Lesson 4 of 5 · Ellipses & Kepler’s laws
The patterns in an orbit
8 minute lesson · 3 checks along the way
Your goal for this lessonUse orbital shape, changing speed, and orbital period to interpret the three patterns described by Kepler’s laws.
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A circle is one kind of ellipse
Johannes Kepler used observations of planetary motion to identify patterns that improved predictions. His first law describes a planet’s orbit as an with the Sun at one focus. An ellipse has two foci, which are geometric reference points inside it. The Sun is not generally at the ellipse’s center, and the other focus does not need to contain an object.
A circle is the special case in which the two foci coincide. Eccentricity describes how far an ellipse departs from a circle: zero for a circle, between zero and one for a noncircular closed ellipse. A very stretched textbook drawing need not resemble a particular planet’s actual orbit. Labels and measured eccentricity are more informative than a dramatic illustration.
Cassini’s view of Saturn on March 27, 2004, combines red, green, and blue exposures; the source team brightened the small moons. We see the rings at an angle, so their apparent oval shape is also affected by perspective.
Look closely
The rings look like a narrow oval. Does that prove the particles follow equally stretched orbits?
See what to notice
A nearly circular ring viewed obliquely can look like a narrow ellipse. The main rings contain many particles following their own, mostly near-circular paths around Saturn. To determine an orbit’s true shape, scientists account for viewing angle and measure motion over time.
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Equal times do not mean equal distances
Kepler’s second law says that a line from the Sun to a planet sweeps out equal areas in equal times. The planet therefore moves faster near the Sun and more slowly farther away along the same elliptical orbit. The distance traveled in a week need not be the same everywhere. The equal quantity is the swept area, not the length of the path.
Imagine marking a comet’s location on a map of its orbit at equally spaced times. In a strongly elliptical solar orbit, marks cluster near its distant, slower part and spread apart near its close, faster passage. That does not mean the clock has changed. A sequence of timed observations reveals a pattern that a single photograph of the comet cannot show.
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A larger orbit takes longer
The is the time for one complete circuit. The is half the ellipse’s longest diameter and describes its size. For small bodies orbiting the same dominant mass, Kepler’s third law relates these quantities: a larger semi-major axis means a longer period. Shape and size are different properties, so a more stretched drawing alone does not establish a longer year.
Optional math makes the relationship precise: the period squared is proportional to the semi-major axis cubed. For small bodies around the Sun, using years and astronomical units, a hypothetical orbit with a semi-major axis of 4 AU has a period of about 8 years. Comparing bodies around different central masses requires accounting for that too; the same orbit size need not mean the same period.
Work through a decision
Predict, compare, and refine
These patterns connect many systems. Separate particles in Saturn’s rings follow their own paths around Saturn rather than moving as a single solid disk. Moons and artificial satellites also follow gravitational trajectories. Applying Kepler’s relationships means identifying the relevant central mass and checking the approximation, rather than inserting Saturn into a formula calibrated specifically for the Sun.
Kepler’s laws describe the ideal isolated two-body problem. More precisely, both bodies move around their shared center of mass, or barycenter; treating a much heavier body as fixed is an approximation. Real planets perturb one another’s paths. A disagreement between a simple prediction and observations can therefore motivate a more complete calculation, rather than erase the value of the original pattern.
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