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Sputnik I was launched to reach as high as 939 km above the sea level and made 1440 orbits. Vostok I with Gagarin aboard was launched to 327 km for one orbit. Wouldn't 200 km or even less be enough to make an orbit?

Was this optimal from an orbital mechanics point of view, given the launcher and payload and location of the launch site and whatnot? Or was it also a kind of political show-off that they had margins to do more? Or was it yet unknown how the R-7/Soyuz launcher would perform? Would the same trajectory have been chosen today by a small-sat launching start-up company, in order to demonstrate orbital space flight capability?

~690 days in LEO

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The important number here is the perigee, not the apogee. In order to make this amount of orbits, you need to have a certain height of perigee in the order of 200 to 300 km. A circular orbit of 250 x 250 km would certainly have sufficed their needs.

But: The important word here is 'circular': You can't get directly into a circular orbit. After launch you'll always end up in a elliptical orbit and you need to do another burn of your engine *) to circularize the orbit. Given the state of development of engines and control systems 60 years ago, it was much simpler to design an engine that does one long burn to get a high apogee and at the same time a perigee that is high enough to stay in orbit for some time.

As Uwe comments, the design of the Sputnik launcher was kept extremely simple: It didn't even have an actual second stage as we know it from today's rockets. The central engine as well as the four booster engines were ignited at lift-off. The boosters were dropped after two minutes while the center core continued to orbit.

*) as Russell Borogove points out, a single burn is sufficient, if it is long enough and if the attitude of the spacecraft can be changed during the burn. The Sputnik main stage burned for only 5 minutes, which is too short, and didn't have an advanced attitude control system.

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    $\begingroup$ For a circularisation burn, an engine ignitable in zero gravity is necessary. To avoid the problems of ignition in zero gravity, the boosters and the first stage of the rocket used for launch of Sputnik were ignited on the ground. The resulting orbit was necessary to be the first one with a satellite in orbit. $\endgroup$ – Uwe Feb 21 '18 at 10:52
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    $\begingroup$ You do not need to do a separate burn to circularize; altering the pitch of the rocket over the course of the burn is sufficient. $\endgroup$ – Russell Borogove Feb 21 '18 at 21:20
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    $\begingroup$ @RussellBorogove in the cases shown in the question one might have to pitch by 180 degrees and slow down to circularize, and then you'd be pretty darn low and circular, and decay quickly. I think what's still missing from this answer is that a 211 x 1,659 km orbit will still last many times longer than a 211 x 211 km orbit. $\endgroup$ – uhoh Feb 21 '18 at 22:03
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    $\begingroup$ @vsz The launcher and the satellite are two different things. $\endgroup$ – Russell Borogove Feb 21 '18 at 23:10
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    $\begingroup$ @RussellBorogove : I doubt they had the means for an accurate enough attitude control for the launcher either. How would they know which way it faced without any computers or control system on board? Not only was the technology (and knowledge about the upper atmosphere) lacking, their goal was to launch something, anything into orbit, any orbit, as soon as possible. $\endgroup$ – vsz Feb 22 '18 at 7:33

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