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Astronomers believe the Milky Way is moving at approximately 630 km per second relative to the local co-moving frame of reference that moves with the Hubble flow. If the Galaxy is moving at 600 km/s, Earth travels 51.84 million km per day, or more than 18.9 billion km per year, about 4.5 times its closest distance from Pluto. The Milky Way is thought to be moving in the direction of the Great Attractor. The Local Group (a cluster of gravitationally bound galaxies containing, among others, the Milky Way and the Andromeda galaxy) is part of a supercluster called the Local Supercluster, centered near the Virgo Cluster: although they are moving away from each other at 967 km/s as part of the Hubble flow, the velocity is less than would be expected given the 16.8 million pc distance due to the gravitational attraction between the Local Group and the Virgo Cluster.
If you reflected all the heat from the sun back onto itself it would get very very hot. When you heat a gas it will expand, so it would get larger and larger. I think this will cause the nuclear reactions to slow down as the pressure will drop. But if you don't let any heat out it will keep expanding until the pressure gets so low that nuclear reactions essentially stop. Probably by this point it would essentially be evaporating as hot atoms at the surface would have more than enough energy to escape the suns gravitational potential well.