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