[QUIZCM-03005]

Answer the following questions for the two body Sun-Earth system. Assume that the forces due to all other heavenly bodies can be neglected. Use \(\vec{r}_1, \vec{r}_2\) to denote positions, and \(\vec{p}_1, \vec{p}_2\) to denote the momenta of the Sun and the Earth, respectively. Check if the quantities listed below are conserved.

1. The sum of angular momenta of the Sun and the Earth
2. The sum of momenta of the Sun and the Earth
3. The angular momentum \(\vec{r}\times\vec{p}\) where \(\vec{r}=\vec{r}_2- \vec{r}_1\) is the position of the earth relative to the Sun. and \(\vec{p}=\mu \vec{v}\) where \(\mu\) is the reduced mass and \(\vec{v}\) is the velocity of the Earth relative to the Sun.
4. Write approximation, if any, under which the angular momentum of the earth will be a constant of motion.

Using Newtonian mechanics only, give brief arguments in support of your answer