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This answer to Trajectory to get into orbit says:

In most real launches to low Earth orbit, the burn continues from liftoff until orbital insertion, without a coasting phase. Some (like Antares) do coast between the first stage and second stage burn; the exact design of the launcher determines which approach is more efficient.

and links to Spaceflight 101's Antares Launch Vehicle Information where the Antares LEO Flight Profile section says:

Antares lifts off from its launch pad two seconds after the AJ-26 engines of the first stage are ignited to allow some time for them to achieve full thrust and monitor their ignition performance. After a short vertical ascent, Antares performs a roll & pitch maneuver to align itself with its pre-planned ascent trajectory.

The first stage burns for 235 seconds and separates after a brief, 5-second post-burn coast. Stage 1 separation occurs at an altitude of 109 Kilometers and a velocity of 4,547m/s. At that point, the stack enters a 100-second coast period to get close to apogee for the second stage burn. After 100 seconds of coasting, the Payload Fairing is jettisoned at an altitude of 184 Kilometers. Ten seconds later, the second stage begins its engine burn for orbital insertion and circularization. Stage 2 shutdown occurs about 471 seconds into the flight at an altitude of 205 Kilometers and a velocity of 7,521m/s. Payload separation occurs after 120 seconds of maneuvering by the second stage attitude control system. The typical Cygnus insertion orbit is 275 by 250 Kilometers at an inclination of 51.66 degrees. (emphasis added)

Question: What are the orbital-mechanical advantages of a long (100+ second) sub-orbital coast phase between first and second stage burns in some cases? It's pretty rare for a multi-stage launch to LEO to do this.

If possible, a little math showing how this helps would be great!

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    $\begingroup$ Note that in the absence of terrain, atmosphere, and thrust limitations, the optimum ascent would be to accelerate horizontally to zero-altitude circular orbit velocity, then execute a Hohmann transfer -- i.e. coast from zero altitude. Unfortunately it takes a while to gain that first 7900 m/s.... $\endgroup$
    – Russell Borogove
    Commented Jun 2, 2019 at 23:30
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    $\begingroup$ @RussellBorogove this is getting really interesting! With no thrust limitation (I think that means impulse approximation but not sure) wouldn't the first impulse be directly into the transfer ellipse? Or do you just mean use a rocket car to accelerate to circular horizontally to circular velocity? $\endgroup$
    – uhoh
    Commented Jun 2, 2019 at 23:38
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    $\begingroup$ Hohmann assumes you’re starting in circular orbit rather than bolted to the planet, so I broke it out as two separate impulses for “clarity”. Optimally they’re instantaneous and consecutive, i.e. effectively one impulse. If you can’t do them instantaneously, a rocket car/sled/train would be the next best thing in the world of spherical cows. $\endgroup$
    – Russell Borogove
    Commented Jun 2, 2019 at 23:53
  • $\begingroup$ @RussellBorogove okay I see that now, excellent! $\endgroup$
    – uhoh
    Commented Jun 3, 2019 at 0:02
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    $\begingroup$ This flight profile avoids having to restart the second stage engine. $\endgroup$
    – Loren Pechtel
    Commented Jun 3, 2019 at 2:15

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The Antares will have a long coast phase due to engineering related trade offs rather than orbital-mechanical advantages.

However from an an orbital-mechanical view alone, there would always be a long coast phase. Ignoring the atmosphere and engineering limitations, you would apply all your "first stage" impulse as close to instantaneously as possible to avoid gravity losses. You would then coast to whatever your apogee was then apply your circularization impulse in as short a period as possible.

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  • $\begingroup$ concise, yet it seems pretty complete... except there are no supporting links. Is it possible to support your conclusions with a source or two? This way future readers will be better able to judge your answer, and have some further reading material to explore. Thanks!! $\endgroup$
    – uhoh
    Commented Jun 4, 2019 at 3:29
  • $\begingroup$ I have a hunch this is the correct answer and I'd like to accept it. Is it possible to support at least some of this with a link? Thanks! $\endgroup$
    – uhoh
    Commented Jun 16, 2019 at 23:18
  • $\begingroup$ any chance of adding one or more supporting links? $\endgroup$
    – uhoh
    Commented Aug 4, 2019 at 22:06

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