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Assume the Earthâs orbit is circular. No external torque acts on the EarthâSun system, so the angular momentum of the Earth is constant. Challenge Problems 78. 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 Earthâs 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 Ã 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 Ã 1024 kg. You will not be able to easily solve the equation you generate; instead, use a computer to verify that 1.48 Ã 109 m is the correct value. 79. The oldest artificial satellite still in orbit is Vanguard I, launched March 3, 1958. Its mass is 1.60 kg. Neglecting atmospheric drag, the satellite would still be in its initial orbit, with a minimum distance from the center of the Earth of 7.02 Mm and a speed at this perigee point of 8.23 km/s. For this orbit, find (a) the total energy of the satelliteâEarth system and (b) the magnitude of the angular momentum of the satellite. (c) At apogee, find the satelliteâs speed and its distance from the center of the Earth. (d) Find the semimajor axis of its orbit. (e) Determine its period. 80. A spacecraft is approaching Mars after a long trip from the Earth. Its velocity is such that it is traveling along a parabolic trajectory under the influence of the gravitational force from Mars. The distance of closest approach will be 300 km

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