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# Closed-form ab initio solutions of geometric albedos and reflected light phase curves of exoplanets

## Abstract

Studying the albedos of the planets and moons of the Solar System dates back at least a century1,2,3,4. Of particular interest is the relationship between the albedo measured at superior conjunction, known as the ‘geometric albedo’, and the albedo considered over all orbital phase angles, known as the ‘spherical albedo’2,5,6. Determining the relationship between the geometric and spherical albedos usually involves complex numerical calculations7,8,9,10,11, and closed-form solutions are restricted to simple reflection laws12,13. Here we report the discovery of closed-form solutions for the geometric albedo and integral phase function, which apply to any law of reflection that only depends on the scattering angle. The shape of a reflected light phase curve, quantified by the integral phase function, and the secondary eclipse depth, quantified by the geometric albedo, may now be self-consistently inverted to retrieve globally averaged physical parameters. Fully Bayesian phase-curve inversions for reflectance maps and simultaneous light-curve detrending may now be performed due to the efficiency of computation. Demonstrating these innovations for the hot Jupiter Kepler-7b, we infer a geometric albedo of $$0.2{5}_{-0.02}^{+0.01}$$, a phase integral of 1.77 ± 0.07, a spherical albedo of $$0.4{4}_{-0.03}^{+0.02}$$ and a scattering asymmetry factor of $$0.0{7}_{-0.11}^{+0.12}$$.

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## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The code used to compute the models and perform Bayesian inference is available at https://github.com/bmorris3/kelp.

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## Acknowledgements

We acknowledge partial financial support from the Center for Space and Habitability (K.H., B.M.M. and D.K.), the PlanetS National Centre of Competence in Research (B.M.M.) and a European Research Council (ERC) Consolidator Grant awarded to K.H. (project EXOKLEIN; number 771620). K.H. acknowledges a honorary professorship from the Department of Physics of the University of Warwick and an imminent chair professorship of theoretical astrophysics from the Ludwig Maximilian University.

## Author information

Authors

### Contributions

K.H. formulated the problem, combined insights from the historical literature, derived the equations, produced all of the figures except Fig. 4 and led the writing of the manuscript. B.M.M. designed and authored open-source software (named kelp) that implemented the derived equations, performed the analysis of Kepler-7b data, produced Fig. 4 and co-wrote the manuscript. D.K. participated in decisive discussions of the problem with K.H. and read the manuscript.

### Corresponding author

Correspondence to Kevin Heng.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Astronomy thanks Mark Marley and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Validation against previous work.

Validation of calculations against the classic work of Dlugach & Yanovitskij (1974; labelled DY74) for the (a) geometric albedo and (b) spherical albedo for various reflection laws. For the Henyey-Greenstein reflection law, the scattering asymmetry factor has an assumed value of 0.5. For isotropic scattering, (c) shows the comparison of the spherical albedo to the classic work of van de Hulst (1974). For non-conservative Rayleigh scattering, (d) shows the comparison of the spherical albedo to the work of Madhusudhan & Burrows (2012).

### Extended Data Fig. 2 Anisotropic versus isotropic multiple scattering.

Calculations of the geometric albedo comparing anisotropic versus isotropic multiple scattering (MS), which follow Hapke’s approach of utilising the two-stream fluxes. Overlaid as circles are the calculations of Dlugach & Yanovitskij (1974). The discrepancies at low values of the scattering asymmetry factor originate from the use of Hapke’s linear approximation to the Chandrasekhar H function for isotropic multiple scattering.

### Extended Data Fig. 3 Cassini data of Jupiter.

Comparing measurements of the spherical albedo of Jupiter (curve with uncertainties of one standard deviation) with the values inferred from phase curve fitting (circles with uncertainties of one standard deviation). The data used were measured by the Cassini spacecraft.

### Extended Data Fig. 4 Regions of normal and enhanced reflectivity for an inhomogeneous atmosphere.

Regions of normal and enhanced reflectivity for an inhomogeneous atmosphere in terms of the longitude Φ within the observer-centric coordinate system. In the local longitude of the exoplanet (where the substellar point sits at x = 0), the atmosphere has a baseline single-scattering albedo of ω0 across x1xx2. When this region is within view of the observer, it is highlighted with a thick blue line. Regions of enhanced reflectivity (with a total single-scattering albedo of $$\omega ={\omega }_{0}+{\omega }^{\prime}$$) that are within the observer’s view are highlighted with thick red lines.

### Extended Data Fig. 5 Formulae for Computing Regions of Enhanced Reflectivity.

The index i refers to the components S, L and C.

### Extended Data Fig. 6 Maximum-likelihood parameters for Kepler-7b inferred from fitting the Kepler light curve.

A model consisting of the inhomogeneous atmosphere in reflected light, thermal emission, a secondary eclipse and a Gaussian process is employed. 1σ uncertainties are stated.

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Heng, K., Morris, B.M. & Kitzmann, D. Closed-form ab initio solutions of geometric albedos and reflected light phase curves of exoplanets. Nat Astron 5, 1001–1008 (2021). https://doi.org/10.1038/s41550-021-01444-7

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• DOI: https://doi.org/10.1038/s41550-021-01444-7