Step by Step
1
A quick recap of laws 1 and 2
Law 1: orbits are ellipses, with the Sun at one focus rather than the center. Law 2: a line from the Sun to a planet sweeps equal areas in equal times, meaning the planet moves faster near perihelion (closest approach).
2
Law 3, stated precisely
P² = a³, where P is the orbital period measured in years, and a is the semi-major axis (average orbital distance) measured in AU. This specific unit combination (years and AU) is what makes the equation this clean, with no extra constants needed.
3
Worked example: Earth
Earth orbits at 1 AU with a period of 1 year. Checking the law: 1² = 1, and 1³ = 1 — confirming the relationship holds exactly for Earth by definition, since Earth's distance and period were used to define the AU and year units in the first place.
4
Worked example: Mars
Mars orbits at 1.52 AU with a period of 1.88 years. Checking the law: 1.88² ≈ 3.53, and 1.52³ ≈ 3.51 — these values match closely, confirming the law holds for Mars as well, with only minor rounding differences.
Applied Walkthrough
1
Using Kepler's third law, if you know a planet's average distance from the Sun in AU, you can calculate its orbital period in years — or vice versa — without needing any other information about the planet's mass or composition.
2
For Earth, this relationship is almost trivially satisfied: at 1 AU with a 1-year period, both sides of the equation (1² and 1³) equal exactly 1.
3
For Mars, the relationship holds just as well with real, non-trivial numbers: at 1.52 AU with a 1.88-year period, 1.88² (about 3.53) closely matches 1.52³ (about 3.51).
4
This same equation, first discovered purely empirically by Kepler through careful observation, was later shown by Newton to follow directly from the mathematics of gravitational attraction — connecting an observational pattern to a deeper physical law.
Exam Application
Exams test whether you can correctly apply P² = a³ to calculate a missing value (period or distance) given the other, and whether you understand why this equation takes such a clean form specifically when using years and AU as units.
⚠ Common Trap
The most common trap is forgetting the specific units required for the clean P² = a³ relationship — this simple form only works when P is measured in years and a is measured in AU; using other units would require an additional constant in the equation.
✓ Quick Self-Check
1. What is Kepler's third law, in equation form?
P² = a³, where P is in years and a is in AU.
Tap to reveal / hide
2. Why does this equation take such a clean form for Earth?
Because Earth's orbit (1 AU, 1 year) was used to define these very units, so both sides trivially equal 1.
Tap to reveal / hide
3. Verify Kepler's third law for Mars, given a = 1.52 AU and P = 1.88 years.
1.88² ≈ 3.53, and 1.52³ ≈ 3.51 — closely matching, confirming the law.
Tap to reveal / hide
4. What units must be used for the simple P² = a³ form to work?
Years for period, and AU for semi-major axis.
Tap to reveal / hide
5. Who later derived this law from a deeper physical principle, and what was it?
Newton, from the law of universal gravitation.
Tap to reveal / hide