Astronomy and astrology in Middle-Earth?

[Macalaure]

Let me throw in two quotes.

They took me and they set me alone on the pinnacle of Orthanc, in the place where Saruman was accustomed to watch the stars.
~The Council of Elrond

Childless lords sat in aged halls musing on heraldry; in secret chambers withered men compounded strong elixirs, or in high cold towers asked questions of the stars.
~The Window on the West

I think these two quotes point towards the existence of (some kind of) astrology.

So, Saruman was accustomed to watch the stars. But is there any way in which Saruman would benefit from astronomy? I mean, his main plan for most of his time in Middle-earth was learning about the Ring and then finding it - how could astronomy help him with that? Saruman was a learned man with a variety of scholar interests, of course, but I find it unlikely that he merely watched the stars for leisure.
The Gondorians, then, didn't only watch the stars, but asked questions of them. Especially being named right after "compounding strong elixirs", I don't think Tolkien had a "proper science" in mind when he wrote it.

The way I read both of these, the Gondorians and Saruman were not trying to learn about the stars, but from them, which would point towards some form of astrology. And if there is astrology in these places, then I don't see why the other occurrences of people watching the stars should point exclusively to astronomy.

It is, however, interesting to note that in these more obvious quotes it is people of a doubtful character who practice astrology. It is also interesting to observe that in all occurrences (as far as I'm aware) of "watching the stars" etc. Tolkien does not say what people learned from them, so in case that we do have astrology here, he carefully leaves it open whether it gives valid answers or is a mere superstition.