This second auto-repair phase applies only to ships that have neither moved nor attacked during this turn, and which have not suffered any additional damage due to enemy fire, collateral damage, or ion storms.  Thus, a ship that does not move, fire, or receive damage effectively doubles its AR during that turn.  This phase is otherwise identical to the first auto-repair phase.
 
Orbitals and any ship in the same sector as a friendly or allied starbase always receives the additional AR during this phase, even if they have moved, fired, or received damage.
 
Note that this auto-repair phase is not affected by blockades and interdiction -- it represents the additional repairs that your ship's crew can effect with the materials on hand.