As an arts student, I really find those cosmological questions hard to understand and hence come here to seek your kind help.
The Hubble constant $H_0$ is estimated to be about 65 km/s/Mpc, where 1 Mpc (megaparsec) is around 3.26 million light-years. At what distance would galaxies be moving away at exactly the speed of light? (I found that there is something called Hubble Radius, but is this the same as Hubble Radius?) If there were galaxies farther than the Hubble radius, how would they appear to us?
Answer
That's a very, very good question! Actually, the point you are addressing is the reason why physicists coined the term "observable universe". Those galaxies moving away from us with a speed bigger than the speed of light will never be visible (in the light they emit right now) to us and are outside the so-called "Hubble sphere". The distance you are looking for is $\frac{c}{H_0}$.
So as sad as it may sound, every second more and more galaxies and astronomical objects are leaving the observable universe, never to be seen again.
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