For those unaware of this quote, it comes about 4 pages into
The Ring Goes South and says (emphasis mine):
"
So the days slipped away as each morning dawned bright and fair, and each evening followed cool and clear. But autumn was waning fast; slowly the golden light faded to pale silver, and the lingering leaves fell from the naked trees. A wind began to blow chill from the Misty Mountains to the east. 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 that glared above the trees on the brink of the valley."
It could be, but, Tolkien did a fair bit of writing on Middle-earth's cosmology. It is possible that it is
Carnil, which is equivalent to our Mars; it is also claimed that it could be
Borgil, which is often equated with either Betelgeuse in the constellation of Orion, or Aldebaran in the constellation of Taurus. To be honest, though, we can't know and I don't know if I can help you any more than I what I've said here.
There was a discussion about this on The Barrow Downs a few years ago which you can view
here. Similarly Mellonath Daeron have done a bit about
Celestial Objects in Middle-earth. Another link of interest might be
The Astronomy of Middle-earth.
.
