Announcements Please pick up HW#1, due Friday Notes on HW Please show your work Please give focused answers to the question EC opportunity – Solar System Planetarium Show, Jan 27 th, sign up now Please turn off all electronic devices Don’t forget to sign the attendance sheet

Retrograde motion a)is explained by the geocentric model as planets moving backwards sometimes b)is explained by the heliocentric models as planets passing each other. c)is seen for the planets, but not the Sun and Moon d)was studied in ancient Greece e)All the other answers are correct f)Copernican system did.

Lecture 5: Kepler & the Harmony of the Planets Astronomy 1143 – Spring 2014

Key Ideas Kepler – mathematician w/Tycho’s data Kepler described planetary orbits by 3 Laws 1 st Law: planets move in ellipses with the Sun at one focus 2 nd Law: line connecting Sun and planets sweeps out equal areas in equal time 3 rd Law: The square of a planet’s orbital period is proportional to the cube of the semimajor axis of the orbit.

Key Ideas Important Consequences: Excellent predictions for positions of planets Fit Tycho’s data Heliocentric model Measured ratios of distances in the solar system Measure one distance = get them all Showed what gravity does (Newton) Take these expectations when observing galaxies

After Copernicus Copernicus died in 1543, the year De Revolutionibus was published Astronomers became familiar with his work for its mathematics, even if they did not accept his cosmology But certain astronomers who read his work thought that heliocentrism was correct Galileo Kepler

Johannes Kepler ( ) Brilliant German Mathematician Staunch Copernican Convinced the Universe was governed by physical laws. Obsessed with finding harmony in the heavens. Had a genius for data analysis Inherited Tycho’s data & post as the Imperial Mathematicus in 1601.

The Motions of Mars Mars was the key to unlocking the secrets of planetary motion. Kepler began analyzing the orbit of Mars in It took him 4 years: Started by determining the orbit of the Earth using successive oppositions of Mars. Fit an off-center circle (i.e., not centered on the Sun) to the first 4 data points. A 5 th test data point did not fit by 8 arcminutes

Kepler's calculations of the orbits of the Earth & Mars, from the Astronomia Nova (1609)

Kepler listened to the data: Knew Tycho’s data were accurate to 1  2 arcminutes. But never off by as much as 8 arcminutes. Kepler questioned his assumptions: Forced to abandon uniform circular motion. Concluded Mars’ orbit was not a circle, but instead an ellipse with the Sun at one focus. Published results in 1609 (Astronomia Nova).

The 1 st Law of Planetary Motion The orbits of the planets are ellipses with the Sun at one focus. Ellipses are characterized by two numbers: Semimajor Axis (a): ½ of length of the longest axis Eccentricity (e): shape of the ellipse. Orbit of Mars: a = AU, e =

F2F2 F1F1 Semimajor Axis Center Focii

The 2 nd Law of Planetary Motion The line joining the Sun and the planet sweeps out equal areas in equal times. Planets move fastest at Perihelion Planets move slowest at Aphelion Kepler’s Second Law provides a geometric description of the change in speed. Completely eliminates epicycles.

T=0 d T=10 d T=0 d T=10 d Equal Areas in Equal Times

Let’s go to the animation….

The 3 rd Law of Planetary Motion The square of a planet’s orbital period is proportional to the cube of the semimajor axis of the orbit. Expressed Mathematically: P = Period in years, a = Semimajor axis in AU

The Astronomical Unit Distance between Earth and Sun Length of the semi-major axis of Earth’s orbit Number of kilometers = ?

The Third Law applies to all bodies orbiting the Sun: planets, comets, rocks, & spacecraft! Planeta (AU)P (yr)a3a3 P2P2 Mercury Venus Earth Mars Jupiter Saturn

Empirical Laws Kepler’s Laws are Empirical Laws: They describe how the planets move. They don’t explain why they move that way. Not yet Physical Laws: Kepler made a start, but he had incorrect ideas about forces. Kepler’s thinking was strongly motivated by his notions of universal harmony. Correct explanation had to wait until the work of Isaac Newton.

Kepler’s Law’s at Work!

Heliocentrism and Kepler Kepler’s belief that the Sun was the driving force in the Solar System was one of the reasons that he could figure out his laws. His laws are very Sun-centric. For example, 2 nd Law arose because he thought that the Sun’s rays pushed along the planets in their orbits – farther away = fewer rays = slower speeds

Important Consequences This is not the last we will hear about Kepler in this course Accuracy of predictions of positions of planets important for convincing people that heliocentrism was correct Third Law gave the ratios of the distances of planets from Sun – know one distance, know them all Laws apply to systems other than our own – except when they unexpectedly don’t…

Rudolphine Tables In 1627, Kepler published the Rudolphine Tables, based on his theory of how the planets moved around the Sun. Predicted positions clearly better than tables based on the models of Ptolemy or Copernicus Definitely noticed by astronomers & part of the reason for the gradual switch to heliocentrism

Rudolphine Tables

Kepler’s Third Law

Speeds in the Solar System

Kepler and Dark Matter As Kepler showed (and Newton explained), the speeds of the planets are lower for the most distant ones Diminishing force of gravity as distance from the largest mass increases We expect that should also be the case for stars orbiting the distant edges of galaxies But……

Spoiler Alert: Non-Keplerian speeds ahead!