2.6.4.1. Kepler's Laws

Table of Contents

From observation to law

Kepler's laws describe how planets move around the Sun. They were discovered from careful astronomical observations, long before Newton explained why they work. In this chapter, the goal is to understand the three laws as statements about planetary motion itself.

Kepler's laws apply very well to planets, moons, and many satellites when one body is much more massive than the other. They describe the shape of the orbit, how the speed changes during the motion, and how the orbital period depends on the size of the orbit.

The first law, elliptical orbits

Kepler's first law says that a planet moves in an ellipse, with the Sun at one focus of the ellipse.

An ellipse is a closed curve that looks like a stretched circle. A circle is a special case of an ellipse. The important point is that the Sun is not at the center of the ellipse in general, it is at one of the two foci.

Rendering scientific sketch…

Ellipse with the Sun at one focus

Two important points on the orbit are perihelion and aphelion. Perihelion is the point where the planet is closest to the Sun. Aphelion is the point where it is farthest from the Sun. The planet does not stay at the same distance from the Sun during its orbit unless the orbit is a circle.

A useful quantity for describing how stretched an ellipse is called the eccentricity, written as $e$. For a circle, $e = 0$. For an ellipse, $0 < e < 1$. The larger the eccentricity, the more elongated the orbit.

Kepler's first law states:
A planet moves in an elliptical orbit with the Sun at one focus.

Basic geometry of the ellipse

Although a full mathematical study of ellipses belongs elsewhere, a few orbit terms are very useful here.

The semimajor axis, written as $a$, is half the longest diameter of the ellipse. It is one of the most important orbit parameters because it sets the overall size of the orbit.

The perihelion distance is often written as $r_p$, and the aphelion distance as $r_a$. For an ellipse,

$$
r_p = a(1 - e)
$$

and

$$
r_a = a(1 + e)
$$

So the same orbit can be described by its size, through $a$, and its shape, through $e$.

Orbit type	Eccentricity $e$	Shape
Circle	$0$	perfectly round
Ellipse	$0 < e < 1$	stretched closed orbit

For an elliptical orbit,
$$
r_p = a(1-e), \qquad r_a = a(1+e)
$$
where $a$ is the semimajor axis and $e$ is the eccentricity.

The second law, equal areas in equal times

Kepler's second law says that the line joining the planet to the Sun sweeps out equal areas in equal time intervals.

This law tells us how the speed changes as the planet moves. When the planet is near the Sun, it moves faster. When it is far from the Sun, it moves slower.

Imagine drawing a line from the Sun to the planet. As the planet moves, that line sweeps across the orbit and creates a sector-like region. Kepler's second law says that if you compare two equal time intervals, the areas swept out are equal, even though the shapes of those swept regions may look different.

Rendering scientific sketch…

Equal areas in equal times

Near perihelion, the planet covers a longer arc in the same time because it is moving faster. Near aphelion, it covers a shorter arc in the same time because it is moving slower.

Kepler's second law states:
Equal areas are swept out in equal times.
This means the planet moves faster when it is closer to the Sun, and slower when it is farther away.

What the second law means physically

The second law is not just a geometric curiosity. It gives a direct qualitative rule for orbital speed.

If a planet moves from aphelion toward perihelion, its speed increases. After passing perihelion and moving outward again, its speed decreases. So orbital motion is generally not motion at constant speed.

This law also shows that the Sun has a special role in the orbit, because the swept area is measured from the Sun's position at the focus, not from the center of the ellipse.

The third law, period and orbit size

Kepler's third law connects the orbital period to the size of the orbit.

The orbital period, written as $T$, is the time needed for one complete orbit. Kepler found that for planets orbiting the Sun, the square of the period is proportional to the cube of the semimajor axis:

$$
T^2 \propto a^3
$$

This means that planets farther from the Sun take much longer to complete one orbit.

In proportional form, this law tells us the pattern. For two planets orbiting the same central body, we can write

$$
\frac{T_1^2}{a_1^3} = \frac{T_2^2}{a_2^3}
$$

This is often the most useful form for comparing two orbits.

Kepler's third law states:
$$
T^2 \propto a^3
$$
For two bodies orbiting the same central mass,
$$
\frac{T_1^2}{a_1^3} = \frac{T_2^2}{a_2^3}
$$

Understanding the third law with examples

Suppose one planet is farther from the Sun than another. Its semimajor axis is larger, so its orbital period must also be larger.

If planet B has a larger orbit than planet A, then planet B moves more slowly on average and takes longer to go around once.

For example, if an orbit has semimajor axis $a$, and another orbit has semimajor axis $2a$, then

$$
\frac{T_2^2}{T_1^2} = \frac{(2a)^3}{a^3} = 8
$$

$$
T_2 = \sqrt{8}\,T_1 = 2\sqrt{2}\,T_1
$$

So doubling the semimajor axis does not just double the period, it increases it by a larger factor.

Comparing the three laws

Each law answers a different question about planetary motion.

Kepler's law	What it tells us
First law	the shape of the orbit
Second law	how the speed changes along the orbit
Third law	how orbital period depends on orbit size

Taken together, they give a complete basic picture of planetary motion around the Sun.

Historical importance

Kepler's laws were a major step in physics because they replaced the old idea that planets had to move in perfect circles. The laws were based on observation and mathematics, not on philosophical preference.

Later, Newton showed that these laws follow from gravity. That deeper explanation belongs to later topics, but it is important to remember the order of discovery. Kepler first found the pattern, Newton later explained the cause.

Limits and usefulness

Kepler's laws are extremely accurate for many situations, but they are idealized. Real planets slightly disturb one another, so actual orbits are not perfectly fixed ellipses forever. Even so, Kepler's laws remain one of the most useful first descriptions of orbital motion.

They also apply beyond planets. Moons around planets, artificial satellites around Earth, and many binary systems can be described approximately by the same ideas.

Core summary:
The orbit shape is an ellipse.
The central body is at one focus.
The orbiting body moves faster when closer, slower when farther.
Larger orbits have longer periods, according to
$$
T^2 \propto a^3
$$

Final picture

Kepler's laws give a simple and elegant description of planetary motion. A planet follows an ellipse, the Sun sits at a focus, the planet speeds up near the Sun and slows down far away, and the time for one orbit grows predictably with orbital size. These three rules are among the most important observational laws in classical mechanics.

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