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Instructor notes: Parallax Distance: Measuring the Stars

Public notes (v0.2 policy). Not linked from primary navigation.

Exhibit: /cosmic-playground/exhibits/parallax-distance/

Overview

Navigation

This guide is instructor-facing Student demo: /play/parallax-distance/
Main code: apps/demos/src/demos/parallax-distance/main.ts
UI markup: apps/demos/src/demos/parallax-distance/index.html
Model code (tested): packages/physics/src/parallaxDistanceModel.ts
Data: packages/data-astr101/src/starsNearby.ts

Where to go next

  • Model + math + assumptions: model.md
  • In-class activities (MW + Friday lab + station version): activities.md
  • Assessment bank (clickers + short answer + exit ticket): assessment.md
  • Future enhancements (planning backlog): backlog.md

Why this demo exists

Why This Matters Parallax is the first “rung” of the distance ladder: a distance measurement built from geometry rather than assumptions about a star’s brightness or physics. This demo helps students see the core astronomy move: use a baseline (Earth’s orbit), measure a tiny angle, and infer an otherwise unreachable distance.

This demo is built to emphasize cause → observable → inference:

  • Cause: Earth moves along its orbit and changes the observing geometry.
  • Observable: the target’s detector position shifts against fixed background stars.
  • Inference: two captures provide $\Delta\theta$ and $B_{\rm eff}$, yielding $\hat p$ and $\hat d$.

Learning goals (ASTR 101)

By the end of this demo, students should be able to:

  • Explain parallax as an apparent shift caused by viewing geometry, not a “property of the star.”
  • State the direction of the relationship: smaller parallax angle → greater distance.
  • Use the parallax-distance relationship conceptually (and optionally numerically) in the parsec system.
  • Describe why parallax measurements are limited by angular resolution/precision.

Learning goals (ASTR 201 stretch)

Students should be able to:

  • Use $d(\text{pc}) = 1/p(\text{arcsec})$ to convert between parallax and distance (including mas and $\mu\mathrm{as}$ units).
  • Interpret the parsec as a geometry-defined unit tied to the measurement.
  • Compare measurement reach for Hipparcos-scale (~mas) vs Gaia-scale (tens of $\mu\mathrm{as}$) astrometry.

10–15 minute live-teach script (projector)

  1. Warm start: human parallax. Have students do the thumb demo (alternate eyes). Ask: “If your thumb were farther away, would the shift look bigger or smaller?” (Prediction before observation.)

  2. Introduce cause and measurement axis. In the orbit panel, point to target direction and parallax axis. Ask: “What changes when Earth moves?” (The line-of-sight from Earth to the same star.)

  3. Use distance-first setup. Set $d_{\rm true}=10,\mathrm{pc}$. Capture A, then move to a separated phase and capture B. Read $\Delta\theta$ and $B_{\rm eff}$, then show inferred $\hat p$ and $\hat d$.

  4. Make inverse scaling explicit. Increase $d_{\rm true}$ to $100,\mathrm{pc}$, repeat captures at similar phases, and compare shifts. Ask: “Why did $\Delta\theta$ shrink and $\hat d$ grow?”

  5. Measurement limits = knowledge limits. Increase $\sigma_p$ and ask: “When does $\hat p$ become comparable to $\sigma_{\hat p}$?” Use $\hat p/\sigma_{\hat p}$ and $\sigma_{\hat d}$ to connect tiny angles to weak inference.

  6. Close the story. Say explicitly: “Parallax gives us distances that calibrate everything else. When the angle is too tiny, we need other methods — but those methods are anchored to this geometric rung.”

Misconceptions + prediction prompts

Use these to surface and correct common wrong models:

  • Misconception: “Parallax is a property of the star.”
    Prompt: “If the star stayed exactly the same, could its parallax change?” (Yes: change baseline or observer location.)

  • Misconception: “Closer stars have smaller parallax.”
    Prompt: “Thumb at arm’s length vs across the room: which shifts more?” Then run near/far distance capture pairs and compare $\Delta\theta$ with inferred $\hat d$.

  • Misconception: “Any two captures are equally good.”
    Prompt: “What happens when captures are close along the parallax axis?” Use $B_{\rm eff}$ warning to make ill-conditioning concrete.

  • Misconception: “Light-years are more ‘natural’ than parsecs.”
    Prompt: “Which unit is defined by the measurement itself?” (Parsec.)

Suggested connections to other demos

  • Telescope resolution: parallax is an angle measurement; resolution/precision sets which angles are measurable.
  • Cosmic Distance Builder (activity): use parallax as the anchor for scaling up the distance ladder language.

Activities

Navigation

MW Quick (3–5 min)

Type: Demo-driven
Goal: Make “closer → larger parallax” a prediction students test.

  1. Open: /play/parallax-distance/
  2. Point to target direction and parallax axis in the orbit panel. Ask: “What changes when Earth moves?” (Line-of-sight.)
  3. Set $d_{\rm true}=10,\mathrm{pc}$, capture A and B, and read $\Delta\theta$ and inferred $\hat d$.
  4. Increase distance to $100,\mathrm{pc}$, repeat captures at similar phases. Prediction prompt (10–20 s): “Will the measured shift be larger or smaller?”
  5. Reveal by comparing readouts. Say explicitly: smaller inferred parallax → greater inferred distance.

MW Short (8–12 min)

Type: Demo-driven (pairs)
Goal: Practice inverse scaling and connect to measurement precision.

Student worksheet (pairs)

Fill in the table by using distance-first captures:

$d_{\rm true}$ (pc)Capture phases A/B (deg)$\Delta\theta$ (mas)$B_{\rm eff}$ (AU)$\hat p$ (mas)$\hat d$ (pc)$\hat p/\sigma_{\hat p}$
100 / 180
1000 / 180
10030 / 150
10080 / 100

Instructions:

  1. Keep $\sigma_p=1,\mathrm{mas}$ at first, then increase it and record how $\hat p/\sigma_{\hat p}$ changes.
  2. Compare cases with similar chord but different $B_{\rm eff}$ to see geometry effects.
  3. Use difference mode to interpret the signed A→B shift direction.

Synthesis prompt (2 minutes): “If we want distances across the whole Milky Way, why can’t parallax be the only method?”

Friday Lab (20–30+ min)

Type: Demo-driven investigation (small groups)
Goal: Do claim–evidence reasoning about what is measurable and why.

Driving question

“How far can we directly measure stellar distances with parallax, and what sets the limit?”

Protocol

  1. Pick two uncertainty settings (e.g., $\sigma_p=1,\mathrm{mas}$ and $\sigma_p=10,\mathrm{mas}$).
  2. For each, find an approximate “reach” distance where inference becomes difficult (use $\hat p/\sigma_{\hat p}\lesssim 1$ as a discussion threshold).
  3. Create a short poster (or shared doc) with:
    • Claim: “With $\sigma_p=_$, captures with $B{\rm eff}=__$ are reliable out to about ____ pc.”
    • Evidence: at least 3 capture sets with $\Delta\theta$, $B_{\rm eff}$, $\hat d$, and $\hat p/\sigma_{\hat p}$.
    • Reasoning: connect tiny shifts and weak baseline projection to measurement challenge.

Extension (if time)

Use the discussion prompt: “What if we could observe from Jupiter’s orbit?” Write one paragraph predicting what would change and what would not.

Station version (6–8 min)

Station card: Parallax Distance (6–8 minutes) Artifact: one capture-based inference with quality statement.

At the station, produce:

  • A chosen true distance $d_{\rm true}$ and captures A/B,
  • Measured $\Delta\theta$, inferred $\hat p$, inferred $\hat d$,
  • One note: “This estimate is [strong/weak] because $\hat p/\sigma_{\hat p}$ is ____ and $B_{\rm eff}$ is ____.”

Word bank + sanity checks Word bank:

  • Parallax: apparent shift caused by a change in viewpoint.

  • Measured shift $\Delta\theta$: detector-space difference between captures.

  • Effective baseline $B_{\rm eff}$: baseline component along the parallax axis.

  • Inferred parallax $\hat p$: measured quantity used to infer distance.

  • Parsec (pc): defined so that:

    $$d,(\mathrm{pc})=\frac{1}{p,(\mathrm{arcsec})}$$

Sanity checks:

  • If $d=1,\mathrm{pc}$, then $p=1,\mathrm{arcsec}$.
  • If distance increases by $10\times$ for similar capture geometry, inferred parallax should drop by about $10\times$.
  • Inference degrades when $B_{\rm eff}$ is tiny, even if capture phases differ.

Assessment

Navigation

Observable keys used in prompts

  • deltaTheta: measured A→B detector shift (mas)
  • B_eff: effective baseline along the parallax axis (AU)
  • p_hat: inferred parallax from captures (mas)
  • d_hat: inferred distance from p_hat (pc)
  • inferred uncertainty: read from sigma_{p_hat}, sigma_{d_hat}, or p_hat/sigma_{p_hat}

Clicker questions (with distractors + explanation)

Clicker 1 — Direction of the relationship

Demo setup: set $d_{\rm true}=10,\mathrm{pc}$, capture A/B at $0^\circ/180^\circ$, and record deltaTheta, B_eff, p_hat, d_hat. Repeat with $d_{\rm true}=100,\mathrm{pc}$ at the same phases.

Question: With similar B_eff, the second run has smaller deltaTheta. Which readout change is expected?

A. p_hat decreases while d_hat increases
B. p_hat increases while d_hat increases
C. p_hat decreases while d_hat decreases
D. p_hat and d_hat both stay the same

Correct: A
Why: With similar geometry, smaller measured shift implies smaller inferred parallax and larger inferred distance.
Misconception targeted: “Closer stars have smaller parallax.”

Clicker 2 — Parsecs from parallax

Demo setup: use any high-quality capture pair (B_eff near $2,\mathrm{AU}$) and read p_hat.

Question: If p_hat = 100\,\mathrm{mas} = 0.1", what d_hat should the class expect?

A. 0.1 pc
B. 1 pc
C. 10 pc
D. 100 pc

Correct: C
Why: $\hat d(\text{pc}) = 1/\hat p(\text{arcsec}) = 1/0.1 = 10$.
Misconception targeted: inverse scaling confusion.

Clicker 3 — Which capture geometry improves inference?

Demo setup: hold $d_{\rm true}$ fixed. Compare captures at $0^\circ/180^\circ$ versus $80^\circ/100^\circ$. Record B_eff and inferred uncertainty.

Question: Which pair should give lower inferred uncertainty in d_hat?

A. $80^\circ/100^\circ$, because captures are closer in time
B. $80^\circ/100^\circ$, because small B_eff stabilizes inference
C. $0^\circ/180^\circ$, because larger B_eff strengthens inference
D. Both pairs, because d_{\rm true} is unchanged

Correct: C
Why: Large effective baseline produces a stronger geometry signal and smaller inferred uncertainty for the same target distance.
Misconception targeted: “Any two captures are equally informative.”

Clicker 4 — Precision and confidence

Demo setup: keep one capture pair fixed (e.g., $0^\circ/180^\circ$), then increase the uncertainty control and watch inferred uncertainty plus p_hat/sigma_{p_hat}.

Question: If inferred uncertainty grows while p_hat stays similar, what should happen?

A. p_hat/sigma_{p_hat} increases and confidence in d_hat increases
B. p_hat/sigma_{p_hat} decreases and confidence in d_hat decreases
C. p_hat/sigma_{p_hat} stays fixed while confidence in d_hat decreases
D. p_hat/sigma_{p_hat} decreases but confidence in d_hat is unchanged

Correct: B
Why: Larger uncertainty lowers signal-to-noise and weakens distance inference reliability.
Misconception targeted: “Measurement precision doesn’t matter.”

Clicker 5 — Geometry versus star property

Demo setup: keep one $d_{\rm true}$ value, then compare two capture geometries with different B_eff. Record deltaTheta, p_hat, and d_hat.

Question: Which statement is most accurate?

A. Parallax is a physical property of the star.
B. deltaTheta depends on capture geometry, but p_hat and d_hat are the geometry-corrected inference.
C. B_eff only changes visuals, not measured quantities.
D. Changing capture geometry should never change any readout.

Correct: B
Why: The measured shift changes with baseline projection; the inference uses B_eff to recover parallax/distance.
Misconception targeted: “Parallax is a property of the star.”

Clicker 6 — Bigger baseline thought experiment

Demo setup: use any completed capture pair as a reference, then consider a larger physical baseline thought experiment with the same star and measurement precision.

Question: With a larger baseline, which change is expected in the same observable framework?

A. smaller deltaTheta and larger inferred uncertainty
B. larger deltaTheta and smaller inferred uncertainty in d_hat
C. unchanged deltaTheta and unchanged uncertainty
D. negative d_hat

Correct: B
Why: Bigger baseline increases measured shift for the same distance, improving inference confidence.
Misconception targeted: “Better measurement is only about better cameras, not geometry.”

Short-answer prompts

  1. From one capture pair, explain how deltaTheta and B_eff combine to produce p_hat and then d_hat.
  2. In your own words, define a parsec and connect it to p_hat in arcseconds.
  3. Describe one case where deltaTheta is measurable but inferred uncertainty is still high.
  4. Why does inferred uncertainty set the distance reach of parallax methods?

Exit ticket (3 questions)

  1. Two runs have similar B_eff; one has deltaTheta about $10\times$ smaller. What happens to p_hat and d_hat? (One sentence.)
  2. If p_hat = 0.5", what d_hat should you report in pc? (One number.)
  3. Name one geometry factor and one measurement factor that increase inferred uncertainty.

Model notes (deeper)

Navigation

Links Student demo: /play/parallax-distance/
Model code (tested): packages/physics/src/parallaxDistanceModel.ts
UI/visualization code: apps/demos/src/demos/parallax-distance/main.ts

What the demo is modeling (big picture)

This demo models the geometry of parallax and nothing else. It is intentionally “physics-light” because the point of parallax is that it is a distance measurement that does not require knowing a star’s luminosity, temperature, or composition.

The demo links three representations of the same idea:

  • A top-down cause view: Earth moves around the Sun, changing line-of-sight.
  • A detector view: the target shifts relative to fixed background stars.
  • A numeric inference: two captures yield $\Delta\theta$, $B_{\rm eff}$, inferred $\hat p$, and inferred $\hat d$.

Units + conventions used in the code

The demo uses:

  • Distance in parsecs (pc) and light-years (ly).
  • Detector shift and parallax in milliarcseconds (mas) and arcseconds (”).
  • A unit-radius orbit (AU) with explicit axis conventions:
    • Target direction $\hat{\mathbf{s}}$.
    • Parallax measurement axis $\hat{\mathbf{a}}=\mathrm{perp}(\hat{\mathbf{s}})$.

Key relationships to foreground (with meaning + units)

Distance definition: $d(\text{pc}) = 1/p(\text{arcsec})$

$$d(\text{pc}) = \frac{1}{p(\text{arcsec})}$$

Let’s unpack each piece:

  • $d$ is distance, measured in parsecs (pc).
  • $p$ is the parallax angle, measured in arcseconds (”).

What this equation is really saying: parallax is an inverse relationship. When a star is $10\times$ farther away, the parallax angle is $10\times$ smaller.

Sanity checks

  • If $p = 1”$, then $d = 1\ \text{pc}$ (this is the definition of a parsec).
  • If $p$ halves, $d$ doubles (inverse scaling).

Capture-based inference used by the demo

Earth position on a unit orbit:

$$ \mathbf{r}(\phi)=\langle \cos\phi,\sin\phi\rangle \quad (\mathrm{AU}) $$

True detector offset is constrained to the measurement axis:

$$ \mathbf{o}{\rm true}(\phi)=p{\rm true,mas},(\mathbf{r}(\phi)\cdot\hat{\mathbf{a}}),\hat{\mathbf{a}} $$

For captures A and B:

$$ \mathbf{b}=\mathbf{r}_B-\mathbf{r}A,\qquad B{\rm eff}=|\mathbf{b}\cdot\hat{\mathbf{a}}| $$

$$ \Delta\theta_{\rm axis}=(\mathbf{o}_B-\mathbf{o}A)\cdot\hat{\mathbf{a}},\qquad \Delta\theta=|\Delta\theta{\rm axis}| $$

$$ \hat p_{\rm mas}=\frac{\Delta\theta}{B_{\rm eff}},\qquad \hat d_{\rm pc}=\frac{1000}{\hat p_{\rm mas}} $$

The displayed equivalent six-month shift is derived as $2\hat p$ and is not the direct measurement unless captures are opposite in phase along the parallax axis.

Assumptions, limitations, and sanity checks

  • The demo treats Earth’s orbit as circular and uses an idealized baseline.
  • Background stars are treated as effectively fixed for the shift visualization.
  • Inference uses the effective projected baseline $B_{\rm eff}$, not raw chord length.
  • If $B_{\rm eff}$ is below threshold, inference is intentionally suppressed as ill-conditioned.
  • Deterministic capture noise is axis-aligned and applied at capture time.
  • Detector exaggeration is visual only and never changes computed $\hat p$ or $\hat d$.

Backlog

Navigation

P0 (blocking / correctness / teachability)

  • Measurement-limit clarity: add one explicit instructor note (and/or student microcopy) that “measurable” depends on precision and that the visualization is exaggerated for teaching.
  • Unit scaffolding: add a tiny on-page “arcsec <-> mas <-> $\mu\mathrm{as}$” conversion reminder so students don’t treat unit changes as physics changes.
  • Assessment alignment: package the clickers into a quick capture sequence where students record deltaTheta, B_eff, p_hat, d_hat, and inferred uncertainty for each case.

P1 (important)

  • Noise + uncertainty: add an optional “measurement noise” mode and repeated measurements to show how uncertainty averages down (matches README future ideas).
  • Baseline comparison: add an Earth vs Jupiter (or spacecraft) baseline toggle to support the “bigger baseline” discussion question.
  • Proper motion: consider a future extension combining proper motion with parallax (README future ideas), clearly separated as an advanced toggle.

P2 (nice to have)

  • 3D visualization: add a depth/3D view to help students connect “angle” to geometry without relying only on the top-down diagram.
  • Catalog expansions: expand the preset star catalog and include explicit “Gaia measurability” labels for more examples.