Lagrange point L1 The Lagrange point L1 is on a line between the Sun and Jupiter, at approximately 5.31 × 1010 m from Jupiter. The Sun-Jupiter distance is 7.78 × 1011 m, the mass of the Sun is 1.99 × 1030 kg, and the mass of Jupiter is 1.90 × 1027 kg. The period of Jupiter is 4330 days. An asteroid of small mass is located at L1.

(a) Write the equation of motion for the asteroid in equilibrium in the rotating system.

(b) Using the numerical data, show that the equation of motion is satisfied to good accuracy.

(c) There are three Lagrange points on the Sun-Jupiter axis. Show, on physical grounds, where the other two can be found. (Part c is qualitative: the exact solution requires finding the three real roots of a fifth-order polynomial