Lecture 2:

The Shape & Size of Earth

Astronomy 1143 – Spring 2014

Key Ideas:

The Earth is Round• Height of Constellations Above the Horizon• Shadow of Earth during a Lunar Eclipse

Measuring Length -- Meters

Measuring Angles • Degrees, Minutes and Seconds• Angular Distances & Sizes

Measuring the Earth's Size• Angle of Sun at two different locations

Classical Greece & Spheres

The Ancient Greeks were intoxicated by geometry, form, and symmetry.

A sphere is the most perfect geometric solid

500 BCE:• Pythagoras proposed a spherical Earth on

purely aesthetic grounds

400 BCE:• Plato espoused a spherical Earth in the

Phaedra.

Aristotle gets Physical...Aristotle (384-322 BCE) proposed aspherical Earth on geometric grounds, Backed up with physical evidence:

• People living in the south see southern constellations higher above the horizon than people living in the north.

• The shadow of the Earth on the Moon during a lunar eclipse is round.

• Matter settling onto Earth would naturally shape itself into a spherical shape

The Basic Idea

If the Earth is round, then people on different parts of Earth will see stars at different heights above the horizon.

Sees North Star directly overhead

Sees North Star on horizon

The Basic IdeaThis is much more realistic, considering the scale of the solar system.

Sunlight

Looking Southfrom Syene Egypt

Latitude: 24º N

Scorpius

Looking Southfrom Athens Greece

Latitude: 38º N

Scorpius

Orion: North and South

Thanks to a spherical EarthSouthern constellations appear higher in the sky as

you move south

The North Star appears lower in the sky as you move south

Constellations/the Moon/etc appear “upside-down” in the Southern Hemisphere compared to the Northern Hemisphere

Some constellations are not visible in the Northern Hemisphere and vice versa for the Southern Hemisphere

Earth Shadow during Lunar Eclipse

Multiple ExposurePhotograph

No Flat Earth (or Moon)

Aristotle’s demonstration was so compelling that a spherical Earth was the central assumption of all subsequent philosophers of the Classical era.

He also used the curved phases of the Moon to argue that the Moon must also be a sphere like the Earth.

We’ve established its shape, what’s its size?

Need to use GEOMETRY

Units: A Useful Digression

The Metric SystemAstronomers use the Metric System:

Length in Meters

Mass in Kilograms

Time in Seconds

All scientists use Metric UnitsOnly the United States, Liberia & Myanmar

(Burma) still use “English” Units.

If you are not paying attention to units, bad things can happen1. Your roller coaster could fall apart

In 2004, an axle at Tokyo Disneyland’s space mountain broke mid-ride, because of problems in converting the English units to metric units

2. You could lose a $125 billion satelliteIn 1999, NASA lost the Mars Climate Orbiter. It was off course by 60 miles by the time it reached Mars because Lockheed Martin was sending the thruster force calculation in pounds and NASA was expecting Newtons

If you are not paying attention to units, bad things can happen3. Your jet could turn into a glider

In 1983, an Air Canada Boeing 767 flying between Montreal and Edmonton ran out of fuel and had to glide to a landing at a former Air Force base in Gimli, Manitoba. Among other mistakes, the crew had calculated the amount of fuel needed in pounds, rather than kilograms, but thought they had the correct number of kilograms. As a result, they had less than ½ the amount they needed

4. You could lose points on your homework

How many kilometers are in 10,000 meters?Or: convert 10,000 meters to kilometers

How seconds in a year?

Units of Length

The basic unit of length is the meter (m)

Traditional Definition:• 1 ten-millionth the distance from the North Pole

to the Equator of the Earth.

Modern Definition:• The distance traveled by light in a vacuum in 1 /

299792458th of a second.

Commonly use meters and kilometers.

Measuring Angles

A complete circle is divided into 360-degrees

The Babylonians started this convention:• 360 is close to 365, the days in a year.• 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15,

18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 & 180 without using fractions.

• Start by quartering the circle (90 degrees), then subdividing further using geometry.

1 degree of arc

Subdividing the Degree

Degrees are divided into Minutes of Arc ('):• 1 degree divided into 60 minutes of arc• from “pars minuta prima” (1st small part)• 1 minute = 1 / 60th of a degree

Minutes are divided into Seconds of Arc ("):• 1 minute divided into 60 seconds of arc• from “parte minutae secundae” (2nd small part)• 1 second = 1 / 60th of a minute or

1 / 3600th of a degree (very small)

Question: Why 60? Answer: Blame the Babylonians...

60 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, & 30 without using fractions.

The Babylonians subdivided the degree as fractions of 60, for example:

7 14/60 degrees

Claudius Ptolemy introduced the modern notation:

7º 14' 00"

Subdividing the Degree (cont’d)

Eratosthenes of Cyrene

Born in Cyrene (now Shahhat Libya) in 276 BCE, lived until about 195 BCE• 2nd Librarian of Alexandria.

At noon on the Summer Solstice in Syene Egypt (modern Aswan), the Sun was straight overhead and cast no shadows.

On the same day, the noon Sun cast shadows in Alexandria, located north of Syene, 5000 stades away.

Tropic of Cancer

Alexandria

Syene

Shadowless in SyeneNo shadows on the Summer Solstice means

that Syene is on the Tropic of Cancer.

Alexandria is north of Syene along the Earth’s curved surface and shadows are cast.

Measuring the angle of the Sun in Alexandria at noon on the Summer Solstice when it was overhead in Syene lets you measure the circumference of the Earth if you assume that the Sun is very, very far away!

Syene

Alexandria

Earth

High Noonon theSummer Solstice

Sunlight

7 12/60º

Noon on the Summer Solstice

At Syene:• Sun directly overhead, no shadows cast

At Alexandria:• Sun 712/60 degrees south of overhead,

casting a shadow

Since a full circle is 360 degrees, the arc from Alexandria to Syene is

1260Arc 7 360 1/ 50th of a circle

The Road to Syene

The circumference of the Earth is 50 times the distance from Alexandria to Syene.

How far is Alexandria from Syene?• 5000 Stades

How big is 1 Stade?• 600 Greek Feet• Best guess is 1 stade = 185 meters

(Attic stade)

The Circumference of the Earth

Eratosthenes computed the circumference of Earth as:

50 5000 stades = 250,000 stades

250,000 stades 185 meters/stade

= 46,250 kilometers

The modern value:

40,070 kilometers

Eratosthenes' estimate is only ~15% too large

Units matter – historical exampleColumbus was not only convinced that he could

reach the treasures of the East by sailing west, but also that it would be a short, relatively easy trip. Just a few days between Spain and the India!

He presented sponsors, such as Queen Isabella and King Ferdinand, with small numbers from two main mistakes:• Too large estimates for the size of Eurasia• Misinterpreting number of Arabic miles as

number of Roman miles (shrunk Earth by 25%)

The rest, as they say, is history

Describing the SkyWe do not “see” a 3-dimensional night sky

We can describe brightnesses and colors and motions

Stars appear as single points of light

Planets are close to points of light (at least to the naked eye)

Sun and Moon appear as actual extended objects

Describe separation of stars on the sky and the apparent size of objects by angular distance and angular size

Angular Size

Angular Distance & Size

Angular Size Changes with Distance

The angular size of a dime and quarter can be the same, even though their physical sizes are different

Measuring big distances

Measuring distances and physical sizes in astronomy is very difficult

Obvious methods such as meter sticks are out (there’s that whole lack of oxygen thing)

We don’t usually have reference objects here on Earth to help us out

Answer: Use geometry