1. Some large stars end their lives in a supernova explosion, leaving behind a neutron star, a very dense object. Consider such a neutron star with a radius of 10 km and a mass 1.4 times that of the sun (typical figures). These stars have no thermonuclear fuel left, but can be detected if they are paired with another star, and matter from the other star falls on to the neutron star, releasing energy as it crashes to the surface.

 (a) Find the energy of a proton, falling from rest at a great height, as it reaches the surface of the neutron star.

 (b) Find its velocity.

(c) Find the period of a small object going around the neutron star in low orbit (actually this needs general relativity to do it properly, but the answer is not far off).

(d) If matter is falling on to the neutron star at a rate of one billionth of the sun’s mass per year, how bright is the neutron star compared with the sun?

2. We discussed at length how two clocks at the ends of a train could be synchronized by flashing a light located at the midpoint of the train, the clocks began registering when the light flash reached them. We calculated the lack of synchronization of these clocks as observed from the ground.

Suppose now that instead of a light flash, identical small toys are simultaneously released from the middle of the train, crawling at speed u towards the ends, and the clocks are triggered when the toys reach them. Give a fully detailed mathematical analysis of the lack of synchronization (if any) of the train clocks as viewed from the ground with this scenario, assuming as usual that the train moves at steady speed v.

French: Chapter 1: 14, 22. Chapter 4: 17. Chapter 5: 19. Chapter 7: 1.