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Assume the Earth’s orbit is circular. No external torque acts on the Earth–Sun
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 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
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 satellite–Earth system and (b) the magnitude of the angular
momentum of the satellite. (c) At apogee, find the satellite’s 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