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The Scale of the Cosmos Lecture 2 POWERS OF 10 & SCIENTIFIC NOTATION In science and especially in astronomy, you have to deal with very large numbers and very small numbers. For example, the number of kilometers in a light year is

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### The Scale of the Cosmos

Lecture 2

• In science and especially in astronomy, you have to deal

• with very large numbers and very small numbers.

• For example, the number of kilometers in a light year is

• approximately 9,500,000,000,000 (9.5 trillion).

• The diameter of the hydrogen atom is 0.000000013

• centimeters (13 billionths).

To manage large numbers and small numbers, professionals make use of powers of 10 and scientific notation.

10,000,000,000 (10 billion) years is the approximate age of the Milky Way Galaxy (MWG). Instead of writing out the large number in expanded form (i.e., 10,000,000,000) it is written as as a power of 10…1010).

10

9

8

7

6

5

4

3

2

1

10,000,000,000.

1010

The number of place values to move

the decimal behind the 1.

(100 thousand) light years.

5

4

3

2

1

100,000.

105

The number of stars in the MWG is approximately

100,000,000,000 (100 billion).

11

10

9

8

7

6

5

4

3

2

1

100,000,000,000.

1011

Large numbers have a positive exponent when written as a power of 10. Small numbers have a negative exponent when written as a power of 10.

Consider the small number 0.000000001 (1 billionth):

1

2

3

4

5

6

7

8

9

0.000000001

10-9

The number of place values to move

the decimal behind the 1.

Positive Exponents power of 10.

Negative Exponents

100 = One

101 = Ten

102 = One Hundred

103 = One Thousand

104 = Ten Thousand

105 = One Hundred Thousand

106 = One Million

107 = Ten Million

108 = One Hundred Million

109 = One Billion

1010 = Ten Billion

1011 = One Hundred Billion

1012 = One Trillion

100 = One

10-1 = One Tenth

10-2 = One Hundredth

10-3 = One Thousandth

10-4 = Ten Thousandth

10-5 = One Hundred Thousandth

10-6 = One Millionth

10-7 = Ten Millionth

10-8 = One Hundred Millionth

10-9 = One Billionth

10-10 = Ten Billionth

10-11 = One Hundred Billionth

10-12 = One Trillionth

SCIENTIFIC NOTATION power of 10.

A number is put into scientific notation if it is in the form

d x 10n (large number) or d x 10-n (small number) where d is a decimal between 1 and 10 (i.e., 1 ≤ d < 10).

The distance to the Sun is about 150,000,000 km

150,000,000 = 1.5 x 108

8

7

6

5

4

3

2

1

150,000,000.

1.5 x 108

10 power of 10.

9

8

7

6

5

4

3

2

1

38,000,000,000.

3.8 x 1010

1

2

3

4

5

6

7

0.000000478

4.78 x 10-7

1

2

3

4

5

0.000031

3.1 x 10-5

To multiply or divide powers of ten you: power of 10.

Add the exponents when you multiply

Subtract the exponents when you divide

103* 104 = 103+4 = 107

105* 107 = 105+7 = 1012

1012= 1012-3 = 109

103

108= 108-5 = 103

105

ARITHMETIC OF NUMBERS IN power of 10.

SCIENTIFIC NOTATION

(2.1 x 103) . (1.5 x 106) = (2.1 . 1.5) x (103. 106) = 3.15 x 109

Multiply the Decimals

Add the Exponents

7.2 x 1017=7.2x 1017=2.4 x 107

3 x 1010 3 1010

Divide the Decimals

Subtract the Exponents

When it comes to the arithmetic of numbers in scientific notation, let your fingers do the walking…USE YOUR CALCULATOR.

Your calculator will handle all of the decimal calculations and the addition/subtraction of the exponents.

All you have to do is to set your calculator to scientific notation mode then key in the numbers in the arithmetic problem.

Note – DO NOT enter a number as “3” “x” “10” “^4”, it will treat the “3” and the “10^4” as separate numbers. Use the “exp” or “EE” button (for “exponent”) as in “3” “EE” “4”. (Often written in shorthand form as 3e4).

SAMPLE PROBLEMS notation, let your fingers do the walking…USE YOUR CALCULATOR.

Light travels at the speed of c = 300,000 km/s. The distance from Earth to the Sun is 150,000,000 km. How long does it take light travel from the Sun to Earth?

Distance = Speed . Time

Speed =Distance

Time

Time =Distance

Speed

Time =Distance=150,000,000 km

Speed 300,000 km/s

Time = 500 seconds

500 s . 1 min/60 s = 8.3 minutes

The Earth is 8.3 light-minutes from the Sun.

The metric system as opposed to the British system of units is used in science. There are two versions of the metric system:

MKS = Meters (m) Kilograms (kg) Seconds (s)

CGS = Centimeters (cm) Grams (g) Seconds (s)

It is customary to use CGS in stellar astronomy and MKS in the other branches of astronomy. We will be using the MKS system of units.

The meter is the unit of distance in the MKS system. It is approximately one yard in length (39.3 inches).

The kilogram is the unit of mass (weight). It is approximately two pounds of weight at Earth’s surface.

Scales of Size and Time is used in science. There are two versions of the metric system:

Astronomy deals with objects on a vast range of size scales and time scales.

Most of these size and time scales are way beyond our every-day experience.

Humans, the Earth, and even the solar system are tiny and unimportant on cosmic scales.

A Campus Scene is used in science. There are two versions of the metric system:

16 x 16 m

(52 x 52 ft)

A City View is used in science. There are two versions of the metric system:

1.6 x 1.6 km

(1 x 1 mile)

The Landscape of Pennsylvania is used in science. There are two versions of the metric system:

160 x 160 km

(100 x 100 miles)

The Earth is used in science. There are two versions of the metric system:

Diameter of the Earth: 12,756 km

Earth and Moon is used in science. There are two versions of the metric system:

Distance Earth – Moon: 384,000 km

No atmosphere. is used in science. There are two versions of the metric system:

Diameter: 3500 km

Sidereal Period: 27.3 days

Synodic Period: 29.5 days

Distance from Earth:

385,000 km

Maria – Younger Surface

3 billion years old

Highlands – Older Surface

4.5 billion years old

EARTH’S MOON

Earth Orbiting Around the Sun is used in science. There are two versions of the metric system:

Distance Sun – Earth = 150,000,000 km

Earth Orbiting Around the Sun is used in science. There are two versions of the metric system:

In order to avoid large numbers beyond our imagination, we introduce new units:

1 Astronomical Unit (AU) = Distance Sun – Earth = 150 million km

(93 million miles)

SUN is used in science. There are two versions of the metric system:

Sunspots (Magnetic Storms)

Solar Flares & Prominences

Boiling Earth-Sized

Convection Cells

Diameter: 1,400,000 km

Rotation Period: 25 days

Surface Temperature: 5,800 K

Core Temperature: 15,000,000 K

The Solar System is used in science. There are two versions of the metric system:

Approx. 100 AU

MERCURY is used in science. There are two versions of the metric system:

With no atmosphere, Mercury is heavily cratered by

4.5 billion years of meteoritic impacts.

Nearest the Sun of all the planets.

60,000,000 km (0.4 AU)

Diameter: 5000 km

Orbital Period: 88 days

Rotational Period: 58 days

Although not much bigger than Earth’s Moon it is much denser, 5.5 times that of water compared to the Moon’s 3.5 times.

Thick atmosphere of is used in science. There are two versions of the metric system:

carbon dioxide and sulfuric acid.

VENUS

Venus has a hot thick atmosphere. It is so thick that optical-based

telescopes cannot penetrate to the surface. Closer to the Sun than

Earth, the temperature at the surface is a blistering 800o F.

Diameter: 12,100 km (almost a match for Earth)

Orbital Period: 225 days

Rotational Period: 243 days (retrograde)

Distance from Sun: 110,000,000 km (0.7 AU)

Soviet Venra Spacecraft photograph of the surface of Venus.

MARS is used in science. There are two versions of the metric system:

Diameter: 6800 km

Orbital Period: 1.88 years

Rotational Period: 24h

Distance from Sun:

225,000,000 km (1.5 AU)

Gigantic gorge in the surface of Mars (Valles Marinaris)

stretching 3000 km across. It would reach across the

entire continental United States.

Polar Cap

Dry Ice & Water Ice

Thin atmosphere of carbon dioxide and water.

ASTEROID BELT is used in science. There are two versions of the metric system:

Gaspra

Ida

The Asteroid Belt is located between 2 and 4 AU from the Sun. It contains billions of rock boulders. The two at the right are 20-60 km in size.

Dactyl

JUPITER is used in science. There are two versions of the metric system:

Atmospheric cloud bands due to

high winds.

Extensive atmosphere of

methane and ammonia.

Diameter: 143,000 km

Orbital Period: 12 years

Rotational Period: 10h

Distance from Sun: 680,000,000 km (5 AU)

Great Red Spot

Jupiter is the largest of the planets in the solar system. It is large enough to fit all of the other planets inside of it…twice! It can easily engulf over 1000 Earths.

It has an extensive atmosphere tens of thousands of kilometers thick. It is believe to have been the first planet to form in the solar system, 100 million years after the Sun formed from a large cloud of gas and dust in the rotational plane of the Milky Way Galaxy.

Moon-wide ocean encrusted in ice is used in science. There are two versions of the metric system:

Most volcanically-active object

In the solar system

THE GALILEAN SATELLITES

• This composite includes the four largest moons of Jupiter which are known as the Galilean satellites. From left to right, the moons shown are Ganymede, Callisto, Io, and Europa.

• The Galilean satellites were first seen by the Italian astronomer Galileo Galilei in 1610. In order of increasing distance from Jupiter, Io is closest, followed by Europa, Ganymede, and Callisto.

SATURN is used in science. There are two versions of the metric system:

Diameter: 121,000 km

Orbital Period: 29 years

Rotational Period: 10h 30m

Distance from Sun:

1,400,000,000 km (9.5 AU)

Extensive system of rings

Atmospheric clouds is used in science. There are two versions of the metric system:

URANUS

NEPTUNE

Diameter: 50,000 km

Orbital Period: 164 years

Rotational Period: 15h

Distance from Sun: 4,500,000,000 km (30 AU)

Diameter: 51,000 km

Orbital Period: 84 years

Rotational Period: 15h

Distance from Sun: 2,900,000,000 km (19 AU)

OORT is used in science. There are two versions of the metric system:

CLOUD

&

KUIPER

BELT

• Diagram of the Oort cloud, showing a few cometary orbits. Most Oort cloud comets never come close to the Sun. Of all the orbits shown, only the most elongated ellipse represents a comet that will actually enter the solar system (which is smaller than the dot at the center of the figure on this scale) and possibly become visible from Earth.

• (b)The Kuiper belt, the source of the short-period comets, whose orbits hug the ecliptic plane.

PLUTO is used in science. There are two versions of the metric system:

Discovered in the 1930’s by Clyde Tombaugh, Pluto was long considered the 9th planet in the solar system. Although it built up in the same way as the planets, it is now better classified as a Kuiper Belt Object (KBO).

Along with newly-discovered KBO Eris and the asteroid Ceres, it is now also classified as a dwarf planet.

The Kuiper Belt is a region outside Neptune’s orbit where billions of ice boulders (the building blocks of the planets) are located. The second KBO after Pluto was discovered in 1992, although their presence was predicted circa 1950 by Gerard Kuiper.

Pluto

Charon

Diameter: 2,200 km

Orbital Period: 248 years

Rotational Period: 6 days

Distance from Sun: 5,900,000,000 km (40 AU)

(Almost) Empty Space Around Our Solar System is used in science. There are two versions of the metric system:

Approx. 10,000 AU

The Solar Neighborhood is used in science. There are two versions of the metric system:

Approx. 17 light years

The Solar Neighborhood is used in science. There are two versions of the metric system:

New distance scale:

1 light year (ly) =

Distance traveled by light in 1 year

= 63,000 AU = 1013 km

= 10,000,000,000,000 km

(= 1 + 13 zeros)

= 10 trillion km

Approx. 17 light years

Nearest star to the Sun:

Proxima Centauri, at a distance of 4.2 light years

The Extended Solar Neighborhood is used in science. There are two versions of the metric system:

Approx. 1,700 light years

The Milky Way Galaxy is used in science. There are two versions of the metric system:

Diameter of the Milky Way: ~ 75,000 ly

The Local Group of Galaxies is used in science. There are two versions of the metric system:

Distance to the nearest large galaxies: several million light years

The Universe on Very Large Scales is used in science. There are two versions of the metric system:

Clusters of galaxies are grouped into superclusters.

Superclusters form filaments and walls around voids.

For next time is used in science. There are two versions of the metric system:

• Read Units 5 and 6

• Remember that the homework is due on Monday