“exactly.” The word “exactly” is part of the verb “found out.”
“the chronology of the appearing of the star.” Herod called this meeting secretly, to avoid arousing any more interest among the people about the birth of a new king, who many of the people would have suspected was their Messiah. Furthermore, Herod tried to hide his true intention—killing the child—from the magi by not asking, “How old would the child be,” but only seeming to be interested in the star and its various appearings. The magi were almost certainly not fooled; why call a secret meeting to ask about the appearances of a star?
It helps us to properly understand this verse when we remember that in biblical times, before the invention of the telescope, the word “star” was used for many things in the heavens, including stars, planets, novas, and asteroids. Herod wanted to know the timing of the “star,” in order to ascertain how old the child must be (cp. Matt. 2:16). In this verse we find evidence for the view that the “star” was a prolonged astronomical event(s), rather than a one-time past appearance seen in the east, then miraculously appearing again to lead the magi in Matt. 2:9. The text uses the word chronos (#5550 χρόνος) to describe the timing of the star. This word refers to “an indefinite period of time during which some activity or event takes place, time, period of time” (BDAG). Literally, the verse reads, Herod “determined from them the period of time of the appearing star.” Appearing is in the present tense, indicating a continual action; the star was “continuously shining” (phainō [#5316 φαίνω]) over an indefinite period of time. Hence, the way most translations go, “the time the star had appeared,” captures the sense of what Herod wanted to know (when the star first appeared), but unfortunately misses the fact that the star appeared over a period of time, and was still appearing when Herod spoke to the magi. At this point, we believe that the best candidate for the “star” was the planet Jupiter in its various positions (cp. The Star that Astonished the World by E. Martin).