Longman Dictionary of Contemporary English - rightly

~ adv 1 for a good or sensible reason  (The audience was rightly outraged at this suggestion. | quite rightly BrE)  (She insisted, quite rightly, that we all put our seat belts on.) 2 correctly  (As she rightly pointed out, this will do nothing to solve the problem. | if I remember rightly spoken)  (If I remember rightly, Ray's parents emigrated shortly after the war.) 3 rightly or wrongly used to say that whatever you think of someone's action, this is what they did  (Rightly or wrongly, the Italians decided to withdraw from the competition.) 4 and rightly so spoken used to say that a decision or action you have just described is fair and morally right  (Bryan was punished, and rightly so.) 5 I can't rightly say/don't rightly know spoken used to say that you are not sure whether something is correct or not