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Chapter 3

The Origin of Modern

AstronomyMain goals:

• Understand the difference between geocentric and heliocentric cosmologies.

• Know the planetary configurations for the heliocentric cosmology.

• Understand how retrograde motion is explained using the Copernican heliocentric system.

• Define parallax.

• Know how Tycho Brahe showed that the 1572 supernova occurred in the distant heavens.

• Know Kepler’s three laws of planetary motion and be able to explain them.

• Describe how Galileo's telescopic observations of Jupiter and Venus supported a

heliocentric cosmology, and refuted the geocentric cosmology.

• State and understand Newton's three laws of motion.

• Know Newton's universal law of gravitation, and the shapes of orbits it predicts.

• Explain the Moon's role in producing tides on Earth, how often high tides occur, and the

requirements for spring and neap tides.

I. Astronomy before Copernicus

A. Aristotle and the Nature of the Earth

B. The Ptolemaic Universe

II. The Copernican Revolution

A. The Copernican Universe

III. The Puzzle of Planetary Motion

A. Tycho Brahe

B. Johannes Kepler

IV. Galileo Galilei

V. Isaac Newton

Outline

The Roots of Astronomy

• There are no written documents about the significance of stone and bronze age

monuments which seem to have an astronomical significance.

• First preserved written documents about ancient astronomy are from ancient Greek

philosophy.

• Greeks tried to understand the motions of the sky and describe them in terms of

mathematical (not physical!) models.

• Models were generally wrong because in general they were based on flawed “first

principles”, believed to be “obvious” and not questioned:a) Geocentric Universe: Earth at the Center of the Universe.

b) “Perfect Heavens”: Motions of all celestial bodies described by motions involving

objects of “perfect” shape, i.e., spheres or circles moving with constant speed. In the

Pythagorean norm, astronomy was a one of the mathematical arts.

Examples of early models (6-5th century BC):

Anaximander – universe as concentric cylinders, plurality of worlds;

Philolaus – accepted a moving Earth

Program: By the assumption of what uniform and orderly motions can the

apparent motions of the planets be accounted for? (Plato, 4th century BC)

a) Perfect Heavens (described by 55 spheres – idea inspired by Eudoxus).

a) An imperfect, changeable Earth in the center. Aristotle’s model was geocentric since

he couldn’t see any stellar parallax due to such a motion

Aristarchus (310-~230 BC) proposed a heliocentric model but Aristotle’s

reputation was so high that the (correct) idea was ignored

Examples of later models:

Eudoxus of Cnidus (409 – 356 B.C.), a pupil of Plato,

proposed the two spheres geocentric model of the universe:a) A spherical central Earth – geocentric model

b) The Earth is at the center of 27 nested spheres containing the

rotating celestial bodies: planets and fixed stars. These bodies

perform uniform circular motion.

Def: Parallax is the apparent change in position of a nearby object relative to the

distant background when the object is viewed from different perspectives. The

closer the object, the greater its parallax.

Def: Parallax is the apparent change in position of a nearby object relative to the

distant background when the object is viewed from different perspectives. The

closer the object, the greater its parallax.

Aristotle (384 – 322 B.C.), an even more famous pupil of

Plato, was a major authority of philosophy until the late middle

ages: he suggested that the universe can be divided into 2 parts:

Later Refinements (2nd century B.C.)

Deferent: orbit about a center (the

Earth was considered a bit off-center)

Epicycle: orbit about a point moving

along the deferent

Hipparchus (~190 -120 BC): Placed the Earth away from the centers

Ptolemy (83-178 AD): Added further refinements, including epicycles and deferents

Ptolemaic Universe

Epicycles: the Ptolemaic System

• Throughout Middle Ages, the Ptolemaic geocentric system was considered the

“standard model” of the Universe, and his book Almagest (about 150 AD) the main

astronomical reference, until the Copernican Revolution. It inspired the Alfonsine

Tables (1251) – most popular medieval astronomical tables

• The Ptolemaic system per se is not to be judged since Ptolemy was a great and

honest thinker of his times: its persistence should be blamed on those who later on

sustained and imposed it even when evidence started to pile up against it…

The epicycles were introduced to explain

the retrograde (westward) motion of planets

The Copernican Revolution

.icolaus Copernicus (1473 – 1543) revolutionized

astronomy (not necessarily purposely…):

• He proposed a heliocentric Universe (Sun in the Center) in

a book , De Revolutionibus Orbium Coelestium (1453)

• Copernicus was not the first to suggest a heliocentric

system:

Ex: See the Greek Aristarchus or the Indian Aryabhata.

• However, his ideas were the first to be incorporated into a

quasi-modern scientific mainstream.

• His model was not correct in its details (for instance it still

assumed uniform rotations and perfect heavens), but his

heliocentric hypothesis was. It inspired the Prutenic Tables

(1551) which replaced the Alfonsine tables

• It was a breakthrough which dethroned the geocentric

model forever and made way for a unified science for the

earthly and heavenly worlds

Copernicus’ .ew (and Correct) Explanation

for the Retrograde Motion of the Planets

• Recall that the retrograde (westward) motion of a planet

occurs when the Earth passes the planet

• It can be understood as a matter of perspective: since the

outer planets have longer orbital periods (local “years”)

than Earth they lag behind Earth for a portion of their

orbit

• This simple explanation in the

heliocentric system made

Ptolemy’s complex epicycles

unnecessary: an example of how

“Occam’s razor” works in

science…

• Albeit based on a correct hypothesis (heliocentric system), the Copernican model

was based on some false assumptions that made the model incorrect

• As a result, it couldn’t predict the motion of the planets better than the Ptolemaic

model: its subsequent success was due to the germinal observations and analysis of

the planetary motions throughout the 16-17th centuries by early scientists such as

Tycho Brahe, Johannes Kepler, Galileo Galilei and Isaac .ewton

• Let’s take a brief look at their contributions…

Tycho Brahe (1546-1601)

Tycho’s observatory:

Uraniborg

Tyco Brahe was one of the first observational astronomers.

• He collected data exceptionally good for his times but he

didn’t have the necessary conceptual tools to process it.

• One of his early observations, in 1572, lead to the discovery

of a “new star” which was in fact a dying star today called

Tycho’s supernova

• Brahe showed that this supernova had no measurable parallax

when viewed at different times of day, so it had to be in the

distant heavens: this was a startling idea since the Aristotelian

heavens were supposed to be unalterable

• He proposed a world model which was a combination

between the Ptolemaic and Copernican systems – the Tychonic

System:

a) Still geocentric (Earth in the center of a sphere of stars)

b) Sun and Moon orbit Earth; Planets orbit the sun.

c) For a while, it was adopted as the philosophically acceptable system before the final

victory of the heliocentric model

Johannes Kepler (1571 – 1630)

Johannes Kepler used the precise observational tables of Tycho Brahe to study

planetary motion mathematically.

• He found a consistent description by abandoning the old assumptions about the

perfection of the celestial motion. Thus, he came to the conclusion that

a) The planetary motion is not perfectly circular

b) The planets do not move with constant speed

• He concluded that planets move around the sun on elliptical paths, with non-

uniform velocities. Their motion is a subject to the laws of physics.

• Kepler’s work crystallized into three empiric laws of planetary motion.

• They were later proved by Newton as a token of the applicability of his theory of

universal gravitation.

• Albeit correct, Kepler’s laws were the byproduct of his rather mystical quest to

discover the divine plan for the geometry of the universe.

• However, Kepler was one of the forefathers of the scientific method.

Kepler’s Laws of Planetary Motion: 1st Law

1. The orbits of the planets are ellipses with the sun at one focus.1. The orbits of the planets are ellipses with the sun at one focus.

Eccentricity e = c/a ≤ 1

c

How to draw and

characterize an ellipse:

a

semi-major axis

Notice that, if e = 0, the ellipse

becomes a circle, the semi-major axis

becomes the radius and the two foci

merge into the center

Focus

Ex: Eccentricities of Ellipses

e = 0 (circle)

e = 0.1 e = 0.2 e = 0.4 e = 0.6

1) 2) 3) 4) 5)

still resemble a circle

Eccentricities of Planetary Orbits

Planetary orbits are virtually indistinguishable

from circles:

Earth: e = 0.0167Most extreme

example is Pluto:

e = 0.248

Mercury:

e = 0.206

2. A line from a planet to the sun sweeps over equal areas in equal intervals

of time.

2. A line from a planet to the sun sweeps over equal areas in equal intervals

of time.

Kepler’s Laws of Planetary Motion: 2nd Law

• Notice that, in order to sweep the areas in the same time interval the planet must

move faster between points A’-B’ (when it is closer to the sun) than between A-B

(further from the sun)

equal areasAssume that the arc A’-

B’ is traveled by the

planet in the same time

as the arc between A-B.

Then the blue areas are

equal.

3. A planet’s orbital period (P) squared is proportional to its average

distance from the sun (a) cubed:

3. A planet’s orbital period (P) squared is proportional to its average

distance from the sun (a) cubed:

Py2 ~ aAU

3 Py = period in years;

aAU = distance in AU

Kepler’s Laws of Planetary Motion: 3rd Law

Py

aAU

Ex: The relationship is represented

graphically as shown by the adjacent

curve. So, for instance, if a planet orbits

at a distance of 39 AU from the sun, it

needs about 200 Earth years to complete

one orbit.

Galileo Galilei (1594 – 1642)

• Galileo was one of the fathers of the modern view of science: his work marked the

transition from a faith-based “science” to an observation-based science.

• He paved the way toward an explanation for the planetary motions, or a celestial

mechanics – task accomplished later on by Isaac Newton

• He greatly improved on the newly invented telescope technology. (Caution:

Galileo did not invent the telescope!) He was the first to meticulously report

telescope observations of the sky to support the Copernican Model of the Universe.

• His book, Dialogo sopra i due massimi sistemi del mondo (1632), marked the

beginnings of scientifical rebuttals of unsustainable theories: Galileo proved wrong

the geocentric views using elements of the modern scientic method:

a) “thought experiments”

b) well directed experimental observations

• Not all Galileo ideas are correct, but he was incorrect due to the shortcomings of

the technology and scientific means of his times, not due to blind faith or ideology.

Major Discoveries of Galileo (1)

(What he really saw…)

Jupiter and its

system of moons

hinted to the

possibility of other

centers of attraction

than Earth

Moons of Jupiter (4 Galilean moons)

Rings of Saturn

Surface structures on the moon:

• first estimates of the height of mountains on the moon

• concluded that the moon is not perfect and its terrain is similar to the one on

Earth


Sun spots

• proved that the sun is also not perfect

• the motion of the spots suggested that the sun is spinning


Phases of Venus (including “full Venus”)

• proved that Venus orbits the sun, not the Earth!


• Notice that, had the Ptolemaic model been correct, only the new and crescent phases would’ve been visible from Earth.

• Galileo observed all phases: this a typical example of how an experimental observation discriminates between two models

Isaac .ewton (1643 - 1727)

• Newton built on the results of his predecessors and offered a mechanics unique

for the earthly and heavenly bodies unsurpassed for centuries

• He added modern physical interpretations and formal consistency to the

mathematical descriptions of astronomy by Copernicus, Galileo and Kepler: for

instance he used his new mechanics to explain and derive Kepler’s empirical laws

from the laws of motion and universal gravitation

Major achievements:

• Newton was one of the inventors of calculus as a necessary tool to solve

mathematical problems related to motion

• He stated in an integrated context the three laws of motion

• Discovered the universal law of gravitational attraction

• He had many contributions to other fields of Physics such as optics

Velocity and Acceleration

• In the 16th century, appeared a new formalized understanding

of physics, based on a quantitative approach

• Here are two formalized physical quantities:

Velocity (v) is the change of position with time.

a

v

Acceleration (a) is the change of a body’s velocity with time:

change in position

timev =

change in velocity

timea =

The position changes so the object

has a velocity

The velocity increases so the

object has an acceleration

Ex: Freely falling object

.ewton’s Laws of Motion: 1st Law

1. A body continues at rest or in uniform motion in a straight line

unless acted upon by some net force.

1. A body continues at rest or in uniform motion in a straight line

unless acted upon by some net force.

Ex: An astronaut floating freely in space will continue to

float forever in a straight line unless some external force

accelerates him.

• With Newton’s mechanics, the physical sciences entered into a new era. His laws

of motion and the law of universal gravitation are central in the understanding of

celestial mechanics since the same laws that govern things on Earth drive the

celestial objects. Here are the laws:

• This is identical with Galileo’s idea about inertia: the

objects don’t need a force to move. They actually move

identically as long there is no force…

.ewton’s Laws of Motion: 2nd Law

So, a force is necessary in order to change the motion of an object, and

the change (effect) will be proportional to the force (cause).

2. If a net force acts on an object, then the object accelerates with an

acceleration proportional to the net force.

2. If a net force acts on an object, then the object accelerates with an

acceleration proportional to the net force.

Ex: The force of your palm acting on an object initially at rest will change its velocity

from zero to finite values.

• As long as the force continues to act the mass will move faster and faster.

• If the force is removed, the mass will continue to move with the velocity at the moment

of the removal (if there is no other force to slow it down)

Ex: A person pushing with a force into a wall will be pushed

by the wall with a force equal in magnitude and opposite in

direction.

.ewton’s Laws of Motion: 3rd Law

Therefore, any force on an object is the manifestation of the presence of another

object which in turn will fill the same force.

The interaction between two different bodies can be always represented

by two forces: body A acts on the body B with a force (action) equal in

magnitude and opposite in direction with the force (reaction) acted by

B on A.

The interaction between two different bodies can be always represented

by two forces: body A acts on the body B with a force (action) equal in

magnitude and opposite in direction with the force (reaction) acted by

B on A.

The Universal Law of Gravity

Any two bodies attract each other through gravitation: a force

proportional to the product of their masses and inversely proportional

to the square of the distance between them.

Any two bodies attract each other through gravitation: a force

proportional to the product of their masses and inversely proportional

to the square of the distance between them.

2F G

mM

r=

r

mM

F F

Note the presence of Newton’s

3rd law: the masses attract each

other with forces forming pairs

action-reaction

Ex 1: the Earth attracts the Moon

and also the Moon attracts the

Earth.

Ex 2: Your “weight” is the

expression of the attraction that the

Earth acts on you, but you also

attract the Earth with the same

force!

Universal constant of gravity.

• One of the first applications that Newton found for his mechanics was to explain

Kepler’s laws. Moreover, he was able to analyze systematically the orbital motions

in the solar system. For instance:

The Explanatory Power of .ewtonian Mechanics

2. Due to the mutual attraction, the objects orbiting each other have to revolve along

trajectories around their center of massEx: The planets do not orbit around the center of the sun but around the center of mass which

is very close to the center of the sun since its mass is much larger than the planetary masses

Ex: The motion of the Moon orbiting Earth is

very similar with the motion of a mass

connected by a string and rotated in a circle

• The difference is that the force that keeps the

Moon on the orbit is the gravitational pull from

the Earth

• The Moon doesn’t fall on Earth since the force

just changes the direction of its motion along a

trajectory that keeps missing the Earth

Gravity

1. The gravitational attraction that determines the weight of the objects on Earth also

keeps the objects on their orbits in the solar system

Too slow⇒ object falls back down to Earth

Escape speed ⇒ object escapes Earth’s

gravity along parabolic or hyperbolic paths

Satellite speed ⇒ object orbits the Earth

along circular or elliptical paths

Understanding Orbital Motion

• All objects in the universe (from dust to galaxies) are under the influence of a

gravitational attraction. However, their trajectory depends on their speed and the

magnitude of the attraction leading to closed or open profiles.

• In order to stay on a closed orbit, an object has to move within a certain range of

velocities in the vicinity of a center of attraction.

• For instance, what determines the difference between two objects launched from

Earth such that one becomes a satellite and the other one a star ship?

• Imagine that you launch an object with a certain initial velocity.

• So, in order to stay on the orbit, a satellite must move with a speed in between the

satellite speed and the escape speed, while a spacecraft must exceed the escape speed

Geosynchronous Orbits

• An object can be launched with a

speed such that it rotates about the Earth

with the same speed as the planetary

spin

• This will keep the satellite above the

same location on Earth: the respective

orbit is then called geosynchronous

since the orbital period is synchronized

with that of Earth’s axial rotation

The tides are caused by the different gravitational attraction exerted by the moon

onto the water on Earth on different sides of the planet

• Consider the gravitational attraction

by the moon on different points of the

Earth: on the water facing the moon,

the center of the Earth and the water

on the other side of Earth

1. The water bulges away from the

moon on the far side since the Earth is

pulled at its center by a larger force –

it is like the planet is pulled out of its

oceans

2. On the other hand, excess gravity pulls water toward the moon on the near side

• As the Earth rotates on its axis once per day, a person on the earth rotates through

these tidal bulges twice per day, causing two high tides and two low tides per day.

• The tidal forces are very common in the universe and can explain a multitude of

astronomical phenomena, not only at the scale of a solar system but at much larger

galactic scales as well.

Other Explanations by .ewtonian Mechanics: The Tides

Spring and .eap Tides

• The sun is also producing

tidal effects, about half as

strong as the moon.

• Near full and new moon,

those two effects add up to

cause spring tides.

• Near first and third quarter,

the two effects work at a right

angle, causing neap tides.

Spring tides

Neap tides

• The "line of tidal bulges" on Earth are slightly ahead of the moon in its orbit as

shown in this figure.

• This is due to the friction with the ocean beds which carry the bulges ahead. This

leads to:

1. The slowing down of the Earth’s spinning speed (the day grows by 0.0023

s/century)

2. The slow increase in the moon’s orbital radius (~3.8 cm/year)

Other Tidal Effects

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