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For example the agreed distance to the Andromeda Galaxy is 2.5 million light years, and consequentially, we see the galaxy as it was 2.5 million years ago! (A time interval in which the galaxy must indeed have moved away/towards our galaxy by a large distance.) So are the distances we use in astronomy the distance which we perceive or are they considered after taking care of the distance change due to the expansion of the universe?

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Generally speaking they refer to the distances from us when the light as emitted. No correction is usually made to say how far away the object is from us now, because this correction would be very small and inconsequential compared to the uncertainty in the original distance measurement.

For instance, taking the Andromeda M31 galaxy as an example. Riess et al. (2012) use Cepheids to get a distance of 752 +/- 27 kpc. The galaxy moves towards us at 300 km/s. So, the light takes roughly 2.5 million years to get here. During that time M31 moves approximately 0.75kpc, a tiny fraction of the uncertainty in the distance and actually only a small fraction of the size of M31.

For more distant, cosmological distances the difference becomes much more important, and model-dependent. In other words if we observe a "light-travel" distance to a distant galaxy, we can attempt to estimate how far away it is now from us using our knowledge of the expansion of the Universe. In an extreme case - the light from the cosmic background radiation has travelled 13.8 billion years to get here. However, a galaxy that formed at that point of time/space in the Universe is now 46 billion light years away from us due to expansion. There are lots of links on Physics SE about the issue of cosmological light travel distances versus proper distances. e.g. How far has a 13.7 billion year old photon travelled

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