1. Introduction

Kepler’s Laws of Planetary Motion describe how planets move around the Sun.

They were developed by the German astronomer Johannes Kepler based on detailed observations collected by Tycho Brahe.

Although Kepler discovered these laws before the development of modern physics, they were later explained by Isaac Newton using the Law of Universal Gravitation.

Kepler’s Laws provide the foundation for understanding:

• planetary motion
• satellite orbits
• comets
• binary star systems

2. Kepler’s First Law

The Law of Ellipses

Every planet moves in an elliptical orbit with the Sun located at one focus of the ellipse.

Mathematical Statement

The orbit is an ellipse rather than a perfect circle.

An ellipse has:

• a major axis
• a minor axis
• two foci

The Sun occupies one focus, not the center.

Physical Meaning

The distance between a planet and the Sun changes continuously during its orbit.

As a result:

• orbital speed changes
• gravitational force changes

throughout the orbit.

3. Important Terms

Perihelion

The point where a planet is closest to the Sun.

At perihelion:

rmin⁡r_{\min}rmin​

The gravitational force is greatest.

The orbital speed is maximum.

Aphelion

The point where a planet is farthest from the Sun.

At aphelion:

rmax⁡r_{\max}rmax​

The gravitational force is smallest.

The orbital speed is minimum.

4. Kepler’s Second Law

The Law of Equal Areas

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

Illustration

Suppose a planet moves:

• fast near perihelion
• slow near aphelion

Even though the distances traveled are different, the area swept out in equal time intervals is identical.

Physical Meaning

Planets move faster when they are closer to the Sun and slower when they are farther away.

This behavior is a direct consequence of angular momentum conservation.

5. Connection to Angular Momentum

The angular momentum of a planet is:

$$L=mvr$$

when velocity is perpendicular to the radius.

Since gravitational force acts toward the Sun:

$$\tau =0$$

Therefore:

$$L=\text{constant}$$

As the distance decreases:

$$r\downarrow$$

the speed must increase:

$$v\uparrow$$

This explains Kepler’s Second Law.

6. Kepler’s Third Law

The Harmonic Law

The square of a planet’s orbital period is proportional to the cube of its orbital radius.

Mathematical Form

For circular or nearly circular orbits:

T2∝r3T^2 \propto r^3T2∝r3

or

T2r3=constant\frac{T^2}{r^3} = \text{constant}r3T2​=constant

where:

• $T$ = orbital period
• $r$ = orbital radius

7. Newton’s Derivation of the Third Law

For a planet in circular orbit:

Gravitational force provides centripetal force.

FG=FCF_G = F_CFG​=FC​

Using Newton’s Law of Gravitation:

GMmr2=mv2r\frac{GMm}{r^2} = \frac{mv^2}{r}r2GMm​=rmv2​

The mass of the planet cancels.

Substituting:

v=2πrTv = \frac{2\pi r}{T}v=T2πr​

gives:

T2=4π2GMr3T^2 = \frac{4\pi^2}{GM}r^3T2=GM4π2​r3

This is the mathematical form of Kepler’s Third Law.

8. General Form of Kepler’s Third Law

For any object orbiting the same central mass:

T12r13=T22r23\frac{T_1^2}{r_1^3} = \frac{T_2^2}{r_2^3}r13​T12​​=r23​T22​​

This equation is commonly used in AP Physics C problems.

9. Example Problem

A satellite orbits Earth with radius:

r1=Rr_1 = Rr1​=R

and period:

T1=90 minutesT_1 = 90\ \text{minutes}T1​=90 minutes

Another satellite orbits at:

r2=4Rr_2 = 4Rr2​=4R

Find its orbital period.

Solution

Using:

T12r13=T22r23\frac{T_1^2}{r_1^3} = \frac{T_2^2}{r_2^3}r13​T12​​=r23​T22​​

Substitute values:

T12R3=T22(4R)3\frac{T_1^2}{R^3} = \frac{T_2^2}{(4R)^3}R3T12​​=(4R)3T22​​ T22=64T12T_2^2 = 64T_1^2T22​=64T12​ T2=8T1T_2 = 8T_1T2​=8T1​ T2=720 minutesT_2 = 720\ \text{minutes}T2​=720 minutes

10. Circular Orbit as a Special Case

A circle is a special type of ellipse.

For a circular orbit:

$$r=\text{constant}$$

The orbital speed remains constant.

Even in this case, Kepler’s Laws still apply.

11. Applications of Kepler’s Laws

Planetary Motion

Explains how planets move around the Sun.

Artificial Satellites

Used to determine orbital periods and orbital radii.

Binary Stars

Allows astronomers to determine stellar masses.

Space Missions

Used for spacecraft trajectory design.

12. Relationship to Newton’s Law of Gravitation

Kepler discovered the laws empirically through observation.

Newton later explained them using:

FG=GMmr2F_G = \frac{GMm}{r^2}FG​=r2GMm​

Kepler’s Laws are therefore consequences of Newton’s Law of Universal Gravitation.

13. Common AP Physics C Mistakes

Mistake 1

Assuming planets move with constant speed.

Only circular orbits have constant speed.

Elliptical orbits do not.

Mistake 2

Confusing period with orbital speed.

A longer orbital period generally corresponds to a larger orbit.

Mistake 3

Using Kepler’s Third Law for objects orbiting different central masses.

The equation:

T2r3=constant\frac{T^2}{r^3} = \text{constant}r3T2​=constant

is valid only when the same central body is involved.

14. Key Concept Connections

First Law

Describes the shape of orbits.

Second Law

Describes how orbital speed changes.

Third Law

Relates orbital size to orbital period.

Newton’s Gravitation

Explains why all three laws work.

Summary

Kepler’s First Law

Planets move in elliptical orbits with the Sun at one focus.

Kepler’s Second Law

Equal areas are swept out in equal times.

Kepler’s Third Law

T2∝r3T^2 \propto r^3T2∝r3

or

T2=4π2GMr3T^2 = \frac{4\pi^2}{GM}r^3T2=GM4π2​r3

for circular orbits.

Key ideas:

• planetary orbits are elliptical
• planets move faster when closer to the Sun
• orbital period increases with orbital radius
• Kepler’s Laws are explained by Newton’s Law of Gravitation

Kepler’s Laws form the bridge between observational astronomy and Newtonian mechanics, making them one of the most important topics in AP Physics C Mechanics.