Continuity mistake: When the Doctor and Ida descend into the core of Krop Tor, the screen charting their progress shows 39% Oxygen. After cutting away to an overhead shot of the two, the camera cuts back to the same descent display, only now the oxygen percentage (which has been descending constantly all this time) is at 42%.
Factual error: Ida, the science officer, says that the planet is in geostationary orbit around the black hole. However, the word "geostationary" applies very specifically to objects orbiting the planet Earth. Since a black hole is a type of star, this planet's orbit could be described as astrostationary, or even just stationary, but definitely not as geostationary. A science expert on an interstellar mission wouldn't make this mistake, and she wasn't dumbing things down, either, since "geostationary orbit" is already a pretty obscure topic for people unfamiliar with space technology.
Factual error: The Doctor (and the staff on the station) refer to the idea of a planet being in orbit around a black hole as 'impossible'. It is not. Gravity (and physics in general) works perfectly well outside of the event horizon. For all practical purposes - regarding orbiting around it - the black hole might as well be any other object, as long as it has an equal mass.
00:07:15 - 00:08:30
Continuity mistake: In shots of the base's Ood habitation area from above, the Ood are shown in four rows from left to right, with the Ood in the third row standing with their backs against the room's middle partition and the other rows sitting on benches. However, in a shot from inside the lower area when the Doctor and Rose first go to Ood habitation, the third row is shown sitting as well. Not until later in the episode, when Danny is ordering the Ood to remain in the room, is the third row shown standing again.
Continuity mistake: When the underground cavern is first shown, the descent pod is at the bottom of a pile of rubble up against the wall at the bottom of the drilled shaft. However, in the shot of the cavern shown when Ida says "My god, that's beautiful," the pod and pile of rubble is further away from the wall.
Continuity mistake: After Ida retracts the roof to show the black hole, there is a shot where the Doctor and Rose are looking at it, with nothing above them, as he explains about it. However, other shots of the base's control room show that there are things in the room, such as wires connecting the central computer table to the walls, that should have been visible.