New mathematical solutions to an old problem in astronomy

Date:
August 30, 2021
Source:
University of Bern
Summary:
The Bernese theoretical astrophysicist Kevin Heng has achieved
a rare feat: On paper, he has derived novel solutions to an old
mathematical problem needed to calculate light reflections from
planets and moons.

Now, data can be interpreted in a simple way to understand
planetary atmospheres, for example. The new formulae will likely
be incorporated into future textbooks.



FULL STORY ==========================================================================
For millennia, humanity has observed the changing phases of the Moon. The
rise and fall of sunlight reflected off the Moon, as it presents its
different faces to us, is known as a "phase curve." Measuring phase curves
of the Moon and Solar System planets is an ancient branch of astronomy
that goes back at least a century. The shapes of these phase curves encode information on the surfaces and atmospheres of these celestial bodies. In modern times, astronomers have measured the phase curves of exoplanets
using space telescopes such as Hubble, Spitzer, TESS and CHEOPS. These observations are compared with theoretical predictions. In order to do so,
one needs a way of calculating these phase curves. It involves seeking
a solution to a difficult mathematical problem concerning the physics
of radiation.


========================================================================== Approaches for the calculation of phase curves have existed since the
18th century. The oldest of these solutions goes back to the Swiss mathematician, physicist and astronomer, Johann Heinrich Lambert, who
lived in the 18th century. "Lambert's law of reflection" is attributed
to him. The problem of calculating reflected light from Solar System
planets was posed by the American astronomer Henry Norris Russell in an influential 1916 paper. Another well- known 1981 solution is attributed
to the American lunar scientist Bruce Hapke, who built on the classic
work of the Indian-American Nobel laureate Subrahmanyan Chandrasekhar in
1960. Hapke pioneered the study of the Moon using mathematical solutions
of phase curves. The Soviet physicist Viktor Sobolev also made important contributions to the study of reflected light from celestial bodies in
his influential 1975 textbook. Inspired by the work of these scientists, theoretical astrophysicist Kevin Heng of the Center for Space and
Habitability CSH at the University of Bern has discovered an entire
family of new mathematical solutions for calculating phase curves. The
paper, authored by Kevin Heng in collaboration with Brett Morris from
the National Center of Competence in Research NCCR PlanetS -- which the University of Bern manages together with the University of Geneva -- and
Daniel Kitzmann from the CSH, has just been published in Nature Astronomy.

Generally applicable solutions "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." Sara Seager brought the
problem to Heng's attention by her summary of it in her 2010 textbook.

By combining these insights, Heng was able to write down mathematical
solutions for the strength of reflection (the albedo) and the shape
of the phase curve, both completely on paper and without resorting to
a computer. "The ground- breaking aspect of these solutions is that
they are valid for any law of reflection, which means they can be used
in very general ways. The defining moment came for me when I compared
these pen-and-paper calculations to what other researchers had done
using computer calculations. I was blown away by how well they matched,"
said Heng.

Successful analysis of the phase curve of Jupiter "What excites me is
not just the discovery of new theory, but also its major implications
for interpreting data," says Heng. For example, the Cassini spacecraft
measured phase curves of Jupiter in the early 2000s, but an in-depth
analysis of the data had not previously been done, probably because the calculations were too computationally expensive. With this new family of solutions, Heng was able to analyze the Cassini phase curves and infer
that the atmosphere of Jupiter is filled with clouds made up of large, irregular particles of different sizes. This parallel study has just
been published by the Astrophysical Journal Letters, in collaboration
with Cassini data expert and planetary scientist Liming Li of Houston University in Texas, U.S.A.

New possibilities for the analysis of data from space telescopes
"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. It opens up new ways of interpreting data that were previously infeasible. Heng is collaborating with Pierre Auclair-Desrotour (formerly CSH, currently at Paris Observatory) to further generalize
these mathematical solutions. "Pierre Auclair-Desrotour is a more talented applied mathematician than I am, and we promise exciting results in the
near future," said Heng.

In the Nature Astronomy paper, Heng and his co-authors demonstrated a
novel way of analyzing the phase curve of the exoplanet Kepler-7b from
the Kepler space telescope. Brett Morris led the data analysis part of
the paper. "Brett Morris leads the data analysis for the CHEOPS mission
in my research group, and his modern data science approach was critical
for successfully applying the mathematical solutions to real data,"
explained Heng. They are currently collaborating with scientists from the American-led TESS space telescope to analyze TESS phase curve data. Heng envisions that these new solutions will lead to novel ways of analyzing
phase curve data from the upcoming, 10-billion- dollar James Webb Space Telescope, which is due to launch later in 2021. "What excites me most
of all is that these mathematical solutions will remain valid long after
I am gone, and will probably make their way into standard textbooks,"
said Heng.

========================================================================== Story Source: Materials provided by University_of_Bern. Note: Content
may be edited for style and length.


========================================================================== Journal References:
1. 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
2. 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 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2021/08/210830123242.htm

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