Preview
Kepler's Laws
Explore planetary motion with three fundamental laws
For twenty years Tycho Brahe logged the planets night after night, and for twenty more Johannes Kepler interrogated those numbers — expecting circles, because everyone expected circles. Mars refused. The orbit that finally fit was an ellipse, and not even with the Sun at its center: the Sun sits at one focus, off to the side, and the planet's speed is just as lopsided — sprinting when it swings near the Sun, crawling when it retreats. Kepler's second law says the lopsidedness is exact: the line from Sun to planet sweeps equal areas in equal times, which is really angular momentum keeping its books. His third law, published ten years later, turned the whole solar system into a clock: T² ∝ a³, so once you know Earth's year, every planet's distance falls out of its period. In this lab you hold all three laws on one panel — the ellipse shape from two sliders, the equal-areas fan painted live in cyan, and the period priced to the millisecond of a year. And you'll use the Earth's own orbit to demolish the most stubborn astronomy belief there is: the idea that summer is Earth swinging close to the Sun. The ellipse you're about to fly is real — it just can't do what that belief needs it to do.
What you'll be able to do
- Read an orbit as an ellipse with the Sun at one focus — not the center — and read Law 2 off the speed rows: eccentricity e stretches the orbit (e = 0 is the circle), and the planet sprints at perihelion and crawls at aphelion; on the comet preset the panel shows the extreme case, 55.57 km/s at perihelion against 3.55 km/s at aphelion, a 15.7× swing with the swept-area fan keeping the books equal
- Apply Kepler's Third Law quantitatively in the sim's own units (AU, years, M☉ with G·M☉ = 4π², so T = √(a³/M)): the defaults read T = 1.000 yr at a = 1 AU; doubling a to 2 AU gives T = 2.828 yr — 2√2, never 2 (quiz Q1's trap); Mars' 1.52 AU prices 1.87 yr (challenge kl-c1, the Astronomy textbook's own worked example)
- Falsify the distance theory of seasons with the Earth preset's own geometry: e = 0.017 leaves only a ±1.7% distance swing (perihelion 0.983 AU, aphelion 1.017 AU) and a 3.5% speed swing (30.29 vs 29.28 km/s) — 'only about 3%', and Earth is closest to the Sun in January; the cause of seasons is the 23.5° axial tilt, and the swept-area fan (areal velocity = L/2m) is the conservation law underneath Law 2
Formulas
Make a prediction
Earth preset: a = 1.00 AU, e = 0.017 — Earth's real orbit, T = 1.000 yr. A classmate points at the ellipse and says: 'See — Earth's orbit is not a circle. Summer is when the orbit dips us close to the Sun.' What does this ellipse actually deliver over one year?
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Your prediction
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Answer: A nearly circular path — perihelion 0.983 AU, aphelion 1.017 AU: the distance swings only ±1.7% and the speed just 3.5% (30.29 vs 29.28 km/s). Nowhere near enough to drive seasons — and Earth is closest to the Sun in January
The geometry is honest arithmetic: perihelion r_p = a(1−e) = 0.983 AU, aphelion r_a = a(1+e) = 1.017 AU — a peak-to-peak swing of 2e ≈ 3.4%, which is exactly what the astronomy textbook reports: Earth's 'distance from the Sun varies by only about 3%' (OpenStax Astronomy 2e §4.2). A 3% distance wobble changes received sunlight by only ~6% — far too little for summer-to-winter swings — and the panel's speed rows confirm the same mildness from the dynamics side: 30.29 km/s at perihelion versus 29.28 at aphelion, a 3.5% difference. Option B is the classic belief the textbook names outright — 'Many people have believed that the seasons were the result of the changing distance between Earth and the Sun… But the facts don't bear out this hypothesis' — and it collapses on a calendar fact the same section states: 'Earth is actually closest to the Sun in January, when the Northern Hemisphere is in the middle of winter.' Distance theory predicts July = nearest; nature delivers January = nearest. The real cause is the 23.5° axial tilt changing how directly sunlight strikes each hemisphere. Option C over-corrects: e = 0.017 is small, not zero — the ellipse and its two extreme points are real (the sim's own sweep-rate row keeps the equal-area books on it), they are just far too gentle to run a climate. Compare the Mercury preset (e = 0.205): there the speed swings 58.94 vs 38.89 km/s — THAT is what a working ellipse looks like.
Quiz (0/3)
Mars has a = 1.52 AU. What is its orbital period?
At perihelion or aphelion — where is the planet moving fastest?
Prove Kepler's Third Law from Newton's gravity for circular orbits.
You can now
- Read an orbit as an ellipse with the Sun at one focus — not the center — and read Law 2 off the speed rows: eccentricity e stretches the orbit (e = 0 is the circle), and the planet sprints at perihelion and crawls at aphelion; on the comet preset the panel shows the extreme case, 55.57 km/s at perihelion against 3.55 km/s at aphelion, a 15.7× swing with the swept-area fan keeping the books equal
- Apply Kepler's Third Law quantitatively in the sim's own units (AU, years, M☉ with G·M☉ = 4π², so T = √(a³/M)): the defaults read T = 1.000 yr at a = 1 AU; doubling a to 2 AU gives T = 2.828 yr — 2√2, never 2 (quiz Q1's trap); Mars' 1.52 AU prices 1.87 yr (challenge kl-c1, the Astronomy textbook's own worked example)
- Falsify the distance theory of seasons with the Earth preset's own geometry: e = 0.017 leaves only a ±1.7% distance swing (perihelion 0.983 AU, aphelion 1.017 AU) and a 3.5% speed swing (30.29 vs 29.28 km/s) — 'only about 3%', and Earth is closest to the Sun in January; the cause of seasons is the 23.5° axial tilt, and the swept-area fan (areal velocity = L/2m) is the conservation law underneath Law 2