The most common and most likely answer to this crossword clue is the 10 plus 5 letter word ASTRONOMER ROYAL
I've seen this crossword clue in The Daily Telegraph
A
S
T
R
O
N
O
M
E
R
R
O
Y
A
L
Buddy explains!
The straight part
We know that "eminent scientist" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case eminent scientist = astronomer royal.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.