I'm trying to understand how you choose your acceleration/deceleration and trajectory during launch and reentry to minimize aerodynamic heating.*

Ignoring efficiency in terms of delta-v (best leave that for another question), there's a tradeoff between high acceleration/deceleration and low accel/decel. If you go low, you'll be exposing your rocket to a lot less heat, but for a lot longer. If you go high, you'll minimize exposure time, but really increase the heat for the short time you've got to deal with it.

Launch/reentry Speed Tradeoff

The way I see it, your damage is going to be roughly proportional to your velocity at every point in time to some factor integrated over the total exposure time to some factor. Really rough estimate here, please correct me if I'm wrong, I'm here to learn :)

$$\textrm{damage} \propto \int v^a \ dt^b$$

Higher accel/decel means more $v$, less $t$. Lower accel/decel means more $t$, less $v$.

When optimizing for minimal total damage, is there a clear choice here between minimizing accel/decel and maximizing it? Basically, how does the impact of exposure time compare with the impact of higher velocity during exposure?

*Clearly, there are a lot more factors to consider beyond heating, like how much energy each launch trajectory takes, which is likely far more important. Seeing, however, as it would be far too broad a question to ask about all the principles to keep in mind during launch and reentry, I'm focusing specifically on minimizing aerodynamic heating for this question.

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    $\begingroup$ There are dozens of questions here already about entry heating and alternate profiles. Here's one: space.stackexchange.com/q/12634/6944 $\endgroup$ Nov 28, 2020 at 22:20
  • $\begingroup$ You also need to consider total fuel expenditure if you're retro-thrusting, or the sequence of drogue parachutes if attempting passive deceleration. There's a lot more than just heating issues. $\endgroup$ Nov 30, 2020 at 13:51