Describe how modern science operates and distinguish it from philosophy, pseudoscience, and religion. In recognition of these distinctions, what individual (s) and methods mark the beginning of Astronomy as a science?

the feature of modern science is that all it's claims are based upon rigorous experiments and all it's claims are put to through testing before they are accepted.

In comparison to philosophy and religion their claims are often not based on any sound basis whose nature may be determined by experimental testing thus are not subject to the same rigorous testing that the claims of science are put through.

Astronomy as a science has been studies from a very early time. It was initially used for navigation across deserts and seas. Particularly it were the Arabs that framed it into a subject to study. The mathematical foundations of the astronomy could be subsequently established by the works of Al-Tusi, Glaleio, Ptolemy, Copernicus, etc

The rigid mathematical proofs were given in the works of Gauss and Newton among other's.

The major distinction of modern astronomy began by developments in telecopes particularly the hubble space telescope and the x-ray telescopes.