Physically faithful joint inversion of asteroid optical and thermal-infrared photometry: nonconvex shapes, wavelength-dependent emissivity, and the roughness–thermal-inertia degeneracy

The combined (convex-inversion) thermophysical model of Ďurech, Delbó, Carry, Hanuš & Alí-Lagoa (2017) simultaneously fits optical lightcurves and mid-infrared fluxes (from the Infrared Astronomical Satellite, IRAS, and the Wide-field Infrared Survey Explorer, WISE) to derive an asteroid's shape, spin, size, light-scattering parameters, and thermal inertia, but deliberately simplifies the physics: surface roughness is held fixed on a grid because it is strongly degenerate with thermal inertia, emissivity is a constant, long-wavelength (submillimeter/millimeter) data are excluded, and the shape must be convex. Applications of the joint inversion have, to date, used convex shapes, a fixed emissivity of 0.9, and a roughness grid, while the missing physics has been incorporated only in models of individual bodies whose shapes were known independently (Bennu, Ryugu) or in single-body millimeter-wavelength thermophysical modeling. The open problem is to extend the joint inversion itself: to resolve or quantify the roughness–thermal-inertia degeneracy, to include wavelength-dependent emissivity so that sub-mm/mm data can be inverted for depth-dependent thermophysical structure, and to allow nonconvex shapes with global shadowing and inter-facet self-heating, with quantified uncertainties for sparse-data targets and applicability at survey scale.

Background

Asteroid surfaces reradiate absorbed sunlight at thermal-infrared wavelengths, and the way the emitted flux varies with rotation and viewing geometry encodes the body's size, shape, spin, and surface properties. Simple radiometric frameworks (the standard thermal model and the near-Earth asteroid thermal model, NEATM) assume a spherical body; a thermophysical model (TPM) instead computes the temperature of each facet of a known shape by solving the one-dimensional heat-diffusion equation in the subsurface, with the surface boundary condition including insolation and, in refined treatments, radiative heat exchange. The key material parameter is the thermal inertia $\Gamma=\sqrt{\rho C \kappa}$ (with $\rho$ density, $C$ specific heat capacity, $\kappa$ thermal conductivity), which measures how strongly the surface resists temperature change over a rotation; it diagnoses regolith structure (fine dust vs. rock). Macroscopic roughness is usually parameterized by spherical-section craters covering a fraction of the surface with a given opening angle (plus the Hapke mean-slope parameter, a statistical average of surface tilt in the Hapke light-scattering model, for the optical scattering); roughness and $\Gamma$ both control the thermal-infrared 'beaming' and are therefore strongly degenerate in disk-integrated (unresolved) photometry. Classically, shape and spin are first reconstructed from optical lightcurves and then held fixed in the TPM fit; because the photometric inverse problem is ill-posed, the shape is restricted to a convex polyhedron, which guarantees uniqueness of the reconstruction. Ďurech et al. (2017) introduced the combined-inversion thermophysical model (CITPM), which instead optimizes shape, spin, size, Hapke scattering parameters, and $\Gamma$ simultaneously against both optical photometry (dense lightcurves plus sparse photometry from the Lowell Observatory photometric database) and thermal fluxes (e.g., from NASA's Wide-field Infrared Survey Explorer, WISE), joined in a weighted $\chi^2$. The paper validated the method on asteroids with independent ground truth ((21) Lutetia, (2867) Šteins, (85) Io, (220) Stephania) but flagged five physical and methodological gaps. First, the roughness parameters (crater fraction, opening angle, Hapke mean slope) are not optimized but fixed, and the fit is rerun over a grid of their combinations, because their correlation with $\Gamma$ is strong; resolving or marginalizing this continuous degeneracy was left open. Second, the emissivity is a wavelength-independent constant (0.9 in all tests); this is acceptable near 10 $\mu$m but fails at submillimeter/millimeter wavelengths, so such data (e.g., Herschel SPIRE 250–500 $\mu$m) were excluded — yet longer wavelengths probe deeper, colder subsurface layers, and the authors noted that reconciling thermal inertias derived from different wavelength ranges would probe depth-dependent $\Gamma$. Third, only convex shapes are used, with no global shadowing by large-scale topography and no radiative self-heating between facets; the authors stated the generalization to nonconvex shapes is straightforward in principle but did not implement or validate it. Fourth, for targets with sparse or low-quality thermal data (the Šteins example) the derived parameter ranges stay broad and strongly correlated. Fifth, scaling the method to the thousands of asteroids with both photometry of the quality archived in DAMIT (the Database of Asteroid Models from Inversion Techniques, the public archive of lightcurve-derived asteroid shape and spin models) and WISE fluxes, including the objective choice of the relative weight between the optical and thermal data terms, was left as future work.

Problem Statement

Extend the combined inversion of asteroid optical lightcurves and thermal-infrared photometry (the CITPM framework of Ďurech et al. 2017) so that the joint fit is physically faithful rather than deliberately simplified. Specifically, within one jointly optimized inversion of both data modes: (1) treat the macroscopic-roughness parameters (crater coverage fraction and opening angle, Hapke mean slope) and the thermal inertia $\Gamma$ simultaneously, with propagated uncertainties and correlations, instead of holding roughness fixed on a grid, thereby resolving or quantifying the continuous roughness–$\Gamma$ degeneracy in disk-integrated data; (2) replace the constant emissivity with a wavelength-dependent treatment valid into the submillimeter/millimeter regime, so that long-wavelength thermal data can be included in the inversion and used to constrain depth-dependent thermophysical structure, reconciling thermal-inertia estimates derived from different wavelength ranges; and (3) admit nonconvex shape models in the inversion, including global shadowing by large-scale topography and radiative self-heating between facets, while controlling the non-uniqueness that motivates the convex restriction. Two linked scientific subgoals of the extension are to quantify and reduce the broad, correlated parameter ranges obtained for targets with sparse or low-quality thermal data (the (2867) Šteins case), and to make the method scalable to the population of asteroids with both archival photometry and WISE-era thermal fluxes, including an objective determination of the relative weight of the optical and thermal data terms. The extensions should be validated against independent ground truth (spacecraft shape models, stellar-occultation sizes, in-situ thermal measurements).

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