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I have seen that sometimes the second stage of a rocket is turned on and off, then after a while turned on and off again.

What are the main reasons that second burn of a second state is initiated while in Earth orbit?

Here are some examples of missions that do this. Is each unique and unrelated or are there some main, well-recognized reasons that are easily summarized?

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  • $\begingroup$ What about a source or cite for turning off and on again? $\endgroup$ – Uwe Oct 6 '20 at 13:14
  • $\begingroup$ It is in the today's SpaceX launch video - Second Engine Cut Off (SECO) $\endgroup$ – Joe Jobs Oct 6 '20 at 13:20
  • $\begingroup$ That's second stage engine cutoff. Every F9 launch has a SECO. The ones with multiple second stage burns have multiple SECOs...SECO-1, SECO-2, etc. $\endgroup$ – Christopher James Huff Oct 6 '20 at 14:39
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That is how a rocket achieves enough speed for a transfer orbit. For SpaceX specifically, the second stage reaches LEO, then, if the satellite like Starman wants to go beyond LEO, they fire the engines to put them in a transfer orbit about the Earth or Sun. If a satellite wants to go to Geostationary or Geosynchronous orbit, it has to achieve apogee, where the the engine is turned on for a second time to circularize the orbit. If a satellite wants to go to interplanetary space, then the rocket has to achieve orbital velocity, then escape velocity by turning on the engine for a second time, which can put them into an elliptical solar orbit. This is called a Hohmann transfer. Some satellites like New Horizons don't turn on their engines for a second time, but rather achieve escape velocity of earth in one long burn. SpaceX does this for Geostationary transfer orbits, but not for interplanetary spacecrafts; they first achieve parking orbit.

Most of the missions you've listed in your question are either interplanetary spacecraft (like Emrites Mars Mission: Hope or Starman), or are geostationary spacecraft (like JCSAT-16). So a second (and sometimes third depending on the rocket) burn is required.

The recent Starlink launch though is a not Geostationary nor interplanetary but rather LEO. According to Spaceflight Now Falcon 9 Starlink timeline, the second burn is required to circularize the orbit. It should be noted that the second stage was ignited for a second time very briefly (2 seconds).

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    $\begingroup$ Okay. Thank you. Most LEO missions that use 2 burns of the 2nd stage use it to circularize the orbit, which often is not good enough in the first engine burn. $\endgroup$ – Star Man Oct 6 '20 at 14:28
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    $\begingroup$ @StarMan It's not that the first engine burn isn't good enough, it's that that's the most efficient way to get a circular orbit at whatever altitude. You burn to an elliptical orbit with a perigee basically just far enough out of the atmosphere and an apogee around the altitude you need to be at such that you're at perigee at engine cut-off. Then you coast half an orbit and burn again to raise the perigee to circularize. This is somewhat of an oversimplification, but that's how pretty much every launch is. Going straight to circular is not as efficient. $\endgroup$ – Tristan Oct 6 '20 at 14:31

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