Re: How does the movement of the sol system affect the

Time measurement, in special relativity, is done from an inertial 
reference frame; and, the speed of light is the same in every inertial 
reference frame.  One of the difficulties with the scenario you have 
suggested, is the measured times would be dependent upon the inertial 
reference frame used.  An additional difficulty, for the measurement of 
time in the spacecraft making the two runs across the galaxy, is the 
accelerations and deccelerations needed in the course of the trip.

Making the assumption that time measurements are made from the non-
accelerating inertial reference frame of an observer at the launch point, 
it would not take longer to travel one direction across the galaxy as 
opposed to another.  There are also no differences in energy required, 
that are dependent upon the direction of travel; as long as one doesn't 
have to consider mass anomalies traveling in preferred directions 
independent of the direction of travel of the galaxy and a host of other 
difficulties that I am not qualified to even list.

There is an interesting discussion on pages 96 to 97 of Walter 
Scheider's 'Maxwell's Conundrum: A serious but not ponderous book about 
Relativity', Cavendish Press, Ann Arbor, Michigan, 2000, isbn 0-9676944-0-
X.  The discussion is about a racer who distrusts the judges and decided 
to carry his own stopwatch.  The next few pages afterwards are about 
muon's and the differences in time measurements in the reference frame of 
muon versus that for a laboratory on the Earth's surface.  Both examples 
offer insights to the nature of time measurements in special relativity 
and may be helpful in unravelling the mysteries of the scenario you have 
proposed.

Thanks for your question.  In addition to the reference noted, there are 
many postings in the MadSci archives on inertial reference frames.  On the 
web, using search terms such as +special_relativity  will reveal many 
excellent references on all aspects of relativity.

sid