Here is the full list of all 36 Procyon achievements.
Complete Stage 5
Complete all stages on Easy difficulty (or higher)
Complete all stages on Normal difficulty (or higher)
Complete Stage 4
Destroy 1000 enemies using the beam
Complete Stage 3
Destroy 50 enemies with one lightning charge
Kill 3000 enemies
Complete any stage without using the shield
Complete any stage without using bullets
Retry from the Game Over screen
Destroy 1000 enemies using the bullets
Complete Stage 1
Complete Stage 2
Complete any stage without using the beam
Achieve a multiplier of 500x
Complete all stages on Hard difficulty
Complete Stage 3 in Attrition mode
Complete Attrition mode on Easy difficulty (or higher)
Complete Attrition mode on Normal difficulty (or higher)
Complete Attrition mode on Hard difficulty
Complete Stage 3 in Survival mode
Complete Survival mode on Easy difficulty (or higher)
Complete Survival mode on Normal difficulty (or higher)
Complete Survival mode on Hard difficulty
Kill 20,000 enemies
Destroy all of the large armada ships in Stage 3 on Normal difficulty or higher
Block 500 bullets with the shield
Block 50 bullets with a single use of the shield
Keep the beam locked continuously on a single enemy for 30 seconds
Sustain a multiplier of 500x for 30 seconds
Reach the boss of a stage without firing any weapons
Reach the boss of a stage without firing any weapons, using any shields, or dying on Normal difficulty or higher
Lose your last life during the final boss battle
Lose a life to the first enemy of the first stage
Complete Stage 4 without shield-blocking or being hit by any purple homing bullets on Normal difficulty or higher.