Pitam:[1]

What is a Pitam: Many Esrogim grow a wood stem protruding on their top called the pitam or the dad. In many the Pitam begins to grow from within the inside of the esrog. In others it grows from its very top on its outside. On the dad/pitam grows the stigma called the shoshanta.

Esrogim that grow without a Pitam:[2] Many Esrogim grow without either a Pitam or Shoshanta and are nevertheless valid being that this is their normal way of growth as they are formed this way from the beginning of their creation. One can identify an Esrog that grew without a Pitam through witnessing a groove or indentation on the top area where the Pitam usually grows.

If the Pitam fell off: An esrog which grew a Pitam and the Pitam fell off [such due to a blow and the like as opposed to naturally while on the tree] then if any area of the esrog’s top is now revealed due to this area falling off it is Pasul. This applies even if only part of the width of the Pitam fell off and revealed the area of the Esrog under that part while the other part remained on the Esrog.[3] If however only the top part of the Pitam became removed while the bottom part of the Pitam remains and completely covers the area of the esrog that the Pitom grew on, the Esrog is valid.[4] However there are opinions which invalidate an Esrog with a missing Shoshanta as it is not Hadar. Practically although we do not rule like this stringent opinion nevertheless it is proper to suspect for their words and hence an esrog with a missing Shoshanta should not be purchased if an esrog of similar quality and beauty can be found. If however the Esrog with the missing Shoshanata is more beautiful than the other Esrogim one should purchase this Esrog as the main opinion follows the first opinion.

On Chol Hamoed:[5] During Chol Hamoed an esrog which was invalidated due to a fallen Pitam may be used [with a blessing[6]] if absolutely no other esrog is available.

Second day in Diaspora:[7] On the second day of Sukkos in the Diaspora an Esrog which was invalidated due to a fallen Pitam may be used if there is no other esrog available. It is to be used without a Bracha.

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[1] 648/17

[2] See Piskeiy Teshuvos 648/6

[3] The reason for this invalidation is because the Pitam is considered part of the actual Esrog and hence if it is missing the Esrog is considered Chasar. Now although there are opinions which rule that even a missing piece is valid unless the missing area is the size of an Issur, nevertheless by the Pitam since it is the top area of the Esrog which is readily recognizable therefore they were stringent to invalidate even the slightest missing piece from that area. This is because the main Hiddur of the Esrog is dependent on its top which a person sees at first sight and hence if it is missing it is not Hadar. [ibid] 

[4] As since the top of the Esrog is not missing it is not considered missing. [ibid] 

[5] 649/19

[6] Arba Haminim P. 278

[7] 649/21