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Marital Relations on Rosh Hashanah 
One is to avoid marital relations on both nights of Rosh Hashanah, in order not to enter into a state of Keri.
Mikveh Night: If the 1st or 2nd night of Rosh Hashanah is the night of Mikvah then one is obligated to have intercourse that night. [One who refrains from doing so is considered as if he is performing a Mitzvah as a result of an Aveira. If a woman delayed going to Mikveh prior to Rosh Hashanah some Poskim rule she is not to immerse on the night of Rosh Hashanah. However many Poskim rule that she is to immerse.]
Strong desire: Those who feel a strong desire which may prove difficult to withhold are better to have intercourse than to come to doing a sin.
Immersing in a Mikveh the next morning: A man who had marital relations on the night of Rosh Hashanah, such as in the event of Mikveh night or a strong desire, is to immerse in a Mikveh the next morning prior to Shacharis. [In the event that one knows he will be unable to immerse in a Mikveh the next morning he is nevertheless obligated in marital relations if his wife went to Mikveh that night.]
 M”A 581/16; M”B 581/26; based on Seder Hayom; Shlah p. 213 brought in Kaf Hachaim 581/86
 Shlah p. 213 brought in Kaf Hachaim 581/86; Nevertheless one is to do so with fear and trepidation and diminish in his physical enjoyment. [ibid; Beir Heiytiv 581/16]
 Kneses Hagedola brought in M”A 581/15; Elya Raba 581/38; Chayeh Adam 139/7
 The reason: As it is not considered a Tevilas Mitzvah. [ibid] and therefore due to the holiness of Rosh Hashanah it is not allowed. [So is implied from above Poskim that it is not prohibited due to the prohibition of immersing on Shabbos but rather due to the holiness of Rosh Hashanah.]
 Peri Chadash 581; Birkeiy Yosef 18; Shaareiy Teshuvah 581/16; Machatzis Hashekel 581/15 [that so is implied from Shlah]; M”E 581/53; Kaf Hachaim 581/87
Why is this not forbidden due to the prohibition of a woman immersing on Shabbos if she could have immerses before hand? The custom to prohibit a woman to immerse on Shabbos if she could have done so before hand [see Admur 326/8; Rama Yoreh Deah 197/2] does not apply on Yom Tov. [Toras Hashelamim 197/3; Sidrei Taharah 197/2; Chochmas Adam 118/8; Lechem Vesimla 197/10; Levushei Mordechai 93; Shaareiy Tohar 17/4; Divrei Yatziv 104; Nitei Gavriel 31/13]
 M”B 581/26; See Seder Hayom brought in Kaf Hachaim 581/86; Machazik Bracha 581/4; Ben Ish Chai Netzavim 9; Chayeh Adam 139/7; Kaf Hachaim 581/88.
 Seder Hayom, brought in Shlah p. 213; Kaf Hachaim 581/86; M”B 581/26
 Kerem Shlomo, brought in Kaf Hachaim 581/89; Chayeh Adam 139/7