Question

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. Restricting attention to equatorial orbits, the gravity law can be corrected in a simple way to account for the Sun's imperfect shape.

→Fg=−GMmr2(1+3J2R22r2)^rF→g=−GMmr2(1+3J2R22r2)r^

where G=6.67×10−11 N⋅m2/kg2G=6.67×10−11 N⋅m2/kg2 is the universal gravitation constant, M=1990000 kgM=1990000 kg is the mass of the Sun, mm is the mass of the orbiting body, R=6.96×105 kmR=6.96×105 km is the mean radius of the Sun, and J2=4.87×10−7J2=4.87×10−7 is the solar quadrupole moment, a dimensionless parameter that characterizes the Sun's slightly aspherical shape.

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, or the distance of closest approach to the Sun, is rp=2.45 AU (astronomical units).rp=2.45 AU (astronomical units). The aphelion, or its largest distance from the Sun, is ra=7.73 AU.ra=7.73 AU. The comet's mass is m=1.94×106 kg,m=1.94×106 kg, and it orbits the Sun once every 9.5 years.9.5 years. See the hint for help with conversion from astronomical units to meters.

Answer 1