Q Fig. 6.10. Kepler's puffy-cheek orbit. Fig. 6.11. The elliptical orbit.

orbit as the 'puffy-cheek' path because it was broader near aphelion than near perihelion.

From the cosine rule on triangle OSM and a , r1(r - a)

r cos a = ae + r1 cos 0 = ae +--, ae we can derive the expansion for r in powers of e:

— = 1 + e cos a — e2 sin2 a + O(e3), a exactly as for an ellipse. Kepler's calculations confirmed that this new orbit reproduced planetary distances accurately and he could have carried his calculations through for the 'puffy-cheek' path and, hence, find that the anomaly a was also predicted accurately by this orbit together with the area law, but, instead he went on, realizing that with a minor modification he could make his 'puffy-cheek' a perfect ellipse. Kepler kept the equation r = a(1 + e cos 0)but, instead of making 0 the angle between OM and OA, he took it to be the angle between OM' and OA, where M' is the point on the circle having diameter PA (known as the auxiliary circle) which is also on the perpendicular to PA through M (see Figure 6.11). This angle is known as the 'eccentric anomaly'.

It is clear that |NM'\ = a sin6 and a simple application of Pythagoras' theorem yields

|NM| = [r2 - a2(e + cos 6)2]1/2 = a sin6 J1 - e2 = \NM\y/1 - e2, a geometrical property that Kepler knew implied that AMP was an ellipse. It is straightforward to derive the standard polar equation for an ellipse from Kepler's equations.77 Since r cos a - ae = a cos 6 and a cos 6 = (r - a)/e, we immediately have a(1 - e2)

Although Kepler had no good empirical evidence for throwing away the 'puffy-cheek' orbit in preference to the ellipse, he was convinced of the truth of the elliptical orbit. Moreover, although Kepler had only derived this new orbital shape for Mars, he assumed simply that since the cause of the motion was the same for all the planets, all the planets must behave in the same way. Nowadays, Kepler's first law of planetary motion usually is stated as:

Planets move on elliptical orbits with the .sun at one focus but it is worth noting that nowhere in the main body of the New Astronomy is the word 'focus' mentioned, and it was only later in his Epitome of Copernican Astronomy that Kepler emphasized this aspect of planetary orbits. Kepler's physical mechanism by which the planets were forced to move in such a fashion was incorrect, but the idea that the study of celestial motions should be based

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