Chapter 13, Problem 45CP

The Solar and Heliospheric Observatory (SOHO) spacecraft has a special orbit, located between the Earth and the Sun along the line joining them, and it is always close enough to the Earth to transmit data easily. Both objects exert gravitational forces on the observatory. It moves around the Sun in a near-circular orbit that is smaller than the Earth’s circular orbit. Its period, however, is not less than 1 yr but just equal to 1 yr. Show that its distance from the Earth must be 1.48 × 10⁹ m. In 1772, Joseph Louis Lagrange determined theoretically the special location allowing this orbit. Suggestions: Use data that are precise to four digits. The mass of the Earth is 5.974 × 10²⁴ kg. You will not be able to easily solve the equation you generate; instead, use a computer to verify that 1.48 × 10⁹ m is the correct value.