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