Question

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Answer 1

Newton\'s Law of Universal Gravitation strictly applies to perfectly spherical bodies. Many celestial bodies, like the Sun and Earth, are not perfect spheres. This has a measureable effect on the trajectories of orbiting satellites. If we restrict our attention to equatorial orbits, we can correct the gravity law in a simple way to account for the Sun\'s imperfect shape:
\[F _{g}=-\frac{ GMm }{ r ^{2} }(1+\frac{ 3J _{2}R ^{2} }{ 2r ^{2} })\]

Answer 2

where G = 6.67 × 10-11 N·m2/kg2 is the Universal Gravitation Constant, M = 1.99 × 1030 kg is the mass of the Sun, m is the mass of the orbiting body, R = 6.96 × 105 km is the mean radius of the Sun, and J2 = .000000175 is the solar quadrupole moment, a dimensionless parameter that characterizes the Sun\'s slightly aspherical shape.

Answer 3

What is the extra work done by the correction term in the gravity law when a small comet moves from aphelion to perihelion along an equatorial orbit? The comet\'s perihelion–the distance of closest approach to the Sun–is rp = 1.25 astronomical units (AU). The aphelion–its largest distance from the Sun–is ra = 9.13 AU. The comet\'s mass is m = 2590000 kg, and it orbits the Sun once every 6.7 years. See the hint for help with conversion from astronomical units to meters.