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Originally Posted by shelltree
What is "PopIII" and "PopII" referring to?
And are you saying it is possible for this SDSS J102915+172927 star to have formed in the same way that you have explained above? Through a gas cloud of hydrogen and helium that becomes so large that its gravity finally forces it to collapse to form the star? And yet it is estimated to be around 13 billion years old! Could this type of star that is deficient of metals continue to "live" so long and if so, how would it be possible?
I find this all very interesting and if I have for any reason misunderstood anything please forgive me, I'm just trying to wrap my head around it all!
Fascinating stuff I must say 
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PopIII and PopII are designations for stars of a certain type that depend on their metals content and orbital dynamics about the centre of the galaxy (any galaxy, actually). Once a star's metals content drops below a certain percentage relative to the Fe/H ratio of the Sun, it's given the designation PopII or PopIII. Most PopII stars have low to extremely low metals content (10%< relative to the Sun) and PopIII stars, the hypothetical stars that formed only a few hundred million years after the BB, have no metals at all, except for a tiny amount of Lithium (and possibly Beryllium) that formed during the BB.
Most PopII stars either reside in the galactic bulge of galaxies or in their halos. They were amongst the first stars to form within the galaxies.
All stars, whether they're SDSS J102915+172927, a PopIII object or the normal run of the mill ones we see today all form in the same way...the collapse of a cloud of hydrogen and helium. It's just that later generations of stars after the PopIII stars burnt out contain more heavy elements in their makeup than the original stars. All due to those stars and other spewing those elements out into the cosmos after they die. The previous generations of stars made the elements out of which the new generations were born from.
The mechanisms by which a gas/dust cloud collapses to form a star are a little more complicated than just gravity collapsing the cloud, but once it starts to collapse gravity becomes the dominant force.
The lifetime of a star is directly proportional to the inverse 2.5 power of the mass of the star....This value is multiplied by 10^10, or the average MS lifetime of a star like the Sun, to give the star's MS lifetime. The star in question is 0.8 solar masses, which is why it can last for 13 billion year or more, because its MS lifetime is (1/0.8^2.5) x 10^10 or 17.5 billion years. However, for very high mass and low mass stars, the relationship of mass to lifetime is a little more complicated and the exponent of the mass is slightly different. Though, the above example will generally give good approximate ages for stars between 0.1 to 40 solar masses.