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Factual error: In the scene right after the Apollo 11 landing, where Lovell is in the garden with his wife, he is playing with his thumb and the waxing moon. He covers the moon with his thumb repeatedly. Usually the diameter of the moon disk is about half of the diameter of the thumbnail on an outstretched arm of an average adult. Since Lovell's arm is not fully stretched the moon should appear even smaller. In this scene we see a moon approximately of the size of the whole thumbnail. In relation to Lovell's thumbnail that moon is far too large.
Factual error: As Lovell explains the Apollo 1 fire to his son (in 1967), a large RC model aircraft can be seen over the child's shoulder. The N-Number (tail number) for FAA registration is N1976M. The particular aircraft represented on screen is a Cessna 182P - which according to FAA records - wasn't built until 1976.
Revealing: In the final 1/2 hour there's a scene in which Kevin Bacon is drifting weightlessly in the lunar module, with a roll of duct tape floating nearby. Just during the last couple of seconds of this shot, suddenly he & the roll accelerate to the right. As everyone now knows the zero-g scenes were photographed on-board an aircraft which flies free-fall arcs for up to about 20 seconds before having to level off. The sudden acceleration on the set means that shot was filmed just as the plane was leveling off from a dive.
Continuity: After Houston reacquires contact with the spacecraft following re-entry, Gene Kranz sits down in his chair, obviously full of emotion. The camera then cuts to a wide shot of Mission Control before panning in on Ken Mattingly. Before the camera zeroes in on Ken, watch the top left corner of the screen and you can see Gene sit down again.
Factual error: When the cabin temperature drops, an astronaut's breath is visible. His visible breath rises as he exhales. This is an effect of gravity - on earth the water vapor in breath, which is the component that becomes visible in the cold, rises because it is lighter than the surrounding air. Since the astronaut was in a "weightless" environment, his breath should have travelled in a straight path from his mouth into the surrounding atmosphere, rather than rising.
Factual error: It is shown that people are casually having conversation next to the giant rocket-ship transporter, known as the 'crawler'. In fact, the crawler is so noisy that no one could possibly have a conversation near it. All technicians who 'drive' the crawler (an eight man crew) and anyone in the vicinity of an operating crawler wear sound-cancelling headphones whenever near it.
Factual error: The paint scheme shown on the Apollo 13 rocket is incorrect: The paint scheme shown in the movie is what was used on the development versions of the Saturn 1-B first stage, and featured a large black band around the middle. Technicians found it much too hot to work inside the pre-launch first stage from the heat generated from this paint scheme. Subsequently, the paint scheme was much simplified to a near all-white version for the first stage beginning with Apollo 4. All subsequent ship-stacks (including Apollo 13) featured this newer paint scheme.
Factual error: When the LM separates from the CM just prior to re-entry, what you see in the film is an undocking, rather than a jettison. The difference is that in a jettison the CM's docking probe is pulled out of the CM to make way for parachute deployment, while in an undocking the probe (the triangular shaped thing pointing "forward" of the CM) is left to allow for redocking later. And in case you're wondering how the astronauts transferred between the two while docked with the probe in place, the answer is that they didn't - they dismantled it to create the tunnel. As an interesting aside, in real life the CAPCOM (I think it was Joe Kerwin) gave a go for "undocking" then corrected himself moments later using the phrase "..correction, GO for jettison...)
Deliberate "mistake": Near the end of the movie, there is a shot of the "Iwo Jima" with the two recovery helicopters taking off. For a moment, you can see the number 11 on the inboard side of the ship's island, which identifies it as the USS New Orleans (LPH-11), a sister of the Iwo Jima. Admittedly, it was the best the producers could do, since by the time the movie was made, the Iwo Jima (LPH-2) had been decommissioned.
Factual error: During the launch sequence, all nine swingarms on the launch tower are seen retracting, one by one, as the Saturn V reaches full thrust. In real life, only five swingarms would still be attached to the rocket during this phase. These "in-flight" arms would swing away as the rocket lifted off and cleared the launcher.
Factual error: The initial smoke shown coming out of the rocket gimbals during 'ignition sequence start' is not nearly fast or 'enthusiastic' enough. Additionally, there was in fact only a very tiny amount of white smoke/vapor emitted, and it almost instantly developed into a fiery, violent semi-transparent rocket plume. There is just way too much white vapor for too long a period shown in the film.