ASTRONOMY 103

A 103 Survey of Astronomy Text: Chaisson-McMillan, Astronomy: A Beginners’ Guide

Notes, Week 1

Angles, Powers of Ten I. Angles A. Degrees

An angle measures part of a circle. Each year, as the Earth orbits the Sun, the Sun appears to move in a circle relative to the background stars, through the constellations of the Zodiac. The Sun moves about °1 per day. Because a year is 365 1/4 days, early astronomers could have divided the Sun's apparent

path into 365 parts, and, if they had done this, there would have been 365o in one circle. To define a degree this way, however, would make life needlessly difficult, because simple angles would have uneven numbers of degrees. If, instead, one divides a circle into 360 degrees, then there will be an integral number of degrees in a quarter of a circle 90o a fifth of a circle 72o a sixth of a circle 60o an eighth of a circle 45o a ninth of a circle 40o a tenth of a circle 36o a twelfth of a circle 30o

The Moon and Sun have almost the same angular size in the sky, about ½ o. Of course the Sun is much larger and much farther away than the Moon. The angular size of a dime held at arm's length (or of your little finger) is about 1o. That is, a dime held at arm's length is just about twice the angular size needed to cover the Moon or the Sun!

B. Minutes and seconds of arc

The angular size of planets is much smaller than a degree, as is the angular distance between stars that orbit each other (binary star systems). A good telescope lets you see objects or groups of objects whose angular size is about 1 ten thousandth of a degree, and it is helpful to have angular units by which to measure such small angles. The commonly used units are minutes of arc and seconds of arc: °1 = 60 arc minutes = 60’

1’ = 60 arc seconds = 60”. Then a second of arc (or an arc second): °1 = 3600 arc sec = 3600”.

A dime seen from a mile away has an angular size of about 2”

Estimating angles in the sky

ASTRONOMY 103

The Sand Reckoner

by Archimedes

Chapter I There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken. But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe. Now you are aware that `universe' is the name given by most astronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account, as you have heard from astronomers. But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premises lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface. Now it is easy to see that this is impossible. For, since the centre of the sphere has no magnitude, we cannot conceive it to bear any ratio whatever to the surface of the sphere. We must however take Aristarchus to mean this: Since we conceive the earth to be, as it were, the centre of the universe, the ratio which the earth bears to what we describe as the ``universe'' is the same as the ratio which the sphere containing the circle in which he supposes the earth to revolve bears to the sphere of the fixed stars. For he adapts the proofs of his results to a hypothesis of this kind, and in particular he appears to suppose the magnitude of the sphere in which he represents the earth as moving to be equal to what we call the "universe.''

ASTRONOMY 103 3

II. Powers of ten A. Definition

In 250 BC, the mathematician Archimedes devised a compact notation for handling large numbers in order t o c ount t he number of grains of s and on Earth. H e s uggested t hat t he num ber of s tars i n the universe was larger than the number of sand grains on the Earth. It turns out that the number of stars in all t he galaxies one c an see w ith a t elescope—in the "visible universe"—really i s about equal t o the number of s and grains on a ll the beaches of the Earth. ( If one includes t he deserts as well, there are probably m ore grains of s and, but the uni verse pr obably e xtends f ar be yond w hat w e c an s ee). It i s astonishing t hat A rchimedes t hought t hat t here were s o m any s tars, be cause one c an onl y s ee about 3,000 even on a clear dark night with no pollution from city light or city. The w ay Archimedes wrote num bers w as e ssentially t his: Y ou can represent t he num ber 10 0 a s 10 x 10 = 210 , or ten multiplied by itself two times: 101 = 10 1 with one 0 102 = 100 = 10x10 1 with two 0's Similarly, 103 = 1000 = 10 x 10 x 10 1 with three 0s 106 = 1,000,000 = 10 x 10 x 10 x 10 x 10 x 10 1 with six 0s (one million) 109 = 1,000,000,000 1 with nine 0s (one billion) 1011 = 100,000,000,000 1 with eleven 0s (one hundred billion)

One just counts the number of zeroes—or, equivalently, moves the decimal point to the right by the number of spaces given. What is100? 100 = 1 1 with no 0s Similarly, small numbers (numbers close to zero) are written using negative powers of ten: 10-1 = 0.1 = 1/10 10-2 = 0.01 = 1/102 10-6 = 0.000001 = 1/106 Finally, to write other numbers in this notation one proceeds as follows: 3 x 102 = 300 Take 3 and move the decimal point over two places 3.1 x 102 = 310 Take 3.1 and move the decimal point over two places 45,200,000 = 4.52 x 107

ASTRONOMY 103 4

B. Gaining intuition for large numbers One gains understanding of large numbers by using them, first in naming examples of large collections of or dinary obj ects, a nd t hen b y doi ng a rithmetic t o s ee how t o w ork with t hem. S and w as w hat Archimedes used, and with sand it is easy to give examples of numbers up to a hundred billion or 1011. Let's begin with smaller numbers: A single average grain of sand is 1/3 mm in diameter—in a line 1 cm long there are about 30 sand grains. A pinch of sand has about 1,000 grains. A 1-ounce shot glass holds about 1 million = 106 grains of sand. An 8-gallon fish tank holds about 1 billion = 109 grains, equal to the number of stars in a small galaxy. A 1-meter cube holds about 100 billion = 1011 grains, nearly the number of stars in

the Milky Way galaxy. All the world's sand (in deserts and under the water along the coastlines) is about 1024 grains, about the same as the total number of stars in the visible universe.

C. Arithmetic using powers of ten How does one multiply 102 x 106? Mechanically, you can simply count the number of zeroes:

100 x 1,000,000 = 100,000,000

This works because it is counting the number of times one multiplies by ten:

102 x 106 = (10 x 10) x (10 x 10 x 10 x 10 x 10 x 10) = 810 or

102 x 106 = 106+2 = 108 What about division?

103/102 = 1000 x 0.01 = 10 Easier:

103/102 = 103 – 2 = 110 = 10 108/104 = 108-4 = 104

General numbers:

3 x 102 x 3 x 107 = 3 x 3 x 102+7 = 9 x 109

4.1 x 102 x 7.6 x 104 = 4.1 x 7.6 x 102 x 104 = 31.2 x 106 = 3.12 x 107 See Sample Problems for Set 1

Once again:

Multiplication

=× 32 1010

=× 511 1010

1010101010 ××××

510=

3210 +=

51110 +

1055 106103102 ×=×××

1610=

Division

=3

2

1010

1010101010××

× 110or 101 −=

=3

2

1010

=−3210 110−

More Division

=××

103106 3

=××

3

2

107101.2

1010

36 3

× 2102×=

3

2

1010

71.2× 1103. −×=

2103 −×=

ASTRONOMY 103

Geography of the Universe I. Yardsticks A. Short distance: cm, m, km

We'll use the metric system to measure shorter distances, up to the distance from the Earth to the Sun. 1 cm = .39 in 1 m = 100 cm = 39.37 in 1 km = 1000 m = 105cm = .62 mi

B. The AU The average distance from the Earth to the Sun is 1.5x108 km

150 million km = 93 million mi This distance is given a name, the astronomical unit or AU:

1 AU = 1.5x 108 km = distance from Earth to Sun

C. Longer yardsticks Finally the measurement of larger distances is based on how far light travels in a given time. The speed of light is c = 3x 105 km/s. This is fast enough to circle the Earth 7 times in one second. Recall that

distance = speed x time and

time = distance/speed

If we write d for distance and t for time, we have

d = ct

and

t=d/c

ASTRONOMY 103

Let's do an example: The average distance to Pluto is 39.5 AU. Find the time in seconds that it takes light to go from the Sun to Pluto. We first write the distance in km and then find the time using,

cdt =

Because 1 AU = 1.5 x 108 km, a distance of 39.5 AU is d = 39.5 AU x 1.5 x 108 km/AU = 5.9 x 109 km. Then

94

5

5.9 10 km 2 10 s 5.5hr3 10 km/s

dtc

×= = = × =

×

Time in hours:

. hr5.5

s 36001hrs102 4 =××

Even at the speed of light, it takes years to reach the nearest star. The distance light travels in one year is called a light-year. 1 1ight year is about 1310 km (we’ll work that out soon) 1 ly = .9 x 1013 km, and the nearest star is 4 light-years away. Here’s the power-point summary:

Geography of the UniverseYardsticks

1 cm = 0.39 in

1 km = 1000 m = 0.62 mi

1 AU = 1.5x108 km

Distances based on how far light travels:

speed of light = 300,000 km/s = 3 x105 km/s

d = c t (distance = speed x time)

1 year = 365 1/4 days = 3.16 x 107 seconds

distance = .9 x 1013 km

1 light-year = .9 x 1013 km

Check: Use 1 day = 24 hr, 1 hr = 60 min, 1 min = 60 s

1 year = 365 1/4 days

≅ .9 x 1013 km

= 365 x 24 x 60 x 60 s = 3.16 x 107 sd=ct = 3x 105 km/s x 3.16 x 107 s

= 9.48 x 1012 km (For 3-place accuracy, use 1 yr = 3.1558 x107 s and c=2.998x105km/s to find1 ly = 9.46x1012 km)

1 light-year = .9 x 1013 km

NOTE:

A LIGHT-YEAR IS A DISTANCE (1013 km)

NOT A TIME

ASTRONOMY 103

II. Scale model of the solar system With the Sun an 8 1/2" bowling ball in a corner of a lecture hall in the Physics Building, Mercury, Venus and Earth are inside the lecture room, Mars just beyond, and Jupiter, Saturn, Uranus, Neptune and Pluto are all on campus, with Pluto in the courtyard between Mitchell Hall and the Fine Arts building. The next nearest star is the α-Centauri system, with α-Centauri another 8 1/2" bowling ball in London. On the geography sheet below, Proxima Centauri is the closest of the three stars that orbit each other in the α-Centauri system. III. Geography sheet The next page summarizes a journey through the universe, in a sequence of larger and larger steps, magnifying our view by 100 times at each step. This is essentially equivalent to the sequence in the Powers-of-Ten film. The narrator of that film is Philip Morrison, one of the leading astrophysicists of the last 50 years.

ASTRONOMY 103

GEOGRAPHY OF THE UNIVERSE

km ly

10-2 10m = 1/100 km

x100 1 UWM CAMPUS

x100 102 SOUTHEAST WISCONSIN

x100 104 DIAMETER OF EARTH (1.3x104km)

x100 106 DIAMETER OF MOON’S ORBIT ( .8 x106km)

x100 108 DISTANCE FROM SUN TO EARTH (1.5x108km)

x100 1010 DIAMETER OF SOLAR SYSTEM (1.2x1010km)

(1012)

x1000 1013km = 1 ly DISTANCE TO PROXIMA CENTAURI (4 ly)

x100 102 ly DISTANCE TO TYPICAL CLUSTER OF STARS

x100 104 ly DISTANCE TO CENTER OF MILKY WAY (3x104ly)

x100 106 ly DISTANCE TO NEARBY GALAXIES

x100 108 ly DISTANCE TO LARGE CLUSTERS OF GALAXIES

x100 1010 ly SIZE OF VISIBLE UNIVERSE

10 m

Size of a large room -the front of the lecture hall this online course is

not using

UWM Campus

1km

100 km

Diameter of Earth

1.3x104 km

Diameter ofMoon’s Orbit

3/4 x 106 km

This photograph, from a spacecraft far from the Earth-Moon system, shows the right relative sizes of the Earth and the Moon (the Earth’s diameter about 3 times the diameter of the Moon). What you cannot see in the photograph is the fact that the distance between the Moon and the Earth is much larger – the distance from the Earth to the Moon is about 60 times the radius of the Earth. In the photograph, the Moon is nearly along the line of sight from the spacecraft to the Earth. Below is what the Earth-Moon system looks like without that foreshortening - from a craft for which the Moon is not along the line of sight: This view shows the loneliness that is typical of the universe: It’s mostly empty space. It takes light about 1 second (1.3 s) to travel from the Moon to the Earth.

Diameter ofMoon’s Orbit

3/4 x 106 km

This photograph, from a spacecraft far from the Earth-Moon system, shows the right relative sizes of the Earth and the Moon (the Earth’s diameter about 3 times the diameter of the Moon). What you cannot see in the photograph is the fact that the distance between the Moon and the Earth is much larger – the distance from the Earth to the Moon is about 60 times the radius of the Earth. In the photograph, the Moon is nearly along the line of sight from the spacecraft to the Earth. Here’s what the Earth-Moon system looks like without that foreshortening - from a craft for which the Moon is not along the line of sight: This view shows the loneliness that is typical of the universe: It’s mostly empty space. It takes light about 1 second (1.3 s) to travel from the Moon to the Earth.

Earth -Sun Distance 1.5 x 108 km

Venus

Mercury

Earth

Sun 108km

108 km

Now, at 108km, we are most of the way to the Sun. The Sun, is an isolated ball: The radius of the Earth’s orbit is 100 times the radius of the Sun. It takes 8 minutes for light from the Sun to reach the Earth.

1010 km

Diameter of Solar System

The distance from the Sun to Pluto is 40 AU, implying a diameter of its orbit that is nearly100 times the distance from the Earth to the Sun. It takes light just over 5 hours for light from the Sun to reach Pluto, and about the same time for light from Pluto to reach the Earth. We see the Sun as it was 8 minutes ago. We see Pluto as it was 5 hours ago.

1012 km

Nothing new: The galaxy (and the universe) is nearly empty. The solar system is a small disk surrounded by empty space.

Only after traveling 1013km, more than1000 times the size of the solar system, do we find another star.

1013 km = 1 ly

4 ly : Distance toProxima Centauri in the α−Centauri system of 3 stars

The next nearest star after the Sun is Proxima Centari, one of 3 stars that orbit each other in the α−Centauri system. The distance of 4 ly means that we see ProximaCentauri as it was 4 years ago.

102 lyDistance to Pleiades: 440 ly

Stars usually form in clusters, from large clouds of gas that are pulled together by their own gravity. Here is a young cluster of stars, the Pleiades, visible in the west in the constellation Taurus. At a distance of 440 ly from the Pleiades, we see the stars as they were 440 years ago.

104 ly

Distance to Center of Milky Way: 3X104 lyWe are part of the Milky Way galaxy, a collection of over 400 billion (4x108 stars) . We see the stars at its center as they were 30,000 years ago.

104 ly3x104 ly

106 ly

Distance to Andromeda Galaxy: 2.6x106 lyThe Milky Way is part of a cluster of about 45 galaxies, the Local Group. It and the Andromeda Galaxy are the two largest members of this cluster of galaxies. The Andromeda Galaxy is the most distant object that you can see naked-eye (without a telescope or binoculars). The light that reaches us from Andromeda shows the galaxy as it was more than 2 million years ago.

Distance to Virgo Cluster of Galaxies: 1/2 x108 lyThe next step of 100, to 100 million light years, takes us out to a much larger cluster of more than 2500 galaxies: the Virgo Cluster. Almost all stars are in galaxies, and most galaxies are in large clusters like this one. You see the galaxies as they were 500 million years ago, more than 200 million years before the age of dinosaurs.

108 ly

1010 ly

Size of Visible Universe: Distance to furthest observed galaxies

Finally, in this Hubble space telescope photo, the dimmest objects are the most distant galaxies we can see. We see them as they were more than 10 billion years ago, when they were first forming. Young galaxies often have very bright centers (galaxies like these are called quasars), and the evidence is strong that the energy they emit comes from stars being swallowed by black holes. The universe may extend far beyond those in the Hubble photo, but we cannot see galaxies at greater distances, because we would have to look back to a time before they existed: The light emitted by galaxies farther away than these has not yet had time to reach us, because the galaxies formed less than13 billion years ago.