Above that speed, you must consider radiative heating from the plasma sheath surrounding the spacecraft, and source conditions vary all around it. None of the models for this are simple at all.
Coming back from Earth orbit, you are moving in the vicinity of 7.7 km/s at atmospheric interface, about 135 km altitude. You are also moving at a very shallow trajectory angle, unless you are very wasteful of retro thrust fuel. Because you are moving at less than escape speed, you cannot bounce off into space, although you might skip unexpectedly far downrange.
Convective stagnation point heat rate per unit area is proportional to the square root of ambient density, inversely proportional to the square root of the “nose radius” facing the flow, and proportional to the cube of the speed. That equation is entirely empirical, and dimensionally-inconsistent, but it does work quite well to first order. The nose radius dependence is why space capsules have blunt heat shields: the blunter, the lower the peak heating rate at stagnation. The effect is quite dramatic.
You can find this heating correlation in a variety of references. The most recent is the Justus and Braun EDL paper, but I had to chase this back to references from the 1950’s and 1960’s before I found a reliable value for the constant of proportionality.
There is also a simplified entry ballistics analysis presented in Justus & Braun, that traces back to Julian Allen at NACA in the 1950’s. I’ve been using it as a zeroth/first order ballpark design model. I had to correct the heat transfer items in it, but not the basic dynamics items, before I could use it.
The gas/plasma total temperature around the afterbody, or anywhere it has been shocked-down to local subsonic, is very crudely numerically equal in degrees K to the vehicle velocity in m/s. This is another empirical approximation, and it is not strictly correct, but it is in the ballpark. It reflects kinetic energy going into ionization instead of internal energy (temperature).
The drag coefficients of blunt objects are crudely constant over the range from Mach 5-ish to Mach 25+. For a capsule shape, there is a blockage or frontal area associated with the shape that is used as the reference for the drag coefficient. The same area is used in ballistic coefficient: W/CD*A
There were 3 original heat shield concepts in the 1950’s: heat sinks, ablatives, and re-radiative (or refractory) concepts. Heat sinks were then, and still are today, a massively-heavy (and therefore undesirable) solution. The refractories back then were also very heavy (usually tungsten or super-alloy metals and/or monolithic chunks of graphite, or metal-graphite combined), leaving ablatives (as heavy as they were back then) as the lightest-weight and most practical solution. That’s why all the early manned capsules had silica-phenolic (or something closely related) for their ablative heat shields.
Today, we have low-density ceramic refractories (shuttle tile), although these have much lower surface temperature limits than the ablatives, and, they are fragile. We also have lower-density ablatives, most notably PICA-X. (And there is my oddball experimental material, which is much tougher than shuttle tile but not quite as lightweight, although lighter than PICA-X.)
There are also sacrificial-liquid-coolant schemes. These are somewhere between heat sinks and ablatives, and inherently tend to be quite heavy. This weight is driven more by the integrated total heat absorbed (which sets the coolant mass to be expended), than just the peak heating rate (although that sizes the coolant flow rate).
Analysis of surface temperature and survivability of ablatives is a tough calculation problem, driven by empirical models. Although, with the old phenolics, surface temperature typically fell near 3000 F (1920 K, 1650 C). Refractories are a little easier, if they are low-density, as the conduction pathway is cut off, completely unlike the old metallic and graphite refractories. The re-radiated heat simply has to balance the convective input.
What that means is that we can estimate the surface temperature of the low-density refractory, given an estimate of its spectrally-averaged emissivity, and a value for the convective input. I did this for low-density ceramics at peak stagnation convective-input values. I found that low ballistic coefficient shapes under 200 kg/sq.m with very blunt heat shields (and I do mean nearly flat) can reduce peak stagnation heating to 25 W/sq.cm, for entry from LEO, far lower at Mars.
Given a black surface average emissivity of 0.8, the peak skin temperature from that heat balance is about 1290 C returning from LEO. That’s cool enough for the ceramic to survive entry from LEO, even at the stagnation point, unlike the application on shuttle. The failure mode above that skin temperature is not melting, but solid phase change-induced shrinkage cracking.
Capsule shapes have a blunt heat shield, and an afterbody shape that is pretty much arbitrary, except that it must be aerodynamically stable with the cg position. Mercury, Gemini, and Apollo all had nose radius/diameter ratios not very far from 1.0-1.1.
If the afterbody is more-or-less conical, you can “fly” at angle-of-attack to the slipstream and generate some lift for trajectory control during entry, without angling that afterbody into the main slipstream. All this takes is attitude thruster fuel, and not very much of it. This is a well-known and well-proven technique, dating back to Gemini in the 1960’s.
The afterbody is more-or-less immersed in more-or-less transonic plasma, which is quite hot. Conditions at entry interface 7.7 km/s would be in the ballpark of 7700 K, just at extreme low density. As the speed decreases, plasma temperatures fall, but density rises due to the descent. This heat transfer environment is far less severe than that on the forward-facing heat shield.
That is why sheet metal heat-sinking was adequate for afterbody structures on Mercury and Gemini. They had corrugated metal shell panels without any ablative at all. But, had the capsule tumbled, it would have been destroyed. That’s why Apollo had ablatives on its afterbody, along with the demands of the faster, hotter entry coming back from the moon.
Coming back from the moon, speed is very nearly Earth escape at 11 km/s. Radiative plasma effects are becoming important, and there is the definite possibility of bouncing off the atmosphere into deep space, if the trajectory is too shallow. Convective heating rates are way far higher, ruling out ceramics at the stagnation point, leaving ablatives as the only practical choice for stagnation regions. About 2 degrees from horizontal is what Apollo used. Steeper is too much heating at too many deceleration gees. This is precision trajectory control during the moon-Earth transit, no way around that.
Once into entry, there is less need for trajectory control, other than controlling attitude with small thrusters, unless a precision landing point is needed. Apollo used the same angle-of-attack trajectory control during entry as Gemini, and it was quite successful. Circular error probable was around 1 or 2 miles (1.6-3.2 km).
Depending upon the nature of the cargo to be brought from the moon, no afterbody shell may be needed at all. Bulk metals or minerals, for example, can just heat sink their way through 3 minute’s exposure to transonic plasma decreasing from about 11,000 K effective at interface.
A design like that is just a cargo deck with heat shield on one side, and the cargo plus a guidance package on the other. Other types of cargo may require protection from the plasma sheath (such as tanks of liquids). A simple metal or otherwise-minimally-protected back shell would work.
At Mars, the entry heat protection is far easier, just because the velocities are about a third of, to at most half of, what we have to deal with at Earth. For example, even for direct entry from interplanetary trajectories, the entry interface velocity is only around 5.6 km/s. Low density ceramics are thus quite feasible, even at stagnation conditions, and even at high ballistic coefficients.
Entry trajectories must be shallow at Mars, just because the air is so thin. If you come steep, you are still way hypersonic when you smack the surface, even in the lowlands. This more-or-less rules out large delta-vee burns for de-orbit purposes. A big deorbit delta-vee is invariably associated with a steeper entry trajectory angle, that’s just the physics of orbital mechanics.
Now, if you have a structurally-tough, low-density ceramic, then you have a reusable heat shield for a reusable “landing boat” at Mars. That’s where my oddball experimental material has a huge amount of potential. More so there at Mars, than here at Earth.
For entry from LEO, one can use a winged or lifting-body shape instead of a blunt capsule. It still needs a nose and leading edges that are as blunt as one can make them, and these will require ablative protection. Well away from stagnation zones, the low-density ceramic refractories become feasible. The more broadside you can fly during entry, the more effectively-blunt you become, and the lower the peak heating you must deal with.
But (and this is a very, very big “but”!!!), angle of attack is severely limited by the structural problem of airloads ripping the wings off (or simple crushing breakup). For shuttle this was 20-30 degrees. Stray outside that AOA range (or almost any off-angle yaw or roll), and you die. One crew did die, because of a hole in a wing leading edge.
A final thought about LEO. The retro deceleration burn from LEO (or LMO) is a far lower-precision thing than trajectory-adjusting burns during transit. You don’t need precision-controllable liquid propellant thruster rockets for that. The cheapest and simplest solution (and often the lightest-weight in a one-shot situation) is a small solid rocket motor.
The retros on Mercury were solids, and they worked quite well. You get to design like the JATO bottles they used launching overweight airplanes, for an application like that. Which makes a difference. They become simple “wooden rounds”, as far as handling and logistics are concerned. Actually, very cheap.