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In Armageddon, two space shuttles launch from adjacent pads mere seconds after one another.

This seems highly implausible for several reasons:

  1. Pressure, heat, vibration and debris from the first launch might damage the second shuttle's launch mechanisms
  2. Exhaust of the first shuttle would make navigational calculations extremely difficult once the second shuttle is airborne
  3. Command has limited staff and a clean handoff between two teams would be unlikely given the time constraints

Are simultaneous, close-proximity dual shuttle launches really feasible, and do they have historical precedent?

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There's no historical precedent for simultaneous launch, but some early space program rendezvous experiments had flights launched somewhat close together.

Gemini 8, 10, 11, and 12 each launched within a couple of hours of an unmanned Agena spacecraft for a rendezvous exercise. They would launch from separate pads at Cape Canaveral, LC-14 for the Agenas and LC-19 for the Geminis.

The Apollo and Soyuz craft used in the ASTP rendezvous mission launched about 8 hours apart, but obviously from very different launch sites.

All the other rendezvous missions I'm aware of launched days apart.

That said, I don't think it's completely infeasible if there were a compelling reason to do it (but there isn't; safely rendezvousing in space is going to take hours whether the ships are launched simultaneously or not). Of your concerns:

  1. Direct interference between the launches: probably not a problem; the big launchpads at Canaveral are something like a mile apart, and the pressure, heat, and vibration from rocket A are way, way, way more troublesome for rocket A than for rocket B.

  2. Exhaust disturbance in the air: probably not a problem for the same reason; if the craft were close enough to one another for exhaust plume impingement to be a concern, the chance of collision would be a far more significant concern.

  3. Command/coordination issues: probably one of the big reasons it's never been done. A lot of activity has to be coordinated for a rocket launch, and doubling that workload without a damned good reason is pointless and hazardous. According to this Q&A though, the infrastructure to support simultaneous launch exists; the redundancy is mainly there for testing, training, rehearsal and maintenance reasons.

If NASA was determined to do a simultaneous launch of two shuttle-sized rockets it could probably replace a couple of the smaller, idle launch pad complexes at Canaveral with one big one, in order to put more distance between the launch pads. That could get maybe 8 miles of separation between the pads, but I don't think NASA would ever be comfortable with the collision risk implied there.

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    $\begingroup$ It was theoretically possible to control 2 shuttle missions simo (this was addressed in the chapter of the CAIB report on rescue options) but this was never actually done from the MCC. Finding enough flight controllers would have been a challenge, $\endgroup$ – Organic Marble Feb 5 '16 at 12:20
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Most of your concerns can be put to rest by this image:

enter image description here

Image credit: NASA, Source: Wikipedia

For the last time in the Shuttle program, Space Shuttles Atlantis and Endeavour are placed at LC39A and LC39B in preparation for the STS-125 mission to the Hubble Space Telescope.

Space shuttle Atlantis on Launch Pad 39A (left) is accompanied by space shuttle Endeavour after Endeavour's rollout to Launch Pad 39B. This was the last time two shuttles were on launch pads simultaneously. Atlantis mission STS-125 was the final servicing mission to upgrade NASA's Hubble Space Telescope. Endeavour was prepped for contingency support (Launch On Need) standing by at pad 39B in the event Atlantis was damaged during flight and unable to safely return to earth, necessitating an emergency rescue mission (STS-400.)

Space Shuttle Endeavour was on its Launch Pad 39B when the Space Shuttle Atlantis launched on its STS-125 HST servicing mission. That said, while the supporting Space Shuttle Endeavour never launched on that mission, it was in readiness to do so should the need arise. That means also ground support for both launch vehicles. They wouldn't launch in tandem like I think was portrayed in the movie, if I remember correctly, but a back-to-back launch was a contingency (STS-400 contingency support mission).

So while there isn't any exact historical precedent, there is about, oh, shall we say 90% of it as it pertains to portrayal in that movie that you mention.

For easier appreciation of how far from each other the two launch sites are, here's an aerial view:

enter image description here

Photo credit: Cory Huston, Source: Wikipedia

This aerial view looking north shows space shuttle Complex 39 Launch Pads A (foreground) and B at NASA's Kennedy Space Center. To the right is the Atlantic Ocean.

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    $\begingroup$ Notably, these two pads are about 1.5 miles (2.4km) apart. $\endgroup$ – Adam Davis Feb 5 '16 at 15:51
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Launching two rockets would be insane. A rocket is about the size of a commercial airliner. When a commercial airliner uses the same runway, they wait a period of a few minutes between launch, for safety purposes (And to let the turbulence to settle down). The closest this has been done in real life was the Gemini/Agena launches happened. What they did was to launch one, then wait an orbit, then launch the second (See the Wikipedia article)

This is really the only thing that makes sense. Dual launches is difficult for a multitude of reasons, which includes range safety, the boosters being too close to each other, debris fallout, collision, and quite a few other things. As in the movie it was ported to be done due to redundancy, waiting an orbit for the second to launch would have made much more sense. Or even better would be to simply launch each one when it was ready. One of the two would inevitably be ready a bit quicker, so concentrating on launching one, then concentrating on the other, would give the best chance of success.

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