Here is the full list of all 16 Gravity Core - there are Braintwisting Space Odyssey achievements.
Take a look at the tutorial
Complete the second mission (speed control)
Complete the first mission (basic controls)
Complete the first prologue aboard M.O.T.H.E.R.
Complete the third mission (rotation controls)
Complete the sixth mission (full controls)
Complete all missions in the Saharis 3 system
Complete the first mission of the Alpha Draconis system
Complete all missions in the Alpha Draconis system
Complete the first mission of the Sirius II system
Complete all missions of the Sirius II system
Complete the first mission of the Zeta Prokyon system
Complete all missions of the Zeta Prokyon system
Complete the first mission of the Tau Ceti system
Complete all missions of the Tau Ceti system
Complete the Earth mission