaxies (many light-years away)

The demo reports the angular diameter in a human-friendly unit:

• degrees ($^\circ$), arcminutes ($\mathrm{arcmin}$), or arcseconds ($\mathrm{arcsec}$)

Angular diameter (exact)

The demo uses the exact geometric relationship:

$$\theta = 2\arctan!\left(\frac{D}{2d}\right)$$

Let’s unpack each piece:

• $\theta$ is angular diameter (radians in the math; converted to degrees for display).
• $D$ is physical diameter (km in the demo’s internal units).
• $d$ is distance to the object (km in the demo’s internal units).

What this equation is really saying: angular size depends on a ratio $D/d$. If you double the distance, the object looks about half as big.

Dimensional analysis Inside the arctangent, $\frac{D}{2d}$ is a ratio of km/km, so it’s unitless (as required for trig functions).

Inverting the equation (distance from a desired angular size)

Sometimes the teaching question is the inverse one: “If an object has diameter $D$, how far away would it need to be to appear $\theta$ wide?”

Solving the exact equation for $d$ gives:

$$d = \frac{D}{2\tan(\theta/2)}$$

This inversion is implemented in the shared model as:

• AngularSizeModel.distanceForAngularDiameterDeg({ diameterKm, angularDiameterDeg })

The demo uses this idea when it sets “perigee-like” and “apogee-like” endpoints for the Moon’s orbit-variation mode from a chosen angular-size range.

Small-angle approximation (useful for reasoning)

For small angles (most astronomy cases), $\tan x \approx x$ (when $x$ is in radians), so:

$$\theta \approx \frac{D}{d}\quad(\text{radians})$$

This is the mental math version students should take away: angular size scales as $D/d$.

Angle unit conversions (for classroom fluency)

$$1^\circ = 60,\mathrm{arcmin} \qquad\text{and}\qquad 1,\mathrm{arcmin} = 60,\mathrm{arcsec}$$

So:

$$1^\circ = 3600,\mathrm{arcsec}$$

These conversions are what let students compare “planets (arcseconds) vs Moon (degrees).”

Moon special modes (orbit vs recession time)

Mode 1: Orbit variation (perigee ↔ apogee)

The demo includes an orbit-mode control that varies the Moon’s distance between two endpoints chosen to match the course “sanity range”:

• angular diameter $\approx 0.49^\circ$ (small) to $\approx 0.56^\circ$ (large)

Those endpoints imply distances (computed by inverting the exact formula above):

Numbers implied by the demo’s orbit range Using $D_{\text{Moon}} = 3474\ \text{km}$:

• $0.56^\circ \Rightarrow d \approx 355{,}436\ \text{km}$ (perigee-like)
• $0.49^\circ \Rightarrow d \approx 406{,}213\ \text{km}$ (apogee-like)

These are reasonable order-of-magnitude values for the real Moon.

Mode 2: Recession time (toy linear model)

The demo’s recession-time mode uses a deliberately simple linear model:

$$d(t) = d_0 + vt$$

Let’s unpack each piece:

• $d(t)$ is Moon distance (km)
• $d_0$ is today’s distance (km)
• $v$ is the recession rate (km/Myr in the demo)
• $t$ is time from today (Myr)

The code starts from a commonly quoted present-day mean recession rate:

• $v \approx 3.8\ \text{cm/yr}$

and converts it into km per million years:

$$1\ \text{cm/yr} = 10\ \text{km/Myr} ;;\Rightarrow;; 3.8\ \text{cm/yr} \approx 38\ \text{km/Myr}$$

Why this is a ‘toy’ model The real Earth–Moon recession rate varies with time (tidal dissipation depends on ocean basins, etc.). The demo uses a linear model because the teaching goal is scaling: “farther away → smaller angular size.”

Connection to eclipses (total vs annular)

The geometric reason total and annular solar eclipses both exist is an angular-size comparison.

• If the Moon’s angular diameter is larger than the Sun’s ($\theta_{\text{Moon}} > \theta_\odot$), then a well-aligned “central” eclipse can be total somewhere on Earth (the Moon’s umbra can reach Earth).
• If the Moon’s angular diameter is smaller than the Sun’s ($\theta_{\text{Moon}} < \theta_\odot$), then a well-aligned “central” eclipse is annular (the umbra ends before Earth; the antumbra reaches Earth).

This is the conceptual bridge to the Eclipse Geometry demo: Moon distance changes $\theta_{\text{Moon}}$, which changes whether “central” eclipses are total or annular.

Sanity-check anchors (built into presets)

Two especially useful reference points for classroom verification:

• Sun at 1 AU: $\theta \approx 0.53^\circ$ (preset: Sun)
• Moon today: $\theta \approx 0.52^\circ$ (preset: Moon)

The “ISS overhead” preset uses:

• diameter $D \approx 0.109\ \text{km}$ (109 m)
• distance $d \approx 420\ \text{km}$

which puts it around the arcminute scale.

Navigation

• Instructor hub: /demos/_instructor/
• Back to this demo guide: Guide
• Student demo: /play/angular-size/
• This demo: Model · Activities · Assessment · Backlog

How to use this backlog This is a planning guide. Prefer changes that improve reasoning and reduce unit-confusion.

Priority	Impact	Effort	Category	Notes	Code entrypoint
P0	High	Low	UX	DONE (2026-01-29): Added explicit “Internal units = km” label near sliders to prevent silent unit errors.	apps/demos/src/demos/angular-size/index.html
P0	High	Medium	Physics	DONE (2026-01-29): Added a “total vs annular” note tied to Moon distance (connect to eclipse demo’s umbra/antumbra idea).	apps/demos/src/demos/angular-size/main.ts + instructor docs
P1	High	Medium	Pedagogy	Add a guided “Sun–Moon coincidence” mini-challenge with prediction checkpoints.	apps/demos/src/demos/angular-size/main.ts
P1	Medium	Medium	Physics	Replace the linear recession model with a clearly-labeled “toy” vs “geology-informed” option (still keep units explicit).	packages/physics/src/angularSizeModel.ts
P1	Medium	Medium	UX	Add a “compare two presets side-by-side” view for ratio reasoning.	apps/demos/src/demos/angular-size/index.html + apps/demos/src/demos/angular-size/main.ts
P2	Medium	Low	Accessibility	Ensure all controls have aria labels + keyboard focus order (audit).	apps/demos/src/demos/angular-size/index.html
P2	Medium	Low	UX	DONE (2026-01-30): Added shared Station Mode (table + CSV copy + print) and a Help/Keys panel.	apps/demos/src/demos/angular-size/index.html + apps/demos/src/demos/angular-size/main.ts
P2	Low	Low	Pedagogy	Expand assessment items after classroom pilot; add distractors about degrees vs arcminutes.	apps/site/src/content/instructor/angular-size/assessment.md