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Is there any rocket where the first stage is ignited after the boosters? Say at least 30 seconds after the boosters.

Because in general the first stage is ignited in the same time with the boosters.

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How often did launch vehicles have a "Stage 0", and what does it mean exactly?

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Titan IIIE-Centaurs (and some other Titan versions) lifted off on solid rocket motor (SRM) power only and ignited the liquid first stage shortly before SRM jettison.

Titan Stage I ignition occurs at approximately 112 seconds after liftoff. The actual time is controlled by three majority-voted acceleration switches which command ignition and initiate a staging timer when vehicle acceleration reduces to 1.5g. Approximately 12 seconds later the SRMs are jettisoned upon command from the staging timer.

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TITAN IIIE/CENTAUR D-IT SYSTEMS SUMMARY

See also

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  • 1
    $\begingroup$ Is there any term to distinguish such rockets from those that ignite 1st stage at T0? $\endgroup$ – Joe Jobs Nov 19 '20 at 20:38
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    $\begingroup$ Don't know of one. $\endgroup$ – Organic Marble Nov 19 '20 at 21:12
  • $\begingroup$ maybe this can also be added to answer(s) to the 2nd question in this comment $\endgroup$ – uhoh Nov 20 '20 at 0:14
  • $\begingroup$ @JoeJobs: Some people would understand the term Asparagus staging $\endgroup$ – dotancohen Nov 20 '20 at 10:09
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    $\begingroup$ I think asparagus staging it implies fuel cross feed. $\endgroup$ – ikrase Nov 20 '20 at 20:21
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Besides the late model Titans that Organic Marble mentions, India's GSLV-III also does this, as described in this QA.

From a theoretical perspective, if you simply label the boosters as "first stage", and the first stage as "second stage", this is equivalent to ordinary staging.

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