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Newton Principia, Book III, Proposition 7. Newton Principia, Book I, Propositions 57-63.

where r = r2 - ri and r = |r| is the distance between P and S. Adding these equations reveals that the centre of gravity C of the two bodies (whose position vector is rc = (Mr1 + m r2)/(M + m)) must move with constant velocity. If we divide the first equation by M, the second by m, and subtract the results, we obtain, with ¡x = G(M + m), r = -X r. (8.8)

This is exactly the same equation as we get from assuming the Sun to be fixed, except in that case we have x = GM. It follows that in this two-body system, the planet still moves according to Kepler's first two laws (since it is subject to an inverse square central force) but the third law must be modified slightly, since x is no longer the same for each planet. From Eqn (8.7) we have that T2x a a3, and so for two planets with masses m 1 and m2, periods T1 and T2, and semimajor axes a1 and a2, we have

Newton hoped to be able to use this result to make an accurate calculation of the semimajor axes of Jupiter and Saturn, but unfortunately there were still considerable uncertainties over the periods of these planets which were not resolved for another century.

On the assumption that a planet's mass (m) is much smaller than that of the Sun (M), Kepler's third law takes the form T2 GM = 4n2R3 (T is the planet's orbital period and R its semimajor axis). Similarly, for a satellite of the planet, we have t2 Gm = 4n 2r3 (t being the satellite's orbital period and r its semimajor

70 The precise form of the force is not important for this result, of course. It follows directly from Newton's three laws of motion and was stated by Newton in Corollary 4 to the Laws of Motion.

axis). Combining these results gives m r3 T 2 M = Rt2'

and Newton used this formula to calculate the masses (as proportions of the Sun's mass) of those planets with satellites. Over the period of nearly 40 years between the publication of the first and third editions of the Principia, the data to which Newton had access got progressively more accurate, and his values for the masses of the Earth, Jupiter, and Saturn varied considerably. In the third edition, we find the values 1/169 282 for the mass of the Earth divided by that of the Sun (which is almost double the correct value, the error being due to Newton's inaccurate value for the solar parallax), 1/1067 for Jupiter (which is within 2 per cent of the modern value), and 1/3021 for Saturn (which is about 16 per cent too big). This technique did not provide any hint as to the masses of Mercury, Venus, and Mars, however, as there were no known satellites orbiting these planets, but Newton felt confident that the masses (and consequently the gravitational effects) of these other planets were very small.

The seventeenth-century Solar System was a system consisting of the Sun, six planets, and numerous satellites, and Newton's theory implied that they each attracted each other. The treatment of the full dynamical problem was way out of reach, but Newton was able to make some progress by considering the case of three bodies and, in particular, for the case of the Sun's effect on the motion of the Moon.

Lunar theory and perturbations

Newton's lunar theory was developed in the latter half of the 1690s and first

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