Is planet earth a common noun?

The noun 'Earth' is a proper noun as the name of a specific planet.

The noun 'planet' is a common noun, a general Word for a celestial body moving in an elliptical orbit around a star; a Word for any planet.

The term 'planet Earth' (note the capital E) is a noun phrase, a group of Words based on a noun (Earth) that functions as a unit in a sentence.

The noun 'earth' (lower case e) is a common noun, a general Word for the substance of the land surface of our planet.