Kepler's Second Law of Planetary Motion
States that the line joining any planet to the Sun sweeps out equal areas in equal intervals of time, implying conservation of angular momentum.
What is Kepler's Second Law of Planetary Motion?
The line joining a planet to the Sun sweeps out equal areas in equal intervals of time.
Key formula / rule: Areal Velocity
Key points
- State Kepler's Second Law of Planetary Motion.
- Explain the physical implications of the law (varying orbital speed).
- Relate Kepler's Second Law to the conservation of angular momentum.
- Derive or explain the constancy of areal velocity from angular momentum conservation.
Common exam trap
Confusing the law with constant orbital speed; orbital speed varies.
Definitions
- Term
Kepler's Second Law
- Meaning
The line joining a planet to the Sun sweeps out equal areas in equal intervals of time.
- Term
Areal Velocity
- Meaning
The rate at which the area is swept out by the position vector of a planet with respect to the Sun.
- Term
Angular Momentum
- Meaning
A measure of the rotational motion of an object, defined as the cross product of the position vector and the linear momentum (L = r x p).
- Term
Central Force
- Meaning
A force whose direction is always along the line joining the interacting particles and whose magnitude depends only on the distance between them.
Learning objectives
State Kepler's Second Law of Planetary Motion.
Explain the physical implications of the law (varying orbital speed).
Relate Kepler's Second Law to the conservation of angular momentum.
Derive or explain the constancy of areal velocity from angular momentum conservation.
Apply the concept to qualitative and quantitative problems involving planetary motion.
Formulae
- Name
Areal Velocity
- Note
Rate at which area is swept by the position vector.
- Expression
dA/dt = (1/2) |r x v|
- Name
Angular Momentum
- Note
For a central force, angular momentum L is conserved.
- Expression
L = r x p = r x (mv)
- Name
Relation between Areal Velocity and Angular Momentum
- Note
m is the mass of the planet. Since L and m are constant, dA/dt is constant.
- Expression
dA/dt = L / (2m)
Prerequisites
Basic understanding of vectors (position, velocity, force).
Concept of area and rate of change.
Newton's Laws of Motion.
Concept of gravitational force.
Understanding of angular momentum and torque.
Basic calculus (derivatives for rates of change).
Common mistakes
Confusing the law with constant orbital speed; orbital speed varies.
Not understanding the connection to conservation of angular momentum.
Assuming the law applies only to circular orbits (it applies to elliptical orbits).
Incorrectly calculating areal velocity or angular momentum.
Forgetting that the force must be central for angular momentum to be conserved about the center of force.
Keywords
Kepler's Laws
Second Law
Law of Equal Areas
Planetary Motion
Orbital Mechanics
Angular Momentum
Conservation of Angular Momentum
Central Force
Areal Velocity
Perihelion
Aphelion
Gravitation
Practice preview
If the Sun suddenly vanished, what would happen to the Earth's motion according to Kepler's laws?…
hard
If a planet is moving in an elliptical orbit around the Sun, its speed will be maximum when it is:…
medium
Consider a planet revolving around the Sun in an elliptical orbit. Let $v_p$ be its speed at perihelion and $v_a$ be its speed at aphelion. Let $r_p$ be the distance at perihelion and $r_a$ be the distance at aphelion. W…
medium