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Atmospheric reentry, as I understand it, is a balancing act between minimizing delta-v needed and not burning up or breaking up in the atmosphere. The higher the velocity on reentry, the greater the reentry heating and aerodynamic forces experienced by the vehicle. Of course, if you had all the delta-v in the world, you could use repeated burns to minimize velocity all the way down to the ground, but to better understand a real-world model, I'm curious about the optimal non-propulsive trajectory (as in "pick a velocity and drop") from low-earth orbit (any inclination will do, but preferably at least mention the case of an equatorial orbit) to sea level.

Of all possible non-propulsive deorbit and reentry trajectories from LEO, which one minimizes total reentry heating and aerodynamic forces? Notably, there may be different trajectories to minimize each of total reentry heating and aerodynamic forces, so I separated them and tried to use the Euler equation to minimize the Euler equations*, but that didn't get me anywhere as I ended up with nonsensical differential equations.

If someone happens to know the optimal trajectory(ies) to minimize reentry heating and aerodynamic forces, preferably but not necessarily with some mathematical backing, please let me know :)

* Hehehehe, I mean using the Euler-Lagrange equation to minimize a functional derived from the Euler equations for adiabatic flow, incredible to me that Leonhard did all of this back in the 1700s

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    $\begingroup$ There's far too many variables for this to be an answerable question. It depends on the aerodynamics of the vehicle (ballistic coefficient and lift/drag ratio), details of the insulating, ablative, and radiative properties of the heat shielding, other potential thermal protection mechanisms such as film cooling, etc. The optimum trajectory is going to be extremely specific to the vehicle involved. $\endgroup$ Sep 26, 2020 at 15:05
  • $\begingroup$ You might want to look at the trajectories used by Mercury, Gemini, and Apollo (LEO missions). They didn't necessarily minimize heating, but probably tried to come close to minimize weight and risk. $\endgroup$ Sep 26, 2020 at 19:57
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    $\begingroup$ Do you mean minimizing total heat absorbed by the vehicle over the course of reentry, or minimizing peak temperature? Steep reentry minimizes total heat at the cost of very high g-loading and peak temp (e.g. Mercury suborbital flights), shallow reentry minimizes peak temp and provides a comfortable ride at the cost of much more total heating (e.g. space shuttle). $\endgroup$ Sep 27, 2020 at 0:46
  • $\begingroup$ @RussellBorogove Ideally you'd be minimizing the load on the engineers, not the astronauts, but that's fairly hard to plug into an equation. How does engineering difficulty compare between managing total heat and managing peak temp? $\endgroup$ Sep 27, 2020 at 12:43
  • $\begingroup$ @TheEnvironmentalist hmm... I don't think spaceflight ever benefits from "minimizing the load on the engineers" :-) Instead they might try to manage the load on engineers by bringing in even more of them, but definitely the focus is on trying to keep the astronauts alive and uninjured, which is a huge challenge and not always achievable: Is the overall mortality rate for being in a spacecraft in space or bound for space about 4%?. $\endgroup$
    – uhoh
    Sep 28, 2020 at 0:29

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@ChristopherJamesHuff is right, many assumptions need to be made to make this feasible, but assuming a ballistic Crew Dragon entry from a 400 km circular orbit is enough to demonstrate some important concepts.

Working off of the "pick a velocity and drop" mantra, here is a plot of the key metrics as a function of the (atmosphere intersecting) transfer orbit periapsis (i.e., how much you wish to slow down):

plot image

Note the logarithmic scale on the right plots.

The tyranny of the rocket equation pretty much constrains us to the right hand extremes of the X-axis (and the relative squishiness of most payloads limits max g-forces).

As @RusselBorogove points out the maxima/minima of different metrics are generally not on the same trajectory (x coordinate).

Caveat: this is a quick and dirty simulation meant to illustrate the qualitative trends of the different trajectory options for this specific vehicle configuration (i.e., the shape of the curves are generally true, but the specific values are probably off)

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    $\begingroup$ note that the leftmost datapoints of those graphs indicate the vehicle coming to a screeching halt, then falling vertically under gravity only. Needless to say, this is a bit expensive and extreme. :) $\endgroup$ Dec 24, 2021 at 9:07

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