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