Why does a planet race past the Sun and then dawdle far away?

How to use this simulation

1. Start with the defaults: a = 1.0 AU and e = 0.50. The period reads 1.00 yr. The planet moves fast near the Sun and slowly far away.
2. Drag the eccentricity slider to 0. The ellipse becomes a circle and the planet moves at one steady speed. Push it toward 0.9 and the orbit becomes a long, thin ellipse.
3. Look at the shaded sectors. Each one is the area swept in the same time. Near the Sun it is short and wide, far away it is long and thin, yet the two outlined sectors show the same area.
4. Change the number of sectors. More sectors means thinner slices, but each pair still has equal area.
5. Raise the semi-major axis and watch the dot on the graph climb the curve. The period grows faster than the distance, as a^1.5.
6. Look at the speed bars. They show the planet's speed, the speed of a circular orbit at that distance, and the escape speed. The escape bar always sits √2 times above the circular one.

Why a planet speeds up near the Sun

Roll a marble around the inside of a funnel and you will see something odd. When the marble dips toward the centre it zips along. When it climbs to the rim it slows down. A planet does the same thing around the Sun, only with gravity doing the job of the funnel.

Long before gravity was understood, the astronomer Tycho Brahe spent years recording exact positions of Mars. His assistant Johannes Kepler studied those numbers for years and found that no circle could fit them. What fitted was an ellipse, and with it two more regularities. Newton later showed that all three follow from the inverse-square law of gravity.

Why does it matter for exams? Because the three laws give you a short list of reliable tools: where the Sun sits, how speed changes along the orbit, and how the period depends on size. Add escape speed and you can answer almost every question this topic asks.

The three laws in plain words

Learn these three sentences first. Everything else on the page is a careful look at one of them.

First law: an ellipse with the Sun at a focus

An ellipse is a stretched circle. It has two special points called foci. Pick any point on the ellipse and add its distances to the two foci: the total is always the same, and it equals twice the semi-major axis, 2a. A gardener can draw an ellipse by tying a string between two pegs and pulling it tight with a pencil.

The Sun sits at one focus. The other focus is empty. So the Sun is not in the middle of the orbit, and the planet is nearer to it at one end than at the other. The nearest point is the perihelion and the farthest point is the aphelion.

How stretched is the ellipse? That is the eccentricity e. For a circle e = 0. For a very long thin ellipse e is close to 1. The semi-major axis a is half the longest width. From a and e you get everything: the perihelion distance is a(1 − e), the aphelion distance is a(1 + e), and the semi-minor axis is b = a√(1 − e²).

Real planets are almost circles. Earth's e is only about 0.017, so you could not tell its orbit from a circle by eye. The simulation uses e = 0.50 by default on purpose, to make the effects big enough to see.

Take the default orbit, a = 1.0 AU. The perihelion is 1.0 × (1 − 0.50) = 0.50 AU and the aphelion is 1.0 × (1 + 0.50) = 1.50 AU. The table shows how the distances and speeds change as e rises, with a fixed at 1 AU.

Second law: equal areas in equal times

Draw a line from the Sun to the planet. As the planet moves, the line sweeps over a region like the hand of a clock, except that the line gets longer and shorter. Kepler's second law says that in equal times the line sweeps equal areas.

Think about what that demands. Near the Sun the line is short, so to sweep the same area the planet must cover a lot of path. Far from the Sun the line is long, so a short stretch of path already sweeps the same area. Therefore the planet must move fast near the Sun and slowly far away. This is exactly what you see on the stage.

The simulation cuts one period into 8 equal-time sectors. Each sector takes 45.7 days of a 1.00-year orbit. The two outlined sectors, at perihelion in green and at aphelion in red, look nothing alike, one short and fat, one long and thin. Yet each has the area 0.340 AU², which is the whole ellipse area πab divided by 8. The bottom of the stage prints both numbers side by side.

Why is it true? Gravity from the Sun always points along the line to the Sun. A force like that can never twist the planet around the Sun, so the planet's angular momentum stays constant. The rate of sweeping area is exactly half the angular momentum per unit mass, so it stays constant too. Kepler found the law by measurement; the reason came from Newton.

A neat consequence is that speed times distance is the same at perihelion and aphelion, because the velocity there is perpendicular to the line to the Sun. For the default orbit the speed at perihelion is 51.6 km/s and at aphelion 17.2 km/s, a ratio of 3.0. That is exactly (1 + e) ÷ (1 − e).

Third law: bigger orbits take longer, and faster than you think

Kepler's third law connects the size of an orbit to its period. The square of the period is proportional to the cube of the semi-major axis, T² ∝ a³. A planet that is twice as far does not take twice as long. It takes 2^1.5, almost three times as long.

Use units that suit the Sun: distances in AU, time in years, mass in solar masses. Then the law becomes pleasantly simple, T² = a³ ÷ M. For the Sun M = 1, so T = a^1.5. Earth has a = 1 and T = 1. If you know any planet's distance you know its year.

The first table checks this against the planets. The last column, T² ÷ a³, comes out as 1 every time, because the constant is the same for everything going around the Sun. The second table shows the powers for a few round numbers.

Change the central mass with the slider. A heavier star pulls harder, so the same orbit is run faster and the period is shorter by √M. That is the clever side of the law: measure the period and size of any orbit, and you can weigh the thing in the middle. This is how astronomers weigh stars with planets, and planets with moons.

How fast is the planet at each point

The second law tells you the speed changes. A single equation tells you the exact value anywhere on the orbit. It is called the vis-viva equation and it comes from energy conservation: v² = GM (2 ÷ r − 1 ÷ a).

Read it carefully. At a given distance r, the speed depends on r and on a only. A planet at distance r on a big orbit is faster than one at the same r on a smaller orbit. If a = r, the orbit is a circle and the equation reduces to the circular speed v = √(GM ÷ r).

The speed bars at the bottom right of the stage use exactly this. The blue bar is the planet's own speed at that moment. The green bar is what a circular orbit at that same distance would need. Watch the blue bar swing above the green one near the Sun and below it far away: near the Sun the planet is going faster than circular speed there, so it overshoots outward; far away it is going slower, so gravity pulls it back.

Escape speed: the speed that breaks the orbit

How fast must you go to leave for good? Gravity never switches off completely, so the answer is the speed at which you just manage to reach infinity with nothing left over. Set the kinetic energy equal to the energy gravity would take back: ½mv² = GMm ÷ r. Cancel m and you get v(escape) = √(2GM ÷ r).

Compare with the circular speed √(GM ÷ r). The escape speed is exactly √2 times bigger, which is 1.414. That ratio does not depend on the body at all. It is why the escape bar on the stage always sits one fixed step above the circular bar.

In the vis-viva equation escaping means the orbit has no far end, 1 ÷ a = 0. A planet below escape speed is on an ellipse and will come back. At escape speed the path is a parabola. Above it, a hyperbola. Comets with parabolic or hyperbolic paths visit once and leave.

The table gives the speeds at four distances from the Sun, using the Sun's mass. At Earth's distance the circular speed is 29.79 km/s, which is Earth's own orbital speed, and the escape speed from the Sun is 42.12 km/s.

Solved problems

Seven problems, easy to tricky. Each number is computed by the same functions the simulation uses, so you can reproduce them with the sliders.

Mistakes that cost marks

Check this list before the exam. Each item has cost someone real marks.

• Putting the Sun at the centre of the ellipse. It sits at a focus.
• Saying the planet moves at constant speed. Only the areal speed is constant.
• Using km or metres in T² = a³ without converting. The simple form needs AU, years and solar masses; otherwise use T² = 4π²a³ ÷ GM.
• Using the full width of the orbit instead of the semi-major axis, which is half of it, in the third law.
• Forgetting that perihelion and aphelion add up to 2a, not a.
• Thinking escape speed depends on the mass of the escaping object. It does not.
• Mixing up the two equal quantities in the second law: equal areas in equal times, not equal distances in equal times.

Where else these laws work

Nothing in Kepler's laws is special to the Sun. Moons circling planets, artificial satellites circling Earth and stars circling a black hole in the centre of the galaxy all obey the same three rules, with only the central mass changing.

Satellite engineers use the period law to choose an altitude for a given orbit time, and use escape speed to size the rocket that sends a probe away. Astronomers find planets around other stars by measuring tiny wobbles, then use the third law to get the size of their orbits. Comets with highly eccentric ellipses, like the hypothetical one above, visit the inner Solar System for a short time and spend almost all of their year far away, moving slowly, just as the second law demands.

Frequently asked questions