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Johannes Kepler (1571--1630) believed in a heliocentric solar system. Using the huge amount of data on planetary motion collected by Tycho Brahe (1546--1601), he concluded that the assumption of circular planetary orbits with the Sun at their center did not match Brahe's data. The focus of Kepler's efforts was on the orbit of Mars, which has the largest orbital eccentricity (i.e., its orbit is the least circular) of the planets for which Kepler had good data. Kepler was able to explain the data of Brahe by making the planetary orbits \emph{elliptical}, with the Sun at one focus of the ellipse, instead of circular. Over a period of almost 20 years, Kepler was able to formulate his three laws of planetary motion, \emph{based entirely on observational data}.

Johannes Kepler (1571--1630) believed in a heliocentric solar system. Using the huge amount of data on planetary motion collected by Tycho Brahe (1546--1601), he concluded that the assumption of circular planetary orbits with the Sun at their center did not match Brahe's data. The focus of Kepler's efforts was on the orbit of Mars, which has the largest orbital eccentricity (i.e., its orbit is the least circular) of the planets for which Kepler had good data. Kepler was able to explain the data of Brahe by making the planetary orbits \emph{elliptical}, with the Sun at one focus of the ellipse, instead of circular. Over a period of almost 20 years, Kepler was able to formulate his three laws of planetary motion, \emph{based entirely on observational data}.

Johannes Kepler (1571--1630) believed in a heliocentric solar system. Using the huge amount of data on planetary motion collected by Tycho Brahe (1546--1601), he concluded that the assumption of circular planetary orbits with the Sun at their center did not match Brahe's data. The focus of Kepler's efforts was on the orbit of Mars, which has the largest orbital eccentricity (i.e., its orbit is the least circular) of the planets for which Kepler had good data. Kepler was able to explain the data of Brahe by making the planetary orbits \emph{elliptical}, with the Sun at one focus of the ellipse, instead of circular. Over a period of almost 20 years, Kepler was able to formulate his three laws of planetary motion, \emph{based entirely on observational data}.

Johannes Kepler (1571--1630) believed in a heliocentric solar system. Using the huge amount of data on planetary motion collected by Tycho Brahe (1546--1601), he concluded that the assumption of circular planetary orbits with the Sun at their center did not match Brahe's data.
\begin{enumerate}
\item
\label{Kepler1}
The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.
\item
\label{Kepler2}
The line joining a planet to the Sun sweeps out equal areas in equal times as the planet travels around the Sun.
\item
\label{Kepler3}
The ratio of the squares of the orbital periods for two planets is equal to the ratio of the cubes of their semimajor axes
\begin{gather}
\frac{P_{1}^{2}}{P_{2}^{2}} = \frac{r_{1}^{3}}{r_{2}^{3}},
\end{gather}
where $P$ is the period, $r$ is the length of the semimajor axis of the ellipse and the subscripts ``1'' and ``2'' denote two different planets.
\end{enumerate}
The focus of Kepler's efforts was on the orbit of Mars, which has the largest orbital eccentricity (i.e., its orbit is the least circular) of the planets for which Kepler had good data.
Kepler was able to explain the data of Brahe by making the planetary orbits \emph{elliptical}, with the Sun at one focus of the ellipse, instead of circular. Over a period of almost 20 years, Kepler was able to formulate his three laws of planetary motion, \emph{based entirely on observational data}.

Johannes Kepler (1571--1630) believed in a heliocentric solar system. Using the huge amount of data on planetary motion collected by Tycho Brahe (1546--1601), he concluded that the assumption of circular planetary orbits with the Sun at their center did not match Brahe's data. The focus of Kepler's efforts was on the orbit of Mars, which has the largest orbital eccentricity (i.e., its orbit is the least circular) of the planets for which Kepler had good data. Kepler was able to explain the data of Brahe by making the planetary orbits \emph{elliptical}, with the Sun at one focus of the ellipse, instead of circular. Over a period of almost 20 years, Kepler was able to formulate his three laws of planetary motion, \emph{based entirely on observational data}.


\ref{Kepler1} \pageref{Kepler1}

\ref{Kepler2} \pageref{Kepler2}

\ref{Kepler3} \pageref{Kepler3}




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