Purpose of the Service of Vayikra 16/2

The Service of Acharei Mot

Exegetical Approaches

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Overview

Commentators debate what was the main goal of the Yom HaKippurim service. While R. Saadia Gaon puts the people at the center, suggesting that all aspects of the rite aimed to achieve atonement for Israel's sins, the Hoil Moshe views the rite as aimed at the Mikdash itself, understanding it to be a purification ceremony meant to cleanse the Mikdash of impurities. Shadal takes a middle position, suggesting that the ritual had a dual focus, to both purge the Mikdash of impurity and to expiate the sins of the nation.

Atonement for the People

The service described in Vayikra 16 was meant to atone for the nation's sins.

Sources:Mishna ShevuotShevuot 1About the Mishna, Sifra16:1616:33About the Sifra Vayikra, R. Saadia GaonCommentary Vayikra 16HaEmunot VeHaDeiot 3:10About R. Saadia Gaon, RashiShemot 30:10Vayikra 16:1-3, 11, 15-16, 34About R. Shelomo Yitzchaki, RambamHilkhot Shegagot 11:9About R. Moshe b. Maimon, RambanVayikra 16:1Vayikra 16:2Vayikra 16:4Vayikra 16:18Vayikra 16:21About R. Moshe b. Nachman, RalbagVayikra 16About R. Levi b. Gershom, AbarbanelVayikra 16:5About R. Yitzchak Abarbanel, SefornoVayikra 16:4-5, 11-12, 24, 30About R. Ovadyah Seforno, HaKetav VeHaKabbalahVayikra 16:3Vayikra 16:15About R. Yaakov Mecklenburg, Reggio

Focus of the ceremony – According to these sources, the ceremony of Vayikra 16 focuses on the people rather than the Mikdash. This might be supported by the summary statement of the unit which points to atonement of the nation as the central goal of the rite: "וְהָיְתָה זֹּאת לָכֶם לְחֻקַּת עוֹלָם לְכַפֵּר עַל בְּנֵי יִשְׂרָאֵל מִכׇּל חַטֹּאתָם אַחַת בַּשָּׁנָה" (verse 34) .¹

Reaction to the Sin of the Golden Calf – This approach might suggest that Yom HaKippurim was instituted in reaction to the Sin of the Golden Calf, either to commemorate the pardon granted or to facilitate future penitence:

• Commemorative – TanchumaKi Tisa 31About the Tanchuma asserts that the day that Hashem forgave the people for the sin was the tenth of Tishrei² and, in commemoration, Hashem set it to be a day of forgiveness for all future generations as well.³
• Corrective – The sin might have further demonstrated the nation's general need for vehicles of repentance and atonement, leading to both the construction of the Tabernacle and introduction of the sacrificial system, a means to atone for transgressions throughout the year,⁴ and to the institution of Yom HaKippurim, a national, annual day of atonement.⁵ In fact, the very first Yom HaKippurim might have even been meant to atone for the Sin of the Calf specifically.

The sin offerings and inner sprinkling of blood – The sin offerings of Aharon and the nation appear to be patterned after two other unique sin offerings which share a similar protocol:⁶ the bull of the anointed priest who has inadvertently issued an erroneous ruling (פר כהן משיח)⁷ and the goat of the nation who has sinned in the realm of idolatry (שעיר עבודה זרה).⁸ This would support the suggestion that the service of Vayikra 16 was intended to atone for both Aharon's role and the nation's sin in the episode of the Golden Calf.⁹

White clothing – R. Chisda in Bavli Rosh HaShanah suggests that the choice of white clothing is also related to the Sin of the Calf.  The priest does not perform the rituals in his normal golden garb so as not recall the sin ("for a prosecutor cannot become a defender"). R"Y Bekhor ShorVayikra 16:1, 4, 10, 33About R. Yosef Bekhor Shor alternatively suggests that gold, being reddish in color, is symbolic of sin, while white connotes a cleansing of sin. Since this is a day of atonement, the priest symbolizes the process of "אִם יִהְיוּ חֲטָאֵיכֶם כַּשָּׁנִים כַּשֶּׁלֶג יַלְבִּינוּ"‎¹⁰ in his choice of garments.

Why three sacrifices? All these sources agree that both the bull and two goats served to atone for sins but they differ in their explanations of why three distinct sin-offerings¹¹ were necessary and why two are brought in the Mikdash, while one is sent outside:

• Different people – R. Saadia¹² suggests that each is meant to atone for the sins of a different group of people. The bull atones for the sins of the high priest.  The "goat for Hashem", understood by R. Saadia to mean "the goat for the House of Hashem",¹³ atones for the regular priests^{. 14}  Finally, the second goat expiates the sins of the nation as a whole. The first two sacrifices are offered in the Mikdash, abode of the priests, while the second goat is sent outside the sanctuary where the nation resides. 
• Distinct sins – Most of the other commentators, following Mishna Shevuot 1:6Shevuot 1About the Mishna, assume that both the bull and "goat for Hashem" atone for sins related to purity and the Mikdash,¹⁵ such as intentionally entering the Mikdash or eating certain sacrifices while impure,¹⁶ while the goat for Azazel atones for all other sins.¹⁷ The blood of the first two is appropriately sprinkled inside where the sins might have taken place.  The second goat, though, is brought outside the camp, as it is so contaminated by the enormity of the sins it bears that it would be unfitting to be offered in the sanctity of the Mikdash.¹⁸ 
• Two staged process – One¹⁹ might alternatively suggest that the offering of the bull was meant to cleanse the high priest from all his iniquities before he could set out to atone for others. The two goats were then brought to atone for two distinct sets of sins of the nation, those related to impurity in the Mikdash and all other sins.²⁰

"וְכִפֶּר עַל הַקֹּדֶשׁ מִטֻּמְאֹת בְּנֵי יִשְׂרָאֵל" – This verse is somewhat difficult for this position as it implies that the blood of the bull and goat was supposed to purify the Mikdash itself (rather than the people). These sources explain it in one of two ways:

• In the Kodesh – R. Saadia reinterprets the phrase "עַל הַקֹּדֶשׁ" to mean "in the Kodesh" rather than "on/for the Kodesh" and understands the word "טֻּמְאֹת" to mean transgressions rather than impurities.  According to him, then, the verse only states that the priest atoned for the people's sins in the Mikdash and says nothing about purification.
• Regarding the Kodesh – The other sources explain the phrase to mean that the priest atoned for sins regarding the Kodesh and impurities of the nation, ie. טֻמְאַת מִקְדָּשׁ וְקָדָשָׁיו.‎²¹

General role of blood – This position's understanding that the sprinkling of the blood serves to atone rather than purify fits the general role played by blood, as Vayikra 17 teaches, "וַאֲנִי נְתַתִּיו לָכֶם עַל הַמִּזְבֵּחַ לְכַפֵּר עַל נַפְשֹׁתֵיכֶם כִּי הַדָּם הוּא בַּנֶּפֶשׁ יְכַפֵּר".‎²²

Goat to Azazel

Why are the laws linked to the death of Nadav and Avihu? This position might suggest one of two explanations:

• Commanded then – It is possible that these laws were commanded right after the deaths of Nadav and Avihu. This day marked the completion of the Tabernacle's construction, the first vehicle for the nation's atonement.  On that very day, Hashem introduced the second vehicle, Yom HaKippurim. 
• Warning – To achieve atonement for the people, it is required for Aharon to go into the Holy of Holies. If Nadav and Avihu were killed for entering,²³ it is logical that Hashem would preface the protocol with a warning to Aharon of what might occur if he does not follow the right procedures.²⁴

Context of laws of purity – Ralbag explains that since the rite was instituted to cleanse the nation from their sins in the realm of impurity, it is logical that it is placed after these laws of impurity.²⁵

When and how often was the ritual enacted? These sources assume that the ritual was enacted only once a year, on Yom HaKippurim. They understand the directive "וְאַל יָבֹא בְכׇל עֵת אֶל הַקֹּדֶשׁ" to mean that Aharon was not permitted to come into the Inner Sanctum except for once a year, and only after following the protocol described in the chapter. The concluding phrase of the chapter, "וַיַּעַשׂ כַּאֲשֶׁר צִוָּה י"י אֶת מֹשֶׁה", which might initially imply that the ceremony was enacted immediately, is understood to refer only to the future, as Rashi writes, ""כשהגיע יום הכפורים עשה כסדר הזה".

Afflictions on Yom HaKippurim – The obligation to afflict one's self on Yom HaKippurim relates to the atoning nature of the day.  Seforno explains that the sacrifices of the day only serve to downgrade the sin; to receive full pardon and absolution one must also afflict one's self, confess, and repent.

Meaning of כפרה

Purification of the Temple

The various rituals were instituted as a means of purifying the Mikdash from impurity.

Sources:Hoil MosheShemot 30:10Vayikra 16About R. Shelomo Yitzchaki

Focus of the ceremony – According to Hoil Moshe, the ceremony revolves around the Mikdash rather than the people, and the tenth of Tishrei might more accurately be called a Day of Purification or Purging, rather than a Day of Atonement.  Any atonement for personal sin is secondary and needed only to ensure the purification of the Mikdash.

Why are the laws linked to the death of Nadav and Avihu? Hoil Moshe asserts that the entire ceremony was instituted in reaction to the deaths of Nadav and Avihu. The deaths of the brothers inside the Mishkan caused immense impurity which needed to be purged. In addition, it led the nation to fear that the Tabernacle's sanctity had been diminished as a result. To combat this fear and rid the sanctuary of pollution, Hashem instructed Aharon how to purify the Mishkan.

When and how often was the ritual enacted? According to Hoil Moshe the first time the ceremony was enacted was not the tenth of Tishrei, but immediately following the deaths of Nadav and Avihu, when the command was issued.²⁶ [After all, the whole rite was originally intended to purify the Mishkan from their deaths specifically.] Afterwards, the ceremony was set to be an annual one, to cleanse the Mikdash from any other intentional or accidental impurity which might have contaminated it throughout the year.²⁷

"וְכִפֶּר עַל הַקֹּדֶשׁ מִטֻּמְאֹת בְּנֵי יִשְׂרָאֵל וּמִפִּשְׁעֵיהֶם לְכׇל חַטֹּאתָם" – Hoil Moshe understands this verse to mean that the sacrificial blood was intended to purge the Kodesh itself from impurity. He gives two possible explanations for the accompanying mention of atonement from sins:

• The verse might refer to the need to atone for any sins which caused Hashem to be dissatisfied with the nation, allowing the Mishkan to be polluted.
• Alternatively, these words refer not to the initial ceremony, which was exclusively for purification, but to future years when Yom HaKippurim also incorporated atoning aspects.

Goat for Azazel – The goat for Azazel was also connected to the purification of the Mikdash:

• Appeasement to prevent future contamination – Hoil Moshe suggests that the nation erroneously believed in a demonic creature named Azazel whom they thought might contaminate the Mikdash and thereby sabotage the purification accomplished through the rituals of the Day of Atonement. To calm the nation's (baseless) concern, a gift is sent to appease (the non-existent) Azazel.²⁸ This part of the ceremony, too, then, revolves around purification and not atonement. [For more on this understanding of the rite, see Why is the Goat Sent to Azazel.]
• Purging and disposing of past contamination – One could alternatively suggest, as does RashbamVayikra 16:1-2, 10, 34About R. Shemuel b. Meir, that the sending of the goat is similar to the purification ceremony of the leper described in Vayikra 14. In both cases, two animals are brought, one of which is killed while the other is sent away alive. In both cases, it is possible that the slaughtered animal's function is to purge the individual/sanctuary from impurity while the dispatched animal is meant to carry that impurity away.

Why two sin offerings? Hoil Moshe does not explain why separate sacrifices were needed for Aharon and the nation as a whole, nor why one was a bull and one a goat.²⁹

Internal sprinkling of blood – As the blood was meant to purify the Mikdash itself, it is logical that the blood is sprinkled inside.³⁰

Context of laws of impurity – Since the chapter revolves around purification of the Mishkan it is logical that it follows other laws of impurity. With regards to each of these, the text teaches what must be done to both purify the impure individual and how to atone for the impurity.  Our chapter then follows with a similar explanation of how to purify and atone for the impurity of the Mikdash itself.

Can the Mikdash become impure? Several verses in Torah outside of or chapter similarly suggest that the Mikdash can indeed be contaminated. At the conclusion of the laws regarding one who has an emission, in the chapter immediately preceding ours, Hashem warns, "וְלֹא יָמֻתוּ בְּטֻמְאָתָם בְּטַמְּאָם אֶת מִשְׁכָּנִי אֲשֶׁר בְּתוֹכָם".  See also Bemidbar 19:20, " וְאִישׁ אֲשֶׁר יִטְמָא וְלֹא יִתְחַטָּא וְנִכְרְתָה הַנֶּפֶשׁ הַהִוא מִתּוֹךְ הַקָּהָל כִּי אֶת מִקְדַּשׁ ה' טִמֵּא". These verses imply that the Mikdash itself can become impure, and moreover, that an action which causes such impurity is a capital crime.

"כִּי בַיּוֹם הַזֶּה יְכַפֵּר עֲלֵיכֶם לְטַהֵר אֶתְכֶם מִכֹּל חַטֹּאתֵיכֶם" – This verse is somewhat difficult for Hoil Moshe as it suggests that the annual service was intended mainly for the purpose of atonement of the nation rather than purification of the Mikdash.³¹

White clothing – As white is a symbol of purity, it was appropriate for Aharon to wear white while performing the ceremony.

Purity and Atonement

The service was dual focused, meant both to purge the Temple from impurity and to atone for the nation's sins.

Sources:perhaps R"Y Bekhor Shor,Vayikra 16:1, 4, 10, 33About R. Yosef Bekhor Shor³²  R"Y KaraVayikra 16About R. Yosef Kara,³³ ChizkuniVayikra 16:2Vayikra 16:6Vayikra 16:8Vayikra 16:21About R. Chizkiyah b. Manoach, ShadalVayikra 16:1-2, 8, 16, 30About R. Shemuel David Luzzatto, ?RDZH, J. Milgrom³⁴

The various offerings – These sources suggest that both the priest's bull and the nation's "goat for Hashem" serve a purification role,³⁵ while the goat sent to Azazel serves an expiatory function.

"וְכִפֶּר עַל הַקֹּדֶשׁ מִטֻּמְאֹת בְּנֵי יִשְׂרָאֵל וּמִפִּשְׁעֵיהֶם לְכׇל חַטֹּאתָם " – This approach might understand this verse in one of two ways:

• Most of these sources explain that the Mikdash had been defiled by those who entered it while impure and that the ceremony was meant to purge the Mikdash from this impurity. The sins mentioned refer specifically to these purity related transgressions which had caused the pollution.
• Milgrom, in contrast, suggests that the verse is implying that not only do physical impurities defile the Mikdash but so do all of the nation's iniquities.  When any individual in the nation sins, it has a polluting effect on the Mikdash.³⁶ The priest, thus, must purify the sanctuary both from impurities (מִטֻּמְאֹת בְּנֵי יִשְׂרָאֵל) and all general transgressions ( וּמִפִּשְׁעֵיהֶם לְכׇל חַטֹּאתָם).

Why distinct offerings for Aharon and the nation? Considering that both the bull and initial goat served the same function, to purge the Mikdash from impurity, it is not clear why two distinct sacrifices were necessary.

• Those who maintain that the verses refer to cleansing of the Mikdash only from ritual impurity might explain that the priests, being in daily contact with the Mikdash, had more occasion to defile it and thus an obligation to bring their own distinct offering to rectify the error.³⁷
• Milgrom alternatively suggests that the priest had to first purge the sanctuary from his own impurities before he could act on behalf of the nation.³⁸

Biblical parallels – Milgrom suggests that all "Chatat" offerings similarly serve to purify the Mikdash rather than to atone for the person. They are brought both by the physically impure such as a zav (one who had an emission) or a metzora, and the the spiritually impure, those who sinned, because both sources of impurity defile the Mikdash. As evidence he points to many places in Tanakh where the word לחטא means to purify³⁹ and points out that the blood of such offerings is sprinkled not on the person but in the Mikdash.⁴⁰

The Goat of Azazel – In contrast to the sin offerings which serve to cleanse the contaminated Mikdash, this goat is meant to rid the nation of its sins, ""וְהִתְוַדָּה עָלָיו אֶת כׇּל עֲוֺנֹת בְּנֵי יִשְׂרָאֵל ".‎⁴¹  Shadal explains that the sins were transferred onto the goat so they could be dispatched to the barren Wilderness, leaving the people clean of sin.

When and how often was the ritual enacted? This approach might follow the position developed by the Gaon of Vilna⁴² who suggests that Aharon was actually allowed to enter the Holy of Holies any time he desired, as long as he followed the protocol laid out in this chapter.⁴³  It is possible that the rite was an emergency measure, undertaken by Aharon any time he thought the Mikdash had been desecrated and needed purifying. After Aharon's death, though, the ritual was limited⁴⁴ and was performed only once a year, on the tenth of Tishrei.⁴⁵

Connection to Nadav and Avihu? The directive is connected to the deaths of Nadav and Avihu since their sin and death in the Mikdash necessitated the first "emergency" cleansing of the Mikdash (and it might have been what pointed to the need for annual purification as well.)

Context of impurity – The ceremony logically follows the various laws related to impurity in chapters 11-15 since it comes to explain how to cleanse the Mikdash of all such ritual impurity. Vayikra 15 ends with a warning against defiling the Mikdash; our chapter continues with the protocol of how to purify it.  [See R"Y Bekhor Shor]

Meaning of "כפרה" – This position might suggest that the word "כפר" has two connotations and can mean both to "purge/wipe away"⁴⁶ and to "atone/expiate".

ANE parallels

White linen clothing – Since white symbolizes both purity and innocence, it was an appropriate choice for a ritual whose purpose was to achieve both.⁴⁷

¹ See also verse 30, "כִּי בַיּוֹם הַזֶּה יְכַפֵּר עֲלֵיכֶם לְטַהֵר אֶתְכֶם מִכֹּל חַטֹּאתֵיכֶם לִפְנֵי י"י תִּטְהָרוּ" which similarly focuses on purification of the nation from sin rather than a purging of the Temple from impurity.

² The Midrash assumes that there were three sets of "forty days" during which Moshe was on the mountain.  During the first set, he received the initial tablets. During the second set, he prayed for forgiveness, and at the end of the third set he received the second tablets and was told that the nation was forgiven. If Moshe ascended the mountain after Revelation on the sixth of Sivan, spent 120 days on the mountain and a couple of days in between the various sets within the camp, the final day would be the 10th of Tishrei.

³ If so, one could even suggest that the directive to fast is commemorative as well.  Since Moshe fasted while on the mountain, the nation, too, is told to fast.

⁴ See Purpose of the Mishkan and Purpose of the Sacrifices for elaboration.

⁵ Cf. Ralbag who points out that the nation needed an elaborate atonement process that could serve as a tangible expression of Hashem's forgiveness of not only inadvertent sins (which sin offerings atone for) but also intentional ones. Without such a ceremony the people would not be convinced that their sins had indeed been forgiven.  This could potentially lead to despair, where people would stop trying to improve, thinking it a lost cause regardless. [See Ralbag's similar explanation for the sacrificial system as a whole and the prohibition of blood in particular in Purpose of the Sacrifices and Prohibition of Blood.]  See also R"Y Bekhor Shor Shemot 30:1About R. Yosef Bekhor Shorwho explains the need for both sacrifices and Yom haKippurim similarly.

⁶ The blood of these two offerings (like that offered on Yom HaKippurim) is similarly sprinkled in the Sanctum and on the Incense Altar, while their flesh is burned outside the camp. This stands in contrast to most sin offerings whose blood is sprinkled on the outer altar and whose flesh is eaten by the priests.

⁷ See Vayikra 4:3-12 and commentators there that a priest who inadvertently issues an erroneous ruling leading the nation to sin must bring a bull as a sin offering.

⁸ See Bemidbar 15:22-26. For this transgression the people are told to bring both a goat as a sin offering and a ram as an Olah, just as in our unit. Though the verses do not clarify either the transgression that requires this offering or the procedure that must be done to the goat, Sifre Bemidbar15:22About Sifre Bemidbar teaches that it refers to a national act of inadvertent idolatry and Bavli ZevachimZevachim 39bAbout Bavli Zevachim teaches that it shares the protocol of the bull of the anointed priest.

⁹ For various understandings of the sin, and whether or not it constituted idolatry, see Sin of the Golden Calf.

¹⁰ See Yeshayahu 1:18.

¹¹ Not all view the goat for Azazel as a sin-offering. In verse 5 it is grouped with the second goat and both are said to be taken "לְחַטָּאת", but subsequently (see verses 9,15,25,27) the term "חַטָּאת" is used to describe only the first goat offered to Hashem and not the goat designated for Azazel.

¹² This is how he explains the ritual in his philosophical work, HaEmunot VeHaDeiot.

¹³ It seems that R. Saadia is partially motivated by a desire to demonstrate that "Azazel" does not refer to a demonic being but a place. The parallel terms "a goat for Hashem" and "a goat for Azazel" might imply that Azazel is some sort of supernatural being like God. R. Saadia, thus, prefers to explain that both the term "‎לה'‎‏‎‏‎‎‏" and "לַעֲזָאזֵל" refer to a place, either the House of Hashem or a rocky mountain. See Why is the Goat Sent to Azazel for elaboration.

¹⁴ This is somewhat difficult considering that the verse refers to the goat as "שְׂעִיר הַחַטָּאת אֲשֶׁר לָעָם" (v. 15). R. Saadia might suggest that it so called because the goat is paid for by the nation. [Nonetheless, one might wonder why the nation should pay for the goat if it is meant for the priests.] Even more difficult is the fact that verses 16-17 state that the goat atones "בְעַד כׇּל קְהַל יִשְׂרָאֵל" / "מִטֻּמְאֹת בְּנֵי יִשְׂרָאֵל", again implying that it is meant for them.

¹⁵ The fact that the bull and goat serve a similar function is supported by the shared protocol of what is done to each sacrifice and the fact that their blood is mingled before being sprinkled on the altar. It is not clear, though, why there needs to be two distinct offerings for this sin, one for Aharon and one for the nation, considering that for other sins, one goat (the goat for Azazel) suffices for both groups.  It is possible that in his role as high priest serving in the Mikdash, Aharon was more responsible for purity-related transgressions that took place there. See Shemot 28:38, which states, "וְנָשָׂא אַהֲרֹן אֶת עֲוֺן הַקֳּדָשִׁים" which could be understood to mean that Aharon bore the iniquity of holiness-related transgressions.  If so, it is understandable why a personal priestly sacrifice might have been necessary.

¹⁶ These are referred to as "טֻמְאַת מִקְדָּשׁ וְקָדָשָׁיו". See below that these sources reach this conclusion from the verse's explanation that through the sprinkling of blood: "וְכִפֶּר עַל הַקֹּדֶשׁ מִטֻּמְאֹת בְּנֵי יִשְׂרָאֵל".

¹⁷ These sources do not adequately explain why a distinct offering for the purity-related sins was necessary and why these could not be atoned for together with the rest of the sins of the nation.

¹⁸ See Rambam, Ralbag and Seforno who all make this point.

¹⁹ Ralbag explains similarly, but suggests that the priest offered his sacrifices separately from the nation, not in preparation for his purifying role, but simply because his sins were of a different caliber. The priests, as a whole, tended to be more careful and less sinful than the nation.  He suggests that it is for this reason, too, that the verse describing their atonement does not list all their various sins as do the verses describing the nations sins (mentioning: "עֲוֺנֹת / טֻּמְאֹת"  and "כׇּל פִּשְׁעֵיהֶם לְכׇל חַטֹּאתָם") but simply says, "וְכִפֶּר בַּעֲדוֹ וּבְעַד בֵּיתוֹ".

²⁰ The advantage of this reading is that the phrase "וְכִפֶּר עַל הַקֹּדֶשׁ מִטֻּמְאֹת בְּנֵי יִשְׂרָאֵל" refers specifically to the impurities of the nation rather than the priest.  There is no equivalent phrase by the bull of the high priest by which it is written only, "וְכִפֶּר בַּעֲדוֹ וּבְעַד בֵּיתוֹ", implying that all his sins were atoned for through the bull, and not only those related to impurity. Similarly, the goat for Azazel is brought by the nation and said to atone for "כׇּל עֲוֺנֹת בְּנֵי יִשְׂרָאֵל וְאֶת כׇּל פִּשְׁעֵיהֶם לְכׇל חַטֹּאתָם", suggesting that it, too, was meant to atone for the nation and not the priest. On the other hand, a disadvantage of this reading is the fact that the blood of the goat and bull are mixed during the ceremony, suggesting that they shared a purpose.  If they are for totally distinct sins and for distinct people, it is not clear why the ceremony combines the two.

²¹ See HaKetav VeHaKabbalah who writes, "נ״ל כי מלת על כאן הוראתו על עסק וענין".

²² See Prohibition of Blood for elaboration.  The main difference between the sprinkling of blood in this rite and other rituals is in its location in the Inner and Outer Sanctum rather than on the outer altar.  See above that this might be related to the nature of the sin being atoned (Mikdash related impurities) or, according to R. Saadia, the people receiving atonement - the priests.  Ralbag adds a further explanation, suggesting that due to the loftiness of the ritual, its atoning for even intentional sins, part of it was performed in the loftiest of places, the Inner Sanctum.

²³ See Why Were Nadav and Avihu Killed for elaboration on this reading of the story.

²⁴ Ralbag suggests that even if their sin was of a different nature, the juxtaposition still serves as a warning that disobedience and laxity in cultic procedures might result in death.

²⁵ Thus, even if one maintains that the directive was issued immediately after the deaths of Nadav and Avihu, it is understandable why the laws of purity are recorded in the intervening chapters.

²⁶ This might be supported from the concluding words of the chapter, "וַיַּעַשׂ כַּאֲשֶׁר צִוָּה י"י אֶת מֹשֶׁה" which imply that Aharon did as commanded immediately, and not only in the future. The phrase "וְאַל יָבֹא בְכׇל עֵת אֶל הַקֹּדֶשׁ" is understood to mean that Aharon cannot come whenever he wants, but only on this specific occasion, and on future Days of Atonement.

²⁷ Hoil Moshe emphasizes the nation's fear of this impurity more than the impurity itself, implying that it was to combat this fear (even more than any actual ritual pollution) that the ceremony was instituted.

²⁸ Hoil Moshe emphasizes that there is no such demonic power except in the minds of the people.  Nonetheless, as a concession to the people, Hashem allowed this ritual.  Hoil Moshe is consistent in suggesting that numerous commandments were given as a concession to the nation's erroneous beliefs or low spiritual level. See discussion of the Hoil Moshe's positions in "עַיִן תַּחַת עַיִן" – An Eye for an Eye, Altars of Earth, Stone, and Wood, and About: R. Moshe Yitzchak Ashkenazi.

²⁹ He could suggest that one purification rite was performed on behalf of Aharon and his sons, since they were the immediate relatives of Nadav and Avihu, and one by the rest of the nation who were unrelated.

³⁰ Hoil Moshe raises the possibility that Nadav and Avihu's sin lay in entering (or attempting to enter) the Inner Sanctum. If so, it is clear why it, too, needed purification.

³¹ The afflictions mandated also are not easily understood if the day revolves around purification rather than atonement.

³² R"Y Bekhor Shor is not explicit but in his comments to verse 33 he appears to distinguish between cleansing the Mikdash from the nation's impurities, and cleansing the priests and nation from their sins.

³³ As we only have fragments of R"Y Kara's commentary to this chapter, is difficult to know his full position. In his comments to verse 16 he implies both that the people need atonement (ויתכפר להם על אהל מועד) and that the Mikdash needs purification for having been defiled by those who entered into it while impure .

³⁴ See J. Milgrom, The Anchor Bible, Leviticus 1-17 (New York, 1991): 1009-1079.

³⁵ R"Y Kara and Chizkuni appear to maintain that these served both to purify the Mikdash itself and also to atone for those who had defiled it by entering while impure.

³⁶ See below that Milgrom suggests that different sins affect different sections of the Mikdash, depending on their severity. As such, on Yom HaKippurim all areas of the Mikdash are purified, the Inner Sanctum (verses 15-16a), the Outer Sanctum (verses 16b and 17) and the Outer Altar (verses 18-19).  In this he differs from the Sages and most commentators who claim that the altar mentioned in verse 18 in the Incense Altar (as it is referred to as "הַמִּזְבֵּחַ אֲשֶׁר לִפְנֵי י"י") and that the rite was confined to the Sanctum itself.  See, though, Ibn Ezra, who also assumes that the altar of verse 18 is the Outer Altar. [This might be supported from the language of "וְיָצָא אֶל הַמִּזְבֵּחַ" which suggests that the priest exited the Ohel Moed.]

³⁷ This explanation works best for R"Y Kara and Chizkuni who maintain that the bull and goat not only served to purify the Mikdash but also to atone for the people who had defiled it to begin with.

³⁸ If so, though, one would have expected that he should need to similarly first atone for his own sins before acting to atone for the sins of the nation in the continuation of the rite. Yet, with regards to expiation, one sacrifice suffices for all.

³⁹ See, for example,  Shemot 29:36, Vayikra 8:14-15, Vayikra 14:52, Bemidbar 19:19 and Yechezkel 45:18.

⁴⁰ In cases of sin, the severity of the sin determines which parts of the Mikdash were affected, and hence where blood is sprinkled.  Inadvertent sins of individuals are the least defiling and affect only the outer altar. Inadvertent sins of the community are more serious and affect also the outer sanctum, requiring purging of the incense altar. Thus, the blood of the bull of the anointed priest (Vayikra 4:1-12) and the community (Vayikra 4:13-21) are sprinkled there.  Brazen sins penetrate to even the inner sanctum, and this is purified through the blood of the Yom HaKippurim offerings.

⁴¹ This difference in purpose is reflected in the language of the verses.  With regards to the initial goat, the verse declares that it will atone "מִטֻּמְאֹת בְּנֵי יִשְׂרָאֵל וּמִפִּשְׁעֵיהֶם לְכׇל חַטֹּאתָם" emphasizing impurity.  The second goat, in contrast, bears "אֶת כׇּל עֲוֺנֹת בְּנֵי יִשְׂרָאֵל וְאֶת כׇּל פִּשְׁעֵיהֶם לְכׇל חַטֹּאתָם", highlighting the people's sins.

⁴² See also

⁴³ He understands the verse "וְאַל יָבֹא בְכׇל עֵת אֶל הַקֹּדֶש" to mean not that Aharon could never enter the Inner Sanctum, but only that Aharon could not enter "at all times"; to enter he needed to perform the procedures laid out in the chapter ("בְּזֹאת יָבֹא אַהֲרֹן אֶל הַקֹּדֶשׁ"). If those were performed, though, he could then enter whenever he wanted.

⁴⁴ The reason for the change might have been twofold.  It is possible that the priests following Aharon were not at a high enough spiritual level to merit multiple entries into the Inner Sanctum. In addition, it is possible that it was only in the Wilderness period, when the nation was camped in such close proximity to the Mishkan that there was heightened danger of contamination.  After arrival in Israel, when most of the nation lived at a distance form the Mikdash, there was less concern of pollution by physical impurities.

⁴⁵ Shemot 30:10 which states that the sprinkling of blood on the Incense Altar took place but once a year ( "וְכִפֶּר אַהֲרֹן עַל קַרְנֹתָיו אַחַת בַּשָּׁנָה מִדַּם חַטַּאת הַכִּפֻּרִים אַחַת בַּשָּׁנָה יְכַפֵּר עָלָיו לְדֹרֹתֵיכֶם") is understood to refer to either the post Wilderness Period, or to the one  time per year that the rite was obligatory and not voluntary.

⁴⁶ See Yeshayahu 27:9 and Yirmeyahu 18:3 where the word is paired with both "מחה" and "הסר" (meaning to erase and remove) and Vayikra 12:7-8, 14:18-20 and Yechezkel 43:26, 45:18-20 where it is paired with the verbs "טהר" or "חטא", both meaning to purify. [In the latter cases, as there is no sin to be atoned, it would be difficult to sustain that meaning of the word.]  In our chapter (and in Yechezkel), the fact that the verb sometimes takes a direct object followed by an inanimate object ("וְכִפֶּר אֶת מִקְדַּשׁ הַקֹּדֶשׁ וְאֶת אֹהֶל מוֹעֵד וְאֶת הַמִּזְבֵּחַ יְכַפֵּר"), again suggests that the word would mean purge rather than atone, as the Mikdash and altar had not transgressed to need atonement. Milgrom further notes that in Akkadian, the word kuppuru means to rub or wipe off and is sometimes found in ritual texts in the context of removing impurity (though it is not always clear if the word itself refers to the concrete action of rubbing or the more abstract notion of purification).

⁴⁷ Milgrom suggests that the choice might have been practical as well.  As the ceremony required more sprinkling of blood than usual, the distinct clothing might have simply been meant to ensure that the priest's regular clothing did not get ruined.