History of Modern Astronomy Chapter 5

Topics Major characters in the development of our understanding of the motions of planets. Kepler’s three laws of planetary motion Newton’s three laws of motion and the law of gravitation

A little drama Characters in the great drama Claudius Ptolemy (140) Nicolaus Copernicus (1473-1543) Tycho Brahe (1546-1601) Johannes Kepler (1571-1630) Galileo Galilei (1564-1642) Isaac Newton (1642-1727)

Aristotle’s shoes It seems natural that our first hypothesis regarding the “structure” of the Solar System would be geocentric. Being more philosophical and less empirical, we would hope to see harmony and perfection in the heavens that fit our philosophy--thus, the motions of bodies are perfect circles.

Then came Ptolemy A good theory should explain what is observed and be able to make predictions. Planets move in circles called epicycles. The center of the epicycle moves in a circle called a deferent. To make theory match prediction, Earth isn’t exactly at the center of the deferent. The test of all knowledge is measurement Ptolemy’s theory explained the retrograde motion of the planets Predicted future locations of the planets

1400 years later - heliocentric idea Copernicus, for philosophical reasons, sought to explain the retrograde motion of planets using a heliocentric solar system. (animation) He still assumed perfect circles for the orbits of planets (with the Sun at the center of the orbits). He could calculate the relative distances to the planets the orbital periods of the planets Predictions of the future positions of planets were not much better than those from the Ptolemaic model

We need better data! Tycho Brahe had ideas for a new model but recognized the need for more precise measurements. He devoted his life to making more precise measurements of the positions of stars and planets. He built the first modern observatory He amassed records of planetary positions from 1576 to 1591 His observations were 2.5 times more accurate than any previous records

Finding a needle in a haystack Kepler believed the Copernican model and sought to prove that it was correct using Brahe’s data for the positions of the planets. He found that Planets orbit in elliptical paths (not circles!) with the Sun at one focus of the ellipse. A line from the Sun to a planet will sweep out the same area in a certain time interval, regardless of where the planet is in its path. The ratio of the (period)2 to (semi-major axis)3 was the same for every planet. He described the planets’ orbits, but could they be explained? Kepler answered “What?” but didn’t know “Why?”

Standing on the shoulders of giants Isaac Newton formulated three laws of motion and a law of gravitation. This model for understanding motion (how motion is related to forces) and gravitation explained Kepler’s three laws. When “Why?” matches “What?” (theory matches observation), we must reexamine our dearly held beliefs. This happened again in 1911 with Einstein’s publication of the General Theory of Relativity an entirely different explanation of gravity explained phenemena that Newton’s law of gravitation could not explain. has been verified by experiment to this day

Kepler’s laws of planetary motion details

Kepler’s first law planet’s orbit the Sun in ellipses, with the Sun at one focus. the eccentricity of the ellipse, e, tells you how elongated it is. e=0 is a circle, e<1 for all ellipses e=0.02 e=0.4 e=0.7

Experiment and theory

eccentricity of the planets Mercury 0.206 Saturn 0.054 Venus 0.007 Uranus 0.048 Earth 0.017 Neptune Mars 0.094 Pluto 0.253 Jupiter

Kepler’s second law The line joining the Sun and a planet sweeps out equal areas in equal time intervals. As a result, planets move fastest when they are near the Sun (perihelion) and slowest when they are far from the Sun (aphelion). simulation 1

If it sweeps out equal areas in equal times, does it travel faster or slower when it is far from the Sun?

If is sweeps out equal areas in equal times, does it travel faster or slower when far from the Sun?

If is sweeps out equal areas in equal times, does it travel faster or slower when far from the Sun? Same Areas

Kepler’s Third Law Period of a planet, P Average distance from the Sun (semimajor axis of ellipse), R P2/R3 = 4p2/(G(m1+m2)) Approximately, P2earth/R3earth = P2planet/R3planet Sometimes we use Earth-years and Earth-distance to the Sun (1 A.U.) as units. The constant of proportionality depends on the mass of the Sun--and that’s how we know the mass of the Sun. We can apply this to moons (or any satellite) orbiting a planet, and then the constant of proportionality depends on the mass of the planet.

Practice While gazing at the planets that are visible with the naked eye, you tell a friend that the farther a planet is from the Sun, the longer its solar year is. Your friend first asks what a solar year is. After explaining that it’s the time required for a planet to return to its same position relative to the Sun, your friend then asks, “Why does it take longer for the outermost planets to orbit the Sun?” What is your reply?

Practice What is the best method for determining the mass of Astronomical objects? Kepler’s Third Law For distant stars, this doesn’t work very well. Fortunately, there is a relationship between mass and brightness that will help us out.

Newton’s laws of motion Newton’s first law an object will have a constant velocity (constant speed, moving in a straight line) unless a net force acts on it Newton’s second law the acceleration of an object is proportional to the net force on the object divided by its mass Newton’s third law if object A exerts a force on object B, B exerts a force of equal magnitude back on A

Newton’s law of gravitation mass attracts mass the magnitude of the force of attraction is proportional to the product of their masses and the inverse of the square of the distance between them ForceB on A ForceA on B mA mB

Gravitational force and distance If the bodies are twice as far apart, the gravitational force of each body on the other is 1/4 of their previous values. This is called an “inverse-square law.” ForceB on A ForceA on B mA mB

Practice The Earth exerts a gravitational force on an orbiting satellite. Use Newton’s third law to compare the force of the satellite on the Earth. Draw a picture similar to the ones I drew for object A and object B. According to Newton’s second law, compare the accelerations of the satellite and Earth as a result of their interaction.

Deriving Kepler’s laws Newton’s law of gravitation, and Newton’s second law (net force = mass x acceleration) can be used to derive Kepler’s three laws of planetary motion.

Summary Understand the importance of experiment. when theory does not explain measurements, a new hypothesis must be developed; this may require a whole new model (a way of thinking about something). know why the geocentric view was abandoned. know what experiments verified the heliocentric view. Understand the roles of the “characters” in the revolution from a geocentric to a heliocentric model. Understand Kepler’s three laws of planetary motion these described the planet’s motions Understand Newton’s law of gravitation and the three laws of motion these explain why Kepler’s three laws are “true”