Analysing Optical Thickness in Simulated Giant Impacts to Determine Observational Signatures of Post-Impact Bodies

Introduction

Giant impacts are a crucial stage of planet formation, capable of altering the properties of planets and the structure of planetary systems [Raymond et al., 2018]. As such, garnering a better understanding of the dynamics of these incredibly violet and energetic events is essential to the study of planetary formation pathways. Unfortunately, the observational constraints we have on the outcomes of giant impacts are the marks left in the present-day properties of planetary systems, such as the chemistry of the Earth-Moon system or the density of exoplanets. This is a limited data set with degeneracies between the initial conditions, processes and final outcomes. However, we have recently entered a new frontier in the study of giant impacts. Recent observations of an infrared excess followed by a visible transit of the star ASASSN-21qj are thought to be the result of an exoplanetary collision [Kenworthy et al., 2023]: the infrared excess could be the result of direct emission from the substantially inflated post-impact body. To leverage this powerful new constraint on the dynamics of giant impacts, we need to be able to relate observed data to models of giant impacts. However, large sections of post-impact bodies are optically thick and observed emission likely comes from the very edge of the post-impact structure where material density is very low. Due to computational expense, it is not currently possible to conduct giant impact simulations that resolve such low densities, inhibiting our ability to use astronomical observations to constrain impact processes.

We present a new method of processing the results of existing smoothed particle hydrodynamics (SPH) impact simulations to emulate the true physical properties of outer regions of post-impact body candidates to compare to astronomical observations.

Going Beyond the Density Floor

We simulate giant impacts using SWIFT [Schaller et al., 2024]. Simulated colliding planets are represented as collections of parcels of mass (particles); the number of particles used is referred to as the simulation’s resolution. Typically, higher-resolution simulations are considered more accurate, but are far more computationally expensive. The radius of these fixed-mass particles, proportional to a “smoothing length” (h),  evolves during the simulation and allows the code to interpolate properties such as the density at the central location of each particle, and calculate forces acting between particles. SWIFT iteratively solves Equation 1 to find a particle’s smoothing length [Price 2012]. However, for computational efficiency, SWIFT imposes an upper limit on how large SPH particles are allowed to become, or a “maximum smoothing length”. This truncation prevents these particles from recording sufficiently low densities at each of their locations and enforcing a lower density limit or “density floor”.

(Equation 1: n is number density, η is a smoothing parameter, x vectors are multi-dimensional locations and particle positions, W is a smoothing kernel described in [Dehnen & Aly 2012].)

Although the density floor is essential to increasing SWIFT’s computational efficiency, it presents an unfortunate problem in the analysis of optical thickness and the process of locating photic surfaces of simulated structures. Post-impact structures can span tens to hundreds of Earth radii, are substantially vaporised and heated, and likely optically thick to very low densities [Kenworthy et al., 2023]. However, high-resolution, well-smoothed simulations of computational feasibility typically have density floors above 10⁻⁵ kg m⁻³, forcing particles in the outer regions of simulated post-impact structures to store the same density values which could be several orders of magnitude higher than reality, and misrepresenting the simulation’s material record of thermodynamic properties.

To overcome this issue, we have implemented a post-processing pipeline which re-solves Equation 1 without enforcing efficiency-induced truncation, and so recovers the extremely low densities expected at the edge of this expansive vaporised structure (Figure 1). We then account for the effect of smoothing length truncation on the rest of the structure’s thermodynamic properties, and simultaneously untangle properties of multiple materials (e.g. forsterite from the impactor/ target mantles and iron from their cores) at different phases. Finally, we model the absorption of silicate vapour [Kraus et al., 2012] and condensates to locate the photic surface of the post-impact structure.

[Figure 1: A histogram showing original interpolated particle densities when smoothing lengths are truncated (green), vs. reprocessed interpolated particle densities (purple). Note the large spike in the original distribution of densities at approximately 10^-3 kgm^-3 – this is the density floor imposed by truncation. The original and processed distributions are near identical for higher density values where assigned smoothing lengths are accurate. For lower density values, particles originally at the density floor are assigned lower density values once their smoothing length has been recalculated.]

Conclusions

We find that giant impacts can easily produce bright, hot, chaotic objects with optically thick regions than span hundreds of Earth radii. The size of the photic surface varies with particle resolution, since it is essential that individual SPH particle masses are low enough to accurately model the relatively low-mass outer regions of post-impact structures. However, for collisions between two Earth-mass bodies we begin to see convergence in the shape of optically thick regions for resolutions above 10⁶ particles.

[Figure 2: Extracting the optically thick region of a post-impact structure, viewed 23 hours after a simulated impact event perpendicularly to the collisional plane. (Black indicates optical thickness, teal optical thinness.) SPH particle locations are superimposed (white). The pipeline identifies that not all SPH particle locations are contained within the optically thick region. This post-impact structure is the result of a collision between two identical planetary objects 1R_⊕ in radius and 0.9M_⊕ in mass, each made up of 10^6.5 particles.]

We are now using this pipeline to relate giant impact simulations to the detectability of post-impact objects, both via direct detection and the transit method, for a variety of telescopes with different technical specifications. The scale of the optically thick regions of post-impact bodies, along with their associated flux values, suggest that a wide range of impacts may feasibly detectable, allowing powerful new constraints to be placed on planetary formation models.