Where does “kāter” come from?

kāter (German) comes from German Katarrh, from Latin catarrhus, from Ancient Greek κατάρροος, from Ancient Greek καταρρέω, from Ancient Greek κᾰτᾰ-, from Ancient Greek κατά, from Ancient Greek καταστερίζω, from Ancient Greek ἀστήρ — to be/become dry; to burn, to glow; hearth.

Ancestry of “kāter”, step by step

Step	Language	Word	Meaning
1	German	Katarrh	Alternative form of Katarr
2	Latin	catarrhus	catarrh, rheum
3	Ancient Greek	κατάρροος	catarrh, head cold
4	Ancient Greek	καταρρέω	flow down; stream, rush; drip
5	Ancient Greek	κᾰτᾰ-	fully; down; denoting motion downwards
6	Ancient Greek	κατά
7	Ancient Greek	καταστερίζω	to place among the stars; I put among the stars
8	Ancient Greek	ἀστήρ	a celestial body; illustrious person; starfish
9	Proto-Hellenic	astḗr	star
10	Proto-Indo-European	h₂stḗr	star
11	Proto-Indo-European	h₂eHs-	to be/become dry; to burn, to glow; hearth

Every word from Proto-Indo-European h₂eHs- →