# An Old Astronomy Problem Has New Mathematical Solutions

People have been observing moon phases for thousands of years. Measuring the phase curves of the Moon and planets of the Solar System is an ancient branch of astronomy going back at least a century. The rise and fall of daylight reflected off the Moon creates a "phase curve." It is believed that the phase curves of these celestial bodies provide information regarding their surfaces and atmospheres. The phase curves of exoplanets have been measured in recent decades by astronomers using space telescopes like Hubble, Spitzer, TESS, and CHEOPS. A comparison is made between observational data and theoretical predictions. A method for calculating these phases curves is needed for this. There is a mathematical problem involving physics of radiation that needs to be solved.

The 18th century saw the introduction of approaches for calculating phase curves. Johann Heinrich Lambert, a Swiss mathematician, physicist, and astronomer, originated the earliest solution in the 18th century. He is credited with the "Lambert law of reflection." Henry Norris Russell wrote an influential 1916 paper about the issue of calculating the reflected light from planets in the solar system. An American lunar scientist named Bruce Hapke's 1981 solution is also well known. He built upon the 1960 classic work by Indian-American Nobel laureate Subrahmanyan Chandrasekhar. Mathematical solutions of phase curves advanced the age of research into the Moon under Hapke. A textbook published by Soviet physicist Viktor Sobolev in 1975 provided important insights into the study of reflected light from celestial bodies. Kevin Heng of CSH at the University of Bern discovered a set of mathematical solutions to the problem of calculating phase curves that have been inspired by the work of these scientists.

**Solutions That Apply Generally**

Heng became aware of the problem through Sara Seager's textbook summary from 2010 stating, "I was fortunate that this rich body of work had already been done by these great scientists. Hapke had discovered a simpler way to write down the classic solution of Chandrasekhar, who famously solved the radiative transfer equation for isotropic scattering. Sobolev had realised that one can study the problem in at least two mathematical coordinate systems."

By consolidating these bits of knowledge, Heng had the option to record numerical answers for the strength of reflection (the albedo) and the state of the stage bend, both totally on paper and without depending on a PC. "The pivotal part of these arrangements is that they are legitimate for any law of reflection, which implies they can be utilized in exceptionally broad ways. The extremely important occasion came for me when I contrasted these pen-and-paper computations with what different analysts had done utilizing PC estimations. I was passed up how well they coordinated," said Heng.

**Suitable Analysis of Jupiter's Phase Curve**

Besides discovering new theories, interpreting data can be greatly impacted by what Heng's discoveries mean. In the early 2000s, Cassini measured phase curves of Jupiter, but an in-depth analysis of the data had not yet been performed, probably due to the computational costs. The Cassini phase curves were analyzed by Heng based on this new class of solutions, and the inference was made that the atmosphere of Jupiter consists of clouds made up of large irregular particles of varying sizes.

**Data from Space Telescopes can be Analyzed in New Ways**

"The ability to write down mathematical solutions for phase curves of reflected light on paper means that one can use them to analyze data in seconds," said Heng.* A previously impossible task can now be accomplished through new methods of interpreting data. To generalize these mathematical solutions further, Heng is collaborating with Pierre Auclair-Desrotour (formerly CSH, currently at Paris Observatory).

Citations

University of Bern. "New mathematical solutions to an old problem in astronomy." ScienceDaily. ScienceDaily, 30 August 2021. <www.sciencedaily.com/releases/2021/08/210830123242.htm>. Accessed 22September 2021.

Materials provided by **University of Bern**. *Note: Content may be edited for style and length.*

Kevin Heng, Brett M. Morris, Daniel Kitzmann. **Closed-form ab initio solutions of geometric albedos and reflected light phase curves of exoplanets**. *Nature Astronomy*, 2021; DOI: 10.1038/s41550-021-01444-7

Kevin Heng, Liming Li. **Jupiter as an Exoplanet: Insights from Cassini Phase Curves**. *The Astrophysical Journal Letters*, 2021; 909 (2): L20 DOI: 10.3847/2041-8213/abe872