### Other Pages

Can I invoke the standard statement which used to be used for freeware ;

You are, of course, welcome to consult and make use of my pages at will. However, if you have found them useful in any appreciable way, please consider sending a small disbursement (even something as small as 1 Euro/Pound/Dollar/etc. would be received with appreciation).

B. Daugherty, 31 MH, Portsmouth PO5 3JG, Britain.

## Kepler's Laws

### Introduction

With these laws, Kepler both confirmed the Copernican idea that the Sun was at the center of the Solar System and extended Copernican ideas, so that by the time of Galileo's problems, the theory had a considerable foundation.

### Kepler's First Law

#### Planetary orbits are ellipses, with the Sun at one focus

The distance of a planet from the Sun therefore varies. When the planet is furthest away it is at aphelion (apogee, in general), and when it is closest it is at perihelion (perigee, in general). The terms described as 'in general' are those terms that can be used whatever the central attracting body is. The terms perihelion and aphelion apply only when the central attracting body is the Sun.

### Kepler's Second Law

#### An imaginary line between a planet and the Sun will sweep out the same area in equal periods of time.

This law is equivalent to the present-day Law of Conservation of Angular Momentum. The classic example of this latter law is an ice-skater who rotates faster when (s)he draws in initially-outstretched arms. In the above diagram, the two shaded regions have identical areas. The period of time taken for the planet to travel between A and B is equivalent to the period of time it takes to travel between C and D.

The path between C and D could represent the path of our Earth at some time during our Summer in Britain. The path between A and B could represent the Earth's path at some time during Australia's Summmer. This ellipse would be highly distorted in comparison with the Earth's true orbital shape, but the general ideas can be seen. All other things being equal, our Summers would be cooler than Australia's but would last longer. These effects are modified by the atmosphere - note that Midsummer's Day in Britain is around June 21, but atmospheric effects ensure that the hottest month is usually August.

### Kepler's Third Law

#### The square of the period of a planet is equal to the cube of its average distance from the Sun.

This is a very simplified version of the law. In its full version it is applied to binary systems. We have carried out two main simplifications - thus

• Strictly speaking, a Sun/planet system is a binary system rotating about the common center of the system. The common center of the system is so close to the center of the Sun that we can approximate the situation by considering the Sun to be static and ignore the planet's mass as being negligible in comparison with the Sun's. This will simplify the original equation.

• By selecting the inserted values to be in specific units (namely years and Astronomical Units), we can eliminate any constants in the full equation, and state the law as we have done at the top.

 In mathematical form, the law is now stated as τ2 = a3 where τ is the period of a planet a is the average distance of the planet from the Sun Remember, τ must be in years and a must be in AUs (Astronomical Units).

 Quick Quiz       a)   What would be the orbital period of a planet with a mean orbit radius of 8 AU ?   b)  The mean distance of Mars from the Sun is 1.5 AU. Calculate its orbital period.   c)   A satellite orbits a planet in 18 hours. It is moved to a new orbit with a period of 4.5 hours. The orbital radius decreases by what factor ?   d)    e)

For a circular orbit, at a height of 200 km above the Earth's surface, the speed will be 7.8 km/s. At 36 000 km, it will be 3.1 km/s. Mir was put into orbit at 350 km. with no eccentricity.  