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).
Maximum dynamic pressure (max Q)
The main advantage of the air-to-orbit configuration is that the vehicle is not exposed to most of the atmosphere. We made a rocket flight trajectory simulator, and depending on the parameters of the vehicle, an approximation of the max Q can be calculated. For example, for a 1.7 thrust-to-weight ratio 652 kg vehicle given by our rocket mass program, with a 30km altitude and 270m/s release speed, the max Q of 4664.36 Pa is reached after 54 seconds of flight, at a speed of Mach 3.40 and at an altitude of 34.8km.
This program uses the International Standard Atmosphere and a very rough estimate of the trajectory that needs to be hand-corrected for any parameter change, so it's not yet of publishable quality.
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 (after aircraft release). 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).
Another particularity of air-to-orbit vehicles is the high angle of attack. Indeed, contrary to balloon or ground launches, we already have an horizontal velocity on ignition. If the beginning of the trajectory is optimized to avoid staying in the low altitude atmosphere, gravity will need to be countered by pitching up aggressively. The body of the rocket will thus be at a high angle of attack, and same thing if supersonic wings are mounted on the rocket, they will not provide enough lift for the first few tens of seconds to change the velocity angle (real pitch).
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.
This Configuration Aerodynamics class may be useful.
If you're new to CFD, the following book is for you: John D. Anderson, Jr. Computational Fluid Dynamics: The Basics With Applications, 1995. This is a beginners guide to CFD, quite outdated for direct software application but still well explained with lots of examples (poor quality nearly complete pdf here).
RASAero created a nice tool for aerodynamics analysis running on Windows. It is free (costs no money): Rogers Aeroscience RASAero Aerodynamic Analysis and Flight Simulation Software.
AeroRocket, a company of John Cipolla, has created several useful aerodynamics analysis tools, like VisualCFD or AeroFinSim. However, these tools are not free (it costs money) and also only work with Windows.
For speeds below Mach 3.0 and angle of attack below 25 degrees, RocketCalculator as described in the following paper can be used: Dahalan, Md. Nizam and Su, Vin Cent and Ammoo, Mohd. Shar (2009). Development of a computer program for rocket aerodynmic coefficients estimation. Jurnal Mekanikal, 28. pp.28-43 (link). This program was requested several times using different communication ways, and no reply was received. Is it fake research?