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Having thought much about this question since I proposed it, I am going to suggest a somewhat abstract answer:

A happy ending is one in which perfection meets desire. That is, the ending which is the perfection, the rightful completion, of the story told, is also the one which the reader desired. 

A sad ending is one in which perfection does not meet desire. That is, the ending which is the perfection, the rightful completion, of the story told, is not the one which the reader desired. 

Both such endings are satisfactory to the reader because they feel that the story ended as it should (perfection) even thought they would have preferred a different outcome (desire).

This leaves two other cases to consider: 

* An imperfect story in which the ending meets the desire. What happens is what the reader wanted to have happen, but it feels false, as if the story should not have come out that way. 

* An imperfect story in which the ending does not meet desire. 

Neither of these is satisfactory. The first is bathos, at least. The second is futility. 

One corollary of these definitions is that the happiness of the ending depends on the desire of the reader and different readers may have different desires. Thus an ending may be happy for some and sad for others. 

But whether an ending is satisfactory or not should be more objective. If the story is perfect (in the sense that it ends as it should) then it should be satisfactory.