Orbits and Planetary Motion: Tycho Brahe and Johannes Kepler

Historical Context of Planetary Motion

• Around the same time Galileo Galilei began his experiments with falling bodies, scientific understanding of planetary motion advanced through observational and mathematical breakthroughs.

• Tycho Brahe provided comprehensive observational data, while Johannes Kepler derived the mathematical principles governing planetary movement.

• Together, their work established Nicolaus Copernicus' heliocentric model on a firm mathematical foundation, paving the way for Sir Isaac Newton in the following century.

Tycho Brahe and the Hven Observatory

• Tycho Brahe was born into Danish nobility three years after the publication of Nicolaus Copernicus' De Revolutionibus.

• Brahe developed an early passion for astronomy and conducted detailed observations of an exploding star (supernova) that brightened significantly in the night sky.

• King Frederick II of Denmark became Brahe's patron, enabling him at age $30$ to establish an astronomical observatory on the island of Hven in the North Sea.

• Brahe was Europe's last and greatest observational astronomer before the invention of the telescope.

• Using large curved instruments (quadrants) to measure precise angles above the horizon, Brahe systematically recorded celestial positions.

• For nearly $20\,\text{years}$ at Hven, Brahe maintained a continuous record of the positions of the Sun, Moon, and planets.

• His measurements revealed consistent discrepancies between observed planetary positions and the traditional tables derived from Claudius Ptolemy.

• While Brahe possessed extraordinary observational precision, he lacked the mathematical ability to analyze his extensive data and develop an alternative theoretical model.

• Following the death of King Frederick II in $1597$, Brahe lost political favor due to his extravagant and cantankerous nature, which had earned him numerous enemies among government officials.

• Brahe relocated to Prague to serve as court astronomer to Emperor Rudolf of Bohemia.

• In $1600$, a year before his death in $1601$, Brahe hired young German mathematician Johannes Kepler as his assistant to help discover a model of planetary motion consistent with the Hven dataset.

• Brahe restricted Kepler's access to his complete dataset out of concern that Kepler would independently solve the mechanics of planetary motion and claim sole credit.

Johannes Kepler

• Johannes Kepler was born into poverty in the German province of Württemberg and spent much of his life during the Thirty Years' War.

• While studying for a theological career at the University of Tübingen, Kepler learned the principles of the Copernican system and adopted the heliocentric hypothesis.

• Following Brahe's death in $1601$, Kepler obtained full access to Brahe's extensive observational records, spending more than $20\,\text{years}$ analyzing the data.

• Kepler derived three fundamental rules describing planetary behavior based on their physical paths through space, known as Kepler's three laws.

• The first two laws were published in $1609$ in his work The New Astronomy.

Geometry of Ellipses and Conic Sections

• An orbit is defined as the path of an object through space.

• Kepler initially assumed planetary orbits were circular, but circular paths could not be reconciled with Brahe's observational data for Mars.

• By analyzing the data for Mars, Kepler determined that the planet's orbit is an ellipse—a somewhat flattened circle.

• An ellipse belongs to a family of curves called conic sections, formed by the intersection of a plane with a cone.

• Properties of an Ellipse:

  • Foci: Two internal points (singular: focus) where the sum of the distances from any point on the curve to these two points is always constant.

  • Construction Method: Constructed by placing two tacks at the foci, looping a piece of string around them, pulling the string taut with a pencil, and tracing the path around the tacks. The total length of the string remains constant, ensuring the sum of the distances from the pencil to both foci stays equal.

  • Major Axis: The widest diameter across the ellipse, with a total length of $2a$.

  • Semimajor Axis ($a$): Half of the major axis, representing the distance from the center of the ellipse to one edge. This distance corresponds to a planet's average distance from the Sun.

  • The semimajor axis of Mars' orbit is $228 \times 10^6\,\text{km}$ ($228\,\text{million kilometers}$).

• Eccentricity ($e$): The ratio of the distance between the two foci to the length of the major axis.

  • When the foci are in the exact same position, the distance between them is $0$, giving an eccentricity of $0$, which defines a perfect circle. For a circle, the semimajor axis is equal to the radius.

  • Moving the foci farther apart increases the eccentricity toward a maximum value of $1.0$, which represents a completely flattened ellipse.

  • The size and shape of an ellipse are completely specified by its semimajor axis and its eccentricity.

  • Mars has an orbital eccentricity of approximately $0.1$. Scale drawings of Mars' orbit are almost indistinguishable from a circle, yet this slight eccentricity was vital for revealing planetary mechanics.

Kepler's Laws of Planetary Motion

Kepler's First Law: The Law of Ellipses

• The orbit of every planet is an ellipse with the Sun located at one focus (the second focus is empty space).

• This law demonstrated that circular motion is not required for a stable cosmological system, replacing traditional Greek philosophical assumptions.

Kepler's Second Law: The Law of Equal Areas

• Describes how a planet's orbital speed varies along its elliptical path.

• A planet speeds up as it approaches the Sun and slows down as it moves farther away.

• Conceptual Model: Imagine an elastic line connecting the Sun to a planet. Near the Sun, the line is short and moves rapidly; far from the Sun, the line stretches and moves slowly.

• Formal Statement: A line connecting a planet and the Sun sweeps out equal areas of space in equal intervals of time ($t$).

• In equal time intervals ($t$), Area $B$ (swept out near perihelion between positions 1 and 2) equals Area $A$ (swept out near aphelion between positions 3 and 4).

• While circular orbits yield constant orbital speeds, elliptical orbits result in continuously variable orbital speeds.

Kepler's Third Law: The Harmonic Law

• Discovered in $1619$ while seeking mathematical relationships between planetary spacing and orbital periods ("harmony of the spheres").

• Orbital Period ($P$): The time required for a planet to complete one full revolution around the Sun.

• Proportionality: The square of a planet's orbital period is proportional to the cube of its semimajor axis:   $P^2 \propto a^3$

• Units and Exact Equality:

  • When orbital period ($P$) is measured in Earth years and semimajor axis ($a$) is measured in Astronomical Units (AU), the proportion becomes an exact equality:     $P^2 = a^3$

  • Astronomical Unit (AU): Defined as the average distance between Earth and the Sun, approximately $1.5 \times 10^8\,\text{km}$.

• Kepler's third law applies to all objects orbiting the Sun, including Earth.

Sample Calculation using Kepler's Third Law

• Problem: Calculate Mars' average distance from the Sun ($a$) given its orbital period of $1.88\,\text{Earth years}$.

• Step 1: Square the orbital period ($P$):   $P^2 = (1.88)^2 = 3.5344 \approx 3.53$

• Step 2: Equate $P^2$ to $a^3$ according to Kepler's Third Law:   $a^3 = 3.53$

• Step 3: Calculate the cube root of $3.53$ to find $a$:   $a = \sqrt[3]{3.53} \approx 1.52\,\text{AU}$

• Conclusion: Mars orbits at an average distance of $1.52\,\text{AU}$, meaning it is approximately $50\%$ farther from the Sun than Earth.