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 central goal of the service described in Vayikra 16 is to achieve atonement for the nation's sins.
Focus of the ceremony: "לְכַפֵּר עַל בְּנֵי יִשְׂרָאֵל מִכׇּל חַטֹּאתָם" – This approach looks to this description of the goal of the ritual as evidence that the rite was aimed at achieving penance and that the ceremony was focused on the people rather than the Mikdash.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:
Commemorative – 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.
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 (a means to atone for transgressions throughout the year) and a national, annual Day of Atonement. In fact, the very first Yom HaKippurim might have even been meant to atone for the Sin of the Calf itself.
The Sin Offerings – The sin offerings of Aharon and the nation appear to be patterned after two other unique sin offerings which share a similar protocol,3 the bull of the anointed priest who has inadvertently transgressed and the goat of the nation who has sinned inadvertently in the realm of idolatry. 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. [For various understandings of the sin, and whether or not it constituted idolatry, see Sin of the Golden Calf.]
White, linen clothing – R. Chisda in Bavli Rosh HaShanah suggests that he Yom HaKippurim service was performed in white clothing rather than the priests' normal golden garb so as not recall the sin ("for a prosecutor cannot become a defender").
"וְכִפֶּר עַל הַקֹּדֶשׁ מִטֻּמְאֹת בְּנֵי יִשְׂרָאֵל" – This verse implies that the blood of the sin offerings not only atoned for the people but also played a role in purifying the Mikdash from impurities. These sources explain it in one of two ways:
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Why start with 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 Tabernacles's construction, the first vehicle for the nation's atonement. On that very day, Hashem introduced teh second vehicle, Yom HaKippuriim.
Warning – To achieve atonement for the people, it is required for Aharon to go into the Holy of Hollies. Since according to R. Saadia Gaon1 Nadav and Avihu were killed for going into the Holy of Hollies, Hashem used them as an example for what could happen if you go into the Holy of Hollies without the accompanying service.
When and how often was the ritual enacted?
What sins are being atoned for? The atonement from the service is for all of the sins of Israel.
Role of blood
Is the purification for the people or the place? The purification is for the people exclusively. Any reference to purifying the Kodesh or any part of the Mikdash is not to be understood as purifying the place or objects themselves; instead these objects are facilitating the atonement through the service of the day.
What is the function of various rituals of the service? All the rituals are for the atonement of the people, and even the text says that specific actions, for example:
And he shall make atonement for the holy place, because of the uncleannesses of the children of Israel, and because of their transgressions, even all their sins; and so shall he do for the tent of meeting, that dwelleth with them in the midst of their uncleannesses.
The blood libations, which come to purify the sanctuary, really use the sanctuary and the utensils to purify the sins of Israel
Purification of the Temple
The various rituals were instituted as a means of purifying the Mikdash from impurity.
Why start with Nadav and Avihu? Since the death of Nadav and Avihu caused the Midash and certain utensils to be impure, there is a need to purify them through this service.
What sins are being atoned for? It depends on whether the impurity is a Halakhic or Metaphoric one:
Halakhic - There is only atonement for sins directly dealing with the laws of impurity of the Mikdash. (come back) i.e., Entering the Mikdash impure, eating sacrifices in an impure state, etc.4
Metaphoric - When Israel sins they are causing a There must be atonement for all the sins of Israel that are causing the Mikdash to be 'impure'.5
Is the purification for the people or the place? The purification is for the Mikdash that has become impure. Any reference to the atonement of Israel's sins is a means to purify the temple.
Is there a greater atonement? The biggest issue with claiming the service on Yom Kippur is solely about purification of the temple is a later verse of the chapter (16:30):
For on this day shall atonement be made for you, to cleanse you; from all your sins shall ye be clean before the Lord.
The verse seems to say that there is a universal atonement that happens on this day which is connected to the service of the day. One way of solving this issue is to say that there is a separate aspect of atonement on Yom HaKippur that has nothing to do with the service done by the Kohen Gadol.6
What is the function of various rituals of the service? The function of the various rituals of the service depends on whether the purification of impurity was on a metaphoric or Halakhic level. According to the Hoil Moshe all of the sacrifices, including the Azazel, were to get rid of Halakhic Tumah and did not atone for any sins. Milgram, on the other hand, says that there was atonement in specific offerings like the Azazel that would facilitate purification of the Mikdash from the metaphoric impurity
Purity and Atonement
The service was dual focused, meant both to purge the Temple from impurity and to attain atonement for the nation.