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### Kepler’s laws

- Drawing ellipses to represent orbits
- Illustration of an elliptical orbit
- Kepler's Second Law on ice
- Kepler's Second Law with a whirling bung
- Kepler's Second Law using a dry ice puck
- Kepler's Second Law using a centripetal force kit
- Kepler’s Second Law using the rotation of a stool
- Guidance on Kepler's Laws
- A (very) brief history of astronomy

## Kepler’s laws

for 14-16

Kepler extracted from Tycho Brahé’s punctilious measurements the three great planetary laws which we now call by his name. These experiments provide ways of explaining the vital second law.

**Class practical**

A simple method for drawing an ellipse, using the two foci.

Apparatus and Materials

*For each student or group of students*

- Drawing board
- Drawing pins, 2
- Loop of thread, about 40 cm long
- Pencil

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

- Stick two drawing-pins into a board (though not pressed too far in) about 10 cm apart and slip the loop of thread over them. Insert a pencil point in the loop and position it so that the thread is tight. Move the pencil round the pins, always keeping the thread tight and thus trace out an ellipse.
- Students could be encouraged to draw ellipses with the drawing pins closer or farther apart.

Teaching Notes

- Drawing ellipses is a useful skill for students to master.
- The Earth's orbit around the Sun is not perfectly circular. It follows a slightly elliptical path, with the Sun at one focus.

### Up next

### Illustration of an elliptical orbit

**Demonstration**

Three methods of visualising elliptical orbits.

Apparatus and Materials

**Either Method 1:**

- Retort stand with rod and clamp, or tripod
- Small steel ball approximately 5 mm diameter, or a marble
- Large glass funnel at least 20 cm diameter

**Or Method 2:**

- Rubber sheeting
- Toy hoop or embroidery frame
- Retort stand, rods, clamps, boss head

**Or Method 3:**

Health & Safety and Technical Notes

Read our standard health & safety guidance

For method 2, stretch the rubber sheeting over the rigid horizontal circular frame and secure it tightly with tape. It should be stretched a little, equally in all directions.

Fix a vertical metal rod over the sheet so that it is pushing the sheet into a curved well, thus imitating, roughly, an inverse-square-force potential.

For method 3, the apparatus is available from a number of suppliers: Philip Harris, Griffin Education, or ASCOL.

Procedure

Method 1: Firmly hold the glass funnel vertically and let the ball fall into it. Friction will affect the orbit and make it precess, but the motion around the funnel will be elliptic. Select a ball which will fall right through the funnel.

Method 2: Project the small steel ball across the sheet. By choosing suitable initial conditions the ball can be made to describe an oval like an ellipse with one focus on the axis of the well.

Method 3: Balance the plastic or aluminium hill upside down. Wood blocks can be used to hold it in position. Project the ball across the inverted hill so that it will orbit the centre. The elliptical path will be visible.

Teaching Notes

- Method 1 is not a true representation of Kepler's Second Law. The ball starts in a circular orbit, but it gradually moves lower in the funnel because of friction and so the time for a circular orbit gets smaller. The orbit also precesses as a result of friction.
- Method 3 uses a valley with a 1/r profile. This corresponds to a 1r
^{2}force of gravitational attraction and so will closely model the movement of a satellite around a larger mass.

### Up next

### Kepler's Second Law on ice

**Demonstration**

Demonstrating the law using air or carbon dioxide pucks – or an ice rink.

Apparatus and Materials

- Air table
- Air or CO
_{2}puck - Light spring
- Rope

Health & Safety and Technical Notes

Take care when handling the glass plate.

If taking a class off site, follow your employer's procedures for off-site educational visits.

Read our standard health & safety guidance

Procedure

- Tether the puck to the air table using a light spring.
- Project the puck across the surface so that it orbits the tether point. The elliptical path can be observed.
- Alternatively, if the class is taken to an ice rink, a heavy student, representing the Sun, should be positioned firmly in the centre of the rink, holding a rope, the other end of which is held by another, lighter, student, who skates around the first. As the first student pulls the rope in, the second, the orbiting planet, will speed up.

Teaching Notes

- The change in speed as the puck orbits can be clearly seen and the areas traced out in equal times mapped out.
- When the
arm

from the Sun to an orbiting planet is long, the planet's speed is slower and the arm is short, the planet's speed is faster. - Starting with Mars, Kepler discovered that the orbital path of each planet is an ellipse with the Sun at one focus. The line from the Sun to any planet sweeps out equal areas in equal times. All the sectors shown have equal areas; the planetary orbits are much closer to circles in shape than in this figure. This is Kepler's Second Law.
- If you mark the position of a planet once a month on its elliptical orbit, and draw radii from the Sun to those points, the areas of sectors between those radii are all equal. This demonstrates Kepler’s Second Law.
- Later, Newton showed that Kepler's Second Law can be explained by a central gravitational force (directed towards the Sun) which goes as 1r
^{2}, where*r*is the planet's average radius of orbit.

### Up next

### Kepler's Second Law with a whirling bung

This simple class activity demonstrates the increase in speed with a diminishing radius of orbit.

Apparatus and Materials

*For each student or group of students*

- Rubber bung
- Length of twine

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

- Attach a length of twine securely to the rubber bung. Whirl it round in a circle.
- After a few revolutions, allow the twine to wind itself up around the finger. As the length shortens, so the
planet

moves faster and faster.

Teaching Notes

- Kepler's Second Law states that a planet in its orbit sweeps out equal areas in equal times. The time of orbit of the puck will fall as its radius of orbit decreases.
- You can measure the load by attaching a newton meter to the bottom of the string. As the force on the puck is increased, its orbit gets smaller and its speed increases.
- The arm from the Sun to planet sweeps out equal areas in equal periods of time. More distant planets move more slowly in their orbit. If you mark the position of a planet once a month on its elliptical orbit, and draw radii from the Sun to those points, the areas of sectors between those radii are all equal.
- Kepler replaced constant speed of planetary motion (as with Ptolemy and the Greeks) with constant sweeping out of an area. Developing scientific explanations often involves a search for constant values.

### Up next

### Kepler's Second Law using a dry ice puck

**Demonstration**

The speed of the puck increases as the radius of orbit of the puck decreases.

Apparatus and Materials

- Glass tube and fishing swivel, or freely swivelling pulley
- Block of solid CO
_{2}, Carbon dioxide cylinder (syphon type)

- Glass plate and gantry
- Wedges, 4
- Brass ring pucks, 2

*Or*

Health & Safety and Technical Notes

Dry ice is very cold. Wear thermal gloves to handle it, and wear safety spectacles.

Take care when handling the glass plate.

Read our standard health & safety guidance

Procedure

- The simplest technique is to use a swivelling furniture castor for the pulley. Drill a hole through the shank and cut a groove around the periphery of the wheel (see diagram). Support the pulley centrally just above the large glass plate so that the wheel can turn freely about the two axes. Use oil, if necessary, to allow the system to move freely.
- Fix a cup hook, or other device, to a double puck, so that a length of light string can be fastened to its side. Place the puck on the glass plate, supported on a block of solid carbon dioxide about 1 cm thick and of external diameter about 2 cm larger than the metal block. Pass the string over the groove and through the pulley shank, so that it can be held by the hand.
- An alternative arrangement uses the bar across the gantry to support a glass tube, which acts as the flexible pulley. Connect one end of the twine to a specially shaped puck, and the other end to a fishing swivel, which enables the twine to rotate freely. Connect this in turn to a spring balance which is held by hand. The puck must be specially shaped to give it a wider base, but the same mass as the usual CO
_{2}pucks. - The glass plate should be carefully cleaned with ethanol before it is used.
- Keep the carbon dioxide puck in motion before starting the experiment, so that it does not freeze to the plate.
- Start the puck moving in a circular orbit on the glass plate, applying the centrally directed force through the string. Increase this force and observe the effect.

Teaching Notes

As the force on the puck is increased, its orbit gets smaller and its speed increases. Kepler's Second Law states that a planet in its orbit sweeps out equal areas in equal times.

### Up next

### Kepler's Second Law using a centripetal force kit

**Class practical**

The simple kit demonstrates the increase in speed with a diminishing radius of orbit.

Apparatus and Materials

*For each student or group of students*

- Centripetal force kit

Health & Safety and Technical Notes

Ensure that the bungs are securely attached and that each student has sufficient space to whirl the bung safely.

The centripetal forces kit is available from:

Procedure

- Set the bung whirling around the top of the apparatus.
- Pull on the thread to increase the load.

Teaching Notes

- Kepler's Second Law states that a planet in its orbit sweeps out equal areas in equal times. The time of orbit of the puck will fall as its radius of orbit decreases.
- You can measure the load by attaching a newton meter to the bottom of the string. As the force on the puck is increased, its orbit gets smaller and its speed increases.

### Up next

### Kepler’s Second Law using the rotation of a stool

**Demonstration**

Apparatus and Materials

- Rotating stool or platform
- Masses, 2 kg, 2

Health & Safety and Technical Notes

Supervise students so they don't spin too quickly, or they may topple over.

A freely spinning music stool (or office chair) may be used for these demonstrations. It should be oiled before use.

Procedure

- Sit a student on the stool and ask them to hold a massive object, about 2 kg , in each hand with their arms stretched out sideways. Turn the stool so that it rotates fairly rapidly.
- Ask the student on the stool to draw in their arms, so holding the ‘planets’ tightly to their chest.
- Ask them to stretch their arms out again.

Teaching Notes

- This demonstration shows the conservation of angular momentum when the radius of the rotating mass is changed. As some parts of a spinning object are pulled in to make it more compact, then the object spins faster.
- Newton gave a geometrical proof of Kepler’s Second Law, using changes of momentum (a vector). The Law is simply a statement of the conservation of angular momentum. The only force acting on a planet (gravity) is always directed towards the Sun.
- Other examples of Kepler’s Second Law include the following...
- Suppose a high diver doing a somersault discovers half way down in his flight that he is going to hit the water feet first, when this is not his plan. He can double up, knees against the chest, and spin faster for a carefully judged time. To spin slower, the arms are spread out wide.
- A skater starts spinning slowly on one toe with arms stretched out and the body tilted. She then pulls herself upright with arms folded and spins very fast.
- The much more complicated movement of a falling cat, which always lands on its feet.

*This experiment was safety-tested in March 2008*

### Up next

### Kepler’s Laws

When Tycho Brahe was 17, he observed the conjunction of Jupiter and Saturn and was dismayed to find that the astronomical tables of the time were inaccurate in predicting the event by as much as a month. He decided to devote his life to making better tables, for which purpose he constructed better and better instruments.

The birth of modern planetary astronomy, with the three planetary laws discovered by Kepler, was based on the precise observations resulting from Tycho Brahe’s passion for accuracy.

#### Kepler: Law-giver of the heavens

In the course of his lifetime, Kepler extracted the three great planetary laws which we now call by his name.

- The orbit of each planet is an ellipse with the Sun at one focus.
- The arm from the Sun to a planet sweeps out equal areas in equal periods of time. If you mark the position of a planet once a month on its elliptical orbit, and draw radii from the Sun to those points, the areas of sectors between those radii are all equal.
- If for each planet you take an average radius,
*R*, and the time,*T*, the planet takes to go once round its orbit (its year) then the ratio*R*^{ 3}*T*^{ 2}is the same for all planets

The third law, which binds the movements of the planets together mathematically, Kepler discovered, with tremendous delight, quite late in life.

#### Mapping the Earth’s orbit in space and time

To map the Earth’s orbit around the Sun on a scale diagram you need many sets of measurements, each set giving the Earth’s bearings from two fixed points. Kepler took the fixed Sun for one of these and for the other he took Mars at a series of times when it was in the same position in its orbit.

Kepler proceeded thus: he marked the ‘position’ of Mars in the star pattern at one position (opposite the Sun, overhead at midnight). That gave him the direction of a base line, Sun – Earth – Mars, SE_{1}M. Then he turned the pages of Tycho’s records to a time exactly one Martian year later. (The time of Mars’ motion around its orbit was known accurately from records over many centuries).

Keplers Scheme to plot the Earth

s orbit.

Then Kepler knew that Mars was in the same position, M, so that SM had the same direction. By now, the Earth had moved on to E_{2} in its orbit. Tycho’s record of the position of Mars in the star pattern gave him the new apparent direction of Mars E_{2} M and the Sun’s position gave him E_{2} S. Then he could calculate the angles of the triangle SE_{2}M from the record thus: since he knew the directions E_{1} M and E_{2} M (marked on the celestial sphere of stars) he could calculate angle A between them. Since he knew the directions E_{1} S and E_{2} S he could calculate angle B. Then on a scale diagram he could choose two points to represent S and M and locate the Earth’s position,E_{2} as follows.

At the ends of the fixed base line SM, draw lines making angles A and B and mark their intersection E_{2} . One Martian year later he could find the directions E_{3} M and E_{3} S from the records and mark E_{3} on his diagram. Thus Kepler could start with the points S and M and locate E_{2} ,E_{3} ,E_{4} ..... enough points to show the orbit’s shape.

Knowing the Earth’s true orbit he could invert the investigation and plot the shape of Mars’ orbit. He found that he could treat the Earth’s orbit either as an eccentric circle or as slightly oval but Mars’ orbit was far from circular: it was definitely oval. It was an ellipse with the Sun at one focus – Kepler’s First Law of planetary motion.

#### Planetary data and Kepler’s Third Law

Kepler continued to brood on one of his early questions: what connection is there between the size of the planet’s orbit and the times of its year

?

Students can try and investigate the relationship between the planetary orbit radius, *R* , and the orbital time, *T*, using modern data. These are more accurate than the data available to Kepler. It will become obvious, fairly quickly, that simple proportion will not do. For example as *R* almost doubles in going from Mercury to Venus, *T*, almost triples; as *R* grows almost 10 times from Earth to Saturn, *T* , grows about 30 times.

Kepler wrestled with this for a very long time, trying different combinations, until he found that *R*^{ 3}*T*^{ 2} was a constant. Kepler was overjoyed!

His three laws were clear, simple and powerful and they fitted the facts very accurately. He earned the title law-giver of the heavens

.

### Resources

Data for the planets, for Jupiter’s moons and for objects orbiting the Earth can be downloaded here.

Planetary data.pdf### Up next

### A (very) brief history of astronomy

Early astronomers, in different civilizations, used the observed motion of the stars, the Sun, Moon and planets as the basis for clocks, calendars and a navigational compass. The Greeks developed models to account for these celestial motions.

Copernicus, in the 16th century, was the first to explain the observed looping (retrograde) motion of planets, by replacing a geocentric heliocentric model of the Universe with a heliocentric model. Modern planetary astronomy really began in the 17th century with Kepler, who used Tycho Brahe’s very accurate measurements of the planetary positions to develop his three laws.

Galileo contributed to the development of astronomy by teaching the Copernican view, and by devising a telescope which he used to show Jupiter’s moons as a model for the solar system, among other things.

Newton built on earlier insights with his universal law of gravitation and its fruits: predictions or explanations of Kepler’s laws, the motion of comets, the shape of the Earth, tides, precession of the equinoxes and perturbations in the motion of planets which led to the discovery of Neptune. He also had to invent the mathematics to do this: calculus.