Nick,

Perhaps without realizing it, you're asking questions about "aerocapture". Aerocapture is where your vehicle is coming in from deep space along one leg of a hyperbolic trajectory. The vehicle hits the planet's atmosphere and dumps enough kinetic energy into the atmosphere that the hyperbolic trajectory becomes an elliptical orbit with the periapsis still in the atmosphere.

For this sort of trajectory, Mach number is almost meaningless because you're coming in from a free molecular regime to a continuum. In the free molecular regime the temperatures could be thousands of degrees Kelvin due to solar radiation thus the local "speed-of-sound" would be quite high even though the gas was a near vacuum. The vehicle can be going at faster than orbital velocity but the Mach number still relatively low. It's better to use atmosphere relative velocity is this context. However even here one can get tripped up. At lower altitudes, the atmosphere can be assumed to move as a solid body with the same angular velocity as the planet's surface. However it's clear that at sufficient altitude, the solar wind's velocity would dominate and the assumption of solid body rotation becomes invalid. To do a trajectory correctly, a simulation software must be able to change coordinate frames from planet fixed to inertial. Knowing when to do that change is tricky.

You touched upon the problem of staying within the atmosphere while "turning the corner". This is a well understood problem. The vehicle must have a nonzero "lift over drag ratio" (L/D) and fly a "lift down" trajectory in order to fly along the planet's curvature. This creates a big "gotcha" because it's hard to create lift with a hypersonic vehicle. If you want a decent L/D then typically it is necessary to have a finite lift "L" with a small drag "D". A small drag is a bad thing in hypersonics because it means greater heating. This is non-intuitive but the greater the drag the the less the total heat load and the less the thermal protection requirement. This was the famous discovery that "Allen-and- Eggers" made at NASA Ames in the late 1950s. Jullian Allen was another one of those Olympian aerospace engineers that Ames was blessed with during the Golden Age of Aerospace (1957-1969). Getting back to aerocapture: The Apollo Command Module (CM) essentially flew an aerocapture trajectory when it returned from the Moon. Of course with the CM, the orbit that it captured to was suborbital to land in the ocean. The big problem with aerocapture trajectories is to stay within the "entry corridor". If the vehicle enters too shallow then it skips off the atmosphere and returns to deep space. If it enters too steep then the G loading goes over a hunderd gravities and squashes the crew. Staying within the entry corridor has been done in the past by keeping the vehicle at constant angle-of-attack and using roll modulation. With the CM this was achieved by carefully placing the center-of-gravity and having an aerodynamic shape with a well behaved center-of-pressure (a spherical section). If you look at the CM's roll angle as a function of time, it almost looked like a white noise plot. The autopilot on the Apollo CM was a nice piece of technology designed by some pretty clever guys. I'm sure it was a wild ride for the astronauts. Something people are starting to worry about now is manned aerocapture to Martian orbit. This is actually a much harder problem than aerocapture to Earth because the Martian atmosphere is so thin coupled with a lower gravitational attraction. For some solutions I've discovered that I was whizzing along at Mach 20 at an altitude of 500 meters (not recommended given that Mars is out-of-round by about 3 kilometers). This stuff is lots of fun and some lucky guys have written Ph.D. dissertations around this topic.

Gary