Although his machine is running out of juice and ignoring the orders of his commanding officer, Tanikaze refuses to return to the Sidonia, instead pursuing the rescue of Hoshijiro from her escape pod even though it takes him outside of the "point of no return" from which he won't be able to return home unaided. Unfazed by this and the complete exhaustion of his Frame's supply of Hyggs Particles, Nagate effects a successful rescue of Hoshijiro and brings her back to his Frame.
From here, all they can do is play the waiting game in the hope of rescue or the Sidonia turning around to retrieve them - something which looks ever-more unlikely as the days pass and supplies of food and water dwindle (which is particularly troublesome for Nagate given his inability to photosynthesize), and leading to some desperate measures just to keep the pair alive. Of course, given his brave actions some parties will do anything to ensure Tanikaze's survival, and better late than never help soon arrives.
Although admittedly this week's Knights of Sidonia was a little light on content in the grand scheme of things (aside from some ill-placed "exposition as training/lesson" scenes which felt at odds with what was going on elsewhere and were delvered too dryly to be interesting), its main story worked wonderfully well as a standalone tale in its own right - watching Tanikaze and Hoshijiro trying to survive and wait for their rescue was an atmospheric and subtly enjoyable affair that managed to avoid over-wrought drama and crisis for a more measured depiction of its two seemingly abandoned characters. The lack of movement and facial characterisation required also served the episode's animation in good stead, allowing the show's CG to do what it needed to do in a competent manner all the way through to its feel-good ending. Knights of Sidonia is certainly on the up and up at present, and hopefully it can keep that monentum now that all of its immediate crises have been resolved.