Kepler’s Three Laws of Planetary Motion for HSC Physics

Imagine a planet moving around the Sun. If the orbit were a perfect circle and the planet moved at constant speed, the motion would be easy to describe. But real planetary motion does neither.

A planet’s path is slightly stretched, its speed changes during the orbit, and planets farther from the Sun take much longer to complete one revolution.

Kepler’s three laws organise those observations into one picture.

Before reading further, make a prediction. A planet is moving from the far side of its orbit towards the Sun. What should happen to its speed?

It speeds up.

Kepler’s second law tells us that this must happen. Newtonian mechanics later explains why.

01The three laws at a glance

Kepler’s laws describe three different parts of orbital motion.

The important thing is not to memorise three disconnected statements. They fit together.

A planet follows an ellipse, so its distance from the Sun changes. Because that distance changes, its speed changes. The overall size of the orbit then determines how long the full journey takes.

02Kepler’s first law: planets move in ellipses

A common first guess is that planets travel in circles with the Sun exactly at the centre.

That is close enough for some rough calculations, but it is not exact.

Kepler’s first law states:

A planet moves in an elliptical orbit, with the Sun at one focus of the ellipse.

An ellipse is basically a stretched circle.

The strange part is the word focus.

What is a focus?

An ellipse has two special points called foci. The Sun sits at one of them.

It does not normally sit at the geometric centre of the ellipse.

Picture tying a loose loop of string around two drawing pins. Put a pencil inside the loop, pull the string tight, and move the pencil around. The shape traced by the pencil is an ellipse.

The two drawing pins represent the two foci.

For a planetary orbit, only one focus contains the Sun. The other focus is just a geometric point.

This string model is useful for understanding the shape, but it does not explain what force makes a planet move that way. Gravity provides that explanation later.

03Perihelion and aphelion

Because the Sun is not at the centre of an elliptical orbit, the planet’s distance from the Sun changes.

Two positions are especially useful:

• perihelion: the point where the planet is closest to the Sun
• aphelion: the point where the planet is furthest from the Sun

The prefix “peri-” means near. “Apo-” means away.

For planets orbiting other stars, we can use the more general terms periapsis and apoapsis, but HSC questions involving the Solar System commonly use perihelion and aphelion.

Semi-major axis

The longest diameter through an ellipse is called the major axis.

Half of this distance is the semi-major axis, \(a\).

This value is extremely important because Kepler’s third law uses the semi-major axis to represent the overall size of an elliptical orbit.

For a circular orbit, the semi-major axis is simply the radius.

Check your understanding

A planet has an elliptical orbit around the Sun. At which point is it closest to the Sun?

04Kepler’s second law: equal areas in equal times

Now suppose the planet moves from aphelion towards perihelion.

Would you expect it to:

1. keep the same speed,
2. slow down, or
3. speed up?

It speeds up.

Kepler’s second law describes this precisely:

A line joining a planet to the Sun sweeps out equal areas in equal intervals of time.

This sounds more complicated than it really is.

Imagine drawing a line from the Sun to the planet. As the planet moves, that line sweeps across part of the orbital plane, rather like the hand of a clock sweeping across its face.

Take two equal time intervals, perhaps 30 days each.

During one 30-day period, the planet is near perihelion.

During another 30-day period, it is near aphelion.

Kepler’s second law says the swept areas must be equal.

05Why equal areas mean changing speed

Near perihelion, the planet is close to the Sun. The line from the Sun to the planet is relatively short.

To sweep out a particular area with this short line, the planet must move through a relatively large distance along its orbit.

So it moves quickly.

Near aphelion, the line is longer.

The same swept area can be produced while the planet travels a smaller distance along its orbit.

So it moves more slowly.

Therefore:

• fastest at perihelion
• slowest at aphelion

This is one of the most important consequences of Kepler’s second law.

A useful mental picture

Imagine you and a friend are dancing while holding hands. Your friend represents the Sun, and you represent the planet.

If you are close together, you may need to move sideways quite quickly to sweep across the same patch of floor.

If your arms somehow became enormously long, you could sweep across the same floor area with less sideways movement.

The analogy helps with the geometry of swept area. It breaks down because planets are not physically attached to the Sun, and gravitational orbital motion must be explained using forces and angular momentum.

06The tempting misconception: gravity is stronger, so the planet is simply “pulled forward”

It is true that gravity is stronger when the planet is closer to the Sun.

But saying “gravity gets stronger, so the planet goes faster” is incomplete.

Gravity points approximately towards the Sun, not simply along the direction of motion. As the planet approaches the Sun, gravity changes both the direction and the magnitude of its velocity.

From Newtonian mechanics, Kepler’s second law follows from the conservation of angular momentum.

For a planet of mass \(m\), orbital angular momentum can be written as

\[
L = mrv_{\perp}
\]

where:

• \(L\) is angular momentum in \(\text{kg m}^2\text{s}^{-1}\)
• \(m\) is the planet’s mass in kg
• \(r\) is its distance from the Sun in m
• \(v_{\perp}\) is the component of velocity perpendicular to the radius line, in \(\text{m s}^{-1}\)

The Sun’s gravitational force acts along the radius line, so it produces no torque about the Sun. The planet’s angular momentum is therefore conserved.

As \(r\) becomes smaller, \(v_{\perp}\) must increase.

That gives the same behaviour described by Kepler’s second law.

Question: comparing speeds

A planet is observed at perihelion and aphelion. Its distance from the Sun at perihelion is \(1.2 \times 10^{11}\) m, while at aphelion it is \(1.8 \times 10^{11}\) m.

At these two extreme points, the velocity is perpendicular to the radius line.

If its speed at aphelion is \(24\text{ km s}^{-1}\), calculate its speed at perihelion.

Step 1: Use conservation of angular momentum

Because \(mrv\) is constant,

\[
mr_{\text{p}}v_{\text{p}} = mr_{\text{a}}v_{\text{a}}
\]

The planet’s mass appears on both sides, so it cancels:

\[
r_{\text{p}}v_{\text{p}} = r_{\text{a}}v_{\text{a}}
\]

Step 2: Rearrange for the perihelion speed

\[
v_{\text{p}} = \frac{r_{\text{a}}v_{\text{a}}}{r_{\text{p}}}
\]

Step 3: Substitute the values

\[
v_{\text{p}}
=
\frac{(1.8 \times 10^{11})(24)}
{1.2 \times 10^{11}}
=
36\text{ km s}^{-1}
\]

07Kepler’s third law: larger orbits take longer

Now compare two planets.

Planet A orbits close to the Sun.

Planet B has a much larger orbit.

Which one should take longer to complete one orbit?

Planet B.

That is not surprising. It has further to travel. But there is another effect: planets farther from the Sun also tend to move more slowly.

So the orbital period increases quite rapidly as orbital size increases.

Kepler found the relationship

\[
T^2 \propto a^3
\]

where:

• \(T\) is the orbital period
• \(a\) is the semi-major axis

This means

\[
\frac{T^2}{a^3} = \text{constant}
\]

for objects orbiting the same central body.

Notice that the relationship is not \(T \propto a\).

Doubling the orbital size does not simply double the orbital period.

Taking the square root of the proportionality gives

\[
T \propto a^{3/2}
\]

So if the semi-major axis doubles,

\[
\frac{T_2}{T_1}
=
\left(\frac{a_2}{a_1}\right)^{3/2}
=
2^{3/2}
\approx 2.83
\]

The orbital period becomes about 2.83 times longer.

08Why Kepler’s third law needs a Newtonian upgrade

Kepler discovered his laws from astronomical observations before Newton developed his theory of gravitation.

Kepler’s version works well when comparing planets orbiting the same central object. But it does not explicitly include the masses involved.

Newton showed why the relationship exists.

For two bodies orbiting one another, the more complete form is

\[
T^2 = \frac{4\pi^2a^3}{G(M+m)}
\]

where:

• \(T\) is orbital period in s
• \(a\) is the semi-major axis of the relative orbit in m
• \(G\) is the universal gravitational constant, \(6.67 \times 10^{-11}\text{ N m}^2\text{kg}^{-2}\)
• \(M\) is the mass of one body in kg
• \(m\) is the mass of the other body in kg

This is the Newtonian form of Kepler’s third law.

It reveals something Kepler’s original relationship hides:

Orbital period depends not only on orbital size, but also on the total mass of the two-body system.

09Why HSC questions often use only the central mass

For a planet orbiting a star,

\[
M \gg m
\]

The star’s mass is usually enormously greater than the planet’s mass.

Therefore,

\[
M + m \approx M
\]

and the equation becomes

\[
T^2 = \frac{4\pi^2r^3}{GM}
\]

for a circular orbit of radius \(r\).

Equivalently,

\[
T = 2\pi\sqrt{\frac{r^3}{GM}}
\]

This form is extremely useful for planets, moons, and artificial satellites when the orbiting body’s mass is negligible compared with the central body’s mass.

10Where does the Newtonian equation come from?

For a circular orbit, gravity supplies the centripetal force.

The gravitational force is

\[
F_g = \frac{GMm}{r^2}
\]

The required centripetal force is

\[
F_c = \frac{mv^2}{r}
\]

For a stable circular orbit,

\[
F_g = F_c
\]

so

\[
\frac{GMm}{r^2}
=
\frac{mv^2}{r}
\]

The orbiting mass \(m\) cancels:

\[
\frac{GM}{r}
=
v^2
\]

Therefore,

\[
v = \sqrt{\frac{GM}{r}}
\]

For one complete circular orbit, distance travelled is \(2\pi r\). Since speed is distance divided by time,

\[
v = \frac{2\pi r}{T}
\]

Substituting this into the orbital-speed equation gives

\[
\frac{2\pi r}{T}
=
\sqrt{\frac{GM}{r}}
\]

Squaring and rearranging produces

\[
T^2 = \frac{4\pi^2r^3}{GM}
\]

So Kepler’s observed relationship \(T^2 \propto r^3\) falls directly out of Newton’s laws of motion and universal gravitation.

That connection is important. Kepler described what planets do. Newton explained why they do it.

11A harder comparison: changing orbital radius

A satellite is moved from a circular orbit of radius \(r\) to another circular orbit of radius \(3r\) around the same planet.

How many times longer is its new orbital period?

You could substitute everything into Newton’s full equation, but there is a faster method.

Because the central mass has not changed,

\[
T^2 \propto r^3
\]

Therefore,

\[
\frac{T_2^2}{T_1^2}
=
\frac{r_2^3}{r_1^3}
\]

Step 1: Substitute the radius ratio

Since \(r_2 = 3r_1\),

\[
\frac{T_2^2}{T_1^2}
=
3^3
=
27
\]

Step 2: Take the square root

\[
\frac{T_2}{T_1}
=
\sqrt{27}
\approx 5.20
\]

12Do heavier planets orbit faster?

Suppose two planets of very different masses orbit the same star at the same orbital radius.

A tempting prediction is that the heavier planet experiences a stronger gravitational force, so perhaps it should orbit faster.

The first part is true. The heavier planet does experience a larger gravitational force.

But it also has more inertia by exactly the same factor.

For a circular orbit,

\[
\frac{GMm}{r^2}
=
\frac{mv^2}{r}
\]

The planet’s mass \(m\) cancels.

So

\[
v = \sqrt{\frac{GM}{r}}
\]

The orbital speed does not depend on the orbiting object’s mass, provided the central body is much more massive.

That is why the simplified period equation,

\[
T = 2\pi\sqrt{\frac{r^3}{GM}},
\]

contains the central mass \(M\), but not the satellite or planet mass.

The more exact two-body equation does contain \(M+m\), so the statement is not mathematically exact when the two bodies have comparable masses.

13Putting all three laws together

Consider one planet completing an elliptical orbit.

At aphelion, it is furthest from the Sun and moving most slowly.

As it falls towards the Sun, gravity accelerates it. Its speed increases.

At perihelion, it is closest to the Sun and moving most quickly.

As it travels away again, gravity continues pulling towards the Sun, which now reduces the planet’s speed.

Throughout the orbit, the Sun-planet line sweeps out equal areas in equal times.

That is Kepler’s second law operating inside the elliptical path described by the first law.

Meanwhile, the overall semi-major axis \(a\) determines the timescale of the entire orbit through the third law:

\[
T^2 \propto a^3
\]

Newton then connects all three behaviours to one underlying interaction: gravity.

14Be careful about radius versus semi-major axis

For a circular orbit, the distance from the central body is constant, so using the orbital radius \(r\) is straightforward.

For an elliptical orbit, the planet does not have one constant orbital radius.

Its distance from the Sun changes continuously.

That is why the general form of Kepler’s third law uses the semi-major axis \(a\):

\[
T^2 = \frac{4\pi^2a^3}{G(M+m)}
\]

Do not automatically substitute the perihelion distance or aphelion distance into this equation.

Use the semi-major axis.

If the perihelion distance \(r_{\text{p}}\) and aphelion distance \(r_{\text{a}}\) are known, then

\[
a = \frac{r_{\text{p}} + r_{\text{a}}}{2}
\]

Question: finding the semi-major axis

A comet has a perihelion distance of \(0.50\text{ AU}\) and an aphelion distance of \(9.50\text{ AU}\).

Find its semi-major axis.

Step 1: Use the relationship

\[
a = \frac{r_{\text{p}} + r_{\text{a}}}{2}
\]

Step 2: Substitute

\[
a
=
\frac{0.50 + 9.50}{2}
=
5.00\text{ AU}
\]

15A compact decision rule for exam questions

When you see an orbital-motion question, first ask what the question is really testing.

The next useful step is to connect Kepler’s laws to orbital energy. Kepler’s second law tells you that orbital speed changes, but energy explains where that changing speed comes from: as a planet moves closer to the Sun, gravitational potential energy decreases while kinetic energy increases. That gives a second, very powerful way to analyse the same orbit.