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