Kepler's Third Law of Planetary Motion
States that the square of the orbital period of any planet is proportional to the cube of the semi-major axis of its elliptical orbit.
What is Kepler's Third Law of Planetary Motion?
The time taken by a celestial body to complete one full orbit around another body.
Key formula / rule: Kepler's Third Law (Proportionality)
Key points
- Understand the mathematical relationship between orbital period and semi-major axis.
- Apply Kepler's Third Law to compare orbital characteristics of different celestial bodies.
- Recognize the significance of this law in celestial mechanics.
Common exam trap
Confusing semi-major axis with the radius for circular orbits.
Definitions
- Term
Orbital Period (T)
- Meaning
The time taken by a celestial body to complete one full orbit around another body.
- Term
Semi-major axis (a)
- Meaning
Half of the longest diameter of an ellipse; it represents the average distance of a planet from the Sun in its elliptical orbit.
Learning objectives
Understand the mathematical relationship between orbital period and semi-major axis.
Apply Kepler's Third Law to compare orbital characteristics of different celestial bodies.
Recognize the significance of this law in celestial mechanics.
Formulae
- Name
Kepler's Third Law (Proportionality)
- Note
T is the orbital period, a is the semi-major axis of the orbit.
- Expression
T² ∝ a³
- Name
Kepler's Third Law (with constant)
- Note
k is the constant of proportionality, which depends on the mass of the central body.
- Expression
T² = k * a³
- Name
Kepler's Third Law (derived from Newton's Law for circular orbits)
- Note
M is the mass of the central body (e.g., Sun), G is the universal gravitational constant. For elliptical orbits, 'a' is the semi-major axis.
- Expression
T² = (4π²/GM) * a³
Prerequisites
Basic understanding of elliptical orbits.
Knowledge of proportionality and exponents.
Concept of orbital period and distance.
Common mistakes
Confusing semi-major axis with the radius for circular orbits.
Assuming a linear relationship between period and distance.
Forgetting to square the period or cube the semi-major axis in calculations.
Not considering the mass of the central body when comparing different systems.
Keywords
Kepler's Laws
Planetary Motion
Orbital Period
Semi-major Axis
Elliptical Orbit
Celestial Mechanics
Gravitation
Practice preview
For two planets, Planet A and Planet B, orbiting the same star, if Planet A has a semi-major axis twice that of Planet B, what is the ratio of their orbital periods (T_A / T_B)?…
medium
According to Kepler's Third Law of Planetary Motion, the square of the orbital period (T) of a planet is proportional to which of the following?…
easy
If the semi-major axis of a planet's orbit is doubled, how does its orbital period change, according to Kepler's Third Law?…
medium