eaves, palace, imperial courtesy
宸 centers on the deep, sheltered space of a grand roof: the eaves of a palace, and by extension the palace itself and the imperial presence associated with it.
宸 combines 宀 (roof) with 辰, which likely contributes the sound. The character originally referred to the deep eaves of a grand building, later extending to the palace and the emperor.
Under a grand roof 宀, the deep eaves 辰 shelter the palace. Picture the sweeping eaves of an imperial hall.
For シン, imagine the shin-deep shade under the palace eaves: shin -> シン, and the deep eaves cast a cool shadow.