Instructor notes: Planetary Conjunctions: Alignments in the Sky

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This guide is instructor-facing Student demo: /play/planetary-conjunctions/ Exhibit page: /exhibits/planetary-conjunctions/ Demo source: apps/demos/src/demos/planetary-conjunctions/

Where to go next

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

Controls worth knowing before you teach

• Target planet chips: Venus, Mars, Jupiter, Saturn.
• Speed slider and Reset.
• Readouts: synodic period, days elapsed, conjunctions observed, both longitudes, and the current angular separation.
• A conjunction flash fires each time the two longitudes line up, and the counter increments.

Why this demo exists

Why This Matters “Conjunction” is one of the few astronomy words students meet in the news, usually attached to a claim that two planets are “close together”. They are not. At a conjunction two planets share a direction in our sky while remaining hundreds of millions of kilometres apart. The demo makes the distinction unavoidable by showing the physical positions and the sky alignment at the same moment.

The deeper payoff is the synodic period: the time for the same alignment to repeat. It is the first quantity many students meet that is not a property of any single object but of a relationship between two, and it behaves in a way that is genuinely surprising until you see why.

Learning goals (ASTR 101)

Students should be able to:

• Define a conjunction as a line-of-sight alignment as seen from Earth, and state that it says nothing about the physical separation of the two planets.
• Explain why alignments repeat, in terms of one planet gaining a full lap on the other.
• Read the synodic period from the demo and explain why it differs from either planet’s orbital period.
• Predict which of two planets has the longer synodic period with Earth, using how close its orbital period is to Earth’s.

10 to 15 minute live-teach script (projector)

Open on the Mars target with the speed slider low.

1. Commit to a prediction. “Jupiter takes about 12 years to orbit the Sun. Mars takes about 2. Which one lines up with Earth more often?” Take a show of hands. Most classes vote Mars, reasoning that faster means more often. Record the vote where everyone can see it.

2. Define the observable. Run the demo until the first flash. Ask: “What was true at that instant?” Steer to: the two planets had the same heliocentric longitude, meaning they lay on the same ray out from the Sun. Point at the separation readout going to zero.

   Note the label. With Mars selected the counter reads “Oppositions observed”, because that geometry puts Earth between the Sun and Mars. Switch to Venus and it becomes “Inferior conjunctions observed”. Ask the class why the same alignment gets two different names before you explain it.

3. Kill the “close together” idea now. Freeze at the flash and ask: “How far apart are these two planets in the picture?” They are on completely different orbits. The alignment is about direction, not distance. This is worth thirty seconds of silence while they look at it.

4. Read the synodic period. Let two or three conjunctions accumulate and point at the days-elapsed and conjunctions-observed readouts. Define the synodic period as the time between successive conjunctions.

5. Resolve the prediction. Switch to Jupiter. The synodic period drops to about 399 days, barely over one year. Switch to Mars: about 780 days, more than two years. The class vote was wrong, and it was wrong for an interesting reason.

6. Explain the surprise. Mars is the hardest to catch, not the easiest, because its orbital period is closest to Earth’s. Earth has to gain a whole lap, and when two runners have similar lap times that takes a long while. Jupiter barely moves, so Earth laps it almost as soon as Earth finishes one orbit.

7. Push to the limit. Ask: “What would the synodic period be for a planet with exactly Earth’s orbital period?” Infinite, because Earth never gains a lap. That limit is the whole idea in one sentence.

Common student responses, and what to do with them

Suggested connections to other demos

• Retrograde motion: the same relative-motion clock. A superior planet goes retrograde once per synodic period, at opposition.
• Kepler’s laws: supplies the orbital periods that set the synodic period, so the two demos chain together.
• Angular size: why “close in the sky” and “close in space” are unrelated statements.

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

Materials and setup

• One device per pair. The demo is short, so pairs keep the pace up.
• A calculator or phone for the reciprocal arithmetic in the investigation.
• For the warm-up, a running track, corridor, or just two students walking a circle works.
• Print the station card from /stations/planetary-conjunctions/ for the rotation.

Warm-up without the computer (3 min, whole class)

Kinesthetic: lapping on a track Two students walk a circle at clearly different speeds, starting side by side. Ask the class to call out each time the faster one draws level with the slower one again.

Then ask the question that matters: “To draw level again, how much extra distance does the faster walker have to cover?” Exactly one full lap. Not a fraction, not a half. One whole lap, every time.

Now make it hard: “What if their speeds were almost the same?” The lapping takes forever. Hold that thought and open the demo.

MW Quick Exploration (3 to 5 min, pairs)

TPS: Are they actually close? Think (30 s): “A news headline says Venus and Jupiter are ‘in conjunction’ and appear side by side. How far apart are they in space?”

Pair (60 s): Agree on an answer and a reason.

Share (1 to 2 min): Run the demo to the first conjunction flash and pause.

1. Read the angular separation. It is near zero.
2. Look at where the two dots actually are. They are on different orbits.
3. Say what a conjunction is a statement about.

Debrief script: “A conjunction is a statement about direction, not distance. Two planets share a line of sight from Earth while staying hundreds of millions of kilometres apart. Nothing about them has moved closer.”

MW Short Investigation (8 to 12 min, pairs or triads)

Investigation: Which planet is hardest to catch? Task: For each target, run the demo until you have timed at least two conjunctions, and record the synodic period.

Target	Orbital period (yr)	Synodic period with Earth (days)	Synodic period (yr)
Venus	0.62
Mars	1.88
Jupiter	11.86
Saturn	29.46

Prompt: “Rank the planets from shortest to longest synodic period. Now rank them by orbital period. The two rankings do not match. Why not?”

Expected values (days): Venus about 584, Mars about 780, Jupiter about 399, Saturn about 378.

The pattern to converge on: the synodic period is longest for the planet whose orbital period is closest to Earth’s (Mars), and shortest for the planets whose periods are furthest from Earth’s (Jupiter, Saturn). Two planets with similar speeds take a very long time to lap one another.

Extension for ASTR 201: verify the relationship numerically. Compute $1/P_{\text{Earth}} - 1/P_{\text{target}}$ for a superior planet and take the reciprocal. Compare with the demo readout.

Share-out (2 to 3 min) Ask the group that ranked Jupiter as “hardest to catch” to explain their reasoning before revealing the answer. The intuition that a slow planet is hard to catch is exactly backwards here, and hearing it stated is more useful than hearing it corrected.

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

Astro Lab: Build the synodic period from scratch Deliverable: One page with a derivation, a data table, and a prediction that you then test.

Stage 1 — Reason it out (7 min). Before using any formula: Earth completes $1/P_{\text{Earth}}$ orbits per year and the target completes $1/P_{\text{target}}$. Write down, in words, how much extra angle Earth must gain on the target between one conjunction and the next. Then write down how fast Earth gains that angle.

Stage 2 — Turn it into a formula (5 min). Combine the two into an expression for the synodic period. You should reach

$$\frac{1}{P_{\text{syn}}} = \left| \frac{1}{P_{\text{Earth}}} - \frac{1}{P_{\text{target}}} \right|$$

Stage 3 — Test it (8 min). Pick two targets. Predict the synodic period from the formula, then measure it in the demo. Record both and the percentage difference.

Stage 4 — Push on it (5 min). Answer two questions in writing:

• What does the formula give for a hypothetical planet with exactly Earth’s orbital period, and what does that mean physically?
• Saturn’s orbital period is more than twice Jupiter’s, yet their synodic periods differ by only about three weeks. Explain why.

What good work looks like: the derivation talks about rates rather than periods, and the answer to the last question notices that for a very slow planet the target’s term nearly vanishes, so the synodic period approaches Earth’s own year no matter how slow the planet gets.

Common stopping point: groups that try to subtract periods rather than rates. Let them; the number will be badly wrong and the demo will say so. That failure is the lesson.

Station version (for the Cosmic Playground capstone rotation)

Station card: Planetary Conjunctions (6 to 8 minutes) Demo setup: start on Mars at a moderate speed, then repeat with Jupiter.

Your station artifact (fill in):

1. Control(s): target planet, animation speed

2. Observable(s): angular separation, conjunctions observed, days elapsed, synodic period

3. Governing relationship: write this in words:

   $$\frac{1}{P_{\text{syn}}} = \left| \frac{1}{P_{\text{Earth}}} - \frac{1}{P_{\text{target}}} \right|$$

4. Sanity check: which target gives the longest synodic period, and is that the fastest or the slowest planet?

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

Word bank + sanity checks Word bank:

• Conjunction: two bodies share the same direction in our sky. A statement about line of sight only.
• Opposition: a superior planet sits opposite the Sun in our sky, so it is up all night. Only possible for planets outside Earth’s orbit.
• Synodic period: the time for the same Sun, Earth and planet alignment to repeat.
• Sidereal (orbital) period: the time for one full orbit around the Sun, measured against the distant stars.
• Superior planet: orbits outside Earth’s orbit (Mars, Jupiter, Saturn here).
• Inferior planet: orbits inside Earth’s orbit (Venus here).

Sanity checks:

• A conjunction says nothing about physical distance. Nothing gets closer.
• The synodic period is never the average of the two orbital periods.
• For a very distant, very slow planet the synodic period approaches one Earth year, because Earth does essentially all the work.
• Mars has the longest synodic period of the four targets here, despite being the nearest superior planet.
• The demo is schematic and not to scale. Orbit sizes on screen are for legibility, not measurement.

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How to use this bank Each item has a demo setup you can run live and notes on what each distractor reveals. Synodic values quoted here are computed from real orbital periods and match the demo to within its rounding.

Clicker questions

Clicker 1: What is a conjunction? Prompt: Two planets are “in conjunction”. What does that tell you?

A. They have come physically close to each other in space B. They appear in nearly the same direction as seen from Earth C. Their orbits have crossed D. They are both at their closest approach to the Sun

Correct: B.

Reasoning: A conjunction is a line-of-sight alignment. The two planets remain on their own orbits, typically hundreds of millions of kilometres apart.

Distractors to listen for:

• A: the headline misconception, reinforced by every news story about a “planetary alignment”. Expect it to lead on a first ask.
• C: orbits do not cross; this often comes with a worry about collisions.
• D: confuses an Earth-relative alignment with each planet’s own perihelion.

Demo setup: select Venus and run to a flash, then freeze. Point at the separation readout near zero, then at the two dots sitting on different orbits. Venus is the right target for this item because the demo labels its alignment “Inferior conjunctions observed”, so the word on screen matches the word in the question.

If A leads: do not argue. Freeze at conjunction, say nothing, and ask one student to describe what they see in the orbit view. The picture does the work.

Clicker 2: Which is hardest to catch? Prompt: Which planet has the longest synodic period with Earth, meaning the longest wait between successive conjunctions?

A. Venus, because it is closest to Earth B. Mars, because its orbital period is closest to Earth’s C. Jupiter, because it takes about 12 years to orbit D. Saturn, because it takes about 29 years to orbit

Correct: B. Mars, at about 780 days (2.14 years).

Reasoning: The synodic period depends on how fast Earth gains a whole lap on the target. Mars orbits in 1.88 years, close to Earth’s 1 year, so Earth gains ground slowly and takes over two years to lap it. Jupiter and Saturn barely move, so Earth laps them in only a little over one year (399 and 378 days respectively).

Distractors to listen for:

• C and D: the dominant wrong answers, and both come from the same reasoning, that a slow planet must be hard to catch. It is exactly backwards, and it is worth naming as a reasonable-but-wrong intuition rather than a careless error.
• A: reasons from distance rather than from period.

Demo setup: run Mars and Jupiter back to back and read both synodic periods aloud.

Recovery move if the class splits: ask “what is the synodic period of a planet that orbits in exactly one year, like Earth itself?” The answer is infinite, and it re-anchors the whole idea on similarity of periods rather than on speed.

Clicker 3: The limiting case Prompt: Imagine a very distant planet with an orbital period of 1000 years. What is its synodic period with Earth, roughly?

A. About 1000 years B. About 500 years C. Just over 1 year D. Exactly 1 year

Correct: C. Just over one year (about 365.6 days).

Reasoning: The planet is almost stationary over one Earth year, so Earth only needs to complete roughly one of its own orbits to line up again. The synodic period approaches Earth’s orbital period from above and never quite reaches it.

Distractors to listen for:

• A: still treating the synodic period as a property of the distant planet.
• D: right magnitude, but misses that Earth must travel slightly more than a full orbit because the target has crept forward. Worth a sentence, since “just over” is the physics.

Demo setup: Saturn is the closest available case at 378 days. Ask students to extrapolate from Jupiter (399 d) to Saturn (378 d) and then to something far slower.

Short answer

Short answer 1: Direction versus distance Prompt (2 to 3 sentences): A friend reads that Jupiter and Saturn are in conjunction and says “they must nearly be touching”. Explain what is actually true, and what a conjunction does and does not tell you.

Answer key: A conjunction means the two planets appear in nearly the same direction from Earth. It is a statement about our line of sight. Jupiter and Saturn remain on their own orbits, separated by billions of kilometres, and nothing about their physical separation has changed.

Partial credit: an answer that says “they only look close” without explaining that it is an Earth-based line of sight has the conclusion but not the reason.

Short answer 2: Build the relationship Prompt: Earth completes one orbit per year. Mars completes one orbit per 1.88 years. Explain, in terms of how much angle each planet covers per year, why the time between conjunctions is about 2.1 years and not 0.88 years.

Answer key: Earth covers a full turn per year; Mars covers about $1/1.88 = 0.53$ of a turn per year. Earth therefore gains about $1 - 0.53 = 0.47$ of a turn on Mars each year. To line up again Earth must gain a whole turn, which takes $1/0.47 \approx 2.1$ years. Subtracting the periods (1.88 minus 1 = 0.88 years) is the common error: the periods do not subtract, the rates do.

Watch for: the 0.88-year answer specifically. It is not a careless slip, it is a coherent wrong model, and naming it as “you subtracted periods instead of rates” is more useful than marking it wrong.

Short answer 3 (ASTR 201 extension): The near-degenerate case Prompt: A hypothetical asteroid orbits the Sun with a period of 1.01 years. Compute its synodic period with Earth and comment on what that means for observing it.

Answer key: $1/P_{\text{syn}} = |1 - 1/1.01| = |1 - 0.990| = 0.0099$ per year, so $P_{\text{syn}} \approx 101$ years. Earth gains only about one hundredth of a lap per year, so the same Sun, Earth and asteroid geometry recurs roughly once a century. In practice the object would stay in nearly the same part of our sky, under nearly the same observing conditions, for decades.

Strong answers connect this back to why Mars is the hardest of the four demo targets to catch, and note that a period slightly less than a year gives the same result.

Exit ticket (1 minute)

Exit ticket: Rank and justify Prompt: Rank Venus, Mars and Jupiter by how long you wait between conjunctions with Earth, longest first. Give one sentence of justification.

Expected: Mars (about 780 d), then Venus (about 584 d), then Jupiter (about 399 d). The justification must refer to how close each planet’s orbital period is to Earth’s, not to its distance or its speed alone.

Fast triage: a ranking of Jupiter, Mars, Venus means the student is still ordering by orbital period. A correct ranking with a distance-based justification is worth a follow-up; they may have memorised the demo output rather than the idea.

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Links Student demo: /play/planetary-conjunctions/ Synodic period: packages/physics/src/twoBodyAnalytic.ts Demo logic: apps/demos/src/demos/planetary-conjunctions/logic.ts

What the demo is modeling (big picture)

Two planets move on circular orbits at constant angular speed. The demo tracks each planet’s heliocentric longitude, watches for the moments when the two longitudes coincide, and reports the interval between those moments.

The model is schematic: orbit radii on screen are chosen for legibility, not drawn to scale. Timing, however, is real. The periods come from the actual semi-major axes via Kepler’s third law, so the synodic periods the demo reports are the correct ones for the solar system.

Where the periods come from

Each target’s semi-major axis $a$ feeds Kepler’s third law with the Sun as the central mass:

with $P$ in years, $a$ in AU and $M$ in solar masses, so $M = 1$ and $P = a^{3/2}$.

These derived periods reproduce the observed ones to better than a part in a thousand, which is a nice incidental result to point out: Kepler’s third law and a single number per planet get you the whole timing structure.

The synodic period

The implementation computes

which is algebraically the same as the form students usually derive:

The rate form is the one to teach. It says directly that what matters is the difference in angular rates, and that the alignment repeats when the faster planet has gained exactly one full turn on the slower one. The product form is convenient for computing and actively unhelpful for understanding; students who memorise it tend to produce the “subtract the periods” error because the subtraction is sitting right there in the denominator.

The function returns Infinity when the two periods are equal. That is not a guard against a division by zero so much as the correct physical answer: two planets on the same orbit never lap one another, so the same alignment never recurs.

Why the surprise is a surprise

Students predict that a fast planet conjoins often and a slow one rarely. The rate form shows why that is wrong. Write $P_1 = 1$ year for Earth:

• As $P_2 \to \infty$, the second term vanishes and $P_{\text{syn}} \to 1$ year. A very slow planet is easy to lap, because Earth does all the work in one of its own orbits.
• As $P_2 \to 1$ year, the difference goes to zero and $P_{\text{syn}} \to \infty$. A planet with nearly Earth’s period is nearly impossible to lap.

So the controlling quantity is not the target’s speed but how different its rate is from Earth’s. Mars, the nearest superior planet, has the longest synodic period of the four targets precisely because its period is the closest to Earth’s. Jupiter and Saturn differ by more than a factor of two in orbital period yet their synodic periods differ by only about three weeks, because both are already deep in the slow limit.

Conjunction versus opposition

The demo fires when the two heliocentric longitudes agree to within 5 degrees, meaning both planets lie on the same ray out from the Sun. What that alignment is, as seen from Earth, depends on which side of Earth’s orbit the target sits:

The demo labels this correctly per target. The counter reads “Oppositions observed” for the three outer planets and “Inferior conjunctions observed” for Venus, and the exported results carry the same label. Until 2026-09-04 it called every alignment a “conjunction”, which was correct only for Venus; if you are working from older notes or a printed handout, check the wording.

Opposition is the observationally interesting case: the planet is closest, brightest, up all night, and going retrograde. That last point is the direct bridge to the retrograde-motion demo, which shows the same event from the other side.

Teaching move worth keeping. Even with the labels correct, the why is worth eliciting rather than telling. Run Venus first, establish inferior conjunction, then switch to Mars and ask: “same geometry from the Sun’s point of view — so why does the label change?” The answer, that Earth is now the one in the middle, is the whole idea.

Assumptions and honest limitations

• Circular, coplanar orbits. Eccentricity and inclination are both ignored. Real conjunctions have a non-zero angular separation because the orbits are tilted with respect to one another; this model can drive the separation to exactly zero.
• Not to scale. Orbit radii on screen are compressed so all four targets are visible in one frame. Do not let students measure distances off the display.
• Constant angular speed. A consequence of the circular assumption. Real planets move faster near perihelion.
• No third bodies, no perturbations. Each run involves exactly Earth and one target.
• Detection is on heliocentric longitude, and the event is then labelled from the target’s geometry (opposition for an outer planet, inferior conjunction for Venus). Published conjunction and opposition dates use full geocentric coordinates including inclination, so they will differ slightly.

Teaching note on the inferior case

Venus is the one inferior planet available, and it has two distinct alignments with Earth: inferior conjunction, with Venus between Earth and the Sun, and superior conjunction, with Venus on the far side of the Sun. These are 180 degrees apart in heliocentric longitude, and each occurs once per synodic period of 584 days, alternating.

The demo’s 5-degree test only fires on the same-longitude case, so the counter counts inferior conjunctions only. Superior conjunctions happen halfway between two flashes and are not flagged. If a student notices Venus passing “behind” the Sun without a flash, they have spotted something real and correct.

Scope note: Future improvements for the planetary-conjunctions instrument, split into (A) content correctness, (B) pedagogy and UX, and (C) model extensions that stay honest. Item A1 is a genuine defect found while writing this bundle and should be fixed before the demo is promoted past experimental.

A) Content correctness

A1. The flash is labelled “conjunction” for all four targets, but it is an opposition for three of them. (FIXED 2026-09-04)

The demo now classifies the alignment from the target’s semi-major axis relative to Earth’s and labels it accordingly: “Oppositions observed” for Mars, Jupiter and Saturn, “Inferior conjunctions observed” for Venus. The label drives the readout, the live-region announcement, and the exported results. The classification is geometric rather than a name lookup (alignmentAtSameLongitude in logic.ts), so adding Mercury, Uranus or Neptune under C3 needs no change here.

Covered by unit tests in logic.test.ts and two E2E tests in apps/site/tests/planetary-conjunctions.spec.ts, one of which asserts that no outer-planet label contains the word “conjunction”.

A2. The complementary 180-degree alignment is never flagged. (P2, still open)

Only the same-longitude alignment is detected. The alignment 180 degrees away is also real and also recurs once per synodic period: superior conjunction for Venus, and conjunction proper for the outer planets. A student watching Venus pass behind the Sun sees nothing happen.

Deliberately left out of the A1 fix. Detecting both would double the flash rate, and the counter is what students time to measure the synodic period in activities.md — successive flashes would then be half a synodic period apart and the investigation would break. Doing this properly means counting the two event types separately, which is a UI change rather than a labelling one.

A3. content_verified is false and there is no docs/reviews/planetary-conjunctions.md physics review. (P2)

The validate-play-dirs gate now enforces that content_verified: true requires a review on file, so the metadata is at least honest. A review still needs writing, and A1 should be resolved first so it does not immediately become a finding.

B) Pedagogy and UX

B1. No opposition marker. Opposition is the observationally interesting alignment: the planet is closest, brightest, up all night, and going retrograde. Adding an explicit opposition indicator would connect this demo directly to retrograde-motion and make the A1 distinction visible rather than verbal.

B2. No synodic-period prediction affordance. Students currently read the synodic period off a readout. A “predict before you run” input, compared against the measured value after two conjunctions, would turn a displayed number into a testable claim and give the demo a Predict-Observe-Explain spine it currently lacks.

B3. Only six controls total. This is the thinnest control surface of any demo in the collection, and the UX audit ranked it second-worst overall. Candidates: a second target so students can watch planet-to-planet alignments rather than only Earth-to-planet, and a toggle between the schematic view and a true-scale view.

B4. No Station Mode, no Challenge Mode, no Help dialog. This demo does not call createDemoModes, making it one of only two demos without the shared shell affordances. Students cannot export a data table for the investigation in activities.md without transcribing by hand.

C) Model extensions that stay honest

C1. Optional inclination. Real conjunctions have a non-zero minimum separation because the orbits are tilted. A toggle that adds each planet’s real inclination would let the demo answer the question students always ask: why is a conjunction not an eclipse or a transit? Keep it off by default so the core timing lesson stays clean.

C2. Optional eccentricity. Circular orbits mean constant angular speed. Adding real eccentricities would make successive synodic intervals vary slightly, which is both true and a good discussion of why “the” synodic period is an average.

C3. Extend the target list. Mercury would add a second inferior planet and a short 116-day synodic period, which usefully breaks the impression that Venus is what an inner planet looks like. Uranus and Neptune would push the slow limit further and make the “approaches one year” asymptote convincing.