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I'm no expert, but if both the star and the Earth were moving away from each other at some % of the speed of light all that time, wouldn't it make sense?
In the same way that the sound from a car's horn blowing, once you've already passed that other car going the opposite way down the highway, wouldn't be heard at the expected, let's say, 1 second delay. This would be because even though you were 343 meters away from the car when the horn blew, by the time the sound catches up to you you are some distance greater than 343 meters away....
Add a great deal of speed to that thought, and it seems to make sense - just as that horn would play minutes later if you were traveling at 342 meters/sec.
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