Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

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



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 \(0.0{7}_{-0.11}^{+0.12}\).

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Schematic describing the geometry of the system.
Fig. 2: Atmospheric scattering properties.
Fig. 3: Shapes of reflected light phase curves.
Fig. 4: Fit to Kepler data.

Similar content being viewed by others

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


  1. Bond, G. P. On the light of the Sun, Moon, Jupiter, and Venus. Mon. Not. R. Astron. Soc. 21, 197–202 (1861).

    Article  ADS  Google Scholar 

  2. Russell, H. N. On the albedo of the planets and their satellites. Astrophys. J. 43, 173–196 (1916).

    Article  ADS  Google Scholar 

  3. Horak, H. G. Diffuse reflection by planetary atmospheres. Astrophys. J. 112, 445–463 (1950).

    Article  ADS  Google Scholar 

  4. de Vaucouleurs, G. Geometric and photometric parameters of the terrestrial planets. Icarus 3, 187–235 (1964).

    Article  ADS  Google Scholar 

  5. Sobolev, V. V. Light Scattering in Planetary Atmospheres (Pergamon, 1975).

  6. Seager, S. Exoplanet Atmospheres (Princeton Univ. Press, 2010).

  7. Horak, H. G. & Little, S. J. Calculations of planetary reflection. Astrophys. J. Suppl. 11, 373–428 (1965).

    Article  ADS  Google Scholar 

  8. Dlugach, J. & Yanovitskij, E. G. The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres. Icarus 22, 66–81 (1974).

    Article  ADS  Google Scholar 

  9. Hovenier, J. W. & Hage, J. I. Relations involving the spherical albedo and other photometric quantities of planets with thick atmospheres. Astron. Astrophys. 214, 391–401 (1989).

    ADS  Google Scholar 

  10. Marley, M. S., Gelino, C., Stephens, D., Lunine, J. I. & Freedman, R. Reflected spectra and albedos of extrasolar giant planets. I. Clear and cloudy atmospheres. Astrophys. J. 513, 879–893 (1999).

    Article  ADS  Google Scholar 

  11. Sudarsky, D., Burrows, A. & Pinto, P. Albedo and reflection spectra of extrasolar giant planets. Astrophys. J. 538, 885–903 (2000).

    Article  ADS  Google Scholar 

  12. van de Hulst, H. C. The spherical albedo of a planet covered with a homogeneous cloud layer. Astron. Astrophys. 35, 209–214 (1974).

    ADS  Google Scholar 

  13. Madhusudhan, N. & Burrows, A. Analytic models for albedos, phase curves and polarization of reflected light from exoplanets. Astrophys. J. 747, 25 (2012).

    Article  ADS  Google Scholar 

  14. Hapke, B. Bidirectional reflectance spectroscopy. 1. Theory. J. Geophys. Res. 86, 3039–3054 (1981).

    Article  ADS  Google Scholar 

  15. Chandrasekhar, S. Radiative Transfer (Dover, 1960).

  16. Pierrehumbert, R. T. Principles of Planetary Climate (Cambridge Univ. Press, 2010).

  17. Demory, B.-O. Inference of inhomogeneous clouds in an exoplanet atmosphere. Astrophys. J. Lett. 776, L25 (2013).

    Article  ADS  Google Scholar 

  18. Hu, R., Demory, B.-O., Seager, S., Lewis, N. & Showman, A. P. A semi-analytical model of visible-wavelength phase curves of exoplanets and applications to Kepler-7b and Kepler-10b. Astrophys. J. 802, 51 (2015).

    Article  ADS  Google Scholar 

  19. Shporer, A. & Hu, R. Studying atmosphere-dominated hot Jupiter Kepler phase curves: evidence that inhomogeneous atmospheric reflection is common. Astron. J. 150, 112 (2015).

    Article  ADS  Google Scholar 

  20. Oreshenko, M., Heng, K. & Demory, B.-O. Optical phase curves as diagnostics for aerosol composition in exoplanetary atmospheres. Mon. Not. R. Astron. Soc. 457, 3420–3429 (2016).

    Article  ADS  Google Scholar 

  21. Esteves, L. J., De Mooij, E. J. W. & Jayawardhana, R. Changing phases of alien worlds: probing atmospheres of Kepler planets with high-precision photometry. Astrophys. J. 804, 150 (2015).

    Article  ADS  Google Scholar 

  22. Kipping, D. M. Binning is sinning: morphological light-curve distortions due to finite integration time. Mon. Not. R. Astron. Soc. 408, 1758–1769 (2010).

    Article  ADS  Google Scholar 

  23. Burrows, A., Sudarsky, D. & Hubeny, I. L and T dwarf models and the L to T transition. Astrophys. J. 640, 1063–1077 (2006).

    Article  ADS  Google Scholar 

  24. Heng, K. & Li, L. Jupiter as an exoplanet: insights from Cassini phase curves. Astrophys. J. Lett. 909, L20 (2021).

    Article  ADS  Google Scholar 

  25. Mihalas, D. Stellar Atmospheres (Freeman, 1970).

  26. Toon, O. B., McKay, C. P., Ackerman, T. P. & Santhanam, K. Rapid calculation of radiative heating rates and photodissociation rates in inhomogeneous multiple scattering atmospheres. J. Geophys. Res. 94, 16287–16301 (1989).

    Article  ADS  Google Scholar 

  27. Heng, K. Exoplanetary Atmospheres (Princeton Univ. Press, 2017).

  28. Heng, K., Malik, M. & Kitzmann, D. Analytical models of exoplanetary atmospheres. VI. Full solutions for improved two-stream radiative transfer, including direct stellar beam. Astrophys. J. Suppl. Ser. 237, 29 (2018).

    Article  ADS  Google Scholar 

  29. Lommel, E. Die Photometrie der diffusen Zurückwerfung. Bayer. Akad. Wiss. München Sitzungsber. 17, 95–132 (1887).

    MATH  Google Scholar 

  30. Seeliger, H. Zur Photometrie zerstreut reflectirender Substanzen. Bayer. Akad. Wiss. München Sitzungsber. 18, 201–248 (1888).

    MATH  Google Scholar 

  31. Davidović, D. M., Vukanić, J. & Arsenović, D. Two new analytic approximations of the Chandrasekhar’s H function for isotropic scattering. Icarus 194, 389–397 (2008).

    Article  ADS  Google Scholar 

  32. Henyey, L. G. & Greenstein, J. L. Diffusion radiation in the Galaxy. Astrophys. J. 93, 70–83 (1941).

    Article  ADS  Google Scholar 

  33. Li, L. et al. Less absorbed solar energy and more internal heat for Jupiter. Nat. Commun. 9, 3709 (2018).

    Article  ADS  Google Scholar 

  34. Lightkurve Collaboration Lightkurve: Kepler and TESS time series analysis in Python. Astrophysics Source Code Library (2018).

  35. Kipping, D. M. Efficient, uninformative sampling of limb darkening coefficients for two-parameter laws. Mon. Not. R. Astron. Soc. 435, 2152–2160 (2013).

    Article  ADS  Google Scholar 

  36. Heng, K. & Workman, J. Analytical models of exoplanetary atmospheres. I. Atmospheric dynamics via the shallow water system. Astrophys. J. Suppl. Ser. 213, 27 (2014).

    Article  ADS  Google Scholar 

  37. Cowan, N. B. & Agol, E. The statistics of albedo and heat redistribution on hot exoplanets. Astrophys. J. 729, 54 (2011).

    Article  ADS  Google Scholar 

  38. Zhang, M. et al. Phase curves of WASP-33b and HD 149026b and a new correlation between phase curve offset and irradiation temperature. Astron. J 155, 83 (2018).

    Article  ADS  Google Scholar 

  39. Beatty, T. G. et al. Spitzer phase curves of KELT-1b and the signatures of nightside clouds in thermal phase observations. Astron. J 158, 66 (2019).

    Article  Google Scholar 

  40. Foreman-Mackey, D., Czekala, I., Agol, E., Luger, R. & Barclay, T. dfm/exoplanet: exoplanet v0.2.4. Zenodo (2019).

  41. Agol, E., Luger, R. & Foreman-Mackey, D. Analytic planetary transit light curves and derivatives for stars with polynomial limb darkening. Astron. J. 159, 123 (2020).

    Article  ADS  Google Scholar 

  42. Southworth, J. Homogeneous studies of transiting extrasolar planets—V. New results for 38 planets. Mon. Not. R. Astron. Soc. 426, 1291–1323 (2012).

    Article  ADS  Google Scholar 

  43. Foreman-Mackey, D., Agol, E., Ambikasaran, S. & Angus, R. Fast and scalable Gaussian process modeling with applications to astronomical time series. Astron. J. 154, 220 (2017).

    Article  ADS  Google Scholar 

  44. Foreman-Mackey, D. Scalable backpropagation for Gaussian processes using celerite. Res. Notes AAS 2, 31 (2018).

    Article  ADS  Google Scholar 

  45. The Theano Development Team Theano: a Python framework for fast computation of mathematical expressions. Preprint at (2016).

  46. Salvatier, J., Wiecki, T. V. & Fonnesbeck, C. PyMC3: Python probabilistic programming framework. PeerJ Comput. Sci. 2, e55 (2016).

  47. Gelman, A. & Rubin, D. B. Inference from iterative simulation using multiple sequences. Stat. Sci. 7, 457–472 (1992).

    Article  MATH  Google Scholar 

  48. Hunter, J. D. Matplotlib: a 2D graphics environment. Comput. Sci. Eng. 9, 90–95 (2007).

    Article  Google Scholar 

  49. van der Walt, S., Colbert, S. C. & Varoquaux, G. The NumPy array: a structure for efficient numerical computation. Comput. Sci. Eng. 13, 22–30 (2011).

    Article  Google Scholar 

  50. Astropy Collaboration. Astropy: a community Python package for astronomy. Astron. Astrophys. 558, A33 (2013).

    Article  Google Scholar 

  51. Foreman-Mackey, D. scatterplot matrices in Python. J. Open Source Softw. 1, 24 (2016).

    Article  ADS  Google Scholar 

  52. Astropy Collaboration. The Astropy Project: building an open-science project and status of the v2.0 core package. Astron. J. 156, 123 (2018).

    Article  ADS  Google Scholar 

  53. Luger, R. et al. starry: analytic occultation light curves. Astron J. 157, 64 (2019).

    Article  ADS  Google Scholar 

  54. Virtanen, P. et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nat. Methods 17, 261–272 (2020).

    Article  Google Scholar 

  55. Schwartz, J. C. & Cowan, N. B. Balancing the energy budget of short-period giant planets: evidence for reflective clouds and optical absorbers. Mon. Not. R. Astron. Soc. 449, 4192–4203 (2015).

    Article  ADS  Google Scholar 

Download references


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 and Affiliations



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.

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing