Question Reaching an orbit with a multi-stage rocket on direct ascent

[Cosmic Penguin]

This is more of a theoretical problem, so feel free to move the thread to a more suitable place.

Many rockets in real life have upper stages that cannot be re-started (Soyuz, Ariane 5 etc.), and these are usually used for low Earth orbit launches. In some cases the upper stage re-ignition ability is not used due to higher performance excess (e.g. Atlas V with the X-37B). This poses a constraint in how to fly to a determined orbit - you need to reach the determined altitude quickly and then keep your apogee constant such that the perigee can be raised to the determined altitude. This means that during ascent you must be at the apogee at all times during the final part of the launch.

So for a multi-stage rocket, how can I determine the pitch profile such that I'm always at the apogee once the apogee is almost raised to the planned altitude? For me it has always been a problem of trial and error, so any help is appreciated! :tiphat:

P.S. This is not about the "direct ascent" method of reaching another object in orbit.

 

[dgatsoulis]

In JK's Java Applets for Teaching of AstroPhysics there is a Flight of a Launch Vehicle applet that

allows the computation of the behaviour of a multistage rocket from the launch site into orbit. The user may change the parameters of the rocket, such as the type and mass of the fuel, the thrust of the engines, the satellite weight, the orientation of the thrust vector during the powered flight etc. One can study the characteristics of the ascent trajectory and the achieved orbit.

Is this what you are looking for?