Science 3210 001 : Introduction to Astronomy Lecture 3 ...

Science 3210 001 : Introduction to Astronomy

Lecture 3 : Planetary Orbits, Physics of Motion,

Matter, and Light

Robert Fisher

Items

! Course Webpage -- New Homework

! Questions

! Far Out Friday at Adler Planetarium -- March 9, 2008

Review of Lecture 1

! Astronomy is an ancient subject, passed down from Greek to

Islamic scholars, and transmitted back to the west.

! Our systems of thought evolve with time at an almost

imperceptibly slow pace, and continue to do so today.

! The universe is thought to have begun with a big bang, and is

expanding.

! The cosmic calendar varies over fantastically-long timescales.

We are very recent newcomers onto the cosmic scene.

! We are all stardust.

Review of Lecture 2

! The motion of the night sky can be described by an idealized

celestial sphere which rotates about the Earth once per day.

! The moon revolves around the Earth once per month.

! The sun appears to move inclined with respect to the distant stars

due to the earth’s tilt, by 23 degrees.

! The motion of the planets is more complex -- they appear to

wander against the distant stars, and may even appear to stop

and move backwards in an effect known as retrograde motion.

Today’s Lecture

! I) Planetary Motion

! A) Tycho Brahe

! B) Kepler and Kepler’s Laws

! II) Physics of Motion

! A) Galileo and the Physics of Kinematics

! B) Newton and Newton’s Laws of Motion

! II) Physics of Matter and Light

Planetary Motion

Tycho Brahe (1546 - 1601)

! Tycho Brahe conducted the most accurate measurements of the

stars, planets, and comets in the pre-telescopic era

! In 1572, while still a young astronomer, something absolutely

wonderful happened…

Tycho Supernova Remnant (1572) Imaged by

Chandra Space Observatory in X-Rays in 2005

Computer Simulation of an Exploding Star

Kepler Supernova Remnant (1602) Imaged by

Chandra Space Observatory in X-Rays in 2006

Johannes Kepler (1571 - 1630)

! Kepler was a talented mathematician and astronomer and contemporary

of Galileo. Hired by Tycho as an assistant, he inherited Tycho’s

observations upon his death and used them to formulate his laws of

planetary motion.

! “For if I thought the eight minutes in longitude were unimportant, I could make a

sufficient correction… Now, because they could not be disregarded, these eight

minutes alone will lead us along a path to the reform of the whole of Astronomy,

and they are the matter for a great part of this work.” -- Kepler

! Kepler constructed a geometric model of the solar system, based

on the fact that there exist precisely five regular, platonic solids.

! Ancient Pythagorean idea that the planets created a music of

their own -- the “harmony of the spheres” -- that only Pythagoras

himself could hear

Kepler’s Platonic Solids Model of the Solar

System

Kepler’s Platonic Solids Model of the Solar

System

! Circumscribing each of the five solids with a sphere, Kepler

created a model with six concentric spheres. Each of these six

spheres coincided with the orbit of one of the known planets.

Kepler’s Geometric Model of the Solar System

! Kepler’s ideas based on the harmony of the spheres may appear

as a quaint idea to us today

! This illustrates the great difficulty which the earliest scientists had

in separating their nascent scientific ideas from ancient pre-

scientific ones

! Despite this, the apparent success of his geometric model

helped inspire and motivate him through the long difficult work

(nearly 30 years!!) that was required to analyze Tycho Brahe’s

data for the planetary orbits

Kepler’s First Law of Planetary Motion

! Kepler noted that the orbit of Mars was well-fit by an ellipse.

! His first law of planetary motion states that all planets move in ellipticalorbits, with the sun at one focus.

! He conjectured that the same law applied to the other planets as well.

Ellipses are Conic Sections

Kepler’s First Law of Planetary Motion

! The amount by which an ellipse differs from a circle is

characterized by its eccentricity -- ranging from zero for an exact

circle, to 1 to a highly elongated ellipse.

Kepler’s First Law of Planetary Motion

! All of Kepler’s laws are essentially empirical -- they describe the

properties of planetary orbits extremely well, but do not explain

why they have these particular sets of properties

! The first principles which explain Kepler’s laws were only

uncovered by Newton much later

Kepler’s Second Law of Planetary Motion

! Kepler noted that when further from the sun, Mars appeared to

move more slowly than when closer to the sun.

! He quantified this effect by his second law : planetary orbits

sweep out equal areas in equal times.

Modern Interpretation of Kepler’s Second Law

! Kepler’s second law is a direct consequence of the conservation

of angular momentum.

! If we imagine a ballerina moving her arms inward while spinning

on a frictionless ice surface, she will tend to spin faster and faster.

! Similarly, as a planet moves in closer to the sun, it will rotate

more quickly than when further out.

Kepler’s Third Law of Planetary Motion

! According to Kepler’s Second Law, the further a single planet is

from the sun, the slower it moves in its orbit.

! Kepler’s third law (sometimes referred to as his “Harmonic Law”)

states that the square of a planet’s period P (in years) is equal to

the cube of its semi-major axis a (in astronomical units).

! Mathematically, Kepler’s Third Law states P2 = a3

! Kepler thought of this law as a realization of the ancient

Pythogorean vision of the “Harmony of the Spheres”

Musical Realization of the Harmony of the

Spheres

! “Upon each of its circles stood a siren who was carried around itsmovements, uttering the concords of a single scale.” -- Plato

! Jazz musician and Yale professor Willie Ruff collaborated with geologistJohn Rodgers to produce a realization of the music of the spheres.

! The actual period of Pluto’s orbit is 248 years, so the actual length of asingle “track” of the music of the spheres is far greater than a humanlifespan.

! Ruff and Rodgers contracted the duration of Pluto’s orbit to about 20minutes in duration, which also upshifted the frequencies of the innerplanets to audible frequencies.

! The outer three planets remained beneath the range of audible tones --they are played with rhythms rather than musical tones.

Galileo Galilei (1564 - 1642)

! The quantitative description of motion is the study of kinematics

! Physics as we know it today began with Galileo’s experiments of

the motion of falling bodies, which he recounted in perhaps the

first popular science book ever -- Two New Sciences

The Aristotelean World View

! Aristotle held that the speed of a falling object was directly

proportional to weight -- a heavier body falls faster than a lighter

body

! This view was colored by the influence of atmospheric friction

(imagine a rock and a feather), but even as a description of such

motions, he did not get it quite right -- the atmospheric friction

exerted on a body depends not on a body’s weight, but rather on

its surface area

! The ancient Greeks were brilliant geometers and logicians, but

still lacked the conceptual machinery to create natural laws of

motion which include both cause and effect -- this would have to

wait until Sir Isaac Newton in the 17th century

The Two New Sciences

! Two New Sciences reveals the flourishing of science, both in the

serious questioning of Aristotelean beliefs by experiment, and in

establishing new principles to take their place. Galileo describes

these developments in a dialogue between three characters :

Sagriedo, Simplicio, and Salviati.

! Sagriedo : “I greatly doubt that Aristotle ever tested by

experiment whether it be true that two stones, one weighing

ten times as much as the other, if allowed to fall, at the same

instant, from a height of, say, 100 cubits, would so differ in

speed that when the heavier had reached the ground, the

other would not have fallen more than 10 cubits.”

The Two New Sciences

! “Simplicio : His language would seem to indicate that he had tried the

experiment, because he says: We see the heavier; now the word see

shows he had made the experiment.

! Sagriedo : But I, Simplicio, who have made the test, can assure you that

a cannon ball weighing one or two hundred pounds, or even more, will

not reach the ground by as much as a span ahead of a musket ball

weighing only half a pound, provided both are dropped from a height of

200 cubits.”

An Aside on Idealizations

! Physicists use many idealizations when thinking about nature --

frictionless surfaces, perfect spheres…

! While these idealizations do not exist in nature, they provide a

first approximation to the real world around us.

! Even more importantly, they provide a way of extracting what is

essential in a problem from what is not, and allow us to arrive at

general conclusions to the fundamental principles of how the

world works.

Spherical Cow Jokes…

Galileo’s Experiments on Motion

! Galileo conceived of a series of experiments which allowed him to

determine the basic physics of motion

Galileo’s Experiments on Motion

! The genius of this construction allowed him to realize that, in the

absence of friction and external forces, a moving body will

continue to move in the same direction and with the same speed

Frame of Reference

! Galileo also was perhaps the first scientist to clearly elucidate

why it can be that the Earth is moving around the sun, and yet we

do not feel the effects of its motion.

! Consider the effect of dropping a ball from the highest mast on

board a tall sailing ship -- from the standpoint of someone on the

ship, will it fall vertically, or hit the deck towards the aft side?

Galileo Movie

Conservation Laws

! With this series of experiments, Galileo uncovered one of the first

fundamental principles of physics -- that the motion of a body

(what we would call momentum today) remains constant in time

! These conservation laws are in a sense “The Constitution” of

physics; as new discoveries are made, their meaning is expanded

and amended, but the fundamental principles remain same

Conservation Laws

! Conservation of Momentum. The net total momentum of a

closed system is conserved in the absence of external forces.

! Conservation of Energy. The total energy of a closed system is

conserved.

! Conservation of Angular Momentum. The total amount of

rotation, or angular momentum, of a system, in the absence of

external torques is conserved.

! Conservation of Mass. The sum total of the mass in the system

is conserved… nearly so. In everyday life, this is almost exactly

true, but nuclear interactions can change the total mass of a

system slightly.

A Few Entries in the Dictionary of Physics

! The amount of matter in a body is measured by its mass. This

amount is an intrinsic property of that body, and is the same no

matter where it is measured.

! The force exerted on a body by gravity is its weight. A body’s

weight will depend on where it is measured -- a kilogram of

feathers has more weight on Earth than it does on the moon.

A Few More Entries in the Dictionary of Physics

! The motion of a body is specified by its velocity.

! Velocity has both a direction and a magnitude. For instance, a car

may be traveling 60 miles per hour in an eastwards direction.

! The magnitude of the velocity of a moving body is its speed. The

speed of the car in the previous example is 60 miles per hour.

! The rate of change of velocity of a body is its acceleration.

Example : Circular Motion

Isaac Newton (1643 - 1747)

! The discoveries of Tycho, Kepler, and Galileo culminated in the

work of Isaac Newton

! “If I have seen a little further, it is by standing on the shoulders of

giants.” -- Newton in a letter to fellow scientist Hooke

! Newton is in a sense the “architect” of physics -- he laid down the

fundamental principles of classical physics in his Principia, using

an elegant exposition inspired by Euclid’s Elements

A Page from Newton’s Principia

“I first constructed proofs for myself and then I compared my proofs with those of Newton.

The experience was a sobering one. Each time I was left with sheer wonder at the

elegance, the careful arrangement, the imperial style, and incredible originality…each

time, I felt like a schoolboy admonished by his Master.” -- Subramanyan Chandrasekhar

on Newton’s Principia

The Darker Side of Isaac Newton

! Newton was a highly complex individual -- besides hisfundamental work on motion and optics, he also pursuedextensive research in alchemy and bible studies.

! As President of the Royal Society, he has the Royal AstronomerFlamsteed’s star catalog seized and published against his will.

! Later in life, Newton became head of the British Mint, andpersonally carried out investigations against counterfeiters --including ten who were convicted and sentenced to death. Heprofited enormously from this position, and died a very wealthyman.

! Newton’s body was discovered to have been contaminated bymercury poisoning -- most likely from his alchemical studies.

Newton’s First Law of Motion

! Newton’s first law of motion states that a body in motion will

remain in motion, unless acted upon by an outside force.

! This provides, in a nutshell, the key concept in Newton’s

framework -- that a force is tied to a change in the velocity of a

body -- an acceleration.

! A body in uniform motion experiences no net force.

! An accelerated body must experience a net force.

Newton’s First Law Example -- The Centripetal

Force

! An accelerated body must experience a net force. For circular

motion, this force is called the “centripetal force”.

Velocity

Change in Velocity

Direction of Force

Conceptual Question

Imagine that the force of gravity were to be suddenly turned off.

Would the Earth

! A) Fly off in uniform motion tangential to its orbit.

! B) Continue to orbit the sun in its current orbit.

! C) Spiral inwards towards the center of the sun.

! D) Fall directly inwards towards the sun.

Newton’s Second Law of Motion

! Newton’s second law of motion states that the force acting upon a

body is the product of the body’s mass and its acceleration --

F = ma

! This is the most powerful of Newton’s three laws -- if one knows

the force acting on a body of a given mass, one can predict the

acceleration of the body and therefore the path of its motion

Newton’s Third Law of Motion

! Newton’s Third Law of Motion states that for every action, there

is an equal and opposite reaction

Conceptual Question

! Imagine that you are an astronaut on a space walk, and your jet

pack has run out. You take your jet pack off. How can you use it

to get back to the shuttle?

! A) Throw it towards the shuttle.

! B) Throw it away from the shuttle.

Newton’s Universal Law of Gravity

! Newton realized that gravity was a universal force acting both on

Earth and throughout the cosmos.

! His universal law of gravity states that the gravitational force

exerted on a test mass by a central body is directly

proportional to both the mass of the test body and the mass

of the central body, and inversely proportional to the square

of the distance between them :

F = G M1 M2 / r2

Conceptual Question

! Imagine that the sun were suddenly and instanteously replaced

by an extremely dense black hole of the same mass as the sun,

and that we can neglect any radiation or other effects other than

gravity.Would the Earth’s orbit :

! A) Expand slowly

! B) Contract slowly

! C) Remain unchanged

! D) Be swallowed by the sun

The Path of Modern Physics

! Newton’s second law exposes the fundamental questions that

have occupied physics since the time of Newton --

F = m a

! What are the fundamental forces? What laws govern them?

! What are the fundamental types of matter?

Light and Matter

Physics of Waves

! Energy which comes in the form of waves (water waves, sound

waves, light waves…) can interfere with one another, producing

either larger or smaller waves.

! Interference is a unique fingerprint of wave phenomena.

Is Light a Wave?

! From the time of Newton to the 19th century, scientists debated

the nature of light -- Newton advanced a corpuscular theory of

light based on discrete particles, while others advanced a theory

of light based on waves.

! The corpuscular theory offered a simple explanation for the

reflection of light.

Is Light a Wave?

! In the 17th and 18th centuries, however, scientists discovered

effects which could not be satisfactorily explained by the

corpuscular theory.

! When passed through two narrow slits very close together, light

can be seen to form an image consisting of light and dark bands -

- similar to water waves passing through a breaker.

Is Light a Wave? Yes, and No!

! However, 20th century physicists have discovered that light hasparticle-like properties as well -- individual atoms emit discretepackets of light energy known as photons, similar to the idea ofNewton’s corpuscles. Einstein in fact won his Nobel prize on hisexplanation of the photoelectric effect, which relied on thephoton theory of light.

! In a sense, light is both a particle and a wave. Viewed on a verytiny scale, it is composed of discrete photons. When enormousnumbers of those photons come together, they exhibit wavelikeproperties.

! The age-old question is perhaps so confusing because it isframed in terms of something light is not -- it is a bit akin to askinga question like, “Can computers think?”

Electromagnetic Spectrum

Spectroscopy -- Continuous Emission

! One of the most important tools modern astronomers have is the

usage of the spectrum of light detected on Earth to learn about

distant bodies.

! A single hot source of light from a solid body emits a continuous

spectrum of light energy -- blackbody radiation. The hotter the

body, the more towards the blue the spectrum will be shifted.

Spectroscopy -- Emission Lines

! When a cold cloud of gas emits light energy, it does so at a set of

unique wavelengths which are a kind of fingerprint.

Spectroscopy -- Absorption Lines

! When continuous spectrum of light energy passes through the

dark cloud, light energy is absorbed at precisely the same

wavelengths, resulting in an absorption spectrum.

Structure of Matter

! Atomic hypothesis is ancient, and dates back to at least the time

of Democritus (470 - 380 BC). Still, by 1900, the evidence for the

“reality” of atoms was sparse, and the idea was not universally

accepted.

! In early 20th century, physicists made rapid progress on atomic

structure, and discovered amazing properties of the structure of

matter on small scales.

Yummy Atomic Theory --

The Plum Pudding Model of the Atom

! By the early 20th century, the basicbuilding blocks of matter had beendiscovered -- electrons, protons, andneutrons.

! Electrons carry a negative charge. Theyare 2000 times lighter than protons andneutrons and have no measurable spatialextent.

! Protons carry a positive charge. Theyhave a measurable spatial extent of abouta tenth of a billionth of a meter.

! Neutrons are almost exactly as massiveas protons, but carry no charge. Theywere not discovered until the 1920s.

! Each atom consists of both electrons andprotons, and neutrons.

! On average, matter must be neutral, andso an equal number of protons andelectrons must reside in every atom.

Rutherford Experiment

! About 1910, Ernest Rutherforddevised an experiment to infer thestructure of the atom.

! Alpha particles -- what we now knowto be two protons and two neutrons -- were shot into a gold foil.

! Rutherford and his assistantsmeasured where the alpha particlewound up.

! Much to their surprise, they foundsome alpha particles recoiled andshot back toward the source.

! Rutherford later said, “It was quitethe most incredible event that everhappened to me in my life. It wasalmost as incredible as if you fired a15-inch shell at a piece of tissuepaper and it came back and hit you.”

Most of the Space in the Atom is “Empty”

Helium Atom

A Note on Atomic Terminology

! The atomic number for an element is the number of protons in thenucleus of the atom and is sometimes referred to as “Z”.

! Hydrogen Z = 1

! Helium Z = 2

! Iron Z = 26

! Because each proton carries a single unit of positive charge and theatom as a whole must be neutral, the atomic number is also equal to thenumber of electrons in a neutral atom.

! The atomic mass number for an element is the number of protons inthe nucleus plus the number of neutrons -- sometimes “A”. This variesdepending on the species, or isotope.

! Hydrogen A = 2 - 3, depending on isotope

! Carbon A 12, 13, 14, depending on isotope

The Strange World of Quantum Mechanics

! As physicists began to unravel the structure of the atom, its rulesbecame clearly different than the rules governing physics on much largerscales.

! On such small scales, matter had wavelike properties. When a beam ofelectrons was squeezed through a diffraction grating, it interfered withitself, just like light.

! No matter how hard one tried to pin down the precise position andvelocity of the electron, it is impossible to specify both simultaneously.

! Perhaps most strangely of all, the rules of physics on small scalesproved to be inherently non-deterministic. This was too much for somephysicists to accept -- Einstein said, “God does not play dice with world.”

! In a cloy rebuttal, Niels Bohr replied, “Einstein, do not tell God what to dowith his dice.”

Quantum Transition

! Electrons in the atom make “quantum jumps” between one orbital and another,

and in so doing either emit or absorb a discrete packet of energy.

Quantum Transitions

! These quantum transitions

explain the spectral features of

light.

! Each line we see in a spectrum

is the unique signature of an

electron hopping between two

energy orbitals and emitting a

photon of a characteristic color.

! We can use this spectrum as a

“fingerprint” to identify the atom

which emitted it.

Next Week

! Next week we will apply these principles of planetary motion and

the nature of light to our sun and the inner bodies in the solar

system -- Mercury, Venus, Earth, and Mars.

! We will learn why Venus is too hot to sustain life today, Mars is

too cold, and Earth is just right.