The Eclipse-mapping Null Space: Comparing Theoretical Predictions with Observed Maps

Abstract

High-precision exoplanet eclipse light curves, like those possible with JWST, enable flux and temperature mapping of exoplanet atmospheres. These eclipse maps will have unprecedented precision, providing an opportunity to constrain current theoretical predictions of exoplanet atmospheres. However, eclipse mapping has unavoidable mathematical limitations because many map patterns are unobservable. This “null space” has implications for making comparisons between predictions from general circulation models (GCMs) and the observed planet maps and thus affects our understanding of the physical processes driving the observed maps. We describe the eclipse-mapping null space and show how GCM forward models can be transformed to their observable modes for more appropriate comparison with retrieved eclipse maps, demonstrated with applications to synthetic data of an ultrahot Jupiter and a cloudy warm Jupiter under JWST best-case and extreme-precision observing scenarios. We show that the effects of the null space can be mitigated and manipulated through observational design, and JWST exposure times are short enough to not increase the size of the null space. Furthermore, we show the mathematical connection between the null space and the “eigenmapping” method, demonstrating how eigenmaps can be used to understand the null space in a model-independent way. We leverage this connection to incorporate null-space uncertainties in retrieved maps, which increases the uncertainties to encompass the ground truth for synthetic data. The comparisons between observed maps and forward models that are enabled by this work, and the improved eclipse-mapping uncertainties, will be critical to our interpretation of multidimensional aspects of exoplanets in the JWST era.

Latitudinal Asymmetry in the Dayside Atmosphere of WASP-43b

“…Here, we measure the location of maximum brightness as the "hotspot offset," which is a different measurement than the phase-curve offset reported in previous works, and consistency between those measurements is not necessarily expected (Challener & Rauscher 2023). Therefore, for the purposes of comparison to past results, we compute the phase-curve offset of our best-fitting model and a posterior distribution of phasecurve offsets from our posterior distribution of map models by calculating the rotational angle that results in maximum brightness of the visible hemisphere.…”

“…Eclipse mapping is highly sensitive to system parameters, particularly the orbital inclination i (Challener & Rauscher 2023). Here, we use our data to refine the system parameters.…”

Two-dimensional Eclipse Mapping of the Hot-Jupiter WASP-43b with JWST MIRI/LRS

“…For instance, rotational phase curves of spatially unresolved objects (i.e., exoplanet phase curves) are insensitive to latitudinal structures. If the object is eclipsed, as is the case for WASP-43b, then the shape of eclipse ingress and egress breaks many of these degeneracies, but still leaves a "null space" of unobservable patterns (Challener & Rauscher 2023). This null space means that retrieved eclipse maps will not always match thermal emission output from GCM simulations.…”

“…To illustrate this effect, we separate our GCM maps into their observable and null components following Challener & Rauscher (2023). The second row of Figure 7 shows the observable component for each GCM.…”

“…First row: the modeled outgoing longwave radiation (OLR) in the 5-10.5 μm MIRI/LRS bandpass for each of the four GCM simulations, expressed as a ratio of planetary to stellar flux. Second row: observable emission from the GCM simulations, with the null space removed as described by Luger et al (2021) and Challener & Rauscher (2023). Third row: the OLR represented using spherical harmonics up to = ℓ 3 max , showing how structure is lost; this is the best map that could be achieved by a fit like the = ℓ 3 max fit in Figure 5.…”