Here is the full list of all 29 Star Horizon achievements.
Complete the game on easy difficulty level
Complete the game on normal difficulty level
Decide your fate
Complete the game on Hard difficulty level
Finish the game
Complete level 9
Complete level 8
Complete level 7
Complete level 6
Destroy at least 2000 enemy ships
Complete level 5
Destroy at least 1500 enemy ships
Complete level 4
Destroy at least 1200 enemy ships
Destroy at least 900 enemy ships
Destroy at least 600 enemy ships
Destroy at least 400 enemy ships
Complete level 3
Complete level 2
Destroy at least 200 enemy ships
Complete level 1