[Dark Souls] Why does killing the false sun in Anor Londo plunge it into darkness?

If we assume that the real sun is the one Solaire is staring at from the bridge into the Undead Parish there's no reason it shouldn't continue to illuminate Anor Londo from that direction. It's even roughly where the fake sun shines in from when the city is undisturbed. Furthermore looking up at it from below afterwards reveals the same sunny city as before illuminated by presumably the true sun.