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The Andromeda galaxy is about 600,000 light years away and is doppler blue-shifted. Are there objects closer to us than Andromeda which are red-shifted? Are there objects farther than Andromeda that are blue-shifted?

I'm curious as to whether there is a more or less uniform radius from earth outside of which objects are generally all moving away from us, and inside of which objects are generally all moving towards us.

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No, there is no spherical volume of space within which objects all tend to approach us and thereby show blue-shift. The fact that the Andromeda galaxy is blue-shifted is because at these relatively small distances the cosmological red-shift is much smaller than further away. Andromeda's own (non-cosmological speed) is larger then the cosmological speed away from us and it happens to move towards us and is therefore blue-shifted. It could also have been moving away from us, and be red-shifted as a result, but that red-shift would not be cosmological.

I'd expect that the chance that a nearby (Local Group) galaxy is either moving away from us or moving towards us is 50%.

Furthermore, the Andromeda galaxy, and all other galaxies belonging to the Local Group, are gravitationally bound and would therefore not show any cosmological red-shift anyhow.

NB. The Andromeda Galaxy is actually 2.5 million light years away from us.

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