- You sacrifice Chromatic Star before Perilous Research resolves, in which case it's not there and you have to choose something else.
- You sacrifice Chromatic Star as part of Perilous Research, in which case it's gone before you have the opportunity to activate its own ability.
However, you'll still draw the card from Chromatic Star's graveyard triggered ability, as it doesn't care how it got there.
This is because, in general, you can only activate abilities at certain times (when you have priority). When a spell is resolving, nobody gets priority until it's finished, so you can't activate the Chromatic Star during the decision to sacrifice it. That is, when a spell resolves, it has to finish completely before anyone can do other things.