The word for the 'mind teaser' contest of the day -to which nobody won the prize, an antenna ball -is 'anhedonia' (the inability to feel pleasure). Ironically, this is a condition that Frasier himself exhibited several times in the series, and was the main theme of one of the episodes.
While Roz and Frasier fight in her booth, John Glenn talks about things he saw in space that he was told not to let people know about. Although this is clearly facetious, some real-life "space-hoax" people (those who think the U.S. never went to the moon, among other space exploits) used this episode as "proof" that NASA had covered up the astronauts' experiences in space.
As Frasier not-so-subtly "auditions" for the announcer's spot on Roz's space show, he ends his speech by saying, "Godspeed, Roz Doyle." Seconds before the 1962 launch of John Glenn's Friendship 7 flight, back-up astronaut and capsule communicator Scott Carpenter said, "Godspeed, John Glenn."
Two references are made here to Tang, a citrus-flavored drink mix. Sales of Tang were poor until NASA chose it to use in experiments on eating in orbit, which started with John Glenn's Mercury flight in February 1962.
B. K. suggests that Roz use the music from "2001: A Space Odyssey" for the beginning of her show. The actual name of that piece is "Also sprach Zarathustra" and was written by Richard Strauss.