You need a second burn just to enter orbit. You burn at the top of the sub-orbital arc (opposite side of the earth) to enter orbit.
Normal orbital insertion is a single burn to orbit (with staging). With the correct initial roll and pitch, the spacecraft follows a perfect gravity turn and ends up in a near circular orbit at main engine cut off.
The rocket yaws and pitches in the first seconds of flight while the vehicle is still subsonic, then flies a gravity turn trajectory at zero angle of attack (facing the direction of travel) at near maximum thrust. Any errors accumulated during early part of the flight will be corrected by adjusting the timing of the second stage cutoff based on radar tracking.
The initial pitch over is just a few degrees off vertical, but must be precise to a fraction of a degree (KSP tolerances are higher due to small planet).
You can get a pretty good circular orbit in Kerbal Space Program with one burn if you do a few attempts and trial and error binary search for the optimal initial pitchover angle, but it's very difficult to do without throttling the 2nd stage burn. If I recall correctly, the MechJeb mod can do a precise single burn to orbit.
Finding the correct number of key taps to get the right pitch angle is the key. The throttle can only help so much.
Thank you. Though I don't expect to open KSP any time in the near future, I love to know how the real rockets do it.
There are reasons "apogee" is more recognized word than "apoapsis".
What are you getting at? The -gee suffix means Earth. Apogee is apoapsis of an orbit around Earth. People playing KSP speak of apoapsis because Kerbin isn't Earth.
If you think about it, there can't be an ellipse that intersects a circle while also being fully encompassed by the latter. That's to say the periapsis can't be higher than the maneuver altitude that the off-apsis burn takes place as the other guy is suggesting.
On real rockets they add small shelf-stable stage such as the spinny boi Star-48 or the notoriously narcoleptic Fregat, inside fairings between S2 payload interface to actual payload, or let the payload pull itself into the final orbit at a great expense.
Yes, you can keep the S2 burning all the way to Ap and then burn at -45deg at Ap to deform the ellipse back into a circle, or fly a by the book gravity turn trajectory, but that's not the most energy efficient insertion, only what are situationally beneficial when there is land below and acceleration is limited.
You surely know that, but mathematically it is unavoidable to have a 2nd burn at the apoapsis to have an orbit (with a periapsis > the initial launch altitude, minus gravity and atmospheric losses), albeit it may start earlier (while wasting a slight amount of delta-v).
Not necessarily - it is completely dependent on the ascent profile. For example, during the Apollo program, the Saturn V would fly directly into a parking orbit and only relight the S-IVB for the Trans-Lunar Injection burn.