We have seen with the first approximation that the gravity drag is higher than expected for a single stage air-to-orbit configuration, when aerodynamic effects are ignored. The rocket engine has to be larger to compensate the gravity.
On this page, we will evaluate how the aerodynamic effects can be used to compensate the gravity (lift) without impacting the thrust too much (drag). This will be evaluated for a rocket without wings or fins at first, then we will do the same evaluation with small supersonic wings similar to the Pegasus (wings image).
Evaluating lift and drag for a transonic/supersonic vehicle
The regular and accurate way to study aerodynamics is to use computational fluid dynamics (CFD). Some examples of that method can be seen here for example. We will first look for approximations in standard conditions before trying this way, as we did for Heat transfer, because CFD is quite complicated when you don't know how to use it, and CPU intensive.
We consider here air-to-orbit rockets, so the subsonic part of the flight will be very short. We will ignore it for now, and directly skip to the transonic part. Our not-yet-published and approximative rocket flight and trajectory simulator informs us that the transonic regime lasts no more than 7 seconds if aerodynamic drag is ignored, with a release speed of Mach 0.9 and a thrust/weight ratio of 1.7. Most of the flight is thus supersonic and even hypersonic (Mach 5 should be reached at an altitude of 45km).
It is quite easy to find information for model rockets with tail fins, mainly in subsonic flight. The best found so far is the OpenRocket technical documentation (pdf, 125 pages) from Sampo Niskanen, july 2011, based on his Master thesis. The document is of very good quality and can be very useful even if it's not directly related to our flight conditions.
In the final steps, if we use fins or small wings, we may use an aeroelastic analysis software like AeroFinSim to design their physical properties.