Here is the full list of all 40 Beyond Sol achievements.
Achieve a diplomatic victory
Own 80% of the solar system
Upgrade a weapon to level 10
Upgrade a player ship to level 10
Have a reserve fleet strength of 400
Lose a war
Have a reserve fleet strength of 200
Have an active fleet strength of 70
Have a city population of 2000
Have an active fleet strength of 40
Have a rival
Have a city population of 1000
Unlock all player ships
Unlock all weapons
Achieve an economic victory
Upgrade a player ship to level 5
Destroy a Military Station
Fully upgrade the Exeter Corporation building
Fully upgrade the Tekla Corporation building
Upgrade your Control Center to level 4
Buy a commodity for less than 40 credits
Make an alliance
Have a city population of 500
Win a war
Destroy a city
Declare war
Fully upgrade the Cryos Corporation building
Have a city population of 250
Upgrade your Control Center to level 3
Have an active fleet strength of 10
Have a city population of 100
Sell a commodity for more than 225 credits
Destroy a pirate base
Collect a pirate bounty
Upgrade your Control Center to level 2
Repel a pirate raid
Establish a city
Destroy a level 4 Control Center
Achieve a military victory
Defeat a rival in war