Kepler's Laws of Planetary Motion
Orbits are ellipses with the Sun at a focus; the radius vector sweeps equal areas in equal ×; and T2 is proportional to a3.
What is Kepler's Laws of Planetary Motion?
One half of the major axis of an ellipse; the average distance of a planet from the Sun.
Key formula / rule: Kepler's Third Law
Key points
- Understand the geometry of planetary orbits.
- Relate areal velocity to angular momentum conservation.
- Calculate orbital periods using the Third Law.
- Analyze velocity changes at different points in an elliptical path.
Common exam trap
Using the diameter instead of the semi-major axis in the Third Law.
Definitions
- Term
Semi-major axis
- Meaning
One half of the major axis of an ellipse; the average distance of a planet from the Sun.
- Term
Areal Velocity
- Meaning
The rate at which area is swept out by the radius vector of a celestial body.
Learning objectives
Understand the geometry of planetary orbits.
Relate areal velocity to angular momentum conservation.
Calculate orbital periods using the Third Law.
Analyze velocity changes at different points in an elliptical path.
Formulae
- Name
Kepler's Third Law
- Note
Where 'a' is the semi-major axis and 'M' is the mass of the central body.
- Expression
T2 = (4π2 / GM) * a3
- Name
Areal Velocity
- Note
L is angular momentum and m is the mass of the planet; remains constant.
- Expression
dA/dt = L / (2m)
- Name
Velocity at Apsides
- Note
Applicable at perihelion and aphelion.
- Expression
v1 * r1 = v2 * r2
Prerequisites
Newton's Law of Universal Gravitation
Conservation of Angular Momentum
Basic properties of an Ellipse
Common mistakes
Using the diameter instead of the semi-major axis in the Third Law.
Assuming linear velocity is constant throughout the orbit.
Confusing Kepler's constant 'K' as being the same for all central bodies (it differs for Sun vs. Earth).
Keywords
Ellipse
Areal Velocity
Angular Momentum
Semi-major axis
Perihelion
Aphelion
Practice preview
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