Two-thousand and nine (2009) was the 400th anniversary of the
publication of Johannes Kepler’s book New Astronomy (Astronomia Nova)
announcing the discovery of the elliptical orbit of Mars to the world.
The discovery of the elliptical orbit of Mars and the mathematical
rule of motion for Mars on its elliptical orbit by Johannes Kepler in
1605 is one of the most important advances in astronomy, physics, and
science. This discovery transformed the unproven heliocentric theory
of Copernicus into a rigorous predictive theory that outperformed the
traditional geocentric theory of Claudius Ptolemy and his successors.
The discovery paved the way for Newton’s theory of gravitation. It
remains one of a small number of cases where a simple mathematical
rule for seemingly complex and confusing data has been found. In many
respects, the discovery of the elliptical orbit of Mars and other
planets is more important than the better known work of Kepler’s
contemporary Galileo. In honor of Kepler, NASA has named its recent
mission to look for extra-solar planets, especially possible other
Earths that might support life or even intelligence, the Kepler
mission.

In Kepler’s time the reigning Ptolemaic theory could predict the
position of Mars to within a few degrees, usually less than a one
percent error. How important is such a small error? Space missions
routinely depend on modern orbital dynamics, a lineal descendant of
Kepler’s work, to make far more accurate calculations to succeed. The
Mars Climate Orbiter mission in 1999 failed due to a tiny error. After
traveling about 300 million miles, the Mars Climate Orbiter came in
about 90 miles, a tiny fraction of 300 million miles, too low, burning
up in the Martian atmosphere rather than aerobreaking successfully
into orbit. Successful space missions, the Global Positioning System
(GPS), and other modern applications depend on precision mathematical
models similar to and sometimes directly descended from Kepler’s model
of the orbit of Mars.

Kepler’s story is very different from the story of Galileo and it
offers different lessons for today. Diverse fields ranging from
astronomy and space physics to artificial intelligence are confronted
with similarly complex and confusing data. A mathematical solution to
an outstanding problem comparable to Kepler’s discovery could reveal
long suspected connections between gravity and other forces, perhaps
enabling new power or propulsion systems, enable computers to
recognize objects and spoken words, or solve other problems. This
article will discuss the discovery of the elliptical orbit of Mars in
the context of Kepler’s time. It will also draw some lessons from
Kepler and compare and contrast Kepler’s process of discovery to
modern astronomy, physics, space science and engineering, including a
detailed discussion of dark matter and dark energy.

http://math-blog.com/2009/12/17/keplers-new-astronomy/