Astro 201: Sept. 6, 2010• Do on-line practice quiz #2 by 9/14• Homework #2: posted on web page, due 9/9• Reading: Hester, Chapters 2-3

Gleiser: Sections 3-6• Today: – Astronomical basis for calendars– Why there are seasons– Tycho, Kepler, Newton– Forces; String Theory– Determinism v. Chaos; Fractals

The Astronomical Basis for Calendars

The Earth rotates on its axis once a day.The Earth orbits the Sun with a period of about 365.242190 daysThe Moon orbits the Earth, such that the period between new moon and new moon is 29.5305 days

These periods vary a little bit because of the gravitational forces exerted by the other planets and other factors.

Every 4 years, we take into account the .242190 days by added a leap day, February 29.

Otherwise we would keep slipping, and eventually January would be summer in the northern hemisphere, etc.

Now, the thing is that there are NOT an integral number of months (29.5305 days) in one 365.242190 year.

So if for example today is a new moon, next year it will not be.

SEASONS: Seasons are a result of the tilt of the Earth’s axis with respect to the plane of its orbit. Seasons ARE NOT the result of the Earth being closer to the Sun in summer,and farther in winter – in fact during winter in the northern hemisphere, we are closer to the Sun than we are in the summer.

The Gregorian Calendar (Christian, one we use)

is based on the motion of the Earth around the Sun.

The length and number of months have no connection to the motion of the Moon

It is based on the Julian Calendar, which was introduced by Julius Caesar in 45 BC.

He made Jan. 1 the start of the Year.

During the middle ages, different groups in Europe adopted different start dates for the new years.

In 1582, Pope Gregory reformed the calendar, which is basically what we use today.

The French recognized a different pope for a while and had a different calendar.

The Islamic Calendar

is based on the motion of the Moon, with no connection to the motion of the Earth around the Sun.

Hijri Calendar, Based on the Qur'an.

Used in many countries around the Gulf, e.g. Saudi Arabia.

Purely lunar: 12 months, each with 29.53 days.

Therefore the Islamic year has 12x29.53 = 354.36 days

The Islamic year is shorter than the time it takes the Earth to go around the Sun.

Thus, a particular Islamic month falls during different times of the year , e.g. Ramadan.

Years are counted since the Hijra, which is is the time that Mohammed emigrated to Medina in AD 622.

So AD 2009 is Islamic year 1430.

The Jewish and Chinese Calendars

combine both, so years are linked to the period of the orbit of the Earth around the Sun, AND months are linked to the motion of the Moon around the Earth.

19 x365.24 = an integral number of 29.53 day months.

Jewish Year: A "ordinary" year has 355 days, and 12 months. A "leap" year has 385 days and 13 months.

The length of a particular month varies from year to year by a day, so that if New Year's Day (Rosh HaShannah) is, say, a new moon, then the first day of every month is a new moon.

Years are counted since the creation of the world, taken to be 3761 BC. Thus 1998 is Jewish year 5759.

Mayan Calendar

The ancient Mayans developed a complex and accurate calendar, which wasadopted by other people in meso-america – Aztecs, Tolmec .

Actually had THREE calendars, which they used at the same time:

1.The Haab Civilian calendar

18 months, 20 days each + 5 days to make 365 day years The 5 extra days were considered unlucky, days of mourning Did not count YEARS in the Haab calendar however

2.The Tzolkin Devine calendar, used for divinations

two types of weeks, one with 13 days, the other with 20 days

Each year was 260 days Years were not counted

Mayan Calendars, continued:

3.The Long CountInstead of years, the Mayan used the Long Count calendar system to

keep track of historical events

kin = 1 dayuinal(1 uinal = 20 kin = 20 days)Tun (1 tun = 18 uinal = 360 days = approx. 1 year)katun (1 katun = 20 tun = 7,200 days = approx. 20 years)baktun (1 baktun = 20 katun = 144,000 days = approx. 394 years) pictun = 20 baktun = 2,880,000 days = approx. 7885 yearscalabtun = 20 pictun = 57,600,000 days = approx. 158,000 yearskinchiltun = 20 calabtun = 1,152,000,000 days = approx. 3 million yearsalautun = 20 kinchiltun = 23,040,000,000 days = approx. 63 million years

The Mayan calendar started on our August 11, 3114 BC. With Date: 0.0.0.0.0.

Then date 13.0.0.0.0 is our December 21, 2012, or 12/21/2012.

This coincidence has caused some people to claim that the Mayan Calendar predicts the end of the world on December 21, 2012.

But the Long Count calendar doesn’t end on 13.0.0.0.

Tycho Brahe (1546-1601) Danish noblemanGot drunk as a college student, lost his nose in a duel

Built an observatory, Uraniborg,financed by King Frederic II

Made extremely accurate Observations of the planets

Astronomiæ Instauratæ Mechanica: Tycho’s book about his instrumentation

Tycho made observations of the positions of the stars and planets that were 15x more accurate than what others had made

He “beat down the errors” by repeating a particular measurement many times and averaging the result

Tycho’s cosmology

Tycho found no parallax for the stars and concluded they are very far awayHe made measurements which neither the Ptolemaic model, nor the simple Copernicus model could explain.

Earth stationary in centerSun and moon orbit the EarthOther planets orbit the SunStars orbit the Earth

Johannes Kepler (1564-1630)

Worked as Tycho’s assistantTook Tycho’s observations when Tycho died and spent the next 29 years analyzing them

First, he wrote “Mysterium Cosmographicum”The orbits of the 6 known planets were described By 5 “Pythagorian” polyhedra in spheres:

Eventually abandoned this model and came up with

Kepler’s Three Laws of Planetary Motion

1. Planets move in elliptical orbits (not circles), with the Sun at one focus.

2. An imaginary line connecting the Sun to any planet sweeps out equal areas in equal time..

Area 2

Area 1

Planets move faster when near Sun

3. The square of a planet’s orbital period is proportional to the cube of its semi-major axis.

P2 (years) = A3 (astronomical units)

1 Astronomical Unit = The Earth-Sun Distance

The farther out a planet is in the solar system, the longer it takes for it to go around the Sun.

Kepler’s Laws are empirical descriptions of the data.

The don’t explain WHYthe planets move as they do.

Isaac Newton (1642-1727)

Bubonic Plague 1665:

While home for 2 years with nothing to do he figured out

Laws of motion

Gravity

“Philosophiae Naturalis Principia Mathematica”The laws explained not only why planets move as they do, but why objects in general move as they do.

Newton's First Law of Motion:Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

speed: how many miles per hour velocity: speed and direction of motion acceleration: change in velocity, i.e. change in speed and/or change in direction of motion An object at rest stays at rest; an object in motion at a constant velocity stays moving at that velocity unless a force is exerted on it.

Objects with mass have INERTIA.

Newton's Second Law of Motion:

F = m a a = acceleration, change of velocity with time m = mass F = force

Newton's third Law of Motion:

For every action there is an equal and opposite reaction.

Law of Gravitation: Fg = Force of Gravity

From kepler.nasa.gov

Forces of Nature:

Gravity: very weak, but very long range. always attractive. always acts between any two masses

Electro-Magnetism: Electric & Magnetic fields; light Result of charged particles, or magnets Long range

Weak responsible for radioactive decay,

e.g. beta decay -- n --> p + e + anti-neutrino the interaction of neutrinos

Strong extremely strong on very short distance scales: only really important on scales of 10-13 cm holds neutrons and protons in the nuclei of atoms together

James Maxwell (1831-1879)

“Unified” Electricity and Magnetism Electro-magnetic field, light

Maxwell’s equations:

Glashow, Weinberg and Salaam got the Nobel Prize in Physics in 1979

for "unifying” Weak & EM forces and coming up with one theory

which describes the "Electroweak Force"

Grand Unified Theories: attempt to unify Electroweak and Strong forces

Theory of Everything: one theory to describe all 4 forces Plus quantum mechanics

STRING THEORY

everything is ultimately made of strings(sub-sub-sub atomic particles)

How big are strings?

Smaller than a Planck length,

which is about 10-33 centimeters

or about a millionth of a billionth of a billionth of a billionth of a centimeter.

Strings vibrate

Closed string

Open string

In many string theories the Universe is 10 dimensional,

with the "extra" dimensions COMPACTIFIED.

All HS math geeks read a book called FLATLAND: A Romance of Many Dimensions,by Edwin A. Abbott (1884)

FLATLAND: a "first person" account of life of a 2-dimensional society

a radical named Arthur Square figures out that space is really 3 dimensional.

So what the string theorists say is that in ordinary life we think we live in 3 dimensions, and

we have to think of ways to detect the other 7.

The extra-dimensions in string theory are Calabi-Yau figures:

Knitted Calabi-Yau Figures

• Kepler: Empirical description of the motion of the planets

• Newton: Law of Gravity. • Developed Calculus, derived orbits of the planets• Solved the “Two-body” problem: Sun + one planet• Couldn’t solve the “Three-body” problem

Mechanical Universe: In Newtonian physics, objects move in perfectly determined ways

OrreryMechanical models of planetary motions in the solar system

But is the real solar system accuratelydescribed by an orrery?

"We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes."

— Marquis Pierre Simon de Laplace (1749-1827)

The Equations

Many thanksTo Scott Tremaine’sNotes fromApril 2006

• King Oscar II of Sweden (1829- 1907)- Prize: How stable is the

universe?

• Jules Henri Poincaré (1854-1912)– Sun (large) plus one planet (circular

orbit)• Stable

– Added 3rd body • Strange behavior

– Not periodic

• Modern approach:Solve many-body problem with computer

calculations– Take a distribution of mass– Figure out the gravitational force on each part– F=ma gives you the acceleration on each part– Compute velocity of each part– Move the parts a little– Repeat

Kuiper belt objectsPlutinos (3:2)Centaurscomets

as of March 8 2006 (Minor Planet Center)

Sensitivity to Initial Conditions"A very small cause which escapes our notice determines a

considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still know the situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. But is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible...". (Poincaré)

Can we predict the motion of a single planet

a billion years from now?

• Laplace and Newton – Yes• Poincare’ – No

• Lorenz – 1963 – “Butterfly Effect” If a butterfly flaps its wings in Brazil, does it result in a tornado in Kansas?

Two kinds of dynamical systems

Regular • highly predictable, “well-

behaved”• e.g. baseball, golf, simple

pendulum, all problems in mechanics textbooks, planetary orbits on short timescales

Chaotic• difficult to predict, “erratic”• appears regular on timescales

short compared to Liapunov time

• e.g. roulette, dice, pinball, weather, billiards, double pendulum

• The Solar System

Double Pendulum: a chaotic system

Consequences of chaos

• Positions of planets are not predictable on timescales longer than 100 Myr

• the solar system is a poor example of a deterministic universe• The solar system is “Chaotic”

Fractals• Geometric forms • Define by a recursive rule• Same on all scales

Benoit Mandelbrot

Serpinski Triangle

Von Koch Snowflake

Fractals are “Self-Similar”:

same when you zoom in

The Julia Set

Gaston Julia, French mathematician

The Mandelbrot Set

Fractals in Nature