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How is the bluntness or curvature of a heat shield of a reentry spacecraft designed?
If the answer depends on size and mass, consider a deep space sample return "mini capsule" 50 cm in diameter with a mass of 100 kg returning on an interplanetary trajectory from Mars or the asteroid belt.
Ballistic reentry vehicle design is something that takes super computers, PhDs, etc. so I can't really fit it into a stackexchange post. Here are the highlights: energy management and aerodynamic stability.
Energy management: At the end of the day, it's energy management. Yes, you can do sharp edged vehicles with special materials in special cases with lifting bodies, but when you hit an atmosphere at Mach 20+ and want to land on the ground at Mach 0 (soft touchdown) you have to get rid of the energy somehow. We do this by dumping the energy using aerodynamic drag. There are only 2 places that energy can go: into the vehicle or out of the vehicle (into the atmosphere). All of that energy, if it goes into the vehicle, would 900% melt all of it. Curiosity reentry dumped about 98% of its entry energy into the atmosphere, the aeroshell can still get to be temps greater than the surface of the sun. When things go greater than Mach 1, shocks form. shock waves are near-instantaneous extreme changes in flow conditions that are accompanied by extreme temperature rise. If the leading edge is sharp then the shock is oblique and attached: https://lh3.googleusercontent.com/proxy/0r3d9D8usICkqSqh4FYpLCMurOzJ6AfRCAT7EgFtikO_smOg1YL0gVHy16g7bb-2YTYHDWZZSutRB2nhCeW_BSajc8cTVnCVV3lySiV0z2ikqbgcHHhc79uwR1eagrh3FEv2BK-ki988bqisleWbxX5SeYDnXg
Attached shocks don't take as much energy to make, and are closer to the vehicle, so much more energy gets shoved into the vehicle. When you create a blunt body, the shock detaches and turns into a normal shock, or bow shock: https://upload.wikimedia.org/wikipedia/commons/3/33/Bowshock_example_-_blunt_body.jpg
These are WAY stronger, and since they're detached from the body, that energy gets dumped into the atmosphere, so your vehicle stays cooler and survives.
Stability: Blunt bodies can be formed by spheres, or cones with spheres at their tips (called sphere-cones, google search will show good examples). The larger the cone angle (the larger the half-angle of the cone) the blunter the body is, the better it will be at deccelerating without heating. BUT, the blunter it is, the more likely it is to tumble and blow up. The sharper of a cone, the more stable and easier it is to keep in a certain orientation, but the less heat it'll dissipate. Traditionally, half angles of 45 and 60° are used, when you want more stability vs when you need to dump more energy. Reentry vehicles are highly tailored to their use case, so if you're off by even a degree or two, the whole thing could melt, hence the desire for increased stability.
If you're looking to design a specific reentry vehicle for earth with payload 100kg 50cm in diameter, you'll need to know the reactivity of the atmosphere, the entry velocity, flight path angle, the offset distance between the mass and the center of pressure, the shoulder radius of the heatshield, the heatshield material, the reactivity of the heatshield material itself, the ballistic coefficient or drag coefficient mapped across all mach regimes, among many other variables
