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The Development ofModern Astronomy

It begins with the introduction of the Sun-centered Solar System by Copernicus, and concludes with Newton's synthesis of the laws of motion in the heavens and the Earth, and Einstein's revision of Newton's ideas in the Relativity Theory.

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The Earth-centered Universe of Aristotle and Ptolemy held sway on Western thinking for almost 2000 years. Then, in the 16th century a new idea was proposed by the Polish astronomer Nicolai Copernicus (1473-1543).

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The Heliocentric System

Copernicus proposed that the Sun, not the Earth, was the center of the Solar System. Such a model is called a heliocentric system.

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In this new ordering the Earth is just another planet (the third outward from the Sun), and the Moon is in orbit around the Earth, not the Sun. The stars are distant objects that do not revolve around the Sun. Instead, the Earth is assumed to rotate once in 24 hours, causing the stars to appear to revolve around the Earth in the opposite direction

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Retrograde Motion and Varying Brightness of the Planets Explained

1. The planets in such a system naturally vary in brightness because they are not always the same distance from the Earth.

2. The retrograde motion could be explained in terms of geometry and a faster motion for planets with smaller orbits, as illustrated

in the following animation.

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The Copernican model, with it assumption of uniform circular motion, still could not explain all the details of planetary motion on the celestial sphere without epicycles. The difference was that the Copernican system required many fewer epicycles than the Ptolemaic system because it moved the Sun to the center.

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The Copernican Revolution

3 incorrect ideas held back the development of modern astronomy from the time of Aristotle until the 16th and 17th centuries

1) the assumption that the Earth was the center of the Universe

2) the assumption of uniform circular motion in the heavens

3) the assumption that objects in the heavens were made from a perfect, unchanging substance not found on the Earth.

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Copernicus challenged assumption 1, but not assumption 2.

The Copernican model implicitly questions the third tenet that the objects in the sky were made of special unchanging stuff. Since the Earth is just another planet, there will eventually be a natural progression to the idea that the planets are made from the same stuff that we find on the Earth.

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His ideas remained rather obscure for about 100 years after his death. But, in the 17th century the work of Kepler, Galileo, and Newton would build on the heliocentric Universe of Copernicus and produce the revolution that would sweep away completely the ideas of Aristotle and replace them with the modern view of astronomy and natural science.

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Been There, Done That: Aristarchus of Samos

The idea of Copernicus was not really new! A sun-centered Solar System had been proposed as early as about 200 B.C. by Aristarchus of Samos (Samos is an island off the coast of what is now Turkey). However, it did not survive long under the weight of Aristotle's influence and "common sense":

history tends to forget that he came to this conclusion about 1,750 years before Copernicus did

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Tycho (1546 - 1601)

The Life and Times of Tycho Brahe

He allegedly challenged a fellow student to a duel with swords in a dispute over who was the better mathematician. Brahe's nose was partially cut off, and he was said to wear a gold and silver replacement

He fell out of favor when a new King came to power in 1588, and moved to Prague shortly thereafter

This is of great historical significance because this move would eventually make Brahe's data available to Kepler Brahe is thought to have died when he contracted a urinary infection while attending a banquet hosted by a baron in Prague in which he drank extensively but felt that etiquette prevented him from leaving the table to relieve himself before the host left.

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King Frederick of Denmark gave him the island Hven

 where he built (1576) his observatory "Uraniborg"    

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For 20 years he made the most accurate observations possible with the naked eye

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Summary of Brahe's Contributions

1. He made the most precise observations that had yet been made by devising the best instruments available before the invention of the telescope.

2. His observations of planetary motion, particularly that of Mars, provided the crucial data for later astronomers like Kepler to construct our present model of the solar system.

3. He made observations of a supernova and declared it a stellaractivity and not an atmospheric activity.

4. Brahe made careful observations of a comet in 1577 and proved that it was moving in space and not the atmosphere

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5. He made the best measurements that had yet been made in the search for stellar parallax

Brahe did not believe that the stars could possibly be so far away and so concluded that the Earth was the center of the Universe and that Copernicus was wrong.

6. Brahe proposed a model of the Solar System that was intermediate between the Ptolemaic and Copernican models (it had the Earth at the center).

It proved to be incorrect !

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Brahe's Data and Kepler

The next great development in the history of astronomy was the theoretical intuition of Johannes Kepler (1571-1630), a German who went to Prague to become Brahe's assistant.

Kepler and Brahe did not get along well. Brahe apparently mistrusted Kepler, fearing that his bright young assistant might eclipse him as the premiere astonomer of his day. He therefore let Kepler see only part of his voluminous data.

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He set Kepler the task of understanding the orbit of the planet Mars, which was particularly troublesome. It is believed that part of the motivation for giving the Mars problem to Kepler was that it was difficult, and Brahe hoped it would occupy Kepler while Brahe worked on his theory of the Solar System

In a supreme irony, it was precisely the Martian data that allowed Kepler to formulate the correct laws of planetary motion, thus eventually achieving a place in the development of astronomy far surpassing that of Brahe.

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Kepler and the Elliptical Orbits

Unlike Brahe, Kepler believed firmly in the Copernican system. Kepler’s work with Mars led to the discovery that planets move in elliptical not circular orbits.

Some Properties of Ellipses

1. For an ellipse there are two points called foci (singular:

focus) such that the sum of the distances to the foci from any

point on the ellipse is a constant. In terms of the

diagram shown to the left, with "x" marking the location of the

foci, we have the equation a + b = constant

that defines the ellipse in terms of the distances a and b.

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2. The amount of "flattening" of the ellipse is termed the eccentricity

a. The orbits of the planets are ellipses but the eccentricities are so small for most of the planets that they look circular at first glance.

b. Pluto and Mercury are exceptions: their orbits are sufficiently eccentric that they can be seen by inspection to not be circles.

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3. The long axis of the ellipse is called the major axis, while the short axis is called the minor axis (adjacent figure). Half of the major axis is termed a semimajor axis. The length of a semimajor axis is often termed the size of the ellipse.

It can be shown that the average separation of a planet from the Sun as it goes around its elliptical orbit is equal to the length of the semimajor axis. Thus, by the "radius" of a planet's orbit one usually means the length of the semimajor axis.

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The Laws of Planetary Motion

Kepler obtained Brahe's data after his death despite the attempts by Brahe's family to keep the data from him in the hope of monetary gain. There is some evidence that Kepler obtained the data by less than legal means; it is fortunate for the development of modern astronomy that he was successful.

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Kepler's First Law

The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.

The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse).

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Kepler's Second Law:

II. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse

The line joining the Sun and planet sweeps out equal areas in equal times, so the planet moves faster when it is nearer the Sun.

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Kepler's Third Law:

III. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes:

In this equation P represents the period of revolution for a planet and R represents the length of its semimajor axis. The subscripts "1" and "2" distinguish quantities for planet 1 and 2 respectively. The periods for the two planets are assumed to be in the same time units and the lengths of the semimajor axes for the two planets are assumed to be in the same distance units.

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Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same.

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Galileo: the Telescope & the Laws of Dynamics

Galileo Galilei (1564-1642) was a pivotal figure in the development of modern astronomy, both because of his contributions directly to astronomy, and because of his work in physics and its relation to astronomy. He provided the crucial observations that proved the Copernican hypothesis, and also laid the foundations for a correct understanding of how objects moved on the surface of the earth (dynamics) and of gravity.

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The Telescope

Galileo did not invent the telescope

. His little telescope was poorer than even a cheap modern amateur telescope, but what he observed in the heavens rocked the very foundations of Aristotle's universe and the theological-philosophical worldview that it supported. It is said that what Galileo saw was so disturbing for some officials of the Church that they refused to even look through his telescope; they reasoned that the Devil was capable of making anything appear in the telescope, so it was best not to look through it.

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Three Observations That “Rocked” Astronomy:

1. Sunspots

2. The Moons of Jupiter

3. The Phases of Venus

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Sunspots

Galileo observed the Sun through his telescope and saw that the Sun had dark patches on it that we now call sunspots (he eventually went blind, perhaps from damage suffered by looking at the Sun with his telescope). Furthermore, he observed motion of the sunspots indicating that the Sun was rotating on an axis.

"blemishes" rotate

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The Moons of Jupiter

Galileo observed 4 points of light that changed their positions with time around the planet Jupiter. He concluded that these were objects in orbit around Jupiter.

One of the arguments against the Copernican system (and the original heliocentric idea of Aristarchus) had been that if the moon were in orbit around the Earth and the Earth in orbit around the Sun, the Earth would leave the Moon behind as it moved around its orbit.

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The Phases of Venus

Galileo used his telescope to show that Venus went through a complete set of phases, just like the Moon. This observation was among the most important in human history, for it provided the first conclusive observational proof that was consistent with the Copernican system but not the Ptolemaic system.

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The Accomplishments of Newton

(1642-1727)

We shall concentrate on three developments

1) Newton's Three Laws of Motion

2) The Theory of Universal Gravitation

3) The demonstration that Kepler's Laws follow from the Law of Gravitation.

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The Great Synthesis of NewtonKepler had proposed three Laws of Planetary motion based on the systematics that he found in Brahe's

data.

1. The laws were only applied to planets.

2. The Laws worked, but no one knew why.

The Problem!

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1. Newton demonstrated that the motion of objects on the Earth could be described by three new Laws of motion

2. Kepler’s Laws of planetary motion were special cases of His Three Laws.

3.Neton demonstrated that Kepler was only partly correctAnd corrected his mistakes.

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Newton's First Law of Motion:

I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

This we recognize as essentially Galileo's concept of inertia, and this is often termed simply the "Law of Inertia".

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Newton's Second Law of Motion:

II. The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma. Acceleration and force are vectors (as indicated by their symbols being displayed in slant bold font); in this law the direction of the force vector is the same as the direction of the acceleration vector.

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Newton's Third Law of Motion:

III. For every action there is an equal and opposite reaction.

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What Really Happened with the Apple?

The apple is accelerated, since its velocity changes from zero as it is hanging on the tree and moves toward the ground. Thus, by Newton's 2nd Law there must be a force that acts on the apple to cause this acceleration. Let's call this force "gravity",

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Sir Isaac's Most Excellent Idea

Now came Newton's truly brilliant insight: if the force of gravity reaches to the top of the highest tree, might it not reach even further; in particular, might it not reach all the way to the orbit of the Moon!

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If we increase the muzzle velocity of an imaginary cannon, the projectile will travel further and further before returning to earth. Newton reasoned that if the cannon projected the cannon ball with exactly the right velocity, the projectile would travel completely around the Earth, always falling in the gravitational field but never reaching the Earth, which is curving away at the same rate that the projectile falls. That is, the cannon ball would have been put into orbit around the Earth. Newton concluded that the orbit of the Moon was of exactly the same nature

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the Moon continuously "fell" in its path around the Earth because of the acceleration due to gravity, thus producing its orbit.

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By such reasoning, Newton came to the conclusion that any two objects in the Universe exert gravitational attraction on each other, with the force having a universal form:

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The Center of Mass for a Binary System

Newton, largely as a corollary of his 3rd Law, demonstrated that the sun’s position actually was more symmetrical than Kepler imagined and that the Sun does not occupy a privileged postion; in the process he modified Kepler's 3rd Law.

The center of mass is familiar to anyone who has ever played on a see-saw

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Newton's Modification of Kepler's Third Law

Because for every action there is an equal and opposite reaction, Newton realized that in the planet-Sun system the planet does not orbit around a stationary Sun. Instead, Newton proposed that both the planet and the Sun orbited around the common center of mass for the planet-Sun system. He then modified Kepler's 3rd Law to read,

Which turns out to beWith a small correction

Where P = T = period or orbit

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Two Limiting Cases

1. If one object is so massive that the center of mass between two object is in the larger mass. This would be like earth and the sun.

2. If two objects have about the same mass they both revolve around their center of mass. This would be like a binary star system.

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Weight and the Gravitational Force We have seen that in the Universal Law of Gravitation the crucial quantity is mass. In popular language mass and weight are often used to mean the same thing; in reality they are related but quite different things.

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What we commonly call weight is really just the gravitational force exerted on an object of a certain mass.

Mass and Weight

Mass is a measure of how much material is in an object, but weight is a measure of the gravitational force exerted on that material in a gravitational field; thus, mass and weight are proportional to each other, with the acceleration due to gravity as the proportionality constant.

Therefore, your mass would be the same on earth as it ison the moon but your weight would decrease by 1/6because the gravity of earth is six times greater thanthe moons gravity.