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I heard that the sky wouldn't be dark at night if the light of every star had already reached the earth.

  1. Of which (currently invisible) star will the light reach the earth next and when will that be?
  2. What was the last star that we can now see at night to reach the earth and when was that?
  3. How can we know (1) if the light hasn't even reached us yet? Or can't we know and will a star suddenly appear without anyone being able to predict that?
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I think you may be thinking of Olbers' paradox . This supposes that if the universe were infinite in time and extent, and stars were more or less randomly placed, then every line of sight would end at a star, and so the sky would be as bright as the surface of a star.

The paradox can be resolved by supposing that the universe is not infinite in time, but had a beginning.

We don't see stars suddenly appearing as their light reaches us. What we see is stars in all stages of their development. When we look the nebulae where stars are born we see clumps of gas in the process of collapsing to form stars. These are called protostars. Now the process of stellar formation takes a long time: about a million years for a medium sized star. The start of a star is not a sudden "turning on". Instead, as the core of the protostar shrinks and heats up, nuclear fusion starts; slowly at first, then increasing until it is able to prevent further collapse and heating.

Protostars usually are dim in visible light, they can be found with infra-red telescopes.

Since the process is slow and there isn't a sudden turning on, your questions cannot be directly answered. However, there is an object HOPS383 which is 1400 light years distant, which between 2004 and 2008 had a "growth spurt" and became visible in infra-red for the first time. It can't be seen in visible light yet - so in a sense this protostar answers your question 1 and 2, in different wavelengths.

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To fully answer your question, you have to take into account the fact that light travels at the speed of light ($\approx 300,000 \mathrm{km/s}$). The further you look, the 'earlier' you look. Imagine that you are watching a star that is $3\ \mathrm{Glyr}$ away (the distance traveled by light in 3 billion years), then the light you see has been emitted 3 billion years ago.

To answer your question, you thus have to find the earliest star created in the Universe we can see. The part of the Universe we can see is restricted to the part located at $14\ \mathrm{Glyr}$ and less, because it is the longest distance light could have traveled since the beginning of the Universe.

So now the question is: when did the first star form? A recent paper made by astronomers reported that the first stars (which are also the farthest!) are to be found $800\ \mathrm{Myr}$ after the Big Bang, and they are the in CR7 galaxy. With increasing time, very far and very dim stars will appear in the sky, but you won't be able to see them with your naked eyes.

To be able to see a star appear in the sky with your naked eye, you probably have to think about supernovae. Supernovae are massive stars that explode at the end of their life, increasing dramatically their emitted light in a matter of hours. If the exploding star was too dim before exploding to be visible, it will look as if a bright star appeared in the sky once it has exploded. You can find a sunnary of the observation of these stars on wikipedia.

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