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Why are they not integral to the lower stages, e.g. like the Falcon 9? This applies not only to the ring between the first and second stage and also to the shoulder between the second and third stage.

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    $\begingroup$ I was convinced this has been answered before, but I couldn't find it. $\endgroup$ – Organic Marble Jun 18 at 14:32
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If the angular rates on the first stage are not close to zero there is a chance of the inter-stage contacting the engine bells. (Falcon 1 flight 2?)

With a separate inter-stage the upper stage can hang onto it (protecting the engine bells) until under propulsion and rapidly accelerating away from it reducing the chance of contact.

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    $\begingroup$ I thought with the Falcon 1 the issue was residual fuel in the first stage which caused it to continue accelerate and bump into the second stage? $\endgroup$ – Michael Jun 19 at 5:11
  • $\begingroup$ Yes, but if the second stage had hung onto the inter-stage the engine would have been protected to some extent. It always amazes me how close the Falcon 9 inter-stage gets to the Merlin engine bells - although that may just be from the perspective of the camera. $\endgroup$ – Ames Jun 19 at 15:56
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    $\begingroup$ @Michael You're thinking of the third flight of the Falcon 1, where a timing error caused the first stage to separate before the engines had completely shut down. youtu.be/v0w9p3U8860?t=165 $\endgroup$ – HiddenWindshield Jun 19 at 18:42
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From Stages to Saturn p. 212

The dual-plane separation was an alternative to a method called "fire in the hole," which involved ignition and separation of the S-II while still in contact with the interstage but not attached to it. Designers preferred to avoid this alternative because of possible perturbations and oscillations at the end of the first-stage boost phase. With the S-II accelerating on an even course, it was easier to drop the interstage during that phase, rather than risk hitting a wobbling interstage attached to the S-IC as the S-II pulled out.

enter image description here

Note that on the final Saturn V flight, the launch of Skylab, the interstage failed to separate from the S-II but the launch was still successful. (More on this eventful launch here.)

(Picture from the Skylab Saturn V Postflight Report)

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As Organic Marble's answer points out, avoiding the 'natural' simplest solution of the "fire in the hole" is a good idea. As to why this is different, its worth noting, the falcon has a "pusher" assembly to mechanically separate the stages to avoid the same problem. So its not just that one is better than the other.

What changed? Its a little hard to say exactly, in part because all such decisions are complex and interwoven with other design choices, but the most obvious is re-usability.

Pyrotechnics are light weight and pretty reliable but if you want to fly things again, cutting your rocket into ever smaller pieces is a practice best avoided.

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