LotR -- Book 2 - Chapter 03 - The Ring goes South

[Texadan]

The Hunter's Moon waxed round in the night sky, and put to flight all lesser stars. But low in the South one star shone red. Every night, as the Moon waned again, it shone brighter and brighter. Frodo could see it from his window, deep in the heavens, burning like a watchful eye taht glared above the trees on the brink of the valley.

Any clue what star this is and why it merited being singled out for such emphasis?