Gemara

Version #1 (Shmuel): If a man's entire body does not feel semen exuding, it is not Tamei.

The Torah calls it Shichvas Zera. It must be able to be Mazri'a (conceive).

Version #2 (Shmuel): If semen does not shoot out like an arrow, it is not Tamei.

These versions argue about one who felt the semen start to move, but did not feel it come out.

Rava was unsure about this;

Question (Rava): If he felt when the semen started moving, but not when it came out, what is the law?

Answer (Mishnah): If a Ba'al Keri immersed before urinating, when he next urinates he is Tamei. (A remaining drop of semen comes out with the urine. Even though he does not feel, it is Metamei him!)

Rejection: Since he felt most of the semen come out, the remaining drop is Metamei even if he does not feel it.

Version #3 (Shmuel): If semen does not shoot out like an arrow, it cannot be Mazri'a (impregnate).

Inference: It cannot be Mazri'a, but it is Metamei. "Ki Yiyeheh Vecha Ish... Mikreh Lailah" discusses any case.

Rishonim

Tosfos (43a DH bi'Re'uyah): Semen is Tamei if it can be Mazri'a, even if it exuded without Hargashah. This is clear from the Gemara below. Here, the Gemara connotes what can be Mazri'a is Tamei, even if it did not shoot out. In Nedarim (91a) and Yevamos (65a), we say that what does not shoot out cannot be Mazri'a! We can say that according to this version, it suffices that the semen was initially uprooted with potential to be Mazri'a. Alternatively, it can be Mazri'a, but it cannot impregnate, for it did not shoot out and does not enter her body. It merely drips out.

Poskim

Rema (EH 25:2): One may have Bi'ah with his wife Lo k'Darkah (in the anus) as long as he is not Motzi Zera l'Vatalah. Some permit even to be Motzi Zera occasionally, but not regularly.

Beis Shmuel (2): If blood was found on her Bedikah cloth and there is a Safek whether it is from her or from him, it should be permitted for him to be Motzi Zera l'Vatalah (i.e. Lo k'Darkah) to verify whether it is from him, for there is no Isur to have Bi'ah [Yad Efrayim - Lo k'Darkah] and be Motzi Zera. We can bring a proof from Yevamos 76a. If the Ever was punctured, we put warm bread by the anus so that he will be Motzi Zera. One can dispel this proof. This requires investigation.

Maharsham (7:26): A couple was childless for 15 years. Doctors say that she is fine. They want him to have Bi'ah, but ejaculate outside (into a tube to test his semen). The questioner said that the opinion that permits Bi'ah Lo k'Darkah would permit. I disagree. The lenient opinion permits only to ejaculate into the woman, for also this is called Bi'ah. All forbid ejaculation outside. I challenge the Beis Shmuel's proof from Yevamos, from Imrei Esh (YD 69). He says that the Isur of Zera l'Vatalah depends on ability to father, just like it depends on Peru u'Rvu. According to one answer in Tosfos, if the semen does not shoot out, he cannot father, so there is no Isur of Zera l'Vatalah. If one may not marry until he is tested (lest he is a Petzu'a Daka), he cannot father, therefore one may test him. If so, perhaps one may be lenient about a man who has not fathered in 16 years. However, perhaps he could father through another woman. We are stringent about a Safek mid'Oraisa. The Zohar says that Zera l'Vatalah is worse than all Aveiros. The Beis Shmuel says that this is not literally true. No'am Megadim says that it is not as severe as Arayos. Pnei Yehoshua (2:44) agrees, but says that perhaps it is like murder. Sedei Chemed (Ishus 1:33) forbids in any case like this. However, after Bi'ah she may turn over in order to expel the semen, for in any case she does not absorb it.

Igros Moshe (EH 2:11): If semen is artificially inserted into a woman, there is no Bi'ah, so there is no concern for Arayos, Mamzerus or becoming Asur to her husband. Otzar ha'Poskim brings from Bar Livai and Minchas Yechi'el that (if she was married,) the child is a Mamzer. They cannot argue without a source against all the great Poskim, who say that there is no concern. Bar Livai says that the child is not called his father's son (unlike the Beis Shmuel and Mishneh l'Melech Hilchos Ishus 15:4), for the only way we found that Pasul seed feeds Terumah is a Mamzer born to a Kohen. He permits his grandmother to eat, even after her husband and son died. We do not say that a Mamzer born through a bathhouse feeds! This is wrong. The child is Kosher! The decree lest one marry his sister (Yevamos 37b) applies when the father is a Yisrael, but not if he is a Nochri. If a woman was inseminated through a sperm bank, in this country most men are Nochrim, and Kosher Yisre'elim would never transgress Zera l'Vatalah to donate semen, so this is no concern. If a Yisrael baby (e.g. he is circumcised) is found deserted, when he matures he may marry. We rely on the majority, who are not his sister. All the more so regarding semen, we rely on the majority of Nochrim, since the entire Isur is only l'Chatchilah.

Igros Moshe (DH v'Nish'ar): One reason Rava gave for the decree of Havchanah is lest we err about the father, and the 'son' will do Chalitzah, but really, the Yevamah is still forbidden. This forbids a woman to remarry within three months. We do not forbid raising an orphan in one's house due to this concern. One may marry someone with children and raise them, without concern lest a step-child be established to be from the step-parent. This is not due to publicity. Rav Yitzchak holds that publicity does not help, but he permits raising a step-child. Since the parents know, they will not allow the child to marry a half-sibling. We are not concerned lest the parents die suddenly without informing the child. Likewise, they will not allow Chalitzah if he is not a paternal brother. A daughter from a Nochri through artificial insemination is Kosher to Kehunah, since she was not born through an Isur Bi'ah. The child of a Nochri and a Bas Yisrael has no lineage from his (or her) father. All the lineage is from the mother. There is no Safek about him, so the Shechinah can dwell on him. It is a serious question whether a man may emit semen for the sake of artificial insemination. Perhaps this is Zera l'Vatalah.

Minchas Yitzchak (4:5): Indeed, the Heter to raise step-children is due to publicity! R. Yitzchak forbids due to Mar'is ha'Ayin, for those who do not know. There, those who know do not publicize this, but they will tell the child, to prevent Isur. If only the parents know, and they keep this a secret and call him their son, we are concerned. Avodas ha'Gershoni (1) and the Chasam Sofer (EH 2:125) were concerned for all this. The parents would not be believed later to say that it is not his son! Tashbatz (3:263) says so. If the husband will die, we cannot rely on the mother. A mother (who remarried within three months) will try to convince people that her son is from her new husband, so he will inherit him. If we could rely on her, three days of Havchanah would suffice!

Minchas Yitzchak (4): B'Di'eved we assume that the father is from the majority, but l'Chatchilah one may not create a Safek. The Torah would not say to send away a bird if it were forbidden (lest the finder rely on the majority of birds, which are permitted, and eat it - Chulin 140a). The Birkei Yosef (13:4) says that in the She'altos' text, Rava explains that Havchanah is lest one marry his sister. It is not a decree; rather, the Torah requires Havchanah for this reason. Even if Havchanah is only mid'Rabanan, perhaps this is when there are only two possible fathers, and it is easy to be concerned for the Safek. If the father could be anyone, perhaps Havchanah is mid'Oraisa. Sefer ha'Chinuch (Mitzvah 35) says that the Isur ("Lo Sechalel Es Bitcha Lehaznosah") is lest children not know their relatives, like the Medrash (Rabah Nasa 9. It obligates adulterers (b'Yedei Shamayim) for all Aveiros that result, e.g. the child unknowingly curses his father...) Many of these concerns apply to artificial insemination, even from a Nochri. Also, sometimes the child is a Kohen, and he thinks that he is Yisrael (and he will marry a divorcee or become Tamei), or vice-versa. Even if the mother's husband knows the truth, if he establishes the child to be his, later he is not believed to say that he is not his son (Chavos Ya'ir 92,93). Also, perhaps the father is a Pasul who may not marry a Yisraelis (so also the child is Pasul).

Divrei Malkiel (4, Sof Teshuvos 107 and 108): One may not emit semen into a tube for artificial insemination. At the time, it is Zera l'Vatalah, just he relies on the doctor to inject it into his wife. What is the source to rely on this?! One may not bless on fruits prone to become wormy until opening them and checking them, lest his Berachah be l'Vatalah. All the more so one may not rely on others regarding Zera l'Vatalah, which is very severe. The Rashba says that we do not bless on Mitzvos that depend on others, lest they retract and his Berachah was l'Vatalah, in spite of the Chazakah that Yisre'elim do not lie. The Chazakah that a doctor will fulfill his word and not ruin his reputation is no stronger than this. Often there is a problem with the tube, and the doctor says that he must eject semen again. He is not so experienced with this, to know the right temperature to store the semen, or to enter it properly. We must be concerned lest the doctor suspects that the husband is sterile, so he put in the tube beforehand his own or another's semen, to ensure that she become pregnant, to fulfill his words! The husband will not tell the doctor that he suspects him and wants to inspect the tube! We are concerned lest people switch if they would benefit from this. I forbid, especially if they were not yet married 10 years without children. If we would permit, many Isurim could result, and we would encourage sinners. I did not see Ezer mi'Kodesh who permits, but I do not advise to rely on him.

Shevet ha'Levi (3:175): The Admor of Satmar says that injecting semen into another woman is just like Bi'ah, regarding Kares, Chiyuv Misah and forbidding her to her husband. I agree that it is an awesome Torah Isur, but I do not consider it like Bi'ah.

Yabi'a Omer (2 EH 1:12): The Chazon Ish was asked about a woman with a very short menstrual cycle. Doctors said that she was childless because her days proper for conception are before she can immerse. He said 'we cannot permit artificial insemination', and permitted her to count a day earlier (the days before immersing). I say that if doctors say that the only possibility for pregnancy is through artificial insemination, one who wants to be lenient has whom to rely on. Maharsham and the Chazon Ish forbade, but they did not explain why. Peru u'Rvu is a great Mitzvah. If they have been married 10 years, being stringent will cause divorce. Hash-m allowed His name be erased in order to make Shalom between a man and wife. If necessary, they may do so even when she is Nidah. Great supervision is needed, lest they switch his semen with another man's. If this happened, he must separate from his wife for three months, for Havchanah. Doctors are suspected to switch.