Temat 8: Kepler's Law

Gravitation in the sky
The expression a = GM/r² for the acceleration due to universal gravity also describes the motion of all the planets across the sky.We usually imagine to be located at the centre of the Sun and say that the planets ‘orbit the Sun’. How can we check this?
First of all, looking at the sky at night, we can check that the planets always stay within the zodiac, a narrow stripe across the sky. The centre line of the zodiac gives the path of the Sun and is called the ecliptic, since the Moon must be located on it to produce an eclipse.This shows that planets move (approximately) in a single, common plane.*
The detailedmotion of the planets is not easy to describe. As Figure 129 shows, observing a planet or star requires measuring various angles.

F IGURE 129 Some important concepts when observing the stars at night.

For a planet, these angles change every night. From the way the angles change, one can deduce the motion of the planets. A few generations before Hooke, using the observations of Tycho Brahe, the Swabian astronomer Johannes Kepler, in his painstaking research on the movements of the planets in the zodiac, had deduced several ‘laws’.The three main ones are as follows:
1. Planets move on ellipses with the Sun located at one focus (1609).
2. Planets sweep out equal areas in equal times (1609).
3. All planets have the same ratio T²/R³ between the orbit duration T and the semimajor axis R (1619).
Kepler’s results are illustrated in Figure 130.

F IGURE 130 The motion of a planet around the Sun, showing its semimajor axis d, which is also the spatial average of its distance from the Sun.

The sheer work required to deduce the three ‘laws’ was enormous. Kepler had no calculating machine available. The calculation technology he used was the recently discovered logarithms. Anyone who has used tables of logarithms to performcalculations can get a feeling for the amount of work behind these three discoveries.
Now comes the central point.The huge volume of work by Brahe and Kepler can be summarized in the expression
a = GM/r²
as Hooke and a few others had stated. Let us see why.