Note: Consult your local Orthodox rabbi for any practical halachic advice.
His esrog is pictured to the right. On the second day of Yom Tov, he dropped it on the floor, and the pittom broke. You can click to zoom.
To explain, and define the terms I plan on using in this post: At the top of the esrog is a short stem, called the dad. This is not where it attaches to the tree -- the stem which attaches to the tree is called the uketz, and is on the bottom of the esrog. So we are referring to the dad. The dad is thin, vertical, and yellow, and emerges from the top of the esrog. And at the top of that dad is a decoration, brown in color and perhaps somewhat crusty, alternately called the pittom or the shoshanta.
When Junior dropped it, half of the shoshanta broke off, and a diagonal cross section of the dad broke off as well. You cannot really tell this so well from the image (click to zoom in), because a new brown crusty layer formed over the break.
So what happens, then? Is the esrog now passul?
I looked it up in my Mishnah Brurah, and the conclusion is that it is not pasul bedieved. Even had the entire shoshanta broken off, it would have been kosher bedieved. And as for the stem (the dad), so long as some of it still exists, and it was not broken off in such a way that there is an furrow in the actual fruit below the stem, it is not considered chaser (deficient) and also is kosher bedieved. See Mishna Brurah siman TaRMaHH (548) inside for the details, but they essentially match what I said above, with some nuance in just what bedieved means in this case. Should one ideally borrow (or acquire) someone else's esrog if available? It seems so, based on what he writes... [I am not so convinced by the imperative to be choshesh for all possible opinions, as is the general directive of the Mishna Berurah.]
To see this inside, let us start with the Shulchan Aruch:
7) If its dad was taken off -- and this is the small head upon which the shoshanta exists -- it is invalid. Rema: And some are stringent if its shoshanta -- which is that which we call pitma is taken off (Ran). And it is good to be stringent
in the case where it is possible. However, as a matter of law, one should not invalidate unless the dad, which is the step upon which the top, the pitma is upon it, and the top is called the shoshanta (haMaggid). And all of this is specifically where it is taken off. But, if it never ever had a dad, it is valid. And so are most of the etrogim which they bring in these countries (the Rosh).
[Beer Hetiv: 11) its dad -- this is the step on the top of the etrog, and the shoshanta is upon it. And see the Taz.]
"29: if its dad was taken off: this is the stem on the top of the etrog, like the tip [nipple] of the breast, and it is inserted into it [the etrog] and the shoshanta is upon it.
30: it is invalid: For it is like it [the etrog] is deficient. And behold, from the language of 'the dad was taken off', the implication is that the stem was taken off, even that which is embedded inside the etrog, such that it is made like a depression, and therefore it is invalid. However, if only that which above the etrog is taken off, there is not to be stringent, and so the Taz agrees as a matter of law. However, there are some of the decisors who side with the idea of being stringent, even if only that which is above the etrog is taken off, and they maintain that this is within the scope of 'deficient' as far as it is concerned. And, if there is left from the stem above the etrog even the slightest amount, the opinion of the Magen Avraham is that one should not be stringent in this.
31: where it is possible: this is where it is possible to take [aquire?] a better one than this. But if this is the best, one should not be stringent because of the shoshanta. And still, it appears that this is just if it is missing the shoshanta, but if it is missing as well some part of the stem, even if there is still some part of the stem above the etrog, it is better to take a different one, if it is possible, since there are those who are stringent as well in this."
There is also an interesting discussion of this in Aruch Hashulchan, here. Maybe fodder for a follow-up post.
See also Halachipedia on the topic, which gives a much more negative outlook on it. More encompassing than my post above, but also omitting some important detail and thus nuance in the Rema, for example. And they also use the term pittom differently than I defined it above for the sake of this post.