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