## Guidance on Kepler's Laws

Teaching Guidance for 14-16

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