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Kepler's Three Laws Describe Planetary Orbits · Celestial Mechanics · Kepler's Laws · Elliptical Orbits

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Prerequisite: Complete or review Kepler's Laws of Planetary Motion

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Kepler's Three Laws Describe Planetary Orbits

with Nova

Nova floats in the darkness beyond Mars, arms outstretched, tracing the sweeping curve of an orbital path with a glowing stylus while planets drift past in their silent ellipses.

Identify the geometric shape of a planetary orbit and the location of the Sun within it according to Kepler's First Law.
Explain why a planet moves faster near perihelion and slower near aphelion using Kepler's Second Law.
Calculate the orbital period of a planet given its semi-major axis using Kepler's Third Law (T² ∝ a³).
Compare the orbital properties of two planets to predict which one moves faster at a given orbital distance.
Distinguish between the misconception that planets speed up near the Sun due to proximity alone versus the correct principle of conservation of angular momentum.

Key terms

Ellipse: A closed oval curve with two focal points; the sum of distances from any point on it to the two foci is constant.
Perihelion and aphelion: The points in an orbit closest to and farthest from the Sun, where the planet moves fastest and slowest respectively.
Semi-major axis: Half the length of an ellipse's longest axis, serving as the average orbital distance in Kepler's third law.
Conservation of angular momentum: The principle that a planet's distance times its perpendicular velocity stays constant, forcing it to speed up when nearer the Sun.

The Geometry of the Three Laws

Kepler's first law replaces the ancient circle with the ellipse, placing the Sun at one focus rather than the center, so a planet's distance varies through its orbit. The second law states that the line from Sun to planet sweeps equal areas in equal times, which is geometrically equivalent to conservation of angular momentum: when the radius shrinks near perihelion, the perpendicular speed must rise to keep r × v_perp constant. The third law then ties an orbit's size to its timing, T² ∝ a³, completing a description accurate to the precision of Brahe's data.

Why Equal Areas Mean Changing Speed

Imagine the radius line as the hand of a clock dragging out wedge-shaped slices of area as the planet moves. Because each equal time interval must sweep the same area, a short, fat wedge near the Sun demands fast motion while a long, thin wedge far out demands slow motion. This is not caused by proximity to the Sun heating or pushing the planet; it is the direct consequence of angular momentum conservation. Near perihelion the small radius is compensated by a large speed, and the reverse holds at aphelion.

Worked examples

A planet has a semi-major axis of 4 AU. Find its orbital period in Earth years.

Use Kepler's third law in solar-system units: T² = a³ with a in AU and T in years.
Cube the semi-major axis: a³ = 4³ = 64.
Take the square root: T = √64 = 8 years.

Answer: 8 Earth years — note that the linear answer of 4 years is wrong because the relation is T² ∝ a³, not T ∝ a.

Hey, I'm Nova — and I want you to feel what it's like to move through space on an orbit. For most of history, people assumed planets traveled in perfect circles. Johannes Kepler, using Tycho Brahe's precise observations of Mars in the early 1600s, proved them wrong. He published three laws that describe every planet's motion with stunning accuracy. **First Law — The Law of Ellipses:** A planet's orbit is an ellipse, not a circle. The Sun sits at one of the two foci (focal points) of that ellipse, not at the center. An ellipse has a longest axis (the major axis); half of that is called the semi-major axis, labeled *a*. The point in the orbit closest to the Sun is **perihelion**; the farthest point is **aphelion**. **Second Law — The Law of Equal Areas:** A line drawn from the Sun to the planet sweeps out equal areas in equal amounts of time. This means when a planet is close to the Sun (near perihelion), it moves *fast* to sweep a fat, short triangle. When it is far away (near aphelion), it moves *slow* to sweep a long, thin triangle — but both triangles have the same area. This follows from conservation of angular momentum: the component of a planet's velocity perpendicular to its radius line, multiplied by its distance from the Sun, stays constant (r × v_perp = constant). At perihelion, r is small, so v_perp must be large; at aphelion, r is large, so v_perp must be small. **Third Law — The Law of Periods:** The square of a planet's orbital period (*T*) is proportional to the cube of its semi-major axis (*a*). Written out: **T² = a³**, where *T* is in Earth years and *a* is in astronomical units (AU, where 1 AU = Earth–Sun distance). So if you know how far a planet sits from the Sun on average, you can calculate exactly how long its year is. Example: Mars has a semi-major axis of about 1.52 AU. Then a³ = 1.52³ ≈ 3.51, so T² ≈ 3.51, giving T ≈ 1.87 Earth years — very close to Mars's actual 1.88-year orbit. Hint for the activity: Remember, the Sun does not sit at the center of the ellipse — it sits at one of the two foci. Use that anchor to place everything else correctly.

Activity

Drag each orbital label to the correct position on the planetary orbit diagram

Practice

Planet Y orbits at a semi-major axis of 9 AU around a Sun-like star. Calculate its orbital period using Kepler's third law and compare it to Earth's.

Explain, using conservation of angular momentum, why a comet whips through perihelion quickly but lingers for years near aphelion.

Common mistakes to avoid

The Sun sits at the center of the ellipse.The Sun occupies one of the two foci of the ellipse; the geometric center has no physical body at it.
Planets speed up near the Sun simply because they are closer.The speed-up follows from conservation of angular momentum, which forces perpendicular velocity to rise as the orbital radius shrinks.

Check your understanding

Earth's average distance from the Sun is 1 AU and its period is 1 year. A newly discovered planet has a semi-major axis of 4 AU. What is its orbital period?

According to Kepler's Second Law, a planet sweeps equal areas in equal times. What does this imply about a planet's orbital speed as it moves from aphelion toward perihelion?

Where is the Sun located in a planet's elliptical orbit?

Two planets orbit the same star. Planet X has a semi-major axis of 1 AU and Planet Y has a semi-major axis of 9 AU. How do their orbital periods compare?

Recap

Kepler's three laws describe planetary orbits as ellipses with the Sun at one focus, with planets sweeping equal areas in equal times due to angular momentum conservation, and with orbital periods related to size by T² ∝ a³.

Reflect

Why was Kepler's leap from perfect circles to ellipses such a turning point in how humanity understood the heavens?