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 elliptical orbits, 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 its movements, uttering the concords of a single scale.” -- Plato

Jazz musician and Yale professor Willie Ruff collaborated with geologist John 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 a single “track” of the music of the spheres is far greater than a human lifespan.

Ruff and Rodgers contracted the duration of Pluto’s orbit to about 20 minutes in duration, which also upshifted the frequencies of the inner planets 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

QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

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 his fundamental work on motion and optics, he also pursued extensive research in alchemy and bible studies.

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

Later in life, Newton became head of the British Mint, and personally carried out investigations against counterfeiters -- including ten who were convicted and sentenced to death. He profited enormously from this position, and died a very wealthy man.

Newton’s body was discovered to have been contaminated by mercury 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 has particle-like properties as well -- individual atoms emit discrete packets of light energy known as photons, similar to the idea of Newton’s corpuscles. Einstein in fact won his Nobel prize on his explanation of the photoelectric effect, which relied on the photon theory of light.

In a sense, light is both a particle and a wave. Viewed on a very tiny scale, it is composed of discrete photons. When enormous numbers of those photons come together, they exhibit wavelike properties.

The age-old question is perhaps so confusing because it is framed in terms of something light is not -- it is a bit akin to asking a 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 basic building blocks of matter had been discovered -- electrons, protons, and neutrons.

Electrons carry a negative charge. They are 2000 times lighter than protons and neutrons and have no measurable spatial extent.

Protons carry a positive charge. They have a measurable spatial extent of about a tenth of a billionth of a meter.

Neutrons are almost exactly as massive as protons, but carry no charge. They were not discovered until the 1920s.

Each atom consists of both electrons and protons, and neutrons.

On average, matter must be neutral, and so an equal number of protons and electrons must reside in every atom.

Rutherford Experiment

About 1910, Ernest Rutherford devised an experiment to infer the structure of the atom.

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

Rutherford and his assistants measured where the alpha particle wound up.

Much to their surprise, they found some alpha particles recoiled and shot back toward the source.

Rutherford later said, “It was quite the most incredible event that ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper 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 the nucleus 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 the atom as a whole must be neutral, the atomic number is also equal to the number of electrons in a neutral atom.

The atomic mass number for an element is the number of protons in the nucleus plus the number of neutrons -- sometimes “A”. This varies depending 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 rules became clearly different than the rules governing physics on much larger scales.

On such small scales, matter had wavelike properties. When a beam of electrons was squeezed through a diffraction grating, it interfered with itself, just like light.

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

Perhaps most strangely of all, the rules of physics on small scales proved to be inherently non-deterministic. This was too much for some physicists 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 do with 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.