Why did Hashem make it public that Aharon was bringing a sin offering?
(9,2) “And he said to Aharon: Take for yourself a bull calf as a sin offering, and a ram as a burnt offering, unblemished, and bring them before Hashem.”
Rashi explained that the words “take for yourself a bull calf” mean to make known that Hashem was granting him atonement with this calf for the incident of the golden calf which he had made. But what does it mean "to make known", and why does Rashi emphasize that it was "this calf"?
We can explain it according to the gemora which explains that the posuk says “in the place where you slaughter the burnt offering you shall slaughter the sin offering” in order not to embarass those who are repenting their sins, since others who see them will assume that they are bringing a burnt offering. If so, how could Hashem command Aharon to publicly and openly bring a sin offering? Surely this will embarass him since everyone will know that it is a sin offering, because it is the first sacrifice to be brought there and so no burnt offering had yet been offered in that place.
But the gemora concludes that this concealment is only true for sin offerings brought for anything other than the sin of idol worship, but one who sinned by worshipping idols, on the contrary, we are obligated to inform the public of his sin and embarass him, because this embarassment is his atonement.
This is what Rashi means "to make known", that Hashem was saying that the public should be know that it was a sin offering, because thereby Hashem would grant him atonement for the sin of the golden calf. And this atonement was "with this calf", because it was the first sacrifice to be offered, and thus obvious to everybody that it was a sin offering.