 At what moment in the story did the characters seem to have the least hope?
In the last hours the star ahead had grown brighter by many magnitudes, and had changed its color from a dazzling blue through white to the normal yellow, of a GO sun. They had seen many such stars come near out of the galaxy's glittering backdrop, and had seen them dwindle, turn red and go out as the Quest III drove on its way once more, lashed by despair toward the speed of light, leaving behind the mockery of yet another solitary and lifeless luminary unaccompanied by worlds where men might dwell. "Our hopes were dashed—and I think all of us, even remembering the Centaurus Expedition's failure, hoped more than we cared to admit. If success comes there, a century will have passed before we can return to Earth; friends, relatives, all the generation that launched the Quest ships will be long since dead. "We face the unpleasant realization that what was feared is probably true—that worlds such as the Sun's are a rare accident, and that we may complete our search without finding even one new Earth. This may be man's last chance of escaping his pitiful limitation to one world in all the Universe. Certainly the building of this ship and its two sisters, the immense expenditure of time and labor and energy stores that went into them, left Earth's economy drained and exhausted. "Looked at historically, such super-human outbursts of energy are the result of a population's outgrowing its room and resources, and therefore signalize the beginning of the end.
 When they were mid-communication with the violent ships.
 When they realized the oxygen supply was extremely low.  When they realized it was impossible to contact the other Quest ships.  When they were about to crash into a comet.  When they were mid-communication with the violent ships.