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Kepler's Laws of Planetary Motion

Kepler gave three rules for how planets move. 1) Each planet moves on an ellipse with the Sun at one focus. 2) The line from the Sun to the planet sweeps equal areas in equal times, so the planet moves faster when it is near the Sun. 3) The square of the time for one round (T²) is proportional to the cube of the semi-major axis (a³). The second law is really conservation of angular momentum. The third law follows from Newton's law of gravitation.

🎬 Step-by-step story

  1. Watch the blue planet. Its path is not a circle. It is a stretched circle called an ellipse. The Sun sits at one special point inside it, called a focus.
  2. Look at the speed. Near the Sun the planet rushes. Far from the Sun it crawls. The labels show fast and slow.
  3. Now two coloured slices. Each one is the area swept in the same time. One is short and wide, one is long and thin. Their areas are equal. This is the law of areas.
  4. Two planets now. The outer one has a bigger orbit and takes much longer. For both, T² divided by a³ gives the same number.
  5. A worked example. A planet is 4 times as far as Earth (a = 4 AU). Press the button to find its year, line by line.
  6. Your turn. Change the size a and the stretch e. Watch the time T and the speed change.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

If the orbit is an ellipse, why do we treat planet orbits as circles in problems?

Most planet orbits have a very small eccentricity (Earth's is 0.017), so they are almost round. A circle is just an ellipse with e = 0, so all three laws still hold, with a = radius.

Why does the planet speed up near the Sun if nothing pushes it forward?

As it falls inward, gravity has a part along its motion, so it gains speed, like a ball rolling downhill. Going outward it slows down, like a ball rolling uphill. Angular momentum staying constant tells you exactly how much.

Equal areas, but the slices look so different. Are they really equal?

Yes. The slice near the Sun is short and fat; the slice far away is long and thin. Area = ½ × r² × angle, and the short-r slice has a bigger angle. They balance out exactly.

Does Law 3 depend on the planet's mass?

No. T² = (4π²/GM) a³ has only the Sun's mass M in it. A tiny asteroid and a big planet at the same a have the same period.

What is 1 AU?

One astronomical unit is the average distance from Earth to the Sun, about 1.5 × 10¹¹ m. Using AU and years makes T² = a³ with no extra constant for the Sun's planets.

Who was Kepler and what did he find?

Long ago, people thought planets moved in perfect circles. Tycho Brahe watched the planets for many years and wrote down very careful numbers. Johannes Kepler studied these numbers. He found three simple rules. We call them Kepler's laws.

Kepler did not know why the rules work. Later, Newton showed that all three come from one force: gravity.

Law 1: the law of orbits

Every planet moves in an ellipse, with the Sun at one focus.

An ellipse is a stretched circle. It has two special points inside called foci (one is a focus). For any point on the ellipse, the distance to one focus plus the distance to the other focus is always the same.

Perihelion distance = a(1 − e). Aphelion distance = a(1 + e).

Law 2: the law of areas

The line joining the Sun and the planet sweeps equal areas in equal intervals of time.

Near the Sun the line is short, so the planet must move a long way to cover the same area. So it moves fast. Far away the line is long, so a short move covers the same area. So it moves slowly.

Why it is true: angular momentum

In a tiny time Δt the planet moves r Δθ sideways. The thin slice is almost a triangle with area ΔA = ½ r² Δθ. So the areal speed is

ΔA/Δt = ½ r² ω = L / (2m), where L = m r² ω is the angular momentum.

Gravity pulls straight toward the Sun, so it gives no torque (τ = r × F = 0, because r and F are along one line). With no torque, L stays constant. So ΔA/Δt stays constant. That is Law 2.

A handy result: at perihelion and aphelion the velocity is at right angles to r, so m v_p r_p = m v_a r_a, or v_p r_p = v_a r_a.

Law 3: the law of periods

The square of the time period of a planet is proportional to the cube of the semi-major axis of its orbit: T² ∝ a³.

So T²/a³ is the same number for all planets of the Sun. A planet twice as far out takes 2^1.5 ≈ 2.83 times as long.

Getting Law 3 from Newton's gravity (circular orbit)

  1. For a planet of mass m moving in a circle of radius r round the Sun (mass M), gravity gives the centripetal force: GMm/r² = m v²/r.
  2. So v² = GM/r.
  3. Time for one round: T = 2πr / v, so T² = 4π²r² / v² = 4π²r² × r / GM.
  4. T² = (4π²/GM) r³. The bracket depends only on the Sun, so T² ∝ r³.

This also lets us find the mass of the Sun (or of any planet with a moon): M = 4π²r³ / (G T²).

Try it at home: draw an ellipse

Push two pins into a card sheet about 8 cm apart. Tie a loop of thread (about 24 cm) around them. Put a pencil inside the loop, keep the thread tight and go round. You get an ellipse. The pins are the two foci. Move the pins closer and the ellipse becomes rounder (smaller e). Put them together and you get a circle.

Key formulas and definitions

Worked examples

1. A planet's orbit has a = 4 AU. Find its period in years (Earth: a = 1 AU, T = 1 year).

T²/a³ is the same: T² = 1 × 4³ = 64. So T = 8 years.

2. Mars is about 1.52 AU from the Sun. Estimate its year.

T = a^1.5 = 1.52^1.5. √1.52 ≈ 1.233, so T ≈ 1.52 × 1.233 ≈ 1.87 years (about 687 days).

3. A comet is 0.5 AU from the Sun at perihelion and 35 AU at aphelion. Its speed at perihelion is 60 km/s. Find its speed at aphelion.

v_p r_p = v_a r_a. v_a = 60 × 0.5 / 35 ≈ 0.86 km/s.

4. An ellipse has a = 10 AU and e = 0.6. Find the closest and farthest distances from the Sun.

r_min = a(1 − e) = 10 × 0.4 = 4 AU. r_max = a(1 + e) = 10 × 1.6 = 16 AU.

5. Two satellites orbit the same planet. The first has radius r and period 2 hours. The second has radius 4r. Find its period.

T₂/T₁ = (r₂/r₁)^1.5 = 4^1.5 = 8. So T₂ = 16 hours.

6. The Moon goes round Earth in 27.3 days (2.36 × 10⁶ s) at 3.84 × 10⁸ m. Find Earth's mass. (G = 6.67 × 10⁻¹¹ N m² kg⁻²)

M = 4π²r³/(GT²). r³ = 5.66 × 10²⁵. 4π² ≈ 39.5, so top = 2.24 × 10²⁷. GT² = 6.67 × 10⁻¹¹ × 5.57 × 10¹² = 371.5. M ≈ 6.0 × 10²⁴ kg.

7. Show that the areal speed of a planet is L/(2m).

In time dt the radius turns by dθ. Swept area dA = ½ r (r dθ) = ½ r² dθ. So dA/dt = ½ r² ω. Angular momentum L = m r² ω, so r²ω = L/m. Hence dA/dt = L/(2m).

Common mistakes

Practice quiz

1. According to Kepler's first law, the Sun is at:
2. Kepler's second law is a result of conservation of:
3. A planet moves fastest at:
4. If the orbit radius is made 9 times, the period becomes:
5. For a circular orbit, T² equals:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What are Kepler's three laws in short?

1) Orbits are ellipses with the Sun at one focus. 2) Equal areas are swept in equal times. 3) T² ∝ a³.

Which conservation law gives Kepler's second law?

Conservation of angular momentum, because gravity is a central force and gives no torque about the Sun.

How is Kepler's third law derived?

Set gravity equal to the centripetal force, GMm/r² = mv²/r, and use v = 2πr/T. You get T² = (4π²/GM) r³.

Where this is taught

RomaniaClasa a IX-aElements of celestial mechanics (optional)
RomaniaClasa a IX-aElements of celestial mechanics (optional)
Spain2º BachilleratoGravitational field
Ukraine11 класCelestial sphere and motion of celestial bodies
CBSE (India)Class 11Gravitation
USA (Common Core, NGSS, AP)Grade 9Space systems
South Korea고등학교 2학년Space-time and motion
South Korea고등학교 2학년Space exploration and planetary systems
South Korea고등학교 3학년Solar system and Earth
South Korea고등학교 3학년Planetary motion
South Korea고등학교 3학년Mechanical interactions
Germany (Bavaria)Jahrgangsstufe 11Independent work on physics topics
China高一Compulsory 2 Ch.7 Gravitation and spaceflight

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