Kenneth McAfee

The thermal loading experienced by a spacecraft during atmospheric entry is the most sought-after parameter to quantify when designing and sizing a thermal protection system. It is also, perhaps, one of the most difficult parameters to quantify for entry spacecraft: knowledge of the atmospheric composition and pressure as well as the vehicle’s attitude and velocity are prerequisites for a ballpark prediction. Add in high-fidelity finite-rate chemistry models, surface catalysis predictions, and correction factors to account for recession (i.e. ablation) of the thermal protection system surface and the resulting uncertainty margins can still be as high as 25%.

Now, if the thermal loads experienced by a spacecraft are so tough to predict, why don’t we just measure them? Directly measuring atmospheric entry heating isn’t as straightforward either- see some of the other work on this site for a more thorough treatment of the limitations of current heat flux sensors (heat flux is the physical quantity that relates to thermal loading). But, to give a TL;DR, all the direct heat flux sensors that have flown on atmospheric entry spacecraft to-date need to be passively cooled. This means that they do not measure the heat loads that a spacecraft actually experiences; to accurately interpret these measurements, substantial time-varying correction factors are needed. So, atmospheric entry heating is both very difficult to predict and very difficult to measure directly. What else can be done?

Alternatively, we can reconstruct the atmospheric entry heat loads from temperature measurements embedded in the spacecraft thermal protection system itself. These techniques are called inverse heat transfer methods, and are quite powerful for a few reasons:

1. They rely on the use of temperature sensors, which are much simpler, more robust, and less invasive than direct heat flux sensors.
2. The reconstructed heating conditions are representative of those that drive the design and sizing of thermal protection systems.
3. The uncertainty margins of the reconstructed heating conditions are due to uncertainties of the material properties of the thermal protection system, which can be refined through ground testing.

Because of their advantages and compatibility with current spacecraft thermal instrumentation suites, inverse heat transfer methods have become the main engine for post-flight analyses of atmospheric entry heating in missions such as Mars Science Laboratory (which landed the Curiosity rover on Mars in 2012), ExoMars Schiaparelli (2016), and Mars 2020 (which landed the Perseverance rover on Mars in 2021).

Of course, there is a catch. The current inverse methods used by the space exploration community are very computationally expensive. There are a few reasons for this, but it really boils down to the preferred use of traditional time-marching finite volume or finite element schemes to model heat transfer in the inverse problem. This inefficiency was what my research targeted in the latter half of my PhD, where I developed fast inverse heat transfer methods tailored for atmospheric entry spacecraft. The approach relied on the use of the Green’s function framework to model heat transfer in spacecraft thermal protection systems, and yielded a greater-than two order of magnitude reduction in computation cost versus the current inverse methods used by the atmospheric entry community. Over the years we presented various algorithm components at AIAA Aviation (2024) and AIAA SciTech (2025), and have a forthcoming journal article coming out detailing the whole algorithm with further adaptations for in-depth pyrolysis and internal convection. This approach is actively being developed to handle even more complex material response behaviors critical for atmospheric entry, and we intend on releasing it under an open source license in the very near future. Stay tuned for future updates!